Frictional behavior of planar and rough granite fractures subjected to normal load oscillations of different amplitudes

Wengng Dng, Kng To, Xinfn Chen

a School of Civil Engineering, Sun Yat-sen University, Zhuhai, 519082, China

b State Key Laboratory of Coal Resources and Safe Mining, China University of Mining and Technology, Xuzhou, 221116, China

c Southern Marine Science and Engineering Guangdong Laboratory, Zhuhai, 519082, China

Keywords:Direct shear test Normal load oscillation fracture roughness Critical oscillation amplitude Dynamic weakening/strengthening

A B S T R A C T

1. Introduction

Discontinuities generally control the properties of rock masses(Barton, 1976; Prassetyo et al., 2017; Meng et al., 2018; Li et al.,2019; Bista et al., 2020) and are generally exposed to dynamic loads in the Earth’s lithosphere (such as injection in geothermal reservoir, mining excavation perturbations, earthquakes, ocean or solid tides) (Stein, 1999; van der Elst and Savage, 2015; Delorey et al., 2017; Beeler et al., 2018; No?l et al., 2019; Ji et al., 2021).Understanding the deformation and failure mechanism of fractures/faults under complex loading conditions is of great significance for the designs of slopes, foundations, underground spaces,and geothermal reservoirs. (Cao et al., 2016; Lian et al., 2020; Liu et al., 2020a; Petr et al., 2020; Dang et al., 2021a). According to previous researches, the friction of rock fracture can be reduced under dynamic loading, leading to the changes in critical equilibrium condition and movement of the rock mass (Stein,1999; van der Elst and Savage, 2015; Delorey et al., 2017; Molinari and Perfettini, 2017; Okada and Naya, 2019). In several cases, under variable normal loads,the corresponding shear force in the affected faulting area may fluctuate periodically (Boettcher and Marone,2004; Dang et al., 2016, 2017, 2018, 2020, 2021b). To investigate the frictional responses of rock fractures under different normal loads, many shear tests under constant/dynamic normal forces have been conducted on laboratory-scale rock samples (Candela et al., 2014; Dang et al., 2016; Kilgore et al., 2017; Meng et al.,2018). Depending on the geological condition of the shear zone in nature,the frictional interaction pattern of rock discontinuities has been simulated by different approaches in laboratory,such as using planar or rough surfaces, with or without filling materials at different filling thicknesses. Many studies investigated the normal load disturbance by water pressurization(e.g.Candela et al.,2014;No?l et al.,2019;Ji et al.,2021),while some others utilized dynamic shearing apparatus to continuously change the normal stress state of rock samples (Boettcher and Marone, 2004; Dang et al., 2016,2017, 2018, 2020, 2021a; Sheng et al., 2020).

One of the most commonly used methods to describe the dynamic mechanical response of discontinuity is the characterization of an interface law, namely the rate- and state-friction (RSF) law(Dieterich,1979; Ruina,1983; Marone,1998). Its initial description implicitly assumes that the contact surface is totally flat and has no thickness (Dieterich,1979). Subsequent works also confirmed that the RSF law can be used in fault gouge and roughness-related analysis. Although it is a phenomenological description, the microstructure explanation has been detailed (Ikari et al., 2016;Molinari and Perfettini, 2017). In addition, the correction factor α proposed by Linker and Dieterich (1992) can deal with the abrupt and continuous changes of normal force, which has a strong influence on the nucleation and triggering of earthquake slips. The normal load oscillation amplitude is strongly linked to the frictional behavior of rock fractures(Boettcher and Marone,2004;Dang et al.,2017). The minimum friction coefficient increases with the decreasing normal load oscillation amplitude,while the maximum friction coefficient keeps nearly constant.

Roughness plays a key role in fault motion (Amitrano and Schmittbuhl, 2002). Previous seismic observations indicate that roughness controls the first-order distribution of coseismic slip(Peyrat and Olsen, 2004). The geometrical complexities of friction surfaces may cause the discontinuities to be oriented unfavorably for movement and stress concentration (Allam et al., 2019). As an important factor governing the shear resistance, it has been well studied in both laboratory experiments and field investigations(Amitrano and Schmittbuhl,2002;Badt et al.,2016;Allam et al.,2017,2019). Over the past few decades, quantitative statistical analyses describing surface geometry have been proposed, such as joint roughness coefficient (JRC) (Barton, 1976) and 3D morphology parameters (Grasselli et al., 2002). Many direct shear tests under constant normal force or constant normal stiffness in laboratory have been conducted to investigate the relationship between shear strengthandroughnessindexofjoints(TseandCruden,1979;Liuet al.,2017,2020b;Zhao et al.,2020).Meanwhile,the relationship between the roughness condition of tectonic faults or fractures and the friction properties have been analyzed numerically(Wang et al.,2020).

