INTRODUCTION
Frictional stick–slip instability along pre-existing faults has been accepted as the main mechanism of earthquakes for about 60 years, following the publication of laboratory results by Brace and Byerlee (1966). The gist of their work is the conclusion that the stick–slip instability along pre-existing faults in rock specimens reflects all the features inherent in earthquakes, such as connection with pre-existing faults, repetitive dynamic instability, low shear stresses activating instability (equal and lower frictional strength), and low stress drop, in contrast to the failure of intact specimens, characterized by high strength and large stress drop. The significance of the work carried out by Brace and Byerlee lies in the experimental proof that stick–slip is the only mechanism known at that time that can explain the combination of the inherent features of earthquakes mentioned above. Their work initiated the modern era of earthquake mechanism studies (Scholz, 2002). Researchers around the world embraced this idea and focused their efforts on a detailed study of the stick–slip processes under natural and laboratory conditions, which has led to significant progress in understanding the mechanics of earthquakes (Ben-Zion, 2001; Dieterich, 1979; Heaton, 1990; Rice, 2006; Rubino et al., 2017; Ruina, 1983; Scholz, 2002; Xia et al., 2004). However, besides the positive influence, the idea of Brace and Byerlee had a negative influence, consisting of diverting the interest in the study of the dynamic rupture of intact rocks as a source of earthquakes for half a century.
This paper shows that, contrary to the generally accepted approach, the vast majority of earthquakes at seismogenic depths of the earth's crust are due to dynamic rupture of intact rocks, while stick–slip along pre-existing faults plays a secondary role. These earthquakes are caused by the recently discovered fan-hinged shear rupture mechanism, which operates in intact rocks at seismogenic depth near pre-existing faults and is characterized by all the specific properties that are currently attributed solely to the frictional stick–slip mechanism. The fan-hinged mechanism is named after the shape and role of the fan structure of the head of dynamic shear ruptures consisting of an echelon of rock slabs formed by the intensive tensile cracking at the rupture tip.
The preference of the fan mechanism over the stick–slip mechanism is clear due to the outstanding properties of the fan mechanism, which consist of the ability to generate new faults in intact dry rocks even at shear stresses that are an order of magnitude lower than the frictional strength; to provide extremely low shear resistance close to zero, which makes rocks superbrittle, with the release of abnormally large elastic energy at spontaneous rupture; to cause low stress drop; to use a new physics of energy supply to the rupture tip, providing supersonic rupture velocity; and to provide a previously unknown interrelation between earthquakes and volcanoes. All these properties make the fan mechanism the most dangerous rupture mechanism at the seismogenic depths of the earth's crust. The depth range of the fan-mechanism operation is approximately 2–40 km. The stick–slip mechanism generally operates beyond this range.
Since extraordinary features of the fan mechanism and the properties of rocks determined by it are contradictory to the modern ideas, the fan-hinged approach is viewed with great skepticism and distrust. The problem is that in the brief papers published so far, it was impossible to present this extremely complex and extensive topic with clear arguments, reliable evidence, and detailed discussion (Tarasov, 2008, 2010, 2011, 2013, 2014, 2016, 2017, 2019; Tarasov & Guzev, 2013; Tarasov & Ortlepp, 2007; Tarasov & Potvin, 2013; Tarasov & Randolph, 2008, 2011, 2016; Tarasov & Sadovskii, 2016; Tarasov et al., 2016, 2017).
To make the subject more comprehensible, two extended companion papers have been written to summarize previously published findings on the fan mechanism, which have been significantly modified, further developed, and supplemented with new theoretical and experimental data. The current paper is Paper 2. Paper 1 entitled “New physics of supersonic ruptures,” published in DUSE (Tarasov, 2023), discusses the theory of the fan mechanism and analyzes the role of the fan mechanism in generating extreme ruptures in an extremely smooth interface. It is believed that the publication of these papers will lead to a breakthrough in understanding the key mechanism of dynamic events at seismogenic depths and usher in a new era in the study of natural and induced earthquakes. Further study of this subject is a major challenge for deep underground science, earthquake and fracture mechanics, physics, and tribology.
PRELIMINARY ILLUSTRATION OF THE FUNDAMENTAL DIFFERENCE IN THE UNDERSTANDING OF EARTHQUAKE MECHANISMS BASED ON THE STICK–SLIP AND FAN-HINGED APPROACHES
To facilitate a grasp of the fan theory, this section demonstrates in advance the fundamental difference in explaining some facts related to earthquakes in terms of fan and stick–slip mechanisms and reveals the main error made by Brace and Byerlee (1966) in assessing the role of intact rocks in creating earthquakes.
The essence of the experimental results of Brace and Byerlee (1966) as the basis for the modern understanding of the earthquake mechanism
Figure 1 demonstrates schematically the essential findings obtained by Brace and Byerlee (1966), on the basis of which they ushered in the modern era of earthquake mechanism studies (Scholz, 2002). Experiments were conducted at triaxial compression σ1 > σ2 = σ3. Two types of granite specimens (originally unfractured and with initial fracture or sawcut) were tested at high confining pressures σ3 corresponding to seismogenic depths. The two schematic stress–displacement curves in Figure 1 show the difference in the failure process between the two types of specimens. The spontaneous fracture of intact specimens started at high stresses corresponding to rock strength τu, and further at faulting (dotted line on the specimen above the curve), there was a large stress drop Δτ = τu – τf, where τf denotes the residual frictional strength.
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Specimens with initial fracture or sawcut (solid line on the specimen above the curve) demonstrated instability at low shear stresses corresponding to the static frictional strength and were accompanied by a small stress drop Δτ = τfs – τfd of about 100 bars, which is typical for major earthquakes. Here, τfs denotes the static frictional strength and τfd represents the dynamic frictional strength. The sliding begins when the static frictional strength τfs has been reached. This jerky sliding could be continued almost indefinitely on the fault, with the stress building up and then being released. Thus, the results obtained on specimens with initial fracture or sawcut reflected the main intrinsic features of earthquakes: connection with pre-existing faults, repetitive dynamic instability, low shear stresses activating instability (equal and lower frictional strength), and low stress drop.
