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On the mechanics of granular shear: The effect of normal stress and layer thickness on stick-slip properties
Tectonophysics 763 (2019) 86–99
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On the mechanics of granular shear: The effect of normal stress and layer thickness on stick-slip properties
T
Zheng Lyua, , Jacques Rivièreb, Qiang Yanga, Chris Maronec, ⁎
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a
State Key Laboratory of Hydroscience and Hydraulic Engineering, Tsinghua University, Beijing 100084, China Department of Engineering Science and Mechanics, Pennsylvania State University, University Park, PA 16802, USA c Department of Geosciences, Pennsylvania State University, University Park, PA 16802, USA b
ARTICLE INFO
ABSTRACT
Keywords: Granular material Normal stress Layer thickness Stick-slip
Seismic stress drop is one of the most important earthquake source parameters, playing a key role in elastic energy release and in scaling relations for seismic moment and fault length. While stress drop does not scale systematically with earthquake size, it varies greatly within seismic catalogs and there is much broad interest in understanding how such variations relate to fault zone properties. Here, we address connections between stress drop and fault zone properties via laboratory experiments that investigate the role of normal stress and layer thickness during frictional sliding. We sheared granular layers at normal stresses from 4 to 22 MPa and document both elastic and inelastic processes that couple with layer dilation to determine the granular fragility and stress drop during stick-slip failure. Stick-slip stress drop scales directly with fault normal stress and inversely with layer thickness. Thicker layers exhibit, greater dilation during shear loading, however shear driven dilatant volume strain is independent of layer thickness. We posit that force chains form rapidly after a dynamic slip event and that bulk inelastic creep occurs via formation and destruction of force chains, interparticle slip, and rolling. Stick-slip recurrence time and stress drop vary with fault normal stress and stiffness, which increases with shear strain, consistent with a model in which stiffness increases as porosity decreases and fault zone density increases. We propose a micromechanical model that accounts for force chains and spectator regions where granular processes are dominated by inelastic slip. We document the role of elastic and inelastic processes during three stages of the stick-slip cycle and show how these change systematically as a function of normal stress and layer thickness. Our work shows that dilation during shear loading contributes to frictional strength via volume strain and that apparent friction scales inversely with the granular thinning ratio.
1. Introduction Tectonic faults produce granular wear material, known as fault gouge, during slip, which gives them different physical properties than the surrounding country rock. Fault gouge is also produced during brecciation associated with fracture for energy production and waste storage. The properties of fault gouge dictate the frictional behavior of faults and thus the seismic properties of earthquakes. To improve our understanding of tectonic- and human-induced earthquakes it is essential to understand how the characteristics of granular fault gouge affect the evolution of frictional strength. Laboratory experiments have revealed that characteristics of fault gouge such as particle angularity and size distribution, environmental humidity, fluids, and shear fabric all play an important role in dictating the frictional behavior of faults (Mair et al., 2002; Frye and Marone,
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2002; Anthony and Marone, 2005; Marone et al., 2008; Ikari et al., 2011, 2015; Scuderi et al., 2014, 2015; Jiang et al., 2016; Dorostkar et al., 2017a, 2017b; Lieou et al., 2017; Gheibi and Hedayat, 2018; Hedayat et al., 2018). Existing works also show that vibration has a strong influence on stick-slip instability (Lieou et al., 2015; Lieou et al., 2016; Johnson et al., 2016; Kothari and Elbanna, 2017). However, the influence of layer thickness and normal stress on stick-slip properties such as recurrence time, stress drop, layer dilation, pre-seismic slip and healing rate remain poorly understood. Force chains play an important role in determining the mechanics of granular materials (Miller et al., 1996; Cates et al., 1998; Nasuno et al., 1998; Aharonov and Sparks, 2002; Peters et al., 2005; Kondic et al., 2012; Barés et al., 2017). Force chains have also been suggested as key agents in unstable stick-slip during motion within sheared granular layers used to simulate fault gouge (Biegel et al., 1989; Sammis and
Corresponding authors. E-mail addresses: [email protected] (Z. Lyu), [email protected] (C. Marone).
https://doi.org/10.1016/j.tecto.2019.04.010 Received 12 October 2018; Received in revised form 28 March 2019; Accepted 10 April 2019 Available online 12 April 2019 0040-1951/ © 2019 Elsevier B.V. All rights reserved.
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shear, impact earthquake stress drop and characteristics of the seismic cycle such as recurrence time, pre-seismic slip, fault dilation and healing rate. We carried out a series of 32 complete experiments, with different initial layer thickness, normal stress, and shearing velocity (Table 1). In addition, we used data and results from another 30 experiments conducted under similar conditions to verify our results and conclusions. Our data indicate that properties of stick-slip vary systematically as a function of layer thickness and normal stress. We focus in particular on the relationship between granular friction and changes in layer thickness during the laboratory seismic cycle and interpret our results in terms of the role of force chains during granular shear.
Table 1 Experiments table. Exp
Gouge Thickness, mm
Normal Stress, MPa
Velocity, μm/s
p4561 p4562 p4578 p4579 p4581 p4583 p4588 p4589 p4688 p4690 p4692 p4700 p4701 p4702 p4709 p4711 p4712 p4767 p4768 p4769 p4770 p4780 p4781 p4782 p4783 p4784 p4899 p4900 p4901 p4904 p4905 p4906
5 5 5 5 5 5 5 5 5 5 5 5 4 4 5 3 3 5 5 5 5 7 9 2 1 5 7 1 5 3 3 5
5 5 5 5 2–8-2 2–3-2 2–4 4 5 2–7 2–7-2 5 2–8-2 2–8-2 2–7 2–7 2–7 6–20 2–20 2–8 2–8 4–22 4–22 4–22 4–22 4–22 2 2 2 2 2 16
6–300 1–100 0.3–300 0.3–300 5 5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 650 10 10 10
2. Experiment methods We sheared granular gouge layers composed of glass beads using a biaxial, servohydraulic testing machine and the double direct shear (DDS) configuration. Experiments were conducted at room temperature and humidity. Normal stress was maintained constant and shear stress was applied via a displacement rate boundary condition imposed at the top of the DDS center block. Both normal and vertical forces were measured with strain gauge load cells. Normal and shear displacements were measured using direct current displacement transducers (DCDTs, measurement precision to ~0.1 μm) located between the moving pistons and the fixed frame (Fig. 1). Data were recorded using a 24-bit analog to digital converter at a rate of 10 kHz and averaged to the desired sampling rate from 1 Hz to 10 kHz. In the DDS configuration, two layers are sheared simultaneously between three forcing blocks. This arrangement involves two stationary side blocks, supported from below, and a central forcing block that applies shear to the layer boundaries (Fig. 1). Our forcing blocks were constructed from 316 Stainless steel and equipped with grooves perpendicular to the shearing direction on the surfaces in contact with the gouge (0.8 mm in height, 1 mm spacing). The grooves ensure that shear occurs within the layer, rather than at the boundary, and promote a transition from pervasive shear to progressively localized shear as a function of increasing net strain. Our roughness is a crude approximation of natural fault zone roughness, which is of course much more complex and includes longer wavelength features and striations parallel to slip. The nominal frictional contact area of each layer is 10 × 10 cm2, and this is maintained constant during shear by using a central block (15 cm in length) that is longer than the side blocks. Although two granular layers are sheared simultaneously in DDS, all results shown in this paper are for one single layer. We sheared granular layers of soda-lime glass beads with particle size from 105 to 149 μm in diameter. This material has been used in
Steacy, 1994; Cates et al., 1998; Morgan and Boettcher, 1999; Mair et al., 2002; Anthony and Marone, 2005; Johnson et al., 2008; Tordesillas et al., 2012; Griffa et al., 2013; Scuderi et al., 2015). Although the role of force chains in granular shear has been discussed extensively, many important questions remain such as what is the effective width of a force chain, do force chains play a major role in shear zones dominated by angular or clay rich gouge, and how do force chains operate when shear bands form and divide fault zones into principal slip zones and spectator regions where particle motion and shear strain are near zero. The purpose of this paper is to investigate how fault zone properties and characteristics of gouge particles, modulated by normal stress and
Fig. 1. (a) Biaxial deformation apparatus and double direct shear (DDS) configuration (center). Red arrows show fault normal stress and shear stress, as applied via horizontal and vertical rams, respectively. Displacements and strains are measured via DCDTs attached to each ram. (b) Sample assembly showing DDS configuration, with two gouge layers between three forcing blocks. Thin latex membrane is visible at the bottom. Guide plates at front and back cover the layers. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Fig. 2. Complete run details showing coefficient of friction versus shear load point displacement for five experiments with different initial layer thickness: 1 mm (p4783), 2 mm (p4782), 5 mm (p4784), 7 mm (p4780) and 9 mm (p4781). Shearing rate was 10 μm/s for each experiment. Normal stress was stepped from 4 to 22 MPa, as noted. All experiments show repetitive stick-slip instabilities. Friction varies systematically with layer thickness, showing a clear negative correlation. Fig. 3. Details of experiment p4784. (a) Stick-slip stress drop, Δμ, defined by the maximum and minimum friction for each failure event. (b) Continuous measurements of gouge layer thickness. Note overall layer thinning, associated with geometric spreading, compaction, and dilation for each cycle of load increase and failure. (c) Detail of layer thickness changes for the first five events after removing the thinning ratio. Note that the magnitude of dilation during loading is similar to that of compaction upon failure.
previous work and exhibits highly reproducible stick-slip frictional sliding (e.g., Mair et al., 2002; Anthony and Marone, 2005). Layers were built by placing the side blocks on a leveling jig and carefully packing a measured mass of material to produce a desired layer thickness and initial porosity. We studied a range of layer thicknesses
from 1 to 9 mm and conducted a large suite of closely related tests to verify the results of the experiments presented here. Table 1 provides details of the main set of experiments. Once the layers were constructed, the central block was placed above one side block using a squaring jig to ensure alignment and the blocks were secured with 88
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Fig. 4. Evolution of friction and layer thickness with shear displacement for a complete stick-slip event from experiment p4784 (see Fig. 3). (a) Three stages are apparent in the friction data: elastic loading where friction increases linearly with load point displacement, an inelastic region associated with creep and granular rearrangement, and finally failure, associated with rapid stress drop and radiation of elastic waves. (b) Detrended layer thickness (see Fig. 3) showing dilation during loading followed by compaction during failure. Note the break in slope associated with the onset of inelastic creep.
