Stick-slip Events Research Articles

Thermo-poromechanics of chemically active faults—enriching Anderson’s theory of faulting in sedimentary rocks

SUMMARY A suggested model to explain the episodic nature of slow earthquakes involves shear zones exhibiting rate- and temperature-dependent frictional behaviour hosting fluid-release chemical reactions. In this work we extend the considerations of that approach, coupling the effects of the mechanics at different faulting regimes to the chemically induced fluid pressurization inside the fault. By introducing a pressure and temperature dependence of the mechanical response in an elasto-viscoplastic model we are able to correlate the inclination angles of those specific faults with their dynamical response and enrich their faulting regimes with kinematic characterization. We retrieve that steeply dipping (normal) faults exhibit a simple response of either being locked or slip at fast seismic velocities; shallow dipping (reverse) faults on the other hand exhibit a much richer behaviour where episodic stick-slip instabilities can be encountered. When present, their magnitude depends on the (reverse) fault’s angle with faults dipping at around 45° exhibiting a maximum, whereas sub-horizontal thrusts exhibit episodic stick-slip events as low velocities and magnitude. These findings position slow earthquakes and episodic tremor and slip sequences as a natural response of shallow dipping (thrust) faults, in a regime that according to rate-and-state friction considerations is intrinsically stable.

Prediction of Dynamic Stick-Slip Events in Belt-Drives Using a High-Fidelity Finite Element Model

AbstractA high-fidelity dynamic finite element model of a one-pulley belt-drive system is used to study how Coulomb friction coefficient and adhesion stress affect belt stick-slip events over driver and driven pulleys. The model shows that at Coulomb friction coefficient below a certain critical value, the belt undergoes pure sliding in the slip arc on driver and driven pulleys with no stick-slip dynamic events. Above that critical friction coefficient value dynamic stick-slip events start to occur in both driver and driven pulleys resulting in torque pulses, which for practical belt-drives may cause nonsmooth operation, excessive belt noise and/or excessive belt wear. For driver pulleys the dynamic stick-slip events are in the form of Schallamach waves, while for driven pulleys the dynamic stick-slip events are in the form of expansion pulses. The model results are validated using recently generated experimental results.

Cross-effects of loading rate and cumulative fault slip on pre-seismic rupture and unstable slip rate of laboratory earthquakes

The fact that the same fault may host different rupture or slip modes was partly attributed to the change in strain rate of the fault in previous studies. However, contradictory observations were obtained in the previous experimental studies, that is, the loading rate can promote or inhibit the rapid instability of faults. Although the two opposite effects of the loading rate were separately explained in the previous studies, the contradiction remains unresolved. Based on an experiment conducted on a granodiorite sample, we use the cumulative fault slip to quantitatively describe the slip-driven changes in fault properties and study the cross-effects of loading rate and cumulative fault slip on the pre-seismic rupture speed, unstable slip rate, and friction coefficient drop of the laboratory earthquakes (stick-slip events). The experimental results show that under the same loading rate, the exponential propagating rupture speed, the unstable slip rate, and the friction coefficient drop of the stick-slip events increase with increasing cumulative fault slip. As a result, the loading rate has opposite effects on the exponential propagating rupture speed, the unstable slip rate, and the friction coefficient drop under different amounts of the cumulative fault slip. This provides an explanation for the observation contradiction that the loading rate can promote or inhibit the rapid instability of the fault. The experimental results also reveal that the steady rupture speed is approximately proportional to the loading rate but weakly depends on the cumulative fault slip. Finally, the significance of the experimental results for understanding the earthquake and faulting mechanics is discussed.

Coexisting neuronal coding strategies in the barrel cortex.

During tactile sensation by rodents, whisker movements across surfaces generate complex whisker motions, including discrete, transient stick-slip events, which carry information about surface properties. The characteristics of these events and how the brain encodes this tactile information remain enigmatic. We found that cortical neurons show a mixture of synchronized and nontemporally correlated spikes in their tactile responses. Synchronous spikes convey the magnitude of stick-slip events by numerous aspects of temporal coding. These spikes show preferential selectivity for kinetic and kinematic whisker motion. By contrast, asynchronous spikes in each neuron convey the magnitude of stick-slip events by their discharge rates, response probability, and interspike intervals. We further show that the differentiation between these two types of activity is highly dependent on the magnitude of stick-slip events and stimulus and response history. These results suggest that cortical neurons transmit multiple components of tactile information through numerous coding strategies.