For investigating fractures, the saw-cut sample can maintain a constant friction when sliding for a certain distance (Dang et al.,2016), that is, the friction coefficient does not change with time or distance.However,the shear strength of a rough fracture varies with the shear distance, and there is a peak. Before reaching the peak shear strength, the increasing shear force on the specimen may influence the overall sliding friction(Dang et al.,2021a).In this study,planar or rough granite fractures have been used to simulate tectonic discontinuities in lithosphere. A large direct shear apparatus was employed to create the periodic normal loading with different amplitudes on the slipping rock fractures.This study could provide some new insight into understanding of dynamic fault weakening/strengthening at changing fracture roughness and normal load amplitudes.

2. Experimental procedure

2.1. Sample preparation and testing equipment

The DJZ-500 shear apparatus (Dang et al., 2022) with dynamic loading modules was utilized to perform the experiments. Displacements and forces were measured by linear variable differential transformers (LVDTs) and load cells, respectively (Fig.1a). The red point in Fig.1a is the position where the slip displacement is measured. As shown in Fig. 1b, the rock fractures were located between the upper and lower parts of shear boxes.The upper shear box is fixed and the lower part can move horizontally forward at a constant shear velocity. The stiffness of loading frame is about 5 GN/m,with the total mass of 5000 kg approximately.The normal pressure is transmitted to the sample through the upper loading plate, including constant normal load (CNL) and dynamic normal load (DNL). The normal and shear forces and normal and shear displacements were recorded at a frequency of 50 Hz.

Fig.1. Shear box apparatus and experimental configuration: (a) Illustration of DJZ-500 shear box apparatus and installation of its main components including vertical and horizontal LVDTs; and (b) Illustration of the direct shear test under different load conditions.

The medium-coarse-grained granites, sampled from Hunan Province, China, were used for the laboratory test. The mineral composition and mechanical parameters of the granite are shown in Table 1. As shown in Fig. 2a, the cubic rock sample with side length of 100 mm was cut into two pieces with a saw and polished to get the planar joint sample(sample A),and the other cubic rock sample was split with a press machine to obtain a rough surface(sample B).Fig.2b shows the distribution of granite minerals under polarized light microscope. For the rough jointed rock mass, a 3D scanner (Fig. 2c) was used to obtain the spatial coordinates of the surfaces to quantify its roughness from which a 3D topographic map can also be drawn (Fig. 2d).

Table 1 Mineral compositions and mechanical properties of the granite in this study.

For shear experiments with rough surfaces, the repeatability of the experiment is verified by the 3D scanning results.We calculated the JRC of sample B based on root mean square slope,Z2(Tse and Cruden, 1979), which can be obtained from our scanned data.Before the test, the JRC was 8, while it was 7.8 after the test. According to the shear strength law of Barton (1976), this reduction had little influence on the shear strength. Furthermore, Fig. 2e corresponds to the profile at the cross-section in Fig.2d(red line).From the elevation of this profile before and after the test, only a small portion of the asperities were worn off,so the degradations of the slip surfaces were very tiny.

Fig. 2. (a) The testing samples (A-planar surface, B -rough surfaces); (b) Microscopy images showing the main mineral components, Pl: plagioclase, Qtz: quartz, Bt: biotite, Ms:muscovite; (c) The 3D surface scanner; (d) The upper and lower topography of sample B; and (e) The profiles before and after the test.