Fig. 5. (a) Details of friction versus time for a stick-slip event. Recurrence time is the time between failure events. (b) Shear displacement at the gouge layer boundary. Note that slip of ~ 40 μm occurs during failure, after which there is a brief ‘stick' period of zero motion followed by inelastic creep prior to the next failure event.
most pronounced for the experiments on thinner layers. The steady state frictional strength scales inversely with layer thickness (consistent with previous work, e.g. Anthony and Marone, 2005; Knuth and Marone, 2007)) and is linearly proportional to normal stress, as indicated by the roughly constant value of friction over our range of normal stresses (Fig. 2). Especially for the thinnest layers, we observe strain hardening at larger displacements, which is likely due to particle spalling and changes in particle size distribution. We increase normal stress after a few mm of slip and we do not remove the shear stress; thus the friction curves drop and recover over a few stick slip cycles for each step change in normal stress (Fig. 2).
adhesive tape. This procedure was repeated for the other layer. A thin rubber membrane was attached to the bottom of the sample to avoid loss of material during shear. Guide plates were attached over the front and back of each layer (Fig. 1) to avoid extrusion of material there (e.g., Savage and Marone, 2007). The adhesive tape remained on the sample until normal stress was applied, in the apparatus, after which it was cut from the central block. All experiments began by applying a normal stress of 4 MPa. Once initial layer compaction was finished (typically 30 min) the vertical piston was driven at a constant displacement rate of 10 μm/s and this shearing rate was maintained constant throughout the experiment. After the shear stress reached a peak value, stick-slip events stabilized. Then the normal stress was varied in 2 MPa steps (Table 1). Using this approach, we studied normal stresses from 4 to 22 MPa. We also ran experiments at a single normal stress and for a variety to normal stress histories, in order to understand the effect of net strain and loading history (Table 1).
3.1. Stick-slip instability Stick-slip events are apparent during the later section of the initial loading curves (Fig. 2). They grow in amplitude and reach a steady state, with some fluctuation, after shear displacement > 4 mm. Because there are many stick-slip events in each experiment, we developed an automated algorithm to pick the beginning and end of each event. This algorithm also collects other parameters, including peak friction, friction drop, stick-slip recurrence time, and changes in layer thickness. For a given normal stress, the failure events are quite similar and define a consistent stress drop for many cycles (Fig. 3). To compare our range of conditions, we report shear stress drop normalized by layer normal stress, which is friction drop (Fig. 3). We also measure changes in layer thickness during loading and failure. In simple shear under our stress conditions, the layers thin with net shear due to geometric spreading (Scott et al., 1994). The rate of thinning, dh, with shear displacement, dx, defines a thinning ratio dh/dx, which is locally linear
3. Results Upon initial application of shear displacement, the shear stress on the granular layer increased along a linear elastic loading curve (Fig. 2). Normal stress across the layers was maintained constant during shear, so we plot the ratio of shear stress over normal stress, which facilitates comparison over our range of conditions (Fig. 2). The coefficient of friction is given by the ratio of shear stress to normal stress during slip. For simplicity, we refer to this ratio as friction. Stick-slip failure events generally began prior to reaching steady state shear and thus prior to the peak stress seen in most experiments (Fig. 2). The peak stress is 89
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Fig. 6. Evolution of the maximum and minimum friction for each stick-slip event (See Fig. 2) shown as a function of load point displacement. Note that μmax and μmin vary systematically with initial layer thickness, showing a clear negative correlation, and net shear displacement, which increases with event number.
but decreases with net shear (Fig. 3). To first order, dh/dx is entirely due to geometric spreading, however, it may vary with shear due to porosity loss and layer densification. To evaluate thickness changes during a given stick-slip cycle, we first remove the linear thinning trend (e.g. Scuderi et al., 2014), dh/dx (Fig. 3 inset). Note that the detrended layer thickness measurements show that dilation during loading is proportional to compaction during failure and thus the net change in layer thickness over a stick-slip cycle is roughly zero, consistent with expectations from critical-state soil mechanics (Fig. 3 inset). For a layer that is initially 5-mm thick, the compaction during stick-slip failure is about 5 μm. Stick-slip events can be divided into three stages (Fig. 4). First friction increases linearly with shear displacement, defining the elastic stage of loading. This stage can be described by a linear stiffness K, which we report as a normalized quantity: K′ = K/σn, where σn is the layer normal stress (Fig. 4). Thus K′ has units of friction per shear displacement. As shear load increases granular layer thickness also increases slowly due to dilation (Reynolds, 1885). After the stage of elastic loading, the stress displacement curves become convex and exhibit non-linearity that defines an inelastic segment (Fig. 4). Inelastic loading is associated with an increase in the dilatancy rate, determined
from the detrended layer thickness data, which becomes approximately linear until the end of this stage, when failure occurs (Fig. 4). We define the failure stage by the maximum and minimum friction and observe that it is associated with abrupt layer compaction (Fig. 4). Our experiments are run by servo-controlling a load point displacement at the top of the DDS central block (Fig. 1). To get the slip imposed at the boundary of the granular layers, we calculate and remove the displacement associated with elastic deformation of the apparatus, using the loading stiffness which is known from calibrations. After correcting our shear displacement measurements for apparatus elasticity, we have the net slip at the layer boundary for each stage of the stick-slip cycle (Fig. 5). Note that slip is zero during the elastic loading stage, just after failure, and then increases gradually until failure (Fig. 5). We define the stick slip recurrence interval, tr, as the time between each stress minimum (Fig. 5). For each experiment, we record several hundred stick-slip events during net shear displacement of almost 40 mm (Fig. 2). To show the evolution of stick-slip cycles with shear displacement and normal stress, we plot the maximum μmax and minimum μmin friction as a function of event number in Fig. 6. Maximum friction shows a clear negative correlation with layer thickness (consistent with previous work, e.g. 90
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Fig. 7. Systematics of stress drop and layer dilation for all failure events. Each symbol represents the mean ± 1 std. dev. Stress drop (normalized by layer normal stress to produce friction drop) varies inversely with layer thickness and shows a maximum at a normal stress of about 18 MPa. Layer dilation is larger for thicker layers and the variation with normal stress is consistent with that for stress drop.
Anthony and Marone, 2005). Note that the maximum and minimum friction values for stick-slip in thin layers (Fig. 6) reflect a small degree of shear strain weakening at normal stresses below 16 MPa. All of our samples show shear strain hardening for normal stresses above 16 MPa, which occurs at shear displacements above 25 mm (Fig. 6). Note that both the maximum friction, at the beginning of a stick slip event, and the minimum friction, at the end of each stick-slip event, increase with shear strain, reflecting shear strain hardening toward the end of our experiments. While our experiments are conducted at stresses below the particle crushing strength (Mair et al., 2002), spalling and particle fracture does occur for the upper range of our normal stresses, particularly for the thinner layers. The increase in frictional strength and minimum friction after stick slip failure (Figs. 2 and 6) is consistent with layer densification associated with shear induced increases in packing density and changes in particle size distribution.
Dilation scales directly with layer thickness, such that thicker layers generally show greater dilation, with some scatter (Fig. 7). The basic observation is straightforward and consistent with Reynolds (1885) dilation in a dense granular media sheared at stresses below the particle fracture stress. For small strains, associated with an individual stick-slip event, thicker layers dilate more because dilation is a strain hardening process and therefore the whole layer is involved. After strain localizes, dilation is restricted to the principal slip surface and spectator regions of the granular shear zone do not dilate (e.g., Frank, 1965). Our data on stick-slip stress drop show a negative correlation with layer thickness and, especially for the thinnest layers, a complex variation with normal stress (Fig. 7). We report the mean values ± 1 standard deviation for all events in our dataset (Fig. 7). Note that the range of stick-slip dilation values scales with layer thickness and that this range is reflected in the friction drop data. Stick-slip friction drop shows a pronounced increase with normal stress for the thinnest layers and a maximum at 18 MPa (Fig. 7). Part of these changes in friction drop could be associated with spalling and grain fracture, because the highest normal stresses occur near the end of our experiments, however, previous works show that particle crushing
3.2. Effect of initial layer thickness and normal stress on stick-slip events We quantify the variation in stick-slip stress drop and layer dilation for each event as a function of layer thickness and normal stress (Fig. 7). 91
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Fig. 8. Stick-slip recurrence interval and creep slip during inelastic loading for all failure events (see also Fig. 7). Each symbol represents the mean ± 1 std. dev. Within the measurement uncertainty, recurrence time is independent of layer thickness but varies systematically with normal stress, decreasing from 4 to 8 MPa and increasing to a peak value at 18–20 MPa. The magnitude of inelastic slip is independent of layer thickness, within the measurement uncertainty, and decreases with normal stress.
Fig. 9. Variation of the minimum friction with layer thinning ratio dh/dx for all stick-slip failure events. Both thinning ratio and μmin vary with layer thickness, but for the majority of our data they define a trend that is independent of layer thickness (see best fit linear relation). 92
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Fig. 10. Friction drop as a function of layer dilation (a) and normalized dilation (b) for all failure events. Each symbol represents the mean ± 1 std. dev. Thicker layers show systematically lower stick-slip friction drop and greater dilation than thinner layers. The data collapse to a single line when dilation is normalized by thickness, which represents a measure of volumetric strain.
and grain angularity lead to smaller stress drop and stable sliding (Mair et al., 2002), which is the opposite of what we observe (Fig. 7). Thus, while particle angularity may increase somewhat during shear we do not believe that particle crushing plays a major role in our experiments. Also, post experiment inspection of the material confirms that grain crushing and changes in particle size distribution are insignificant. We quantify stick-slip recurrence interval and inelastic slip for each event (Fig. 8) to complement the data of Fig. 7. Within the measurement uncertainty, recurrence time is independent of layer thickness but varies systematically with normal stress, showing a peak value at 18–20 MPa. For normal stresses below 10 MPa, the magnitude of inelastic slip varies little with layer thickness but decreases with normal stress. While for normal stresses above 10 MPa, the magnitude of inelastic slip decreases with decreasing layer thickness, especially for 2 mm and 1 mm layers, but is independent of normal stress. Taken together, these data show that larger events have longer recurrence
interval and that inelastic creep decreases at higher normal stress, consistent with the idea that layers are more compact and stiffer (Figs. 7 and 8). We find that in each experiment, dilation shows a different trend below a normal stress of 10 MPa, between 10 and 18 MPa and above 18 MPa (Fig. 7a). Dilation during a stick-slip event decreases slightly with normal stress below 10 MPa, but it increases for normal stresses above 10 MPa and decreases again above 18 MPa. This trend is clearest for the thicker layers. A similar trend is observed for friction-drop (Fig. 7b), except here, thinner layers show clearer and larger amplitude increases from 10 to 18 MPa. The largest mean friction-drop reaches ~0.17 at 18 MPa for 1-mm layers, which is almost 50% larger than friction-drop at normal stress lower than 10 MPa. In contrast, for 9-mm thick layers the mean friction-drop is roughly 0.09 and shows no clear trend with normal stress.
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Fig. 11. Shear stress drop increases with stick-slip recurrence time for normal stress > 8 MPa. At lower normal stresses, a weak negative trend is apparent (see also Fig. 8). Inset shows data for 1 mm layers, indicating the negative healing rate before normal stress exceeds 8 MPa.