As has been pointed out, in contrast to pre-existing faults, the fracture process of intact rocks does not reflect the above-mentioned earthquake features. The spontaneous fracture starts at very high shear stresses and the magnitude of stress drop can reach about tens of kilobars, which is not observed during earthquakes. If the rocks are initially fractured, it is difficult to observe how stress could be built up a second time to produce an earthquake by fracturing the intact rock in the same area. This means that the formation of a new fault in the intact rock with strength τu near the pre-existing fault is impossible, since the shear stress τ in the field cannot exceed the frictional strength τsf, as shown in Figure 1. Therefore, it is postulated that the conducted experiments are the laboratory analogs of natural earthquakes, which are a frictional, rather than fracture, phenomenon.
Brief introduction to the fan-hinged mechanism and the role of intact rocks in causing earthquakes in contrast to the conclusion of Brace and Beyrlee
Extraordinary properties of the fan mechanism
First of all, it should be noted that the mode of macroscopic rupture of hard rocks under high σ3 is always a shear rupture (fault). To understand the rupture mechanism operating in hard intact rocks at high σ3, it is necessary to determine the genesis and role of a specific fault structure characteristic of all extreme ruptures associated with natural earthquakes, shear rupture rockbursts, and laboratory earthquakes. Photographs in Figure 2a illustrate the structure of dynamic faults that caused a severe shear rupture rockburst in a South African ultra-deep mine and the San Andreas earthquake (Ortlepp, 1997; Scholz, 2002). This structure consists of an echelon of inclined tensile cracks and inter-crack slabs (plates). To study the structure Ortlepp (1997) dismantled the ruptures and found that the layer of slabs looked like fish scales or rows of tiles. Notably, the principles of formation of such structures are different for primary ruptures and segmented faults. However, the mechanism that controls this process is the same. This mechanism is fan-hinged shear. Features of the fan-structure formation on the basis of segmented faults have been considered in previous studies (Tarasov, 2014a, 2014b; Tarasov & Randolph, 2016). In this section, some of the outstanding properties of the fan-hinged mechanism are briefly introduced based on the primary rupture. The detailed analysis of this mechanism has been presented in the companion Paper 1.
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The tensile cracking at the rupture tip that accompanies any type of extreme rupture is a key process in the fan mechanism. Sequential formation of frontal tensile cracks at the tip of a propagating rupture occurs in the direction of the major stress σ1, at an inclined angle α0 to the fracture plane (see the model in Figure 2b). Thereafter, the relative slip Δ of the fracture faces, which increases with the distance from the fracture tip (see the graph above), causes the slabs to rotate at correspondingly increasing angles and thus form a fan structure. Figure 2b shows the pulse-like rupture mode, in which the slip occurs only in the rupture head, represented by the fan structure, whereas in front of and behind the fan zone, the slip is locked. Seismic inversions have indicated that most earthquake ruptures propagate in the pulse-like mode (Heaton, 1990; Noda et al., 2009; Perrin et al., 1995; Zheng & Rice, 1998). For simplicity, the model of the fan structure in Figure 2b only includes about 30 slabs, while the fan structure of real fractures typically includes thousands of slabs (see the companion Paper 1). The geometry of the slabs depends on the level of σ3, and at high σ3, the slabs acquire a certain slenderness ratio (the ratio between length and width) that allows them to withstand rotation without collapse. In this case, they play the role of hinges between the rupture faces, which slide as a wave along the fan structure propagating through intact rocks. It can reduce the friction between the sliding rupture faces to almost zero.
The diagram under the model reflects some of the extraordinary properties of the fan structure. At this stage of the description of the fan mechanism, these properties will be briefly introduced. The physics of all these properties has been described in the companion Paper 1. The most important properties are as follows:
- 1.
Extremely low shear resistance: The red curve indicates the distribution of shear resistance along the propagating rupture. In front of the fan, the shear resistance is equal to the material strength τu and behind the fan, it corresponds to the static frictional strength τfs. The breakdown strength τr decreases sharply, as it is determined by the creation of the frontal tensile crack, after which the frontal slab is subjected to rotation. Shear resistance of all rotating slabs involved in the fan structure τfan is very low and can be close to zero.
- 2.
High amplified shear stress: The black curve shows the distribution of shear stresses. Here, τ0 is the initially applied shear stress; τ1 denotes the shear stress behind the rupture head; and τamp and τcon are the shear stresses generated by the fan structure, which represents a natural extremely powerful stress amplifier. The maximum of the amplified shear stress τamp(max) is at the rupture tip and can significantly exceed the material strength τu. Due to this, the fan mechanism can provide the shear rupture propagation through hard intact rocks with the strength τu at shear stresses applied even significantly lower than the frictional strength, such as τ0 ≈ 0.1τfs. It should be noted that the stress concentration τcon in front of the rupture tip is also provided by the fan structure. The physics of stress τcon in the fan mechanism is fundamentally different from the Griffith theory (Griffith, 1921).
- 3.
Steady self-sustaining disbalanced stress state: Due to the huge difference Δτdis between the amplified shear stress τamp represented by the black curve and the shear resistance represented by the red curve, including breakdown τr and frictional τfan parts, a steady self-disbalanced stress state arises in the fan zone, which forces the fan structure to propagate spontaneously with a sharp acceleration just behind the rupture tip (which is typical for supershear and supersonic ruptures).
- 4.