2.2. Test procedure

The same experimental process was used for testing two types of rock fractures. Unlike the previously used sinusoidal wave loading mode (Dang et al., 2016, 2017, 2021a), we used the triangular wave loading mode to ensure the constant loading and unloading rate. As shown in Fig. 3 and Table 2, in these tests, only the amplitude of normal load oscillation was altered. In all experiments, the average normal force was kept constant at 40 kN. The normal load oscillation amplitude (A) were 5 kN, 15 kN, 20 kN,25 kN and 30 kN, respectively. Hence, the corresponding normalized amplitudes (ε) were 12.5%, 37.5%, 50%, 62.5% and 75%,respectively. The frequency of normal force oscillation was 0.5 Hz.The slip velocity has a great influence on the friction of fractures(Wang et al., 2020; Dang et al., 2021a). The shear rate was 5 mm/min, close to the speed of some slow slip events in nature (Rice,2006; No?l et al., 2019). During the slip process, this speed remained constant, with the total slip distance (u) of 6 mm and total time of 72 s. The additional direct shear test was performed under a CNL of 40 kN, for comparison purpose. All the tests were performed at room temperature of 15°C,with relative humidity of about 50%. All experiments were conducted within a short period of time, such that the external environment of all tests can be maintained almost the same.

Table 2 Test scheme for different normal force oscillation amplitudes.

3. Results

3.1. Characteristics of shear force, coefficient of friction and shear work

3.1.1. Shear force

Fig. 4 reports all experimental data of shear forces for the two samples. Since the normal force followed a triangle waveform pattern,the variation of shear forces also followed the same pattern but more serrated. The fracture shear stiffness can be calculated from the shear force-displacement relationship, which is about 0.1456 GN/m for sample A and about 0.0582 GN/m for sample B.Combined with the stiffness (5 GN/m) of our shear loading framework,the stiffness of the whole friction system with sample A and sample B is 0.1414 and 0.0576 GN/m,respectively.The machine effect, i.e. ‘self-oscillation’ (Pasternak et al., 2020), was not considered. Dτ is the variation of shear force in its loading or unloading process. As can be seen in Fig. 5a and b, for both the planar and rough fractures, the peak of shear force first increases and then decreases, while the minimum keeps decreasing with increasing normalized amplitudes.

Fig.3. The input loading mode.The normal load changes as a triangular wave with time,and the increase of horizontal displacement is linear.The normal load vibrates 36 times in each experiment.

Fig. 4. The evolution of recorded shear force with the slip displacement for (a) Sample A and (b) Sample B. The images on the right are the partial enlargements.

Fig. 5. The maximum, minimum and average friction forces in one loading/unloading cycle as function of normalized amplitude for (a) Sample A and (b) Sample B.

The variation pattern of normal force followed a triangular wave, with a stable loading/unloading rate in each test. Fig. 6a shows the evolution of normalized shear force change,i.e.the ratio ofDτ to the normal force range(2A),with the normalized amplitude during the loading/unloading stage. The overall changing trend ofDτ/(2A) was downward, and there was similarity between the loading and unloading stages. Fig. 6b shows the changing pattern between the shear (un)loading rate (SLR) and the normal (un)loading rate (NLR) at both loading and unloading stages. It should be noted that the loading and unloading trends were different.With the increase of normal loading rate, the shear loading rate increased first and then decreased slightly. However, the shear unloading rate increased with increasing normal unloading rate.As shown in Fig.6c,the(SLR/NLR)at the loading and unloading stage were also different. In the loading stage, SLR/NLR decreased with the increasing normalized amplitude. However, SLR/NLR mainly kept unchanged at the unloading stage. Meanwhile, it can be seen from Fig. 6 that all the values of the two samples were similar,which demonstrates that the fracture surface condition has slight influence on its loading/unloading behavior.

3.1.2. Friction coefficient variation

The apparent friction coefficient (μ) defined as the real-time shear force divided by real-time normal force (μ =FS/FN) has been calculated in the slip displacement range from 3.5 mm to 4.5 mm, as shown in Fig. 7a and b. Amontons-Coulomb’s friction law assumes a constant friction coefficient (μss) (Byerlee, 1978),which is the static friction coefficient under the CNL condition.However, the simple constitutive law would have limitations for materials exposed to variable loading conditions (Linker and Dieterich,1992; Perfettini et al., 2001). As shown in Fig. 7a and b,it is obvious that the DNL caused the variation of friction coefficient,and it varied periodically with the same period as the normal force did. In addition, it can be seen that the variation range of friction coefficient gained with increasing normalized amplitude. This range was larger for the planar fracture than that for the rough one.The larger magnitude of normal load oscillation led to the larger peak and the smaller valley values. The valley value of the friction coefficient was apparently smaller than the μss. Therefore, traditional Coulomb’s friction law may overestimate the shear strength of the fracture under dynamic loading conditions. In addition, as illustrated in Fig.7c,for planar fracture(sample A),the peak value of friction coefficient (μpeak) increased, while the valley value of friction coefficient (μvalley) decreased with increase of the normalized amplitude. For the rough fracture (sample B), the μpeakwas almost constant (approximately equals to μss), and the μvalleydecreased with the increase of amplitude.