4. Discussion
drop, which varied with event recurrence time. Their experiments explored a range of fault slip velocities and humidity. We observe similar trends for each of our layer thickness (Fig. 10a) and the data define a single trend, with some scatter, when friction drop is compared to layer volume strain (Fig. 10b). Thus, we posit that stick-slip friction-drop is modulated by the normalized dilation, which is essentially a layernormal strain. The idea is that normal strain is a proxy for energy potential, with larger values associated with a more fragile granular structure, thus allowing for greater friction-drop.
4.1. Layer thickness influence on friction behavior Our observations show that friction and significant aspects of the stick-slip cycle vary systematically with layer thickness (Figs. 2 and 6). Thicker layers have relatively lower apparent friction, consistent with previous work (Scott et al., 1994; Ikari et al., 2011). The previous analysis for the relationship between layer thickness and apparent friction (e.g. Scott et al., 1994) is based on the contribution of volume strain to frictional strength and it predicts that apparent friction should scale inversely with the thinning ratio. Following their results, we consider the relationship between μmin and the layer thinning ratio (Fig. 9). Here, the discrete thinning ratio is calculated for each stick-slip event, using the layer thickness change and shear displacement. The data of Fig. 9 show a clear trend for a wide range of conditions. A line fit to the main data concentration (Fig. 9) is consistent with the idea that thinning contributes to frictional weakening (e.g. Scott et al., 1994). Looking at the data outside the main cloud (Fig. 9), we see that minimum friction is high when the thinning ratio is low, which happens for our thin layers. At the other extreme, minimum friction shows a long tail for large values of thinning ratio, suggesting that other factors dictate the minimum friction under these conditions. We also observe a relationship between layer thickness and the magnitude of dilation (Figs. 7). We extend this to evaluate the relationship between stick-slip friction drop and shear driven dilation during the stick-slip cycle (Figs. 4 and 10). The raw data show that thicker layers have larger dilation during loading and smaller stress drop during failure (Fig. 10a). However, when we normalize dilatant changes in layer height by the layer thickness just prior to stick-slip, we see that all events collapse to a single line, with some scatter (Fig. 10b). The data of Fig. 10b connect with previous work on the relationship between dilation, stick-slip stress drop, and frictional healing. Scuderi et al. (2014) conducted stick-slip experiments and showed changes in friction drop as a function of dilation for a range of event recurrent times. They found that larger dilation was associated with larger stress
4.2. Normal stress influence on frictional healing and recurrence time The role of shearing rate on stick-slip recurrence time is well established (e.g., Karner and Marone, 2000; Beeler et al., 2014). Here, we find that normal stress also plays a role in determining recurrence time for granular shear (Fig. 8). Part of this could arise from increased frictional strength at higher normal stress but the major effect is due to changes in the elastic vs. inelastic segments of loading. We further explore this relationship by plotting stress drop vs. stickslip recurrence interval (Fig. 11). The main trend of these data is an increase in stress drop for larger recurrence intervals. This is consistent with the idea that for a given loading rate, a larger stress drop requires a longer loading duration to reach the failure stress (Fig. 11). Our data also show another, weaker, trend for lower normal load. For normal stresses below 8 MPa, a negative rate can be seen (Fig. 11). This behavior is especially clear for the 1-mm layer (Fig. 11 inset). At low normal stress, larger values of recurrence interval are associated with smaller stress drop. This occurs when the elastic stage of loading is short and loading is dominated by inelastic processes, which produces a lower overall loading rate and lesser dilation which in turn produces a lower potential energy, less fragile granular state and thus smaller stress drop. Taken to the extreme, where inelastic creep dominates and elastic loading is negligible, sliding would be stable, which is consistent with a simple extrapolation of our data trend to zero stress drop (Fig. 11). To further examine these ideas, we plot recurrence time vs. normal stress for one complete experiment (5-mm layer thickness) and break tr 94
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Fig. 12. Detail of the stick-slip recurrence time tr as a function of normal stress for one complete experiment (p4784). (a) Breakdown of tr into the elastic and inelastic loading segments of the stick-slip cycle (see Fig. 5): te is the time of elastic loading, ti is the time of inelastic creep loading. (b) Elastic and inelastic fractions of the loading time. Note that the elastic portion of the loading curve becomes greater with normal stress while the time of inelastic loading becomes proportionally less: extrapolation of these trends indicates that they will become equal at roughly 25 MPa, slightly above our highest normal stress.
into the intervals associated with elastic and inelastic loading (Fig. 12). These data are representative of our suite of experiments (Table 1) and provide a clear picture of why stick-slip recurrence time first decreases with normal stress, up to 8 MPa, and then increases with increasing normal stress (Fig. 12). For low normal load (lower than 8 MPa), shear loading is dominated by inelastic processes and although the elastic fraction increases with increasing normal stress, the rate of increase is smaller than the rate at which the inelastic portion decreases, resulting in an overall decrease in recurrence time (Fig. 12a). For our range of conditions, shear loading is dominated by inelastic processes. Elastic processes become increasingly important at higher normal loads, and our data indicate that they exceed inelastic processes at a normal stress of roughly 25 MPa.
1976; Ruina, 1983; Gu et al., 1984). The effective elastic stiffness for our experiments is determined by the testing machine, the DDS sample assembly, and the granular layers. That is, the net shear stiffness K can be described as: 1/Kmachine+1/KDDS + 1/Klayer = 1/K. The machine stiffness and the stiffness of the DDS assembly are well known from calibration and the net stiffness K is readily measured in-situ during the stick-slip cycles. The stiffness of the granular layer is also readily measured (e.g., Leeman et al., 2016; Scuderi et al., 2016), however, previous works have typically ignored its role and/or assumed that it does not change with shear. Here, we address layer stiffness and its variation with normal stress, layer thickness, and net strain. The changes in Kmachine and KDDS are negligible for our range of conditions, thus all changes in K can be attributed to Klayer. We measure the net stiffness K from the elastic portion of the loading curves (Fig. 4). Following Leeman et al. (2016) we use an automated routine to measure stiffness and track its evolution throughout shear (Fig. 13). We find that K varies with both normal stress, layer thickness, and net shear strain. The symbols in Fig. 13 are colour coded
4.3. Elastic stiffness A simple spring-slider model has been used widely in studies of frictional stick slip (e.g., Dieterich, 1978, 1979; Johnson and Scholz, 95
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small. The overall trends of stiffness vs. layer thickness, normal stress, and shear strain indicate a stress and shear driven densification process that decreases in rate as density increases. 4.4. Micromechanical model for force chain formation and breakage during shear Force chains are known to play an important role in granular processes (Sammis et al., 1987; Cates et al., 1998; Morgan and Boettcher, 1999; Aharonov and Sparks, 1999; Mair et al., 2002; Geng et al., 2003; Anthony and Marone, 2005; Johnson et al., 2008; Griffa et al., 2013; Scuderi et al., 2014; Dorostkar et al., 2018; Gao et al., 2018). Existing models show that force chains become less brittle as layer thickness increases, leading to a transition from unstable to stable sliding (Mair et al., 2002; Anthony and Marone, 2005). This is consistent with the idea that fragility scales with the number of particles in a chain (Cates et al., 1998). Moreover, these ideas suggest that longer force chains are more compliant and may undergo significant creep prior to failure. In a study of frictional healing for a granular medium, Scuderi et al. (2014) suggested that force chains form directly after a dynamic slip event and that pre-seismic slip and dilation are due to elasto-plastic deformation within force chains associated with rolling and interparticle slip. Here we propose a synoptic model (Fig. 14) based on granular force chains to explain our results. We envision that elastic loading is initially associated with pervasive strain across the layer and that forces are distributed equally among many if not all of the particles within the layer (Fig. 14). During this stage, the whole layer deforms as a granular solid, as indicated by the decreasing slope of inverse stiffness with increasing normal stress (Fig. 13). This means that the shear modulus increases with increasing packing density caused by increasing normal stress. This stage results in elastic deformation at contact junctions and also particle rolling. The rolling motion is small and reversible, such as commonly observed in dilatant granular systems. With accumulated shear load, force chains begin to form and resist shear motion. Many grains must be fully mobilized in the process of building long force chains. In the context of the stick-slip cycle, this stage begins at the end of elastic loading. At the onset of inelastic creep stage, the dominant force chains start to rotate simultaneously causing layer dilation (Fig. 15). At this point, the layers are separated into two regions: region one, which is dominated by the rotating force chains (Fig. 14) and a second region, where force chains are absent. Region 1 is in the center of the layer in Fig. 14, but it may also occur near the boundaries. Of course the measured layer thickness changes are determined by the combined effect of both regions. Region 1 will tend to dilate during the early part of the stick-slip cycle when shear stress is increasing, whereas Region 2 may consolidate via particle rolling and sliding (Fig. 14). If dilation exceeds thinning, the measured layer thickness will show dilation, which is what we observe in most cases (layer thicknesses of 1 to 7 mm Fig. 16a–c). However, when Region 2 dominates, thinning will exceed dilation and we should see an overall thinning. This is the case for our 9-mm thick layers, as seen in Fig. 16d.
Fig. 13. Relationship between inverse elastic loading stiffness (see Fig. 4) and layer thickness h for five experiments, with colour showing normal stress. Note that for a given normal stress, the data define a distinct slope (dashed lines). The slope of these lines decreases systematically with increasing normal stress. Within a given experiment, stiffness increases with shear displacement and layer thinning.
to show normal stress. For a given experiment, stiffness increases along a trend of decreasing layer thickness that becomes increasingly nonlinear. For a given normal stress, the data define a roughly linear relationship between 1/K and layer thickness, with big changes in slopes as a function of normal stress. Within a given experiment, stiffness increases with shear displacement and layer thinning, defining a distinct slope for a given normal stress (Fig. 13). The trend of decreasing K as layer thickness increases (dashed lines) decreases systematically with increasing normal stress. These data are consistent with a model in which layer stiffness increases as a function of decreasing porosity and increasing granular packing density. Granular porosity decreases with normal stress as grains are forced into a closer packing arrangement. This is the main effect seen in Fig. 13, where stiffness increases rapidly at normal stresses below 8 MPa. At higher normal stress, it becomes more difficult to increase packing density and the slope of the lines decrease, indicating smaller changes in stiffness for a given change in layer thickness. Shear enhanced compaction is also an important factor. Shear occurs throughout our experiments and the trends in Fig. 13 show both the effects of normal stress and shear strain. For thicker layers and low normal stress, even a small change in normal stress produces a big change in stiffness for a given layer thickness. As normal stress increases, layers become dense and additional changes in stiffness are
Fig. 14. Schematic model of granular processes during shear with black and red dashed lines showing the positions of a strain marker at the beginning and end of each stage. At the beginning of a stick-slip cycle, shear strain is pervasive across the layer, such that the whole layer deforms as a granular solid. At the onset of inelastic creep, the strongest force chains rotate which means this zone will dilate. At the same time there are regions where force chains are absent. These are regions of inelastic creep due to particle sliding and rolling, which results in local compaction and layer thinning. Once the strong force chains reach a limit and break there is a sudden collapse, with layer compaction and stress drop. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 96
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Fig. 15. Details of change in friction and layer thickness (raw data) during shear loading (data from experiment p4904 with 3 mm layer, 2 MPa normal stress and 10um/s shear velocity). Note that in each stick-slip cycle, at the beginning, the layer is thinning at the predicted thinning ratio (green dashed line). This stage of layer thinning occurs simultaneously with elastic loading. Note that layers begin to dilate at the same point that inelastic creep starts. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Fig. 16. Details of layer thickness (raw data) change during shear loading under 6 MPa normal stress for a) 1 mm, b) 5 mm, c) 7 mm, d) 9 mm layers. Note that for 1 mm, 5 mm, 7 mm layers, layer thickness increased during inelastic loading stage by 2 to 1 μm, while for 9 mm layer thickness decreased by 1.3 μm during this inelastic stage. This represents the “thinning region” described in our model (Fig. 14) are getting thicker with increasing layer thickness.