Extraordinary energy budget of the rupture process: The average diagrams of shear stress and shear resistance versus slip within the fan zone are used to represent the energy budget per unit rupture area. When analyzing the energy budget, the segment OS on the horizontal axis is considered as the total slip between the rupture faces occurring inside the slipping zone. The segment ODc is the breakdown displacement and the segment DcS is the slip displacement. The energy budget includes two sources of driving energy: the first one is related to the total strain energy change due to the rupture motion (area inside the dotted green trapezoid); the second one is the amplified energy (pink area) generated by the fan structure in the fan zone and directly at the rupture tip, which indicates a new physics of energy supply for supersonic ruptures. The dissipated portion of energy includes the breakdown energy Wr (dark gray thin triangle) and the friction energy Wf (light gray area). Both energies can be very low. The released (or seismic) energy Ws corresponds to the yellow area. Due to the high value of the active energy Ws + Wamp compared to the dissipated energy Wr + Wf, the rupture velocity increases considerably, even up to a supersonic level. It should be noted that the fan mechanism also provides a third source of driving power, presented in the companion Paper 1.
- 5.
Low stress drop: The rupture propagation through intact hard rocks with the strength τu is accompanied by low stress drop Δτ = τ0 – τ1, which is typical for earthquakes. This low stress drop is in conflict with the large stress drop obtained in laboratory experiments (see Figure 1). This discrepancy will be explained further.
Figure 2c shows the average diagram of shear stress and shear resistance versus slip, reflecting the energy budget of stick–slip ruptures (for similar models, see Abercrombie & Rice, 2005; Rice et al., 2005; Schmitt et al., 2015). This budget has one source of driving energy related to the total strain energy change due to the rupture motion (area inside the dotted green trapezoid). According to classic theories, spontaneously propagating ruptures are driven by elastodynamic waves, where the energy released by the rupture motion (yellow area) is transferred through the medium to the rupture tip region at the maximum speed equal to the pressure wave speed (Freund, 1998; Needleman, 1999; Rice, 2001; Rosakis et al., 2007). This physics does not allow the achievement of supersonic speeds. However, new experimental results obtained by Gori et al. (2018) demonstrate supersonic ruptures. These ruptures are governed by the fan mechanism, which provides a new physics of the energy supply to supersonic ruptures (for details, see Paper 1).
The main intrinsic features of earthquakes as characteristic of the rupture of intact rocks by the fan mechanism
As has been discussed above, the main intrinsic features of earthquakes—connection to pre-existing faults, repetitive dynamic instability, low shear stresses activating instability (equal and lower frictional strength), and low stress drop—are attributed exclusively to the stick–slip process. However, the fan mechanism produces similar characteristics when rupturing intact rocks, as shown in Figure 3. Figure 3a shows a fragment of the rock mass involving a pre-existing fault (black line [I]) at two stages: stages 1 and 2. Due to the complex geometry of the fault interface, any fault can serve as a local stress concentrator. Zones of concentrated stresses in the intact rock mass are located around the steps and are indicated by red dotted circles in the pictures. Graphs under the pictures characterize the stress condition. Horizontal dotted lines indicate the following stress levels: τu—fracture (or ultimate) strength of intact rock and τfs—static frictional strength of the pre-existing fault. The level of the initial field shear stress τ0 is shown by the green line. It is significantly lower than the static frictional strength of the fault: τ0 < τfs, indicating that the situation is stable in respect of stick–slip. For the formation of the initial fan structure, a local shear stress equal to the fracture strength τu is required (for an explanation, see the companion Paper 1). The fact is that any pre-existing fault serves as a stress concentrator. At stage 1, the level of local concentrated stresses in the intact rock, marked as a red dotted circle, is below the ultimate strength τu.
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However, if the local stress in the intact rock approaches the level τu, as shown at stage 2, the tensile cracking process starts forming the initial fan structure (or double fan structures for bilateral rupture propagation). When the formation of the fan structure is completed, its shear resistance τfan becomes lower than the applied stress τ0. The completed fan structure also generates the amplified shear stress τamp, which can significantly exceed the rupture strength at the fan tip: τamp(max) > τu. The combination of these factors creates the self-disbalancing stress state in the fan zone (Δτdis = τamp – τfan) and the outstanding energy budget (as shown in Figure 2b), which cause the spontaneous rupture propagation at extreme velocity through intact rock loaded by a low shear stress equal to τ0. The new fault (II) involving a step can serve as a stress concentrator for the formation of the next fault (III), causing an earthquake or aftershock, and thereby resulting in repeated rupturing.
After the rupture process, the field stress decreases from τ0 to τ1, causing the stress drop Δτ = τ0 – τ1, which is significantly lower than that in the laboratory experiments. The difference between the value of stress drop in nature and in the laboratory is explained in Figure 3a on the right. The stress–displacement curve here shows that in the laboratory, to form the initial fan structure, the entire specimen must be loaded to the ultimate strength τu, which leads to a large stress drop Δτ = τu – τf. In nature, the initial fan structure is generated by local high stress equal to τu, while the entire rock mass is under low shear stress τ0 << τu. Despite the low shear stress, the completed fan structure can spontaneously propagate through the intact rock, forming a new fault and an earthquake, which is accompanied by a small stress drop Δτ = τ0 – τ1. Thus, the conclusion arrived at by Brace and Byerlee about the role of intact rock in the generation of earthquakes on the basis of laboratory experiments is incorrect.
The analysis shown in Figure 3a reveals that the dynamic rupture process in intact rocks governed by the fan mechanism shows all the features inherent in earthquakes (generation on the basis of pre-existing faults; low field shear stresses activating the instability; low stress drop; repeatable reactivation) that have been considered as exceptional attributes of the stick–slip process along pre-existing faults for the past 60 years. Furthermore, Figure 3 demonstrates that the dynamic rupture of intact hard rocks is preferred over dynamic stick–slip along pre-existing faults because the failure of intact rocks can take place at field shear stresses much lower than the frictional strength τfan << τf. This means that the fan mechanism is the main mechanism for shallow earthquakes in the earth's crust, while the stick–slip mechanism only plays a secondary role. This issue will be discussed further in more detail.