Numerical insight into energy partitioning during stick-slip events based on the framework of rate-and-state friction law

Numerical insight into energy partitioning during stick-slip events based on the framework of rate-and-state friction law

Stick-slip Dynamics in Penetration Experiments on Simulated Regolith

Abstract The surfaces of many planetary bodies, including asteroids and small moons, are covered with dust to pebble-sized regolith held weakly to the surface by gravity and contact forces. Understanding the reaction of regolith to an external perturbation will allow for instruments, including sensors and anchoring mechanisms for use on such surfaces, to implement optimized design principles. We analyze the behavior of a flexible probe inserted into loose regolith simulant as a function of probe speed and ambient gravitational acceleration to explore the relevant dynamics. The EMPANADA experiment (Ejecta-Minimizing Protocols for Applications Needing Anchoring or Digging on Asteroids) flew on several parabolic flights. It employs a classic granular physics technique, photoelasticity, to quantify the dynamics of a flexible probe during its insertion into a system of bi-disperse, centimeter-sized model grains. We identify the force chain structure throughout the system during probe insertion at a variety of speeds and for four different levels of gravity: terrestrial, Martian, lunar, and microgravity. We identify discrete, stick-slip failure events that increase in frequency as a function of the gravitational acceleration. In microgravity environments, stick-slip behaviors are negligible, and we find that faster probe insertion can suppress stick-slip behaviors where they are present. We conclude that the mechanical response of regolith on rubble-pile asteroids is likely quite distinct from that found on larger planetary objects, and scaling terrestrial experiments to microgravity conditions may not capture the full physical dynamics.

Experimental and Numerical Study of the Effect of Surface Patterning on the Frictional Properties of Polymer Surfaces

Abstract We describe benchmark experiments to evaluate the frictional properties of laser patterned low-density polyethylene as a function of sliding velocity, normal force, and humidity. The pattern is a square lattice of square cavities with sub-mm spacing. We find that dynamic friction decreases compared to nonpatterned surfaces, since stress concentrations lead to early detachment, and that stick-slip behavior is also affected. Friction increases with humidity, and the onset of stick-slip events occurs in the high humidity regime. Experimental results are compared with numerical simulations of a simplified 2D spring-block model. A good qualitative agreement can be obtained by introducing a deviation from the linear behavior of the Amontons-Coulomb law with the load due to a saturation in the effective contact area with pressure. This also leads to the improvement of the quantitative results of the spring-block model by reducing the discrepancy with the experimental results, indicating the robustness of the adopted simplified approach, which could be adopted to design patterned surfaces with controlled friction properties.

Empirical Characterization of Friction Parameters for Nonlinear Stick-slip Simulation to Predict the Severity of Squeak Sounds

Squeak and rattle (S&R) are nonstationary annoying sounds inside the car cabin that impose high warranty costs on car manufacturers. The need for taking S&R preventing measures and the maturity level and cost considerations of the physical prototypes during the pre-design-freeze stages justifies the use of virtual simulation methods. Squeak is a friction-induced high-frequency sound that is attributed to the stick-slip friction phenomenon. The importance of the friction parameters in the squeak severity prognosis is analytically and experimentally mentioned in the literature. However, studying the variation of these parameters respecting the changes in loading and driving conditions with the aim of application in virtual simulations has remained limited or too simplistic. In this work, the rate weakening effect of the friction coefficient curve was involved in the nonlinear finite element simulation of stick-slip events by an exponential decay formulation. The approximated squeak severity by the virtual simulations for selected material pairs agreed with the empirical results from a flexural stick-slip test bench. From the empirical stick-slip data, the dependence of the squeak severity on the friction decay coefficient and the difference of static and kinetic friction coefficients at low and high normal loads were observed, respectively. The relativity of friction parameters on the test conditions demands a dynamic updating of the friction model that can be achieved by polynomial or exponential approximations. Also, the observed polynomial relationship between the squeak severity and the operational conditions can be used to estimate the squeak severity from the linear dynamic simulation results. The outcome of this work can help to better understand the influence of the friction modelling parameters and their variation respecting the operational conditions. This can facilitate a more accurate prediction of squeak risk by employing virtual simulation tools in the pre-design-freeze stages of car development.