In agreement with previous works (e.g., Mair et al., 2002; Anthony and Marone, 2005; Scuderi et al., 2014) our data suggest that force chains form immediately upon failure and support elastic stress accumulation. In addition, we posit that force chains continue to strengthen and evolve during inelastic loading in association with particle rearrangement and local changes in packing density. The dominance of granular rearrangement coincides with net shear and inelastic strain within the granular layer (Fig. 5) resulting in dilatancy (Fig. 15). With continued strain, we posit that multiple force chains form and fail as part of the process of forming a support structure for the larger chains
that carry the load. The grain structure of a force chain is presumably maintained dynamically, with particles being continually added and lost during shear (Fig. 14). In our model, stress drop and shear stiffness scale inversely with force chain length (Figs. 13 and 14). Thus, when we consider the total accumulated shear in our experiments (e.g., Fig. 2), our data indicate that shorter force chains form in thinner layers, as suggested by the data of Fig. 13. In the context of the stick-slip cycle, our data suggest that layer dilation should scale with the slip required to form force chains and thus dilation should increase with layer thickness, as we observe 97
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(Fig. 10). In summary, force chains support stresses across the granular layers. They form during the inelastic stage of loading in association with particle motion and dilation (Fig. 14). Upon failure, the force chains break, compaction occurs throughout the layer, and new force chains form during the next cycle.
induced stabilization in stick-slip dynamics of weakly wet, sheared granular fault gouge. J. Geophys. Res. Solid Earth 123 (3), 2115–2126. https://doi.org/10.1002/ 2017JB015171. Frank, F., 1965. On dilatancy in relation to seismic sources. Rev. Geophys. 3, 485–503. Frye, K.M., Marone, C., 2002. The effect of particle dimensionality on granular friction in laboratory shear zones. Geophys. Res. Lett. 29 (19). https://doi.org/10.1029/ 2002GL015709. Gao, K., Euser, B.J., Rougier, E., Guyer, R.A., Lei, Z., Knight, E.E., Johnson, P.A., 2018. Modeling of stick-slip behavior in sheared granular fault gouge using the combined finite-discrete element method. J. Geophys. Res. Solid Earth. https://doi.org/10. 1029/2018JB015668. Geng, J., Reydellet, G., Clément, E., Behringer, R.P., 2003. Green's function measurements of force transmission in 2D granular materials. Physica D 182 (3–4), 274–303. https://doi.org/10.1016/S0167-2789(03)00137-4. Gheibi, A., Hedayat, A., 2018. Ultrasonic investigation of granular materials subjected to compression and crushing. Ultrasonics 87, 112–125. https://doi.org/10.1016/j. ultras.2018.02.006. Griffa, M., Ferdowsi, B., Guyer, R.A., Daub, E.G., Johnson, P.A., Marone, C., Carmeliet, J., 2013. Influence of vibration amplitude on dynamic triggering of slip in sheared granular layers. Phys. Rev. E 87 (1), 012205. https://doi.org/10.1103/PhysRevE.87. 012205. Gu, J.C., Rice, J.R., Ruina, A.L., Tse, S.T., 1984. Slip motion and stability of a single degree of freedom elastic system with rate and state dependent friction. J. Mech. Phys. Solids 32 (3), 167–196. Hedayat, A., Haeri, H., Hinton, J., Masoumi, H., Spagnoli, G., 2018. Geophysical signatures of shear-induced damage and frictional processes on rock joints. J. Geophys. Res. Solid Earth 123 (2), 1143–1160. https://doi.org/10.1002/2017JB014773. Ikari, M.J., Niemeijer, A.R., Marone, C., 2011. 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Nature 451 (7174), 57. https://doi.org/10.1038/nature06440. Johnson, P.A., Carmeliet, J., Savage, H.M., Scuderi, M., Carpenter, B.M., Guyer, R.A., Marone, C., 2016. Dynamically triggered slip leading to sustained fault gouge weakening under laboratory shear conditions. Geophys. Res. Lett. 43 (4), 1559–1565. https://doi.org/10.1002/2015GL067056. Karner, S.L., Marone, C., 2000. Effects of loading rate and normal stress on stress drop and stick-slip recurrence interval. Geophys. Monogr. Am. Geophys. Union 120, 187–198. Knuth, M., Marone, C., 2007. Friction of sheared granular layers: the role of particle dimensionality, surface roughness, and material properties. Geochem. Geophys. Geosyst. 8https://doi.org/10.1029/2006GC001327. (Q03012). Kondic, L., Goullet, A., O'Hern, C.S., Kramar, M., Mischaikow, K., Behringer, R.P., 2012. Topology of force networks in compressed granular media. EPL (Europhys. Lett.) 97 (5), 54001. https://doi.org/10.1209/0295-5075/97/54001. Kothari, K.R., Elbanna, A.E., 2017. Localization and instability in sheared granular materials: role of friction and vibration. Phys. Rev. E 95 (2), 022901. https://doi.org/10. 1103/PhysRevE.95.022901. Leeman, J.R., Saffer, D.M., Scuderi, M.M., Marone, C., 2016. Laboratory observations of slow earthquakes and the spectrum of tectonic fault slip modes. Nat. Commun. 7, 11104. https://doi.org/10.1038/ncomms11104. Lieou, C.K., Elbanna, A.E., Langer, J.S., Carlson, J.M., 2015. Stick-slip instabilities in sheared granular flow: the role of friction and acoustic vibrations. Phys. Rev. E 92 (2), 022209. https://doi.org/10.1103/PhysRevE.92.022209. Lieou, C.K., Elbanna, A.E., Carlson, J.M., 2016. Dynamic friction in sheared fault gouge: implications of acoustic vibration on triggering and slow slip. J. Geophys. Res. Solid Earth 121 (3), 1483–1496. https://doi.org/10.1002/2015JB012741. Lieou, C.K., Daub, E.G., Guyer, R.A., Ecke, R.E., Marone, C., Johnson, P.A., 2017. Simulating stick-slip failure in a sheared granular layer using a physics-based constitutive model. J. Geophys. Res. Solid Earth 122 (1), 295–307. https://doi.org/10. 1002/2016JB013627. Mair, K., Frye, K.M., Marone, C., 2002. Influence of grain characteristics on the friction of granular shear zones. J. Geophys. Res. Solid Earth 107 (B10). https://doi.org/10. 1029/2001JB000516. Marone, C., Carpenter, B.M., Schiffer, P., 2008. Transition from rolling to jamming in thin granular layers. Phys. Rev. Lett. 101 (24), 248001. https://doi.org/10.1103/ PhysRevLett.101.248001. Miller, B., O'Hern, C., Behringer, R.P., 1996. Stress fluctuations for continuously sheared granular materials. Phys. Rev. Lett. 77 (15), 3110. https://doi.org/10.1103/ PhysRevLett.77.3110. Morgan, J.K., Boettcher, M.S., 1999. Numerical simulations of granular shear zones using the distinct element method: 1. Shear zone kinematics and the micromechanics of localization. J. Geophys. Res. 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5. Conclusions We study stick-slip stress drop for laboratory faults composed of synthetic granular fault gouge (glass beads) sheared within rough surfaces. Our experiments focus on a single fault surface roughness and granular particle size, which do not vary as a function of strain. We find that layer thickness and normal stress both have a significant effect on stick-slip sliding such as frictional strength, friction drop, layer dilation, recurrence time, and inelastic slip. We show that stress drop and both minimum and maximum friction decrease with increasing layer thickness. We also show that normal stress has a different effect above and below 10 MPa. Below 10 MPa, dilation, friction-drop and recurrence time decrease slightly and inelastic slip decreases strongly with increasing normal stress, but above 10 MPa, dilation, friction-drop and recurrence time increase and inelastic slip decreases more slowly with increasing normal stress. We note three distinct stages of the stick-slip cycle. In our model we suggest that during elastic loading, the layer is deforming evenly as a whole body with some normal stress controlled shear modulus, then inelastic loading starts and force chains emerge. Acknowledgements We thank S. Swavely for technical help in the laboratory and Paul Johnson for key discussions. We gratefully acknowledge the comprehensive reviews of two reviewers and the Editor as well as support from the DOE geothermal program (DOE EERE DE-EE0006762), the CSES program at LANL, the NSF Geophysics program (EAR-1520760 and EAR-1547286/1547441), and the National Natural Science Foundation of China (grant No. 51739006, 11572174). The first author would also like to acknowledge the financial support from China Scholarship Council (No. 201506210328). References Aharonov, E., Sparks, D., 1999. Rigidity phase transition in granular packings. Phys. Rev. E 60 (6), 6890. https://doi.org/10.1103/PhysRevE.60.6890. Aharonov, E., Sparks, D., 2002. Shear profiles and localization in simulations of granular materials. Phys. Rev. E 65 (5), 051302. https://doi.org/10.1103/PhysRevE.65. 051302. Anthony, J.L., Marone, C., 2005. Influence of particle characteristics on granular friction. J. Geophys. Res. Solid Earth (B8), 110. https://doi.org/10.1029/2004JB003399. Barés, J., Wang, D., Wang, D., Bertrand, T., O'Hern, C.S., Behringer, R.P., 2017. Local and global avalanches in a two-dimensional sheared granular medium. Phys. Rev. E 96 (5), 052902. https://doi.org/10.1103/PhysRevE.96.052902. Beeler, N.M., Tullis, T., Junger, J., Kilgore, B., Goldsby, D., 2014. Laboratory constraints on models of earthquake recurrence. J. Geophys. Res. Solid Earth 119 (12), 8770–8791. Biegel, R.L., Sammis, C.G., Dieterich, J.H., 1989. The frictional properties of a simulated gouge having a fractal particle distribution. J. Struct. Geol. 11 (7), 827–846. https:// doi.org/10.1016/0191-8141(89)90101-6. Cates, M.E., Wittmer, J.P., Bouchaud, J.P., Claudin, P., 1998. Jamming, force chains, and fragile matter. Phys. Rev. Lett. 81 (9), 1841. https://doi.org/10.1103/PhysRevLett. 81.1841. Dieterich, J.H., 1978. Time-dependent friction and the mechanics of stick-slip. In: Rock Friction and Earthquake Prediction. Birkhäuser, Basel, pp. 790–806. https://doi.org/ 10.1007/978-3-0348-7182-2_15. Dieterich, J.H., 1979. Modeling of rock friction: 1. Experimental results and constitutive equations. J. Geophys. Res. Solid Earth 84 (B5), 2161–2168. https://doi.org/10. 1029/JB084iB05p02161. Dorostkar, O., Guyer, R.A., Johnson, P.A., Marone, C., Carmeliet, J., 2017a. On the role of fluids in stick-slip dynamics of saturated granular fault gouge using a coupled computational fluid dynamics-discrete element approach. J. Geophys. Res. Solid Earth 122 (5), 3689–3700. https://doi.org/10.1002/2017JB014099. Dorostkar, O., Guyer, R.A., Johnson, P.A., Marone, C., Carmeliet, J., 2017b. On the micromechanics of slip events in sheared, fluid-saturated fault gouge. Geophys. Res. Lett. 44 (12), 6101–6108. https://doi.org/10.1002/2017GL073768. Dorostkar, O., Guyer, R.A., Johnson, P.A., Marone, C., Carmeliet, J., 2018. Cohesion-