Figure 3b,c shows plan views of shear ruptures generated by the fan mechanism. Figure 3b shows a general view of two shear ruptures propagated from the hypocenter in dry quartzite surrounding an ultra-deep mine in South Africa and generated strong rockbursts seismically indistinguishable from earthquakes (Ortlepp, 1997). Figure 3c shows a schematic representation of the plan view of a shear rupture and its cross-section after failure from the point of view of the fan mechanism. The plan view looks like fish scales represented by the layer of slabs that completed their rotation. Spontaneous rupture propagation is preceded by a local tensile cracking process in the nucleation zone (hypocenter), where the initial fan structures are formed. For ruptures propagating bilaterally, double fan structures are generated, which represent the heads of the rupture moving spontaneously in opposite directions. During spontaneous rupture, all rows of slabs governed by their fan structures propagate simultaneously, forming the rupture front. The rupture fronts at different stages of the fault propagation are shown on the model as dotted red lines. The development of such rupture fronts was observed in experiments by Ben-David et al. (2010); Rubinstein et al. (2004).
Examples of old and new explanations of some earthquake features
Observation reveals that nature prefers the generation of new faults in intact rock masses near pre-existing faults, rather than slipping along weak frictional faults. Figure 4a shows a set of faults that are formed in an intact rock mass at different time points, causing severe earthquakes in New Zealand (Temblor, 2017). The latest faults are shown in green. Figure 4b shows the global distribution of earthquakes in the period from 1900 to 2014, and global plate boundaries (Silva et al., 2017). The points of different colors on the map represent hypocenters of earthquakes nucleated at different depths. As can be seen, the vast majority of hypocenters of shallow earthquakes (red dots representing the depth range 0–30 km) are widely dispersed outside global boundaries and can form earthquake swarms. Today, this fact can be explained by the fact that the earth's crust is riddled with faults and those that are correctly oriented with respect to the acting stresses can be reactivated. This paper proposes the fundamentally different reason for such behaviors of earthquakes: the faults are generated in intact rocks by the fan mechanism. Figure 4b shows that at depths > 30 km, hypocenters are mainly located at global plate boundaries, indicating that the underlying mechanism of deep earthquakes is highly likely to be the stick–slip.
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The situation is similar to volcanic earthquakes. Most volcanoes and related earthquakes are formed at the edges of tectonic plate boundaries and are generally considered to be related to the stick–slip processes. However, some of the earth's most well-known volcanoes (such as those of the Hawaiian Islands and the Yellowstone) arise in the middle of a plate. Figure 5a,b shows the top view of a map of earthquake swarms generated on the basis of the intraplate Yellowstone volcano and the depth distribution of earthquakes along the line AA′ within the seismogenic layer (modified from Smith et al., 2009). The paper proposes a currently unknown relationship between volcanoes and earthquakes induced by the fan mechanism. It will be shown that earthquakes caused by the fan mechanism, under certain conditions, can generate large volumes of molten rock, which can serve as a source of magma for volcanoes.
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The fan mechanism is also responsible for such dynamic events as shear rupture rockbursts in deep mines. Figure 5c shows a symbolic typical depth distribution of earthquake frequency. It will be shown that such a distribution as well as the range of depths of the seismogenic layer are determined by the fan mechanism. When a mine penetrates to the depths of the fan-mechanism activity (below the upper cutoff in Figure 5c), it may encounter a special kind of rockburst, which results in the formation of new powerful dynamic faults in intact rocks surrounding the mine. Photographs in Figure 5d (from Ortlepp, 1997) show that dynamic faults generated severe rockbursts in a mine deep in South Africa. The rockbursts are seismically indistinguishable from earthquakes and share the apparent paradox of failure under low shear stress (Gay & Ortlepp, 1979; McGarr et al., 1979). One remarkable feature of these ruptures is that they nucleate at a considerable distance away from the opening surface and propagate with considerable ferocity. All the features cannot be explained on the basis of conventional frictional rupture mechanisms. Deep mines facilitate observations of dynamic events at seismogenic depths caused by the formation of new faults in intact rocks and direct study of these faults, whereas earthquake faults are usually hidden from view.
One more important shortcoming of the stick–slip approach can be illustrated by a modern explanation of the typical depth distribution of the earthquake frequency (see Figure 6). Almost all earthquakes on the continents are confined within a crustal layer (Maggi et al., 2000). A symbolic histogram in Figure 6a shows a typical shape of the depth–frequency distribution of earthquake hypocenters. According to the stick–slip model, pre-existing faults located in the seismogenic layer of the earth's crust (zone between the upper and lower cutoffs) are characterized by brittle interfaces with velocity-weakening friction. At lower and higher depths, the friction is velocity-strengthening. The important question is: What causes the typical shape of the earthquake distribution? Two most accepted explanations on the basis of the stick–slip approach are as follows:
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The first one considers the focal depth distribution of earthquakes as an indicator of lithospheric strength (Albaric et al., 2009; Brace & Kohlstedt, 1980; Kirby, 1980; Kohlstedt et al., 1995). This approach proposes a similarity between the shape of profiles reflecting the depth distribution of the earthquake frequency and the lithospheric strength (Figure 6b). The upper part of the strength model uses Byerlee's friction law for the limiting strength (Byerlee, 1978). The lower part of the model is determined theoretically on the basis of a high-temperature steady-state flow (Kirby & Raleigh, 1973). The resulting model yields a lower limit of the strength of the lithosphere, because it describes the stress levels required to drive a suitably oriented fault. The similarity of the two profiles implies that within the seismogenic layer, the probability of instability increases with increasing lithospheric strength. This means that the stronger the lithosphere, the greater the probability of failure, which does not sound very logical.