Coulomb’s friction law can evaluate the shear strength under the CNL conditions since the frictional resistance is proportional to the normal stress. However, it is not suitable under the DNL conditions due to the periodically changed friction coefficient. The minimum friction coefficient under higher amplitude is extremely smaller than that in the steady state,indicating an unstable trend of the discontinuity. Moreover, our results are very similar to the ultra-low friction phenomenon and friction reduction effects in deep mining under high in situ stresses and strong excavation disturbance, which were reported to be strongly related to rock burst (Kurlenya and Oparin,1999).

3.1.3. Shear work

The shear work represents a main mechanism of friction energy consumption during the fracture slipping and can be calculated by Eq. (1). In the tests, the shear work can be calculated as the area between the friction curve and the transverse axis(see Fig.4),with the results reported in Fig. 8. For rough fractures (sample B), the total work decreased dramatically with the increasing amplitude.However, the experiments with planar fracture (sample A) had a slight rising trend at small amplitudes in the beginning, and then declined.The shear work of the planar fracture was larger than that of the rough fracture, suggesting the susceptibility of rough fractures to slip instability under the dynamic disturbance.

whereWsis the shear work,andumaxis the maximum slip distance.

3.2. Normal displacement

The vertical LVDTs measured the normal displacement of the samples during the experiments,as illustrated in Fig.9a and b.The normal displacements (set to zero at the beginning of slip) were also changed periodically with the vibration of normal forces. The normal displacement consists of two parts: the compression deformation of the rock block itself and the deformation of the fractures.Due to the large elastic modulus of granite,the recorded normal displacements were mainly the fracture deformation. For the planar fracture,the normal displacement was constant at first,and then decreased slowly. But the change was small, and was basically not more than 0.1 mm,compared with the rough sample.For the sample with rough surface, the upper block showed a decreasing trend,which was related to the geometric morphology of the slip surface. For every cycle of loading and unloading, the fractures between the upper and lower sample parts were compressed and separated. Fig. 9c reports the normal displacements change(Dd),and the normal deformation during one compressionand-separation cycle with the normalized amplitude follows an approximately linear relationship.

Fig.6. (a)Normalized shear force changes,i.e.Dτ/2A,versus normalized amplitude;(b)The relationship between SLR and NLR;and(c)The relationship between SLR/NLR and normalized amplitude.

Fig. 7. Experimental data of friction coefficient as a function of slip displacement: (a)Sample A,(b)Sample B,and(c)Relative variations of friction coefficient,i.e.(μpeak-μss)/μss or (μvalley-μss)/μss, versus normalized amplitude.

3.3. Time shift

The peak (valley) values of the shear and normal forces do not occur simultaneously, so does the friction coefficient. Dang et al.(2016) proposed that the time shift between the normal and shear forces under variable normal loading conditions was caused by the change of fracture stiffness(a certain amount shear displacement is needed to activate (mobilize) shear stress and to reach the peak value).As shown in Fig.10,we define γτPand γτVas the time shifts between the peak and valley values of normal and shear forces,and γfpand γfVas the time shifts between the peak and valley values of coefficientoffriction,aswellastheγdPandγdVastimeshiftsbetween the peak and valley values of normal force and normal displacement.As the slip displacement increases,γτ,γfand γdremain unchanged.

Fig. 8. Shear work (friction energy consumption) of samples A and B with different oscillation amplitudes.

Fig.9. Test results of normal displacement.The downward displacement is indicated as negative.(a)Normal displacement variation versus slip displacement in tests on sample A,(b)Normal displacement variation versus slip displacement in tests on sample B,and(c)The normal displacement change(Dd)versus the normalized amplitude change in a loading cycle.