Identification of different manifestations of nonlinear stick–slip phenomena during creep groan braking noise by using the unsupervised learning algorithms k-means and self-organizing map

Creep groan is a friction-induced, low-frequency vibration and noise phenomenon of a vehicle’s brake system which is excited by a repeating stick–slip effect. Together with high influences of design and operational parameters, the non-linear stick–slip leads to an interesting bifurcation behaviour of creep groan. For objective rating procedures, detection and classification methods considering this bifurcation behaviour are necessary. Within this study, an approach based on acoustic emission is presented. The approach harnesses high-frequency acceleration contents that accompany creep groan’s characteristic stick–slip transitions. Whereas low-frequency vibration contents below 500 Hz are mainly defined by the characteristics of the brake system and the suspension of the vehicle, vibrations in the high-frequency range above 10 kHz exhibit patterns of waveforms similar to the patterns of acoustic emission bursts. By applying non-overlapping high- and low-pass filters, a novel signal, enveloping these bursts, was created. This envelope bursts signal enables a precise detection and quantification of stick–slip transitions directly in time domain, and led to the development of a whole new set of vibration signal features. These nine signal features were used to feed the unsupervised classification algorithms k-means and Kohonen’s self-organizing map, which delivered robust and meaningful results. Four different creep groan classes were detected, where each has shown to be linked to a specific creep groan manifestation: Low-frequency groan, high-frequency groan and two transition phenomena with two/three stick–slip events per cycle were found. Classification results and their linked mechanical behaviour suggest an interaction between two significant vibration patterns during creep groan, probably a longitudinal and a torsional displacement of the axle. Aside of deeper insights in creep groan’s bifurcation behaviour, the presented study enables not only the identification of creep groan, but also the automatic classification of its manifestations in real-time, and therefore provides further possibilities for creep groan control methods.

Experimental investigation of the effects of loading rate, contact roughness, and normal stress on the stick-slip behavior of faults

The tectonic movement rate, in situ stresses (e.g., normal stress) and fault roughness may change during the long evolutionary histories of seismogenic faults. To investigate the effects of these three factors on fault slip behaviors, we conducted direct shear tests by sequentially changing the shear loading rate and normal stress on small-scale granite specimens under three roughness conditions. We found that for a given fault system, the shear loading rate is a primary controlling parameter of the recurrence intervals of stick-slip events, and the correlation between them can be empirically expressed by a power law. Stick-slip recurrence intervals can influence the stress drops of stick-slip events via two different tendencies. The shear stress drops increase quasilinearly with the recurrence intervals at a given shear loading rate, while the combined data from all shear rate tests show that the shear stress drops increase with the logarithm of the recurrence time. In the investigated range of fault roughness, we observed that surface roughness conditions strikingly affect the sliding mode and recurrence frequency of stick-slip events. In detail, the two smooth fault systems slide with regular stick-slip events, and the roughest fault system exhibits bimodal behavior. Namely, at lower normal stresses, stick-slip events cannot be stably generated, a few were occasionally generated, and regular stick-slip events were interrupted by local stick-slip. When the applied normal stress is sufficiently high, the fault slides with continuous regular stick-slip events in the same way as those in the smoother fault systems. We further noticed that the recurrence frequency of stick-slip and friction drop are correlated at a fixed sliding distance, and this relationship can be described by an inverse proportional function. Our results suggest that all three factors, namely, shear loading rate, fault surface roughness and normal stress, can influence the behaviors of fault slip and hence are important for evaluating earthquake hazards.

Nanoscale Wear and Mechanical Properties of Calcite: Effects of Stearic Acid Modification and Water Vapor.

Understanding the wear of mineral fillers is crucial for controlling industrial processes, and in the present work, we examine the wear resistance and nanomechanical properties of bare calcite and stearic acid-modified calcite surfaces under dry and humid conditions at the nanoscale. Measurements under different loads allow us to probe the situation in the absence and presence of abrasive wear. The sliding motion is in general characterized by irregular stick-slip events that at higher loads lead to abrasion of the brittle calcite surface. Bare calcite is hydrophilic, and under humid conditions, a thin water layer is present on the surface. This water layer does not affect the friction force. However, it slightly decreases the wear depth and strongly influences the distribution of wear particles. In contrast, stearic acid-modified surfaces are hydrophobic. Nevertheless, humidity affects the wear characteristics by decreasing the binding strength of stearic acid at higher humidity. A complete monolayer coverage of calcite by stearic acid results in a significant reduction in wear but only a moderate reduction in friction forces at low humidity and no reduction at 75% relative humidity (RH). Thus, our data suggest that the wear reduction does not result from a lowering of the friction force but rather from an increased ductility of the surface region as offered by the stearic acid layer. An incomplete monolayer of stearic acid on the calcite surface provides no reduction in wear regardless of the RH investigated. Clearly, the wear properties of modified calcite surfaces depend crucially on the packing density of the surface modifier and also on the air humidity.