The second explanation (Scholz, 1998, 2002) is illustrated in Figure 6c. Here, the strength of the lithosphere continues to follow the friction law below the lower cutoff, which is in conflict with the first explanation. The seismic properties of faults at different depths are described by the variable rate of the velocity-weakening response, which is determined by the difference between the level of static and dynamic friction. The solid curve corresponds to static strength profiles (long-term), while the dotted curve corresponds to dynamic strength profiles. It should be noted that to induce instability at any depth within the seismogenic layer, the applied stress must overcome the long-term strength. According to this model, despite the fact that the long-term strength of the lithosphere increases linearly with depth, the probability of instability is variable, which also does not seem logical. Thus, it can be concluded that there is no consensus on the cause of the typical depth distribution of earthquake frequency. Both models discussed above offer mechanisms whose physics is not very clear.
A new explanation based on the fan mechanism is illustrated in Figure 6d. Strength profiles of rock here, determining the lithospheric strength, include the frictional strength τf, the fracture strength of intact rock τu, and the fan strength of intact rock τfan (red curve). The fan mechanism will be dealt with in detail in the main part of the article. Here, only two points need to be noted: (1) the fan mechanism operates in a special range of confining pressures σ3, which determines the corresponding depth range of the earth's crust, and (2) in this range of pressure σ3 or depths, the fan mechanism operates with variable efficiency. The efficiency characterizes how much the fan mechanism can reduce the strength of intact rock with respect to frictional strength ψ = τf/τfan. Efficiency ψ increases from the borders (upper and lower cutoffs) to the center of the depth range under the action of the fan mechanism. At the depth of maximum efficiency, the intact rock shows the minimum strength τfan(min).
In contrast to the frictional strength profile τf, which represents the long-term lithospheric strength, the fan strength profile τfan (red curve) represents the transient lithospheric strength. The transient strength comes into play locally when the fan structure is generated in an intact rock mass (as shown in Figure 3). The depth range where τfan < τf represents the seismogenic layer. Here, earthquakes are mainly generated in the intact rock mass because their shear resistance τfan is lower than the frictional strength τf. Such earthquakes can occur at any level of shear stress within the gray zone. The depth distribution of earthquake frequency is determined by the distribution of the transient lithospheric strength: the lower the strength, the higher the probability of earthquakes. The maximum frequency of earthquakes occurs at a depth of minimum transient lithospheric strength τfan(min).
POSTPEAK PROPERTIES OF ROCKS CAUSED BY THE FAN MECHANISM
Postpeak rock properties at high σ3
The fan mechanism determines the rupture process and hence the postpeak rock properties. Since macroscopic spontaneous failure is possible only beyond the peak strength, knowledge of postpeak rock properties at the stage of failure is the key to understanding rupture mechanisms that control dynamic processes both in the laboratory and under natural conditions. In the experimental studies carried out by Brace and Byerlee (1966), postpeak properties of intact rocks were ignored; hence, a proper assessment of the role of intact rocks in the generation of earthquakes was not carried out.
Figure 7 shows the postpeak behaviors of rocks at different levels of confining pressure σ3. Postpeak properties of rocks with uniaxial compressive strength Suc < 250 MPa have been studied comprehensively within a wide range of σ3. All the results demonstrate that the residual strength determined by friction represents the minimum level of rock strength at any level of σ3. Rocks with Suc > 250 MPa can be studied beyond the peak stress at relatively low σ3; however, they remain unexplored at σ3 corresponding to seismogenic depths. The range of seismogenic depths is illustrated symbolically by a typical earthquake depth–frequency histogram below the curves. Due to the lack of experimental results, the postpeak properties of hard rocks under stress conditions corresponding to seismogenic depths are now considered by analogy with well-studied softer rocks (green dotted curves).
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The lack of the postpeak results is explained by the fact that available testing machines, despite their high stiffness and servo control, cannot prevent the highly violent rupture process beyond the peak stress on hard rocks (representative of seismogenic depths) at high confining pressure σ3 (corresponding to seismogenic depths). Further, it will be shown that this “anomalous” behavior of hard rocks is due to the fan mechanism. The term “hard rocks” (and hardness) is used in the paper to characterize the rocks in which the fan mechanism is generated. The fan mechanism operates in rocks with a specific combination of the following properties: high strength, brittleness, and density. Depending on these properties, the efficiency of the fan mechanism can vary. Experiments show that the fan mechanism is usually activated in hard rocks under high σ3. This is due to the fact that a high value of σ3 promotes the formation of the fan structure. It should be noted that a few articles published experimental results obtained on hard rocks at high σ3, where the postpeak control was provided (e.g., Goetze, 1971; Wong, 1982). This can be explained by the fact that the tested specimens were very small (diameter 18–20 mm and length 36–40 mm). In this case, the fan structure could not develop due to lack of space, and the rupture occurred in a different frictional scenario.
Figure 8 shows why the fan structure is formed at high σ3. It shows the evolution of failure mechanisms in hard rock specimens with increasing σ3 along the horizontal axis. It is known that failure of brittle rocks is accompanied by the formation of tensile cracks at any level of σ3. However, the length ℓ of tensile cracks decreases with increasing σ3 in accordance with a symbolic dotted line because σ3 suppresses the tensile crack growth. The length ℓ of tensile cracks in turn determines the macroscopic failure mechanism and the failure pattern shown schematically in rock specimens from (i) to (v).
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Within the pressure range 0 ≤ σ3 < σ3shear, shear rupture cannot propagate in its own plane due to the creation of relatively long tensile cracks at the rupture tip that prevent the shear rupture propagation. Two failure mechanisms may be distinguished under these stress conditions:
- 1.
Splitting by long tensile (or extension) cracks.
- 2.
Distributed microcracking, followed by coalescence of microcracks.
At σ3 ≥ σ3shear, the failure mode is localized shear. Here, tensile cracks generated in the rupture tip become sufficiently short to as to enable shear rupture to propagate in its own plane. According to Reches and Lockner (1994), the dilation of one microcrack induces the dilation of a closely spaced neighboring crack, thereby sequentially forming an array of tensile cracks and inter-crack slabs. These slabs are subjected to rotation at shear displacement of the rupture faces (King & Sammis, 1992; Peng & Johnson, 1972; Reches & Lockner, 1994). However, different behaviors of rotating slabs within different ranges of σ3 determine the following fundamentally different shear rupture mechanisms:
- 3.