The time shifts of two samples are reported in Fig. 11, and obviously, the variation trends are similar. For the friction force(Fig. 11a), γτPincreased with larger normalized amplitude, and there was a time difference of about 23% at the maximum amplitude between the two samples.However,γτVremained zero at all conditions.The minimum friction response and normal load always occurred simultaneously.In other words,an increase of the normal force would lead to an increase of the shear force. For the friction coefficient (Fig. 11b), both the peak-to-peak and valley-to-valley values of time shift (γfPand γfV) were similar and approximately half a period. When the normal force was at its peak value, the friction coefficient was at its minimum,and vice versa.As shown in Fig.11c,γdPand γdVwere 50% for all tests.

Fig. 11. The variation of the specified three kinds of time shifts with normalized amplitude: (a) Time shift of shear force; (b) Time shift of friction coefficient; and (c)Time shift of normal displacement.The solid line represents the peak-to-peak shift and the dashed line represents the valley-to-valley.

Boettcher and Marone (2004) investigated the time shift between shear force and the sinusoidal shaped normal load in double shear experiments.It was observed that a maximum value of time lag between the normal and shear loads exists upon changing the normal oscillation frequency.Perfettini et al.(2001)used analytical method to predict time shifts between shear and sinusoidal normal loads, as described in Eq. (2). ρτis defined as the modulus of the changed normalized shear stress,which has a complex form related to stress conditions and system stiffness. Dang et al. (2016, 2017,2018) indicated that the relative time shift of the shear force on a planar fracture increases with the amplitude of oscillation,which is consistent with the findings in this work. Also, the relative time shifts between the normal force and the apparent friction coefficient are approximately constant(Dang et al.,2021a).In this work,it is also observed that this pattern is irrelevant of the surface geometry.

where τ is the shear stress, ρτis the modulus of the changed normalized shear stress,tis the time,fis the frequency of normal load oscillation, and σ is the normal stress.

4. Discussion

4.1. Rate and state friction law with parameter α

The friction behavior of fractures in this study was quantified using the RSF theory (Dieterich, 1979; Ruina, 1983), and coupled with the empirical Ruina-type“slip law”.It describes the evolution of shear resistance from a reference steady-state value (μ0FN) towards a new steady-state value over a critical slip distance(Dc),see Eq. (3). For the rate dependence of friction, in response to an instantaneous change in slip rate from an initial velocity (v0) to a new velocity (v), the state variable (θ) is commonly viewed as the average time span of a population of micro contacts(Marone,1998).It defines the evolution of contact friction via Eq. (4):

where μ0is the original friction coefficient;v is a new velocity,and v0is the initial velocity;Dcis the critical slip distance;aandbare parameters describing the direct and evolutionary effect;and θ and θ0are the state variables.

However, in its original framework, RSF can only describe the velocity effect, and does not reflect the normal load perturbation.Linker and Dieterich (1992) modified the formula by introducing the correction factor α to consider the abrupt and continuous changes of normal force with the empirical relation. As shown in Eq. (5), the change of θ relates to the change of normal force as controlled by the factor α and time.For an estimation,an empirical relationship between α and friction response was proposed, as shown in Eq.(6).ΔFSis the change of shear force induced by normal load transition fromFN0toFN.

where α represents the parameter effecting the evolution of the state variant under the change of normal force.

To study the effect of DNL on the frictional response in our experiment with RSF law, Eq. (6) has been used to investigate the difference of dynamic parameters α under the same normal load perturbation. We used an intact oscillation cycle to calculate α under different normalized amplitudes, with ΔFS=Dτ (see Fig. 4)andFN-FN0= 2A. As shown in Fig. 12, as the amplitude increases, the value of α decreases linearly from over 0.5 to below 0.15.In addition,except for a very low amplitude,the value of α for planar fractures is generally larger than that for rough fractures.

Fig.12. Relationship between α and the normalized amplitude.

4.2. Dynamic frictional strengthening or weakening

As mentioned in Section 3.1, among all tests, some results of the friction under DNL conditions are larger than those under the corresponding CNL conditions, while some others are not. For this, we defineFyieldas the difference between the maximum friction under the DNL and the friction under the CNL (Fsc) for the same shear distance (Fig. 13a). The positive value ofFyieldindicates the friction enhancement and aseismic slip; correspondingly, the negative value indicates the shear strength of the discontinuities weakened by the dynamic loads. As shown in Fig. 13b, the ratio betweenFyieldandFscis calculated to explain the frictional strengthening/weakening effect among the two series of tests with different normal load oscillation amplitude. εcis the critical normalized amplitude which makesFyield= 0. For the planar fracture,Fyield/Fscfirst increased with increasing normalized amplitude, and reached the peak at ε = 0.38; then it decreased. In the present work, even though no negativeFyield/Fscwas observed, it indicates that εcis about 0.9. For the rough fracture,Fyield/Fscfirst increased with increasing normalized amplitude and reached the peak at ε = 0.42; then it decreased.Fyield/Fscbecame 0 at ε = 0.64. The result indicates that rough fracture can be easier to cause frictional weakening under the normal load disturbance.