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Z. Lyu, et al. Peters, J.F., Muthuswamy, M., Wibowo, J., Tordesillas, A., 2005. Characterization of force chains in granular material. Phys. Rev. E 72 (4), 041307. https://doi.org/10.1103/ PhysRevE.72.041307. Reynolds, O., 1885. LVII. On the dilatancy of media composed of rigid particles in contact. With experimental illustrations. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 20 (127), 469–481. https://doi.org/ 10.1080/14786448508627791. Ruina, A., 1983. Slip instability and state variable friction laws. J. Geophys. Res. Solid Earth 88 (B12), 10359–10370. https://doi.org/10.1029/JB088iB12p10359. Sammis, C.G., Steacy, S.J., 1994. The micromechanics of friction in a granular layer. Pure Appl. Geophys. 142 (3–4), 777–794. https://doi.org/10.1007/BF00876064. Sammis, C., King, G., Biegel, R., 1987. The kinematics of gouge deformation. Pure Appl. Geophys. 125 (5), 777–812. https://doi.org/10.1007/BF00878033. Savage, H., Marone, C., 2007. The effects of shear loading rate vibrations on stick-slip behavior in laboratory experiments. J. Geophys. Res. 112https://doi.org/10.1029/ 2005JB004238. (B02301). Scott, D.R., Marone, C.J., Sammis, C.G., 1994. The apparent friction of granular fault
gouge in sheared layers. J. Geophys. Res. Solid Earth 99 (B4), 7231–7246. https:// doi.org/10.1029/93JB03361. Scuderi, M.M., Carpenter, B.M., Marone, C., 2014. Physicochemical processes of frictional healing: Effects of water on stick-slip stress drop and friction of granular fault gouge. J. Geophys. Res. Solid Earth 119 (5), 4090–4105. https://doi.org/10.1002/ 2013JB010641. Scuderi, M.M., Carpenter, B.M., Johnson, P.A., Marone, C., 2015. Poromechanics of stickslip frictional sliding and strength recovery on tectonic faults. Journal of Geophysical Research: Solid Earth 120 (10), 6895–6912. https://doi.org/10.1002/ 2015JB011983. Scuderi, M.M., Marone, C., Tinti, E., Di, S.G., Collettini, C., 2016. Precursory changes in seismic velocity for the spectrum of earthquake failure modes. Nat. Geosci. 9 (9), 695–700. Tordesillas, A., Walker, D.M., Froyland, G., Zhang, J., Behringer, R.P., 2012. Transition dynamics and magic-number-like behavior of frictional granular clusters. Phys. Rev. E 86 (1), 011306. https://doi.org/10.1103/PhysRevE.86.011306.
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On the mechanics of granular shear: The effect of normal stress and layer thickness on stick-slip properties
T
Zheng Lyua, , Jacques Rivièreb, Qiang Yanga, Chris Maronec, ⁎
⁎
a
State Key Laboratory of Hydroscience and Hydraulic Engineering, Tsinghua University, Beijing 100084, China Department of Engineering Science and Mechanics, Pennsylvania State University, University Park, PA 16802, USA c Department of Geosciences, Pennsylvania State University, University Park, PA 16802, USA b
ARTICLE INFO
ABSTRACT
Keywords: Granular material Normal stress Layer thickness Stick-slip
Seismic stress drop is one of the most important earthquake source parameters, playing a key role in elastic energy release and in scaling relations for seismic moment and fault length. While stress drop does not scale systematically with earthquake size, it varies greatly within seismic catalogs and there is much broad interest in understanding how such variations relate to fault zone properties. Here, we address connections between stress drop and fault zone properties via laboratory experiments that investigate the role of normal stress and layer thickness during frictional sliding. We sheared granular layers at normal stresses from 4 to 22 MPa and document both elastic and inelastic processes that couple with layer dilation to determine the granular fragility and stress drop during stick-slip failure. Stick-slip stress drop scales directly with fault normal stress and inversely with layer thickness. Thicker layers exhibit, greater dilation during shear loading, however shear driven dilatant volume strain is independent of layer thickness. We posit that force chains form rapidly after a dynamic slip event and that bulk inelastic creep occurs via formation and destruction of force chains, interparticle slip, and rolling. Stick-slip recurrence time and stress drop vary with fault normal stress and stiffness, which increases with shear strain, consistent with a model in which stiffness increases as porosity decreases and fault zone density increases. We propose a micromechanical model that accounts for force chains and spectator regions where granular processes are dominated by inelastic slip. We document the role of elastic and inelastic processes during three stages of the stick-slip cycle and show how these change systematically as a function of normal stress and layer thickness. Our work shows that dilation during shear loading contributes to frictional strength via volume strain and that apparent friction scales inversely with the granular thinning ratio.
1. Introduction Tectonic faults produce granular wear material, known as fault gouge, during slip, which gives them different physical properties than the surrounding country rock. Fault gouge is also produced during brecciation associated with fracture for energy production and waste storage. The properties of fault gouge dictate the frictional behavior of faults and thus the seismic properties of earthquakes. To improve our understanding of tectonic- and human-induced earthquakes it is essential to understand how the characteristics of granular fault gouge affect the evolution of frictional strength. Laboratory experiments have revealed that characteristics of fault gouge such as particle angularity and size distribution, environmental humidity, fluids, and shear fabric all play an important role in dictating the frictional behavior of faults (Mair et al., 2002; Frye and Marone,
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2002; Anthony and Marone, 2005; Marone et al., 2008; Ikari et al., 2011, 2015; Scuderi et al., 2014, 2015; Jiang et al., 2016; Dorostkar et al., 2017a, 2017b; Lieou et al., 2017; Gheibi and Hedayat, 2018; Hedayat et al., 2018). Existing works also show that vibration has a strong influence on stick-slip instability (Lieou et al., 2015; Lieou et al., 2016; Johnson et al., 2016; Kothari and Elbanna, 2017). However, the influence of layer thickness and normal stress on stick-slip properties such as recurrence time, stress drop, layer dilation, pre-seismic slip and healing rate remain poorly understood. Force chains play an important role in determining the mechanics of granular materials (Miller et al., 1996; Cates et al., 1998; Nasuno et al., 1998; Aharonov and Sparks, 2002; Peters et al., 2005; Kondic et al., 2012; Barés et al., 2017). Force chains have also been suggested as key agents in unstable stick-slip during motion within sheared granular layers used to simulate fault gouge (Biegel et al., 1989; Sammis and
Corresponding authors. E-mail addresses: [email protected] (Z. Lyu), [email protected] (C. Marone).
https://doi.org/10.1016/j.tecto.2019.04.010 Received 12 October 2018; Received in revised form 28 March 2019; Accepted 10 April 2019 Available online 12 April 2019 0040-1951/ © 2019 Elsevier B.V. All rights reserved.
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shear, impact earthquake stress drop and characteristics of the seismic cycle such as recurrence time, pre-seismic slip, fault dilation and healing rate. We carried out a series of 32 complete experiments, with different initial layer thickness, normal stress, and shearing velocity (Table 1). In addition, we used data and results from another 30 experiments conducted under similar conditions to verify our results and conclusions. Our data indicate that properties of stick-slip vary systematically as a function of layer thickness and normal stress. We focus in particular on the relationship between granular friction and changes in layer thickness during the laboratory seismic cycle and interpret our results in terms of the role of force chains during granular shear.