Frictional shear—a: This mechanism operates within the pressure range σ3shear ≤ σ3 ≤ σ3fan(min). Here, shear displacement of the rupture faces causes the collapse of relatively long slabs during rotation that provides friction in the rupture head. In this range of σ3, the postpeak behavior is of class II, and the failure process can be controlled on stiff and servo-controlled machines.
- 4.
Fan-hinged shear: Within the pressure range σ3fan(min) < σ3 < σ3fan(max), specifically short slabs formed from hard rocks can withstand the rotation caused by shear displacement of the rupture faces. Due to consecutive generation and rotation of slabs and increasing relative displacement between the rupture faces with distance from the rupture tip, the slabs create the fan structure representing the shear rupture head. The extraordinary properties of the fan structure are shown in Figure 2.
It should be noted that within the pressure range σ3fan(min) < σ3 < σ3fan(max), the fan mechanism operates with different efficiencies. The efficiency of the fan mechanism ψ = τf/τfan is determined by the ratio between the frictional strength τf and the fan strength τfan. The fan strength τfan depends on how perfect and uniform the fan structure is and how perfectly and uniformly it operates. The fact that the length ℓ of tensile cracks and corresponding slabs continues to decrease with increasing σ3 makes the fan-mechanism efficiency variable. The green curve in Figure 8 shows graphically the typical variation of the fan-mechanism efficiency ψ = τf/τfan within this pressure range. At the low end of this range (near σ3fan(min)), when the length of the slabs is still relatively large, partial destruction of the slabs occurs as they rotate. In this case, the fan-mechanism efficiency is quite low. At higher σ3, with shorter slabs, this imperfection decreases, rendering the fan mechanism more efficient. The optimal efficiency takes place at σ3fan(opt) when the slabs rotate with minimum destruction. At greater σ3 up to the level σ3fan(max), the efficiency reduces because shorter slabs gradually lose their potential for operation as hinges.
- 5.
Frictional shear—b: At σ3 ≥ σ3fan(max), due to very short tensile cracks and slabs, they lose the capability to operate as hinges completely and the rock behavior reverts to the conventional frictional mode.
Class III postpeak rock behavior
Figure 9 shows the features of the rupture process governed by the fan mechanism in an intact specimen at high σ3. Figure 9a shows the five stages of the rupture growth along the dotted line ON, indicating the future fault. The red and black curves in Figure 9b show the variations in shear resistance τfan and amplified shear stress τamp(pot) caused by the fan structure during its initial formation and further propagation through the specimen between points O and N. Points on the curve indicate identical stages of shear rupture propagation as those in Figure 9a. The tensile-cracking process proves that the nucleation of a localized shear rupture starts at τ* of the prepeak stage close to the peak stress (Lei et al., 2000; Reches & Lockner, 1994). The tensile-cracking process is the basis for the fan-structure formation.
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The formation of the first half of the fan structure is accompanied by an increase in both shear resistance and the amplified stress up to the level τu (the reason for this has been explained in the companion Paper 1). The formation of the first half of the fan governs the ultimate material strength τu (the fan mechanism determines the rock strength). The formation of the second half of the fan and further propagation of the completed fan structure take place at the postpeak stage. Between stages 1 and 2, the developing fan structure induces a decrease in shear resistance up to the minimum level τfan and a marked increase in the potential amplified stress τamp(pot). The increasing difference Δτdis between decreasing τfan and increasing τamp(pot) causes spontaneous rupture propagation. It should be emphasized that the amplified energy generated by the fan structure Wamp (see the energy budget in Figure 2b) is in addition to the elastic energy stored in the specimen and in the loading machine. The total excess energy leads to highly violent failure of hard rocks at high σ3, usually observed in experiments.
To stop the spontaneous rupture, it is necessary to decrease the applied shear stress below the horizontal part of the red curve. At the stress level τfan corresponding to the horizontal part, the amplified shear stress at the rupture tip, generated by the completed fan structure, becomes equal to the material strength τu. Thus, at this very low level of the applied shear stress, the fan structure can propagate through the intact rock, as shown in the specimens between stages 2 and 4. This means that the specimen strength at these stages of failure is equal to τfan, which is significantly lower than the frictional strength τf. This stage of the postpeak rupture is classified as class III (Tarasov, 2019). When the fan crosses the specimen, the strength of the specimen increases and is determined by the residual frictional strength τf.
The actual postpeak properties of the specimen determined by the fan mechanism are reflected by the stress–displacement diagram in Figure 9c. It is worth noting that the descending part of the curve between stages 1 and 2 corresponds to extreme class II at which the postpeak modulus M is close to the unloading elastic modulus E (this will be shown experimentally). Between stages 2 and 4, the specimen strength remains constant and equal to τfan. This part of the postpeak curve corresponds to class III. When the fan crosses the specimen, the strength becomes equal to the residual frictional strength τf. Due to the fact that none of the existing testing machines can provide fast enough unloading of the specimen beyond the peak strength τu to release excess elastic energy and prevent spontaneous rupture, there is no information about the actual postpeak properties of hard rocks at high σ3, as shown in Figure 7. Special testing machines are needed to study the postpeak stage of class III.
NEW APPROACH TO THE EXPERIMENTAL STUDY OF POSTPEAK PROPERTIES OF HARD ROCKS AT HIGH σ3
Special requirements for specimen dimensions and testing machine
Based on the analysis conducted previously, a special method is developed for the experimental study of postpeak properties of hard rocks at high σ3. First of all, the specimen dimensions suitable for such experiments should be determined. The point is that specimens should have sufficiently large dimensions to accommodate the fan structure at its initial formation and for subsequent proper operation. Only under these conditions is it possible to obtain in experiments the true properties of the rock determined by the fan mechanism. In small specimens, the rupture mechanism is conventionally frictional. The length of the fan structure ℓfan can be different for different rocks and levels of σ3. As has been proved, the specimen diameter should be no less than 40 mm and its length should be no less than 80 mm.