Conveyance of texture signals along a rat whisker

Neuronal activities underlying a percept are constrained by the physics of sensory signals. In the tactile sense such constraints are frictional stick–slip events, occurring, amongst other vibrotactile features, when tactile sensors are in contact with objects. We reveal new biomechanical phenomena about the transmission of these microNewton forces at the tip of a rat’s whisker, where they occur, to the base where they engage primary afferents. Using high resolution videography and accurate measurement of axial and normal forces at the follicle, we show that the conical and curved rat whisker acts as a sign-converting amplification filter for moment to robustly engage primary afferents. Furthermore, we present a model based on geometrically nonlinear Cosserat rod theory and a friction model that recreates the observed whole-beam whisker dynamics. The model quantifies the relation between kinematics (positions and velocities) and dynamic variables (forces and moments). Thus, only videographic assessment of acceleration is required to estimate forces and moments measured by the primary afferents. Our study highlights how sensory systems deal with complex physical constraints of perceptual targets and sensors.

Mechanism of avalanche precursors in inclining granular layers using a continuum model obtained by discrete element method

How do large shear stress and stress ratio peaks, which are avalanche precursors, occur in granular layers in an inclining box and why do the precursors have periodicities? The answers are our main objectives. The large and small shear stress peaks which are constituted by the large and small sticks and slips are found in our simulation results. Small sticks and slips, which occur after the large slip, change the granular structure until the stress ratio becomes equal to or larger than the frictional coefficient of granular materials. Here the next large slip with the large stick and slip, which shows the precursor, occurs. The precursors which are constituted by the series of the large stick–slip events have naturally periodicities because the stick–slip events are oscillation phenomena. Our 3-dimensional simulation results using the smoothed particle hydrodynamics (SPH) method based on our constitutive equations obtained by the 3-dimensional discrete element method (DEM) present the precursors in the bulk and on the surface and show the difference between them. The precursors are also found in the longitudinal and the depth directions of the granular layer. The simulated periodicities agree with the published experimental results.

Measurement of Mud Motor Back-Drive Dynamics, Associated Risks, and Benefits of Real-Time Detection and Mitigation Measures

SummaryNorth American shale drilling is a fast-paced environment where downhole drilling equipment is pushed to the limits for the maximum rate of penetration (ROP). Downhole mud motor power sections have rapidly advanced to deliver more horsepower and torque, resulting in different downhole dynamics that have not been identified in the past.High-frequency (HF) compact drilling dynamics recorders embedded in the drill bit, mud motor bit box, and motor top subassembly (top-sub) provide unique measurements to fully understand the reaction of the steerable-motor power section under load relative to the type of rock being drilled. Three-axis shock, gyro, and temperature sensors placed above and below the power section measure the dynamic response of power transfer to the bit and associated losses caused by back-drive dynamics. Detection of back-drive from surface measurements is not possible, and many measurement-while-drilling (MWD) systems do not have the measurement capability to identify the problem. Motor back-drive dynamics severity is dependent on many factors, including formation type, bit type, power section, weight on bit, and drillpipe size. The torsional energy stored and released in the drillstring can be high because of the interaction between surface rotation speed/torque output and mud motor downhole rotation speed/torque. Torsional drillstring energy wind-up and release results in variable power output at the bit, inconsistent rate of penetration, rapid fatigue on downhole equipment, and motor or drillstring backoffs and twistoffs.A new mechanism of motor back-drive dynamics caused by the use of an MWD pulser above a steerable motor has been discovered. HF continuous gyro sensors and pressure sensors were deployed to capture the mechanism in which a positive mud pulser reduces as much as one-third of the mud flow in the motor and bit rotation speed, creating a propensity for a bit to come to a complete stop in certain conditions and for the motor to rotate the drillstring backward. We have observed the backward rotation of a polycrystalline diamond compact (PDC) drill bit during severe stick-slip and back-drive events (−50 rev/min above the motor), confirming that the bit rotated backward for 9 milliseconds (ms) every 133.3 ms (at 7.5 Hz), using a 1,000-Hz continuous sampling/recording in-bit gyro. In one field test, multiple drillstring dynamics recorders were used to measure the motor back-drive severity along the drillstring. It was discovered that the back-drive dynamics are worse at the drillstring, approximately 1,110 ft behind the bit, than these measured at the motor top-sub position. These dynamics caused drillstring backoffs and twistoffs in a particular field. A motor back-drive mitigation tool was used in the field to compare the runs with and without the mitigation tool while keeping the surface drilling parameters nearly the same. The downhole drilling dynamics sensors were used to confirm that the mitigation tool significantly reduced stick-slip and eliminated the motor back-drive dynamics in the same depth interval.Detailed analysis of the HF embedded downhole sensor data provides an in-depth understanding of mud motor back-drive dynamics. The cause, severity, reduction in drilling performance and risk of incident can be identified, allowing performance and cost gains to be realized. This paper will detail the advantages to understanding and reducing motor back-drive dynamics, a topic that has not commonly been discussed in the past.