Table 1 Experiments table. Exp
Gouge Thickness, mm
Normal Stress, MPa
Velocity, μm/s
p4561 p4562 p4578 p4579 p4581 p4583 p4588 p4589 p4688 p4690 p4692 p4700 p4701 p4702 p4709 p4711 p4712 p4767 p4768 p4769 p4770 p4780 p4781 p4782 p4783 p4784 p4899 p4900 p4901 p4904 p4905 p4906
5 5 5 5 5 5 5 5 5 5 5 5 4 4 5 3 3 5 5 5 5 7 9 2 1 5 7 1 5 3 3 5
5 5 5 5 2–8-2 2–3-2 2–4 4 5 2–7 2–7-2 5 2–8-2 2–8-2 2–7 2–7 2–7 6–20 2–20 2–8 2–8 4–22 4–22 4–22 4–22 4–22 2 2 2 2 2 16
6–300 1–100 0.3–300 0.3–300 5 5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 650 10 10 10
2. Experiment methods We sheared granular gouge layers composed of glass beads using a biaxial, servohydraulic testing machine and the double direct shear (DDS) configuration. Experiments were conducted at room temperature and humidity. Normal stress was maintained constant and shear stress was applied via a displacement rate boundary condition imposed at the top of the DDS center block. Both normal and vertical forces were measured with strain gauge load cells. Normal and shear displacements were measured using direct current displacement transducers (DCDTs, measurement precision to ~0.1 μm) located between the moving pistons and the fixed frame (Fig. 1). Data were recorded using a 24-bit analog to digital converter at a rate of 10 kHz and averaged to the desired sampling rate from 1 Hz to 10 kHz. In the DDS configuration, two layers are sheared simultaneously between three forcing blocks. This arrangement involves two stationary side blocks, supported from below, and a central forcing block that applies shear to the layer boundaries (Fig. 1). Our forcing blocks were constructed from 316 Stainless steel and equipped with grooves perpendicular to the shearing direction on the surfaces in contact with the gouge (0.8 mm in height, 1 mm spacing). The grooves ensure that shear occurs within the layer, rather than at the boundary, and promote a transition from pervasive shear to progressively localized shear as a function of increasing net strain. Our roughness is a crude approximation of natural fault zone roughness, which is of course much more complex and includes longer wavelength features and striations parallel to slip. The nominal frictional contact area of each layer is 10 × 10 cm2, and this is maintained constant during shear by using a central block (15 cm in length) that is longer than the side blocks. Although two granular layers are sheared simultaneously in DDS, all results shown in this paper are for one single layer. We sheared granular layers of soda-lime glass beads with particle size from 105 to 149 μm in diameter. This material has been used in
Steacy, 1994; Cates et al., 1998; Morgan and Boettcher, 1999; Mair et al., 2002; Anthony and Marone, 2005; Johnson et al., 2008; Tordesillas et al., 2012; Griffa et al., 2013; Scuderi et al., 2015). Although the role of force chains in granular shear has been discussed extensively, many important questions remain such as what is the effective width of a force chain, do force chains play a major role in shear zones dominated by angular or clay rich gouge, and how do force chains operate when shear bands form and divide fault zones into principal slip zones and spectator regions where particle motion and shear strain are near zero. The purpose of this paper is to investigate how fault zone properties and characteristics of gouge particles, modulated by normal stress and
Fig. 1. (a) Biaxial deformation apparatus and double direct shear (DDS) configuration (center). Red arrows show fault normal stress and shear stress, as applied via horizontal and vertical rams, respectively. Displacements and strains are measured via DCDTs attached to each ram. (b) Sample assembly showing DDS configuration, with two gouge layers between three forcing blocks. Thin latex membrane is visible at the bottom. Guide plates at front and back cover the layers. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Fig. 2. Complete run details showing coefficient of friction versus shear load point displacement for five experiments with different initial layer thickness: 1 mm (p4783), 2 mm (p4782), 5 mm (p4784), 7 mm (p4780) and 9 mm (p4781). Shearing rate was 10 μm/s for each experiment. Normal stress was stepped from 4 to 22 MPa, as noted. All experiments show repetitive stick-slip instabilities. Friction varies systematically with layer thickness, showing a clear negative correlation. Fig. 3. Details of experiment p4784. (a) Stick-slip stress drop, Δμ, defined by the maximum and minimum friction for each failure event. (b) Continuous measurements of gouge layer thickness. Note overall layer thinning, associated with geometric spreading, compaction, and dilation for each cycle of load increase and failure. (c) Detail of layer thickness changes for the first five events after removing the thinning ratio. Note that the magnitude of dilation during loading is similar to that of compaction upon failure.
previous work and exhibits highly reproducible stick-slip frictional sliding (e.g., Mair et al., 2002; Anthony and Marone, 2005). Layers were built by placing the side blocks on a leveling jig and carefully packing a measured mass of material to produce a desired layer thickness and initial porosity. We studied a range of layer thicknesses
from 1 to 9 mm and conducted a large suite of closely related tests to verify the results of the experiments presented here. Table 1 provides details of the main set of experiments. Once the layers were constructed, the central block was placed above one side block using a squaring jig to ensure alignment and the blocks were secured with 88
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Fig. 4. Evolution of friction and layer thickness with shear displacement for a complete stick-slip event from experiment p4784 (see Fig. 3). (a) Three stages are apparent in the friction data: elastic loading where friction increases linearly with load point displacement, an inelastic region associated with creep and granular rearrangement, and finally failure, associated with rapid stress drop and radiation of elastic waves. (b) Detrended layer thickness (see Fig. 3) showing dilation during loading followed by compaction during failure. Note the break in slope associated with the onset of inelastic creep.
Fig. 5. (a) Details of friction versus time for a stick-slip event. Recurrence time is the time between failure events. (b) Shear displacement at the gouge layer boundary. Note that slip of ~ 40 μm occurs during failure, after which there is a brief ‘stick' period of zero motion followed by inelastic creep prior to the next failure event.
most pronounced for the experiments on thinner layers. The steady state frictional strength scales inversely with layer thickness (consistent with previous work, e.g. Anthony and Marone, 2005; Knuth and Marone, 2007)) and is linearly proportional to normal stress, as indicated by the roughly constant value of friction over our range of normal stresses (Fig. 2). Especially for the thinnest layers, we observe strain hardening at larger displacements, which is likely due to particle spalling and changes in particle size distribution. We increase normal stress after a few mm of slip and we do not remove the shear stress; thus the friction curves drop and recover over a few stick slip cycles for each step change in normal stress (Fig. 2).
adhesive tape. This procedure was repeated for the other layer. A thin rubber membrane was attached to the bottom of the sample to avoid loss of material during shear. Guide plates were attached over the front and back of each layer (Fig. 1) to avoid extrusion of material there (e.g., Savage and Marone, 2007). The adhesive tape remained on the sample until normal stress was applied, in the apparatus, after which it was cut from the central block. All experiments began by applying a normal stress of 4 MPa. Once initial layer compaction was finished (typically 30 min) the vertical piston was driven at a constant displacement rate of 10 μm/s and this shearing rate was maintained constant throughout the experiment. After the shear stress reached a peak value, stick-slip events stabilized. Then the normal stress was varied in 2 MPa steps (Table 1). Using this approach, we studied normal stresses from 4 to 22 MPa. We also ran experiments at a single normal stress and for a variety to normal stress histories, in order to understand the effect of net strain and loading history (Table 1).
3.1. Stick-slip instability Stick-slip events are apparent during the later section of the initial loading curves (Fig. 2). They grow in amplitude and reach a steady state, with some fluctuation, after shear displacement > 4 mm. Because there are many stick-slip events in each experiment, we developed an automated algorithm to pick the beginning and end of each event. This algorithm also collects other parameters, including peak friction, friction drop, stick-slip recurrence time, and changes in layer thickness. For a given normal stress, the failure events are quite similar and define a consistent stress drop for many cycles (Fig. 3). To compare our range of conditions, we report shear stress drop normalized by layer normal stress, which is friction drop (Fig. 3). We also measure changes in layer thickness during loading and failure. In simple shear under our stress conditions, the layers thin with net shear due to geometric spreading (Scott et al., 1994). The rate of thinning, dh, with shear displacement, dx, defines a thinning ratio dh/dx, which is locally linear
3. Results Upon initial application of shear displacement, the shear stress on the granular layer increased along a linear elastic loading curve (Fig. 2). Normal stress across the layers was maintained constant during shear, so we plot the ratio of shear stress over normal stress, which facilitates comparison over our range of conditions (Fig. 2). The coefficient of friction is given by the ratio of shear stress to normal stress during slip. For simplicity, we refer to this ratio as friction. Stick-slip failure events generally began prior to reaching steady state shear and thus prior to the peak stress seen in most experiments (Fig. 2). The peak stress is 89
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Fig. 6. Evolution of the maximum and minimum friction for each stick-slip event (See Fig. 2) shown as a function of load point displacement. Note that μmax and μmin vary systematically with initial layer thickness, showing a clear negative correlation, and net shear displacement, which increases with event number.
but decreases with net shear (Fig. 3). To first order, dh/dx is entirely due to geometric spreading, however, it may vary with shear due to porosity loss and layer densification. To evaluate thickness changes during a given stick-slip cycle, we first remove the linear thinning trend (e.g. Scuderi et al., 2014), dh/dx (Fig. 3 inset). Note that the detrended layer thickness measurements show that dilation during loading is proportional to compaction during failure and thus the net change in layer thickness over a stick-slip cycle is roughly zero, consistent with expectations from critical-state soil mechanics (Fig. 3 inset). For a layer that is initially 5-mm thick, the compaction during stick-slip failure is about 5 μm. Stick-slip events can be divided into three stages (Fig. 4). First friction increases linearly with shear displacement, defining the elastic stage of loading. This stage can be described by a linear stiffness K, which we report as a normalized quantity: K′ = K/σn, where σn is the layer normal stress (Fig. 4). Thus K′ has units of friction per shear displacement. As shear load increases granular layer thickness also increases slowly due to dilation (Reynolds, 1885). After the stage of elastic loading, the stress displacement curves become convex and exhibit non-linearity that defines an inelastic segment (Fig. 4). Inelastic loading is associated with an increase in the dilatancy rate, determined
from the detrended layer thickness data, which becomes approximately linear until the end of this stage, when failure occurs (Fig. 4). We define the failure stage by the maximum and minimum friction and observe that it is associated with abrupt layer compaction (Fig. 4). Our experiments are run by servo-controlling a load point displacement at the top of the DDS central block (Fig. 1). To get the slip imposed at the boundary of the granular layers, we calculate and remove the displacement associated with elastic deformation of the apparatus, using the loading stiffness which is known from calibrations. After correcting our shear displacement measurements for apparatus elasticity, we have the net slip at the layer boundary for each stage of the stick-slip cycle (Fig. 5). Note that slip is zero during the elastic loading stage, just after failure, and then increases gradually until failure (Fig. 5). We define the stick slip recurrence interval, tr, as the time between each stress minimum (Fig. 5). For each experiment, we record several hundred stick-slip events during net shear displacement of almost 40 mm (Fig. 2). To show the evolution of stick-slip cycles with shear displacement and normal stress, we plot the maximum μmax and minimum μmin friction as a function of event number in Fig. 6. Maximum friction shows a clear negative correlation with layer thickness (consistent with previous work, e.g. 90
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Fig. 7. Systematics of stress drop and layer dilation for all failure events. Each symbol represents the mean ± 1 std. dev. Stress drop (normalized by layer normal stress to produce friction drop) varies inversely with layer thickness and shows a maximum at a normal stress of about 18 MPa. Layer dilation is larger for thicker layers and the variation with normal stress is consistent with that for stress drop.
Anthony and Marone, 2005). Note that the maximum and minimum friction values for stick-slip in thin layers (Fig. 6) reflect a small degree of shear strain weakening at normal stresses below 16 MPa. All of our samples show shear strain hardening for normal stresses above 16 MPa, which occurs at shear displacements above 25 mm (Fig. 6). Note that both the maximum friction, at the beginning of a stick slip event, and the minimum friction, at the end of each stick-slip event, increase with shear strain, reflecting shear strain hardening toward the end of our experiments. While our experiments are conducted at stresses below the particle crushing strength (Mair et al., 2002), spalling and particle fracture does occur for the upper range of our normal stresses, particularly for the thinner layers. The increase in frictional strength and minimum friction after stick slip failure (Figs. 2 and 6) is consistent with layer densification associated with shear induced increases in packing density and changes in particle size distribution.