The main limitation of all existing stiff and servo-controlled testing machines in studying the postpeak properties of hard rocks at high σ3 is the impossibility to create the necessary high speed of unloading at the rupture stages 1–2 in Figure 9b,c to stop the spontaneous process. A wealth of experience in designing various testing systems for studying properties of rocks in a wide range of experimental conditions has been used to create a new testing machine (Stavrogin & Tarasov, 2001). In this machine, the loading stiffness is maximized, and the inertia mass of all mechanical elements and the volume of fluid in chambers of the actuator are minimized to create the loading–unloading process. The maximum unloading rate obtained is about 25 000 GPa/s. A general view of the machine and the specimen equipped with load, axial, and lateral (diametral) gauges is shown in Figure 10. For failure of class II, diametral strain is used as the control mode.
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The application of the new machine has shown that this unloading rate is not high enough to control the entire fracture process and study class III postpeak behavior. Nevertheless, it is sufficient to penetrate the inaccessible zone below the frictional strength τf at the stage of class III behavior (see Figure 9c). Such studies confirm experimentally the existence of the very low fan strength τfan and the extreme brittleness created by the fan mechanism.
Experimental study of extraordinary postpeak properties of hard rocks at high σ3
Postpeak modulus and postpeak strength
Experiments were conducted on dolerite specimens (Suc ≈ 300 MPa) obtained from a seismically active gold mine in Australia (Tarasov, 2010; Tarasov & Randolph, 2008). The grain size ranged from 0.05 to 0.7 mm. Stress–strain curves in Figure 11a show that increasing the confining pressure changes postpeak rock behavior from class I to extreme class II. At σ3 < 60 MPa, the total postpeak control was provided for both types of rock responses: class I and class II. At σ3 ≥ 60 MPa, control was only possible at the start of the postpeak stage. After points A on the graphs, unstable failure took place, which was characterized by abnormally high dynamics and volatility. The postpeak dynamic failure at high σ3 ≥ 60 MPa is associated with propagation of a thin shear fracture (Figure 11c) with thickness less than 0.1 mm. The spontaneous rupture beyond points A demonstrates that the increased unloading rate provided by the new machine is insufficient for controllable postpeak rupture.
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It should be noted that the controllable postpeak stage ∆σ before the instability (enlarged curves on the right) decreases with increasing σ3. When confining pressure σ3 ≥ 150 MPa, the controllable postpeak stage becomes so small that spontaneous failure starts practically at the peak stress. From enlarged fragments of the stress–strain curves for σ3 = 75 and 60 MPa involving the postpeak parts at stress degradation from σu till σA in Figure 11b, the reason for the instability starting at points A can be analyzed. The fragments are replicated four times and divided into four steps (1–4) with equal intervals of differential stress. Each postpeak step is characterized by elastic modulus E (solid blue lines) and postpeak modulus M (dotted red lines). To determine the elastic modulus, the specimen tested at σ3 = 75 MPa was slightly unloaded just before the peak stress. The same value of elastic modulus was used for σ3 = 60 MPa. Areas located between the E and M lines at each postpeak step indicate the current postpeak rupture (or breakdown) energy Wr associated with shear rupture development.
As can be seen, Wr decreases markedly with the rupture growth from step 1 to step 4. It should be noted that, within the whole stress region between σu and σA, the servo-control system enables reliable control of the static fracture growth. However, after points A, where the fracture energy Wr becomes negligibly small (postpeak modulus M and elastic modulus E practically coincide), rupture control becomes impossible. Based on this analysis, it can be assumed that during spontaneous rupture beyond points A, the values of postpeak modulus M and elastic modulus E continue to remain identical.
The stress–time curves in Figure 12 represent the records provided by the load cell during the spontaneous rupture beyond points A for σ3 = 60, 75, and 150 MPa. This information is recorded by the dynamic data acquisition system that works in parallel with the static system and is activated when the dynamic process starts. Points A on the curves are marked by red asterisks. The curves show that the duration of the rupture process is about 0.1 ms and the maximum unloading rate achieved for σ3 = 150 MPa is about 24 000 GPa/s. Despite the fact that the unloading rate is not high enough to stop the spontaneous rupture growth, it is sufficient to approach the specimen strength σfan caused by the fan mechanism at point B. After point B, the specimen strength increases up to the frictional strength σfs. For σ3 = 150 MPa, the achieved minimum strength σmin ≈ 0.15 σfs. σfs denotes the static frictional strength determined experimentally after the spontaneous stage of rupture.
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For a better understanding of these results, the stress–strain and stress–time curves can be considered together. These curves for σ3 = 150 MPa are shown in Figure 13a. The red curves represent hypothetically the actual postpeak properties caused by the fan mechanism that remain “elusive” at spontaneous rupture. It should be emphasized that in this case, unloading significantly below the frictional strength σf does not stop the rupture process. This is because the actual strength of the specimen is equal to σfan and the rupture can be stopped only at stresses below this level. In contrast to this, in the conventional rupture mechanism, the residual strength σf represents the lower limit of rock strength. At stresses below σfs (or σfd), the rupture process is impossible.
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In the case of the fan mechanism, the actual very low strength of the specimen can only be detected at high unloading rates. In these experiments, the unloading rate of 24 000 GPa/s was found to be sufficient to approach the level of the actual strength of the specimen. However, much higher unloading rates are required to obtain a full postpeak curve shown symbolically in red in Figure 13a. It was found that at a lower unloading rate, the stress–time curves take the form of an AEC curve, and for spontaneous rupture without servo-controlled unloading, they take the form of an AF curve. A more detailed consideration of the duality of the strength properties of hard rocks in the rupture process controlled by the fan mechanism is presented in the findings of Tarasov (2019).