Fig.13. (a) The definition of Fyield, and (b) Evolution of Fyield with normalized amplitude. εc is the critical normalized amplitude making Fyield = 0.

The RSF framework can explain the destabilization of slip process,and the critical amplitude is related to all RSF parameters and mechanical stiffness. Perfettini et al. (2001) proposed Eq. (7) to compute this critical amplitude under the critical (resonant)oscillation period. For the differences of weakening range among two samples, we can see that the planar surface has larger εcthan the rough surface. Previous studies point out that the larger the value of α,the stronger the anti-interference ability of the fault and the more stable the slip (e.g. Boettcher and Marone, 2004; Hong and Marone, 2005). Although we cannot directly calculate the critical amplitude by Eq.(7)because of the mismatched oscillation period,it shows that the smaller critical stiffness and larger α result in a larger critical amplitude according to Perfettini’s description.Eq. (8) presents a common solution of critical stiffness.

where εcis critical normalized amplitude,kcis critical stiffness,kis system stiffness, andDcis the critical slip distance.

5. Conclusions

In this paper,we investigated the friction patterns of laboratory faults under normal load fluctuations. The samples with different surface topographies under the same direct shear testing condition showed different frictional behaviors.

For the dynamic frictional resistance under periodic normal force with different oscillation amplitudes, it is observed that the shear forces, friction coefficient and normal displacements were periodically changed. The DNL first enhanced, and then weakened the peak shear strength of a creeping fracture with increasing oscillation amplitude. A series of phase shift was confirmed,showing a consistency irrespective of the surface geometry. However,some discrepancies caused by roughness also exist.Compared with the rough fracture, the variation of apparent friction coefficient on the planer surface sample is larger. The results also show that the shear work on the planner surface first increased and then decreased with the increasing normal load oscillation amplitude,while the shear work on the rough surface decreased monotonically. The RSF law with the state change parameter, α, was introduced to explain the observed friction results during the normal load oscillation.The results show that a rougher sample has a smaller α during the slip,indicating that the weakening behavior or destabilization can easily occur with rougher slip surface on a fault because of the smaller critical amplitude.

It indicates that the stability of creeping faults is strongly related to the fracture surface topography and DNL oscillation amplitude.As slip velocity and humidity effects also play important roles in fault stability evolution (Rice et al., 2001; Rice, 2006; Dang et al.,2021a, b), further investigations on related topics are required.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

We acknowledge the funding support from the National Natural Science Foundation of China(Grant No.51904359),the Guangdong Provincial Department of Science and Technology (Grant No.2019ZT08G090), and the Open Research Fund of the State Key Laboratory of Coal Resources and Safe Mining, China University of Mining and Technology(CUMT)(Grant No.SKLCRSM20KF002).The authors would like to thank Mr.Junpeng Chen for his experimental assistance.

List of symbols

FSShear force

FNNormal force

CNL/DNL Constant normal load/dynamic normal load

FyieldDifference between the maximum friction under the DNL and the friction under the CNL

FSCFriction under the CNL

ANormal load oscillation amplitude

ε Normalized amplitude

εcCritical normalized amplitude

σ Normal stress

τ Shear stress

tElapsed time

vSlip rate

uSlip distance

dNormal displacement

μ Apparent friction coefficient

μssFriction coefficient under a constant normal force

γτPhase lag between normal force and shear force

γfPhase lag between normal force and coefficient of friction

γdPhase lag between normal force and normal

displacement

ρτModulus of the normalized shear stress change

NLR Normal(un)loading rate

SLR Shear (un)loading rate

WsShear work

kcCritical stiffness

DcThe critical slip distance

aAn empirical friction parameter

bAn empirical friction parameter

θ State variable

α A parameter effecting the evolution of the state variant under the change of normal force