Dilation scales directly with layer thickness, such that thicker layers generally show greater dilation, with some scatter (Fig. 7). The basic observation is straightforward and consistent with Reynolds (1885) dilation in a dense granular media sheared at stresses below the particle fracture stress. For small strains, associated with an individual stick-slip event, thicker layers dilate more because dilation is a strain hardening process and therefore the whole layer is involved. After strain localizes, dilation is restricted to the principal slip surface and spectator regions of the granular shear zone do not dilate (e.g., Frank, 1965). Our data on stick-slip stress drop show a negative correlation with layer thickness and, especially for the thinnest layers, a complex variation with normal stress (Fig. 7). We report the mean values ± 1 standard deviation for all events in our dataset (Fig. 7). Note that the range of stick-slip dilation values scales with layer thickness and that this range is reflected in the friction drop data. Stick-slip friction drop shows a pronounced increase with normal stress for the thinnest layers and a maximum at 18 MPa (Fig. 7). Part of these changes in friction drop could be associated with spalling and grain fracture, because the highest normal stresses occur near the end of our experiments, however, previous works show that particle crushing
3.2. Effect of initial layer thickness and normal stress on stick-slip events We quantify the variation in stick-slip stress drop and layer dilation for each event as a function of layer thickness and normal stress (Fig. 7). 91
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Fig. 8. Stick-slip recurrence interval and creep slip during inelastic loading for all failure events (see also Fig. 7). Each symbol represents the mean ± 1 std. dev. Within the measurement uncertainty, recurrence time is independent of layer thickness but varies systematically with normal stress, decreasing from 4 to 8 MPa and increasing to a peak value at 18–20 MPa. The magnitude of inelastic slip is independent of layer thickness, within the measurement uncertainty, and decreases with normal stress.
Fig. 9. Variation of the minimum friction with layer thinning ratio dh/dx for all stick-slip failure events. Both thinning ratio and μmin vary with layer thickness, but for the majority of our data they define a trend that is independent of layer thickness (see best fit linear relation). 92
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Fig. 10. Friction drop as a function of layer dilation (a) and normalized dilation (b) for all failure events. Each symbol represents the mean ± 1 std. dev. Thicker layers show systematically lower stick-slip friction drop and greater dilation than thinner layers. The data collapse to a single line when dilation is normalized by thickness, which represents a measure of volumetric strain.
and grain angularity lead to smaller stress drop and stable sliding (Mair et al., 2002), which is the opposite of what we observe (Fig. 7). Thus, while particle angularity may increase somewhat during shear we do not believe that particle crushing plays a major role in our experiments. Also, post experiment inspection of the material confirms that grain crushing and changes in particle size distribution are insignificant. We quantify stick-slip recurrence interval and inelastic slip for each event (Fig. 8) to complement the data of Fig. 7. Within the measurement uncertainty, recurrence time is independent of layer thickness but varies systematically with normal stress, showing a peak value at 18–20 MPa. For normal stresses below 10 MPa, the magnitude of inelastic slip varies little with layer thickness but decreases with normal stress. While for normal stresses above 10 MPa, the magnitude of inelastic slip decreases with decreasing layer thickness, especially for 2 mm and 1 mm layers, but is independent of normal stress. Taken together, these data show that larger events have longer recurrence
interval and that inelastic creep decreases at higher normal stress, consistent with the idea that layers are more compact and stiffer (Figs. 7 and 8). We find that in each experiment, dilation shows a different trend below a normal stress of 10 MPa, between 10 and 18 MPa and above 18 MPa (Fig. 7a). Dilation during a stick-slip event decreases slightly with normal stress below 10 MPa, but it increases for normal stresses above 10 MPa and decreases again above 18 MPa. This trend is clearest for the thicker layers. A similar trend is observed for friction-drop (Fig. 7b), except here, thinner layers show clearer and larger amplitude increases from 10 to 18 MPa. The largest mean friction-drop reaches ~0.17 at 18 MPa for 1-mm layers, which is almost 50% larger than friction-drop at normal stress lower than 10 MPa. In contrast, for 9-mm thick layers the mean friction-drop is roughly 0.09 and shows no clear trend with normal stress.
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Fig. 11. Shear stress drop increases with stick-slip recurrence time for normal stress > 8 MPa. At lower normal stresses, a weak negative trend is apparent (see also Fig. 8). Inset shows data for 1 mm layers, indicating the negative healing rate before normal stress exceeds 8 MPa.
4. Discussion
drop, which varied with event recurrence time. Their experiments explored a range of fault slip velocities and humidity. We observe similar trends for each of our layer thickness (Fig. 10a) and the data define a single trend, with some scatter, when friction drop is compared to layer volume strain (Fig. 10b). Thus, we posit that stick-slip friction-drop is modulated by the normalized dilation, which is essentially a layernormal strain. The idea is that normal strain is a proxy for energy potential, with larger values associated with a more fragile granular structure, thus allowing for greater friction-drop.
4.1. Layer thickness influence on friction behavior Our observations show that friction and significant aspects of the stick-slip cycle vary systematically with layer thickness (Figs. 2 and 6). Thicker layers have relatively lower apparent friction, consistent with previous work (Scott et al., 1994; Ikari et al., 2011). The previous analysis for the relationship between layer thickness and apparent friction (e.g. Scott et al., 1994) is based on the contribution of volume strain to frictional strength and it predicts that apparent friction should scale inversely with the thinning ratio. Following their results, we consider the relationship between μmin and the layer thinning ratio (Fig. 9). Here, the discrete thinning ratio is calculated for each stick-slip event, using the layer thickness change and shear displacement. The data of Fig. 9 show a clear trend for a wide range of conditions. A line fit to the main data concentration (Fig. 9) is consistent with the idea that thinning contributes to frictional weakening (e.g. Scott et al., 1994). Looking at the data outside the main cloud (Fig. 9), we see that minimum friction is high when the thinning ratio is low, which happens for our thin layers. At the other extreme, minimum friction shows a long tail for large values of thinning ratio, suggesting that other factors dictate the minimum friction under these conditions. We also observe a relationship between layer thickness and the magnitude of dilation (Figs. 7). We extend this to evaluate the relationship between stick-slip friction drop and shear driven dilation during the stick-slip cycle (Figs. 4 and 10). The raw data show that thicker layers have larger dilation during loading and smaller stress drop during failure (Fig. 10a). However, when we normalize dilatant changes in layer height by the layer thickness just prior to stick-slip, we see that all events collapse to a single line, with some scatter (Fig. 10b). The data of Fig. 10b connect with previous work on the relationship between dilation, stick-slip stress drop, and frictional healing. Scuderi et al. (2014) conducted stick-slip experiments and showed changes in friction drop as a function of dilation for a range of event recurrent times. They found that larger dilation was associated with larger stress
4.2. Normal stress influence on frictional healing and recurrence time The role of shearing rate on stick-slip recurrence time is well established (e.g., Karner and Marone, 2000; Beeler et al., 2014). Here, we find that normal stress also plays a role in determining recurrence time for granular shear (Fig. 8). Part of this could arise from increased frictional strength at higher normal stress but the major effect is due to changes in the elastic vs. inelastic segments of loading. We further explore this relationship by plotting stress drop vs. stickslip recurrence interval (Fig. 11). The main trend of these data is an increase in stress drop for larger recurrence intervals. This is consistent with the idea that for a given loading rate, a larger stress drop requires a longer loading duration to reach the failure stress (Fig. 11). Our data also show another, weaker, trend for lower normal load. For normal stresses below 8 MPa, a negative rate can be seen (Fig. 11). This behavior is especially clear for the 1-mm layer (Fig. 11 inset). At low normal stress, larger values of recurrence interval are associated with smaller stress drop. This occurs when the elastic stage of loading is short and loading is dominated by inelastic processes, which produces a lower overall loading rate and lesser dilation which in turn produces a lower potential energy, less fragile granular state and thus smaller stress drop. Taken to the extreme, where inelastic creep dominates and elastic loading is negligible, sliding would be stable, which is consistent with a simple extrapolation of our data trend to zero stress drop (Fig. 11). To further examine these ideas, we plot recurrence time vs. normal stress for one complete experiment (5-mm layer thickness) and break tr 94
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Fig. 12. Detail of the stick-slip recurrence time tr as a function of normal stress for one complete experiment (p4784). (a) Breakdown of tr into the elastic and inelastic loading segments of the stick-slip cycle (see Fig. 5): te is the time of elastic loading, ti is the time of inelastic creep loading. (b) Elastic and inelastic fractions of the loading time. Note that the elastic portion of the loading curve becomes greater with normal stress while the time of inelastic loading becomes proportionally less: extrapolation of these trends indicates that they will become equal at roughly 25 MPa, slightly above our highest normal stress.
into the intervals associated with elastic and inelastic loading (Fig. 12). These data are representative of our suite of experiments (Table 1) and provide a clear picture of why stick-slip recurrence time first decreases with normal stress, up to 8 MPa, and then increases with increasing normal stress (Fig. 12). For low normal load (lower than 8 MPa), shear loading is dominated by inelastic processes and although the elastic fraction increases with increasing normal stress, the rate of increase is smaller than the rate at which the inelastic portion decreases, resulting in an overall decrease in recurrence time (Fig. 12a). For our range of conditions, shear loading is dominated by inelastic processes. Elastic processes become increasingly important at higher normal loads, and our data indicate that they exceed inelastic processes at a normal stress of roughly 25 MPa.