High-temporal-resolution quasideterministic dynamics of granular stick-slip

We report high-temporal-resolution observations of the spontaneous instability of model granular materials under isotropic and triaxial compression in fully drained conditions during laboratory tests representative of earthquakes. Unlike in natural granular materials, in the model granular materials, during the first stage of the tests, i.e., isotropic compression, a series of local collapses of various amplitudes occurs under random triggering cell pressures. During the second stage, i.e., shearing under triaxial compression, the model granular samples exhibit very large quasiperiodic stick-slip motions at random deviatoric triggering stresses. These motions are responsible for very large stress drops that are described by power laws and are accurate over more than 3 decades in logarithmic space. Then, we identify the quasideterministic nature of these stick-slip events, assuming that they are fully controlled by the cell pressure and solid fraction. Finally, we discuss the potential mechanisms that could explain these intriguing behaviors and the possible links with natural earthquakes.

Analysis of Torsional Stick-Slip Situations from Recorded Downhole Rotational Speed Measurements

Summary The use of recorded downhole rotational speed measurements with a bandwidth up to 9 Hz gives new insights into the conditions under which stick-slip torsional oscillations occur. Observations made while drilling two reservoir sections have shown that, out of all the stick-slip situations identified, 72% of them for one well and 64% for the other well occurred in off-bottom conditions. In these off-bottom conditions, stick-slip was systematically observed while starting the topdrive (TD) until a sufficiently high TD rotational velocity was requested. For these two sections, off-bottom stick-slip was either related to using TD speeds below 120 rev/min or to reaming down during reciprocation procedures. In on-bottom conditions, stick-slip events occurred predominantly when the TD speed was less than 120 rev/min (53 and 32% of the on-bottom cases) but also in association with downlinking to the rotary steerable system (RSS) (23 and 46% of the on-bottom cases), and this, even though the TD speed was larger than 120 rev/min. These on-bottom stick-slip situations did not necessarily occur at a very high weight on bit (WOB) because 98% of them for one well and 46% for the other well took place when the WOB was lower than 10 ton. Downhole measurements have shown that when the drillstring is subject to strong stick-slip conditions, the downhole rotational speed changes from stationary to more than 300 rev/min in just a fraction of a second. Direct observations of downhole rotational speed at high frequency help in discovering conditions that were not suspected to lead to large torsional oscillations. This new information can be used to improve drilling operational procedures and models of the drilling process, therefore enabling increased drilling efficiency.

Probing regolith-covered surfaces in low gravity

The surfaces of many planetary bodies, including asteroids, moons, and planets, are composed of rubble-like grains held together by varying levels of gravitational attraction and cohesive forces. Future instrumentation for operation on, and interacting with, such surfaces will require efficient and effective design principles and methods of testing. Here we present results from the EMPANADA experiment (Ejecta-Minimizing Protocols for Applications Needing Anchoring or Digging on Asteroids) which flew on several reduced gravity parabolic flights. EMPANADA studies the effects of the insertion of a flexible probe into a granular medium as a function of ambient gravity. This is done for an idealized 2D system as well as a more realistic 3D sample. To quantify the dynamics inside the 2D granular material we employ photoelasticity to identify the grain-scale forces throughout the system, while in 3D experiments we use simulated regolith. Experiments were conducted at three different levels of gravity: martian, lunar, and microgravity. In this work, we demonstrate that the photoelastic technique provides results that complement traditional load cell measurements in the 2D sample, and show that the idealized system exhibits similar behaviour to the more realistic 3D sample. We note that the presence of discrete, stick-slip failure events depends on the gravitational acceleration.