1976; Ruina, 1983; Gu et al., 1984). The effective elastic stiffness for our experiments is determined by the testing machine, the DDS sample assembly, and the granular layers. That is, the net shear stiffness K can be described as: 1/Kmachine+1/KDDS + 1/Klayer = 1/K. The machine stiffness and the stiffness of the DDS assembly are well known from calibration and the net stiffness K is readily measured in-situ during the stick-slip cycles. The stiffness of the granular layer is also readily measured (e.g., Leeman et al., 2016; Scuderi et al., 2016), however, previous works have typically ignored its role and/or assumed that it does not change with shear. Here, we address layer stiffness and its variation with normal stress, layer thickness, and net strain. The changes in Kmachine and KDDS are negligible for our range of conditions, thus all changes in K can be attributed to Klayer. We measure the net stiffness K from the elastic portion of the loading curves (Fig. 4). Following Leeman et al. (2016) we use an automated routine to measure stiffness and track its evolution throughout shear (Fig. 13). We find that K varies with both normal stress, layer thickness, and net shear strain. The symbols in Fig. 13 are colour coded
4.3. Elastic stiffness A simple spring-slider model has been used widely in studies of frictional stick slip (e.g., Dieterich, 1978, 1979; Johnson and Scholz, 95
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small. The overall trends of stiffness vs. layer thickness, normal stress, and shear strain indicate a stress and shear driven densification process that decreases in rate as density increases. 4.4. Micromechanical model for force chain formation and breakage during shear Force chains are known to play an important role in granular processes (Sammis et al., 1987; Cates et al., 1998; Morgan and Boettcher, 1999; Aharonov and Sparks, 1999; Mair et al., 2002; Geng et al., 2003; Anthony and Marone, 2005; Johnson et al., 2008; Griffa et al., 2013; Scuderi et al., 2014; Dorostkar et al., 2018; Gao et al., 2018). Existing models show that force chains become less brittle as layer thickness increases, leading to a transition from unstable to stable sliding (Mair et al., 2002; Anthony and Marone, 2005). This is consistent with the idea that fragility scales with the number of particles in a chain (Cates et al., 1998). Moreover, these ideas suggest that longer force chains are more compliant and may undergo significant creep prior to failure. In a study of frictional healing for a granular medium, Scuderi et al. (2014) suggested that force chains form directly after a dynamic slip event and that pre-seismic slip and dilation are due to elasto-plastic deformation within force chains associated with rolling and interparticle slip. Here we propose a synoptic model (Fig. 14) based on granular force chains to explain our results. We envision that elastic loading is initially associated with pervasive strain across the layer and that forces are distributed equally among many if not all of the particles within the layer (Fig. 14). During this stage, the whole layer deforms as a granular solid, as indicated by the decreasing slope of inverse stiffness with increasing normal stress (Fig. 13). This means that the shear modulus increases with increasing packing density caused by increasing normal stress. This stage results in elastic deformation at contact junctions and also particle rolling. The rolling motion is small and reversible, such as commonly observed in dilatant granular systems. With accumulated shear load, force chains begin to form and resist shear motion. Many grains must be fully mobilized in the process of building long force chains. In the context of the stick-slip cycle, this stage begins at the end of elastic loading. At the onset of inelastic creep stage, the dominant force chains start to rotate simultaneously causing layer dilation (Fig. 15). At this point, the layers are separated into two regions: region one, which is dominated by the rotating force chains (Fig. 14) and a second region, where force chains are absent. Region 1 is in the center of the layer in Fig. 14, but it may also occur near the boundaries. Of course the measured layer thickness changes are determined by the combined effect of both regions. Region 1 will tend to dilate during the early part of the stick-slip cycle when shear stress is increasing, whereas Region 2 may consolidate via particle rolling and sliding (Fig. 14). If dilation exceeds thinning, the measured layer thickness will show dilation, which is what we observe in most cases (layer thicknesses of 1 to 7 mm Fig. 16a–c). However, when Region 2 dominates, thinning will exceed dilation and we should see an overall thinning. This is the case for our 9-mm thick layers, as seen in Fig. 16d.
Fig. 13. Relationship between inverse elastic loading stiffness (see Fig. 4) and layer thickness h for five experiments, with colour showing normal stress. Note that for a given normal stress, the data define a distinct slope (dashed lines). The slope of these lines decreases systematically with increasing normal stress. Within a given experiment, stiffness increases with shear displacement and layer thinning.
to show normal stress. For a given experiment, stiffness increases along a trend of decreasing layer thickness that becomes increasingly nonlinear. For a given normal stress, the data define a roughly linear relationship between 1/K and layer thickness, with big changes in slopes as a function of normal stress. Within a given experiment, stiffness increases with shear displacement and layer thinning, defining a distinct slope for a given normal stress (Fig. 13). The trend of decreasing K as layer thickness increases (dashed lines) decreases systematically with increasing normal stress. These data are consistent with a model in which layer stiffness increases as a function of decreasing porosity and increasing granular packing density. Granular porosity decreases with normal stress as grains are forced into a closer packing arrangement. This is the main effect seen in Fig. 13, where stiffness increases rapidly at normal stresses below 8 MPa. At higher normal stress, it becomes more difficult to increase packing density and the slope of the lines decrease, indicating smaller changes in stiffness for a given change in layer thickness. Shear enhanced compaction is also an important factor. Shear occurs throughout our experiments and the trends in Fig. 13 show both the effects of normal stress and shear strain. For thicker layers and low normal stress, even a small change in normal stress produces a big change in stiffness for a given layer thickness. As normal stress increases, layers become dense and additional changes in stiffness are
Fig. 14. Schematic model of granular processes during shear with black and red dashed lines showing the positions of a strain marker at the beginning and end of each stage. At the beginning of a stick-slip cycle, shear strain is pervasive across the layer, such that the whole layer deforms as a granular solid. At the onset of inelastic creep, the strongest force chains rotate which means this zone will dilate. At the same time there are regions where force chains are absent. These are regions of inelastic creep due to particle sliding and rolling, which results in local compaction and layer thinning. Once the strong force chains reach a limit and break there is a sudden collapse, with layer compaction and stress drop. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 96
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Fig. 15. Details of change in friction and layer thickness (raw data) during shear loading (data from experiment p4904 with 3 mm layer, 2 MPa normal stress and 10um/s shear velocity). Note that in each stick-slip cycle, at the beginning, the layer is thinning at the predicted thinning ratio (green dashed line). This stage of layer thinning occurs simultaneously with elastic loading. Note that layers begin to dilate at the same point that inelastic creep starts. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Fig. 16. Details of layer thickness (raw data) change during shear loading under 6 MPa normal stress for a) 1 mm, b) 5 mm, c) 7 mm, d) 9 mm layers. Note that for 1 mm, 5 mm, 7 mm layers, layer thickness increased during inelastic loading stage by 2 to 1 μm, while for 9 mm layer thickness decreased by 1.3 μm during this inelastic stage. This represents the “thinning region” described in our model (Fig. 14) are getting thicker with increasing layer thickness.
In agreement with previous works (e.g., Mair et al., 2002; Anthony and Marone, 2005; Scuderi et al., 2014) our data suggest that force chains form immediately upon failure and support elastic stress accumulation. In addition, we posit that force chains continue to strengthen and evolve during inelastic loading in association with particle rearrangement and local changes in packing density. The dominance of granular rearrangement coincides with net shear and inelastic strain within the granular layer (Fig. 5) resulting in dilatancy (Fig. 15). With continued strain, we posit that multiple force chains form and fail as part of the process of forming a support structure for the larger chains
that carry the load. The grain structure of a force chain is presumably maintained dynamically, with particles being continually added and lost during shear (Fig. 14). In our model, stress drop and shear stiffness scale inversely with force chain length (Figs. 13 and 14). Thus, when we consider the total accumulated shear in our experiments (e.g., Fig. 2), our data indicate that shorter force chains form in thinner layers, as suggested by the data of Fig. 13. In the context of the stick-slip cycle, our data suggest that layer dilation should scale with the slip required to form force chains and thus dilation should increase with layer thickness, as we observe 97
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(Fig. 10). In summary, force chains support stresses across the granular layers. They form during the inelastic stage of loading in association with particle motion and dilation (Fig. 14). Upon failure, the force chains break, compaction occurs throughout the layer, and new force chains form during the next cycle.
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5. Conclusions We study stick-slip stress drop for laboratory faults composed of synthetic granular fault gouge (glass beads) sheared within rough surfaces. Our experiments focus on a single fault surface roughness and granular particle size, which do not vary as a function of strain. We find that layer thickness and normal stress both have a significant effect on stick-slip sliding such as frictional strength, friction drop, layer dilation, recurrence time, and inelastic slip. We show that stress drop and both minimum and maximum friction decrease with increasing layer thickness. We also show that normal stress has a different effect above and below 10 MPa. Below 10 MPa, dilation, friction-drop and recurrence time decrease slightly and inelastic slip decreases strongly with increasing normal stress, but above 10 MPa, dilation, friction-drop and recurrence time increase and inelastic slip decreases more slowly with increasing normal stress. We note three distinct stages of the stick-slip cycle. In our model we suggest that during elastic loading, the layer is deforming evenly as a whole body with some normal stress controlled shear modulus, then inelastic loading starts and force chains emerge. Acknowledgements We thank S. Swavely for technical help in the laboratory and Paul Johnson for key discussions. We gratefully acknowledge the comprehensive reviews of two reviewers and the Editor as well as support from the DOE geothermal program (DOE EERE DE-EE0006762), the CSES program at LANL, the NSF Geophysics program (EAR-1520760 and EAR-1547286/1547441), and the National Natural Science Foundation of China (grant No. 51739006, 11572174). The first author would also like to acknowledge the financial support from China Scholarship Council (No. 201506210328). References Aharonov, E., Sparks, D., 1999. Rigidity phase transition in granular packings. Phys. Rev. E 60 (6), 6890. https://doi.org/10.1103/PhysRevE.60.6890. Aharonov, E., Sparks, D., 2002. Shear profiles and localization in simulations of granular materials. Phys. Rev. E 65 (5), 051302. https://doi.org/10.1103/PhysRevE.65. 051302. Anthony, J.L., Marone, C., 2005. Influence of particle characteristics on granular friction. J. Geophys. Res. Solid Earth (B8), 110. https://doi.org/10.1029/2004JB003399. Barés, J., Wang, D., Wang, D., Bertrand, T., O'Hern, C.S., Behringer, R.P., 2017. Local and global avalanches in a two-dimensional sheared granular medium. Phys. Rev. E 96 (5), 052902. https://doi.org/10.1103/PhysRevE.96.052902. Beeler, N.M., Tullis, T., Junger, J., Kilgore, B., Goldsby, D., 2014. Laboratory constraints on models of earthquake recurrence. J. Geophys. Res. Solid Earth 119 (12), 8770–8791. Biegel, R.L., Sammis, C.G., Dieterich, J.H., 1989. The frictional properties of a simulated gouge having a fractal particle distribution. J. Struct. Geol. 11 (7), 827–846. https:// doi.org/10.1016/0191-8141(89)90101-6. Cates, M.E., Wittmer, J.P., Bouchaud, J.P., Claudin, P., 1998. Jamming, force chains, and fragile matter. Phys. Rev. Lett. 81 (9), 1841. https://doi.org/10.1103/PhysRevLett. 81.1841. Dieterich, J.H., 1978. Time-dependent friction and the mechanics of stick-slip. In: Rock Friction and Earthquake Prediction. Birkhäuser, Basel, pp. 790–806. https://doi.org/ 10.1007/978-3-0348-7182-2_15. Dieterich, J.H., 1979. Modeling of rock friction: 1. Experimental results and constitutive equations. J. Geophys. Res. Solid Earth 84 (B5), 2161–2168. https://doi.org/10. 1029/JB084iB05p02161. Dorostkar, O., Guyer, R.A., Johnson, P.A., Marone, C., Carmeliet, J., 2017a. On the role of fluids in stick-slip dynamics of saturated granular fault gouge using a coupled computational fluid dynamics-discrete element approach. J. Geophys. Res. Solid Earth 122 (5), 3689–3700. https://doi.org/10.1002/2017JB014099. Dorostkar, O., Guyer, R.A., Johnson, P.A., Marone, C., Carmeliet, J., 2017b. On the micromechanics of slip events in sheared, fluid-saturated fault gouge. Geophys. Res. Lett. 44 (12), 6101–6108. https://doi.org/10.1002/2017GL073768. Dorostkar, O., Guyer, R.A., Johnson, P.A., Marone, C., Carmeliet, J., 2018. Cohesion-
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