Stress drop–magnitude dependence of acoustic emissions during laboratory stick-slip

SUMMARYEarthquake source parameters such as seismic stress drop and corner frequency are observed to vary widely, leading to persistent discussion on potential scaling of stress drop and event size. Physical mechanisms that govern stress drop variations are difficult to evaluate in nature and are more readily studied in controlled laboratory experiments. We perform two stick-slip experiments on fractured (rough) and cut (smooth) Westerly granite samples to explore fault roughness effects on acoustic emission (AE) source parameters. We separate large stick-slip events that generally saturate the seismic recording system from populations of smaller AE events which are sensitive to fault stresses prior to slip. AE event populations show many similarities to natural seismicity and may be interpreted as laboratory equivalent of natural microseismic events. We then compare the temporal evolution of mechanical data such as measured stress release during slip to temporal changes in stress drops derived from AEs using the spectral ratio technique. We report on two primary observations: (1) In contrast to most case studies for natural earthquakes, we observe a strong increase in seismic stress drop with AE size. (2) The scaling of stress drop with magnitude is governed by fault roughness, whereby the rough fault shows a more rapid increase of the stress drop–magnitude relation with progressing large stick-slip events than the smooth fault. The overall range of AE sizes on the rough surface is influenced by both the average grain size and the width of the fault core. The magnitudes of the smallest AE events on smooth faults may also be governed by grain size. However, AEs significantly grow beyond peak roughness and the width of the fault core. Our laboratory tests highlight that source parameters vary substantially in the presence of fault zone heterogeneity (i.e. roughness and narrow grain size distribution), which may affect seismic energy partitioning and static stress drops of small and large AE events.

Seismic and Aseismic Preparatory Processes Before Large Stick\u2013Slip Failure

Natural earthquakes often have very few observable foreshocks which significantly complicates tracking potential preparatory processes. To better characterize expected preparatory processes before failures, we study stick-slip events in a series of triaxial compression tests on faulted Westerly granite samples. We focus on the influence of fault roughness on the duration and magnitude of recordable precursors before large stick–slip failure. Rupture preparation in the experiments is detectable over long time scales and involves acoustic emission (AE) and aseismic deformation events. Preparatory fault slip is found to be accelerating during the entire pre-failure loading period, and is accompanied by increasing AE rates punctuated by distinct activity spikes associated with large slip events. Damage evolution across the fault zones and surrounding wall rocks is manifested by precursory decrease of seismic b-values and spatial correlation dimensions. Peaks in spatial event correlation suggest that large slip initiation occurs by failure of multiple asperities. Shear strain estimated from AE data represents only a small fraction (< 1%) of total shear strain accumulated during the preparation phase, implying that most precursory deformation is aseismic. The relative contribution of aseismic deformation is amplified by larger fault roughness. Similarly, seismic coupling is larger for smooth saw-cut faults compared to rough faults. The laboratory observations point towards a long-lasting and continuous preparation process leading to failure and large seismic events. The strain partitioning between aseismic and observable seismic signatures depends on fault structure and instrument resolution.

A modified model for estimating peak shear displacement of artificial joints

The peak shear displacement (PSD) of artificial joints was studied by the shear test of cement mortar duplicate joints under constant normal load. First, the splitting rock joints were produced by the Brazilian splitting method, and the high-precision point cloud of joint surface morphology was acquired by three-dimensional (3D) scanning method. The polylactic acid (PLA) base mold was produced by 3D printing technology. The duplicate joints with the same joint surface as the splitting rock joints were obtained by pouring cement mortar on the PLA base mold. Then, shear tests of 20 cement mortar specimens with 5 groups of joint surface morphologies under 4 different normal loads were carried out, and the joint shear stress-displacement curves were obtained. The influencing factors of the PSD were studied. The results indicate that the PSD is inversely related to the joint roughness and positively related to the normal stress (NS). The limitation of Barton’s empirical formula is that it cannot predict (fit) the PSD of the joints with JRC = 0 and reflect the effect of NS on the PSD. To quantitatively study the PSD of joints, an empirical formula that can comprehensively consider the influence of surface roughness and NS is introduced based on the test results. The modified model overcomes the limitation of Barton’s empirical formula which ignored the NS and can well reflect the sensitivity of joint roughness coefficient (JRC) to PSD. The acquisition of PSD can provide a basis for the numerical simulation of rock engineering and provide quantitative research methods for the analysis of stick-slip events.