As mentioned above, the rock hardness in the paper is characterized by a special combination of strength, brittleness, and density of a microstructure, which determines the possibility of the fan-mechanism operation. High confining pressure increases hardness by limiting the length of tensile cracks and inter-crack slabs. In some rocks with a special microstructure, the fan mechanism can work even under uniaxial compression. Figure 13b,c shows this on quartzite. In general, quartzite is a metamorphic rock formed when quartz-rich sandstone or chert has been exposed to high temperatures and pressures. Such conditions fuse the quartz grains together, resulting in the formation of a dense, hard, equigranular rock. Figure 13b (Raith et al., 2012) shows the features of a quartzite microstructure when recrystallization takes place at different levels of pressure and temperature. Usually, grain boundaries represent the weakest elements of the structure. The structure of quartzite in Figure 13b,c has high hardness. Due to features of recrystallization at high pressure and temperature (about 700°C) associated with grain boundary migration, intensive interfingering sutures are formed (Stipp et al., 2002). With grain boundaries consisting of interfingering sutures, the grains are cemented together so firmly that once induced in the rock, the fractures pass through, rather than around, the grains (Raith et al., 2012).
Specimens from quartzite with the C-type structure show extreme class II behavior at uniaxial compression. Figure 13c shows a set of stress–strain curves presented in different colors. The failure mode is a shear rupture caused by an intense and localized microcracking process. It should be emphasized that the postpeak control is possible up to points A shown in the graphs, after which a spontaneous and violent failure follows. This failure process is governed by the fan mechanism. As has been observed, at uniaxial compression, the controllable postpeak stage (up to points A) is larger compared to the failure at high σ3. The reason for this is the lower efficiency of the fan-mechanism operation due to the higher length of rotating slabs at uniaxial compression. To provide total postpeak control for hard rocks, a new generation of testing machines is required. Some general ideas for new machines are discussed below.
Extreme brittleness of hard rocks at high σ3 and the universal scale of brittleness
Spontaneous macroscopic failure can occur beyond the peak stress only. As demonstrated above, hard rocks at high σ3 show class II behavior. Rocks of class II fail spontaneously even at absolute stiffness of the loading system due to the elastic energy available from the specimen. A brittleness criterion K1 is used to characterize the degree of capability of the rock of self-sustaining failure. This criterion is determined by the ratio between the postpeak rupture energy Wr and the withdrawn elastic energy We (for more details, see Tarasov, 2011; Tarasov & Potvin, 2013; Tarasov & Randolph, 2011). The ratio between these types of energy can be expressed using the elastic and postpeak modules:
This criterion allows design of the universal scale of postpeak brittleness of rocks at compression shown in Figure 14a. It is represented graphically by a series of stress–strain curves, where the postpeak rupture energy Wr corresponds to the gray area, the elastic energy We corresponds to the area in the red triangle, and the released energy Wa corresponds to the yellow area. The elastic modulus E is the same for all curves, while the postpeak modulus is variable and represented by dotted lines. The brittleness increases from ductility at K1 = ∞ on the left to absolute brittleness at K1 = 0 on the right.
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The important point on the scale is K1 = 1, representing the boundary between class I and class II behavior. To the right of this point, the material behavior is class II + III, characterized as spontaneous self-sustaining failure. This zone is shown in pink. Here, the internal elastic energy accumulated in the material We exceeds the rupture energy Wr. The closer the value of K1 is to zero, the more brittle the material and the more violent the failure will be. At absolute brittleness K1 = 0, E = M and the postpeak rupture energy is equal to zero (Wr = 0), which means that the total elastic energy stored in the material body will be transformed into seismic energy. It should be emphasized that all rocks located at seismogenic depths show class II + III behavior. This is because the spontaneous development of a shear rupture in an intact rock mass subjected to triaxial compression can occur only due to the release of internal elastic energy from the failing rock. To the left of the point K1 = 1, self-sustaining failure is impossible, and some additional energy from the loading system is necessary to cause postpeak failure development. This behavior is typical for rocks located at low depths above the seismogenic layer.
The universal scale of brittleness can be used to evaluate how the brittleness of rocks changes with increasing confining pressure σ3 or depth (for details, see Tarasov, 2011; Tarasov & Potvin, 2013; Tarasov & Randolph, 2011). Figure 14b shows the variation of brittleness for four rocks (sandstone, quartzite, dolerite, and granite) versus σ3.
The horizontal axis represents the brittleness index K1 and the vertical axis represents the level of σ3. The self-sustaining failure regime corresponds to 1 ≥ K1 ≥ 0 (pink area). The sandstone curve indicates that an increase in confinement σ3 makes the rock less brittle. This behavior is typical for soft rocks. For the quartzite, an increase in confinement σ3 within the range of 30–100 MPa makes the material more brittle, with the maximum brittleness at σ3 = 100 MPa. At greater σ3, the brittleness decreases. For the granite, at σ3 > 30 MPa, the brittleness increases markedly. The dolerite curve also shows very severe rock embrittlement. At σ3 = 75 MPa, according to the brittleness index K1, the dolerite became 250 times more brittle when compared to uniaxial compression (K1[0] = 1.5; K1[75] = 0.006). At σ3 = 100 and 150 MPa, the brittleness increases, which makes postpeak control impossible even at the very beginning of the postpeak stage. The dotted lines indicate the expected brittleness variation for granite and dolerite at higher values of σ3; the brittleness continues to increase until it reaches a maximum at some level of σ3 and then decreases in a similar pattern as in the case of quartzite. It can be supposed that the maximum brittleness for granite and dolerite can be reached at σ3 ≈ 300 MPa. This variation in rock brittleness is identical to the variation in the efficiency of the fan mechanism with σ3, shown by the green curve in Figure 8, since the fan mechanism determines the postpeak rock properties. The shape of the histogram of the depth distribution of earthquake frequency on the right (modified from Scholz, 2002) is similar to that of the brittleness variation. Further, it will be shown that this shape and the range of depths of the seismogenic layer are determined by the range of action of the fan mechanism.
Complete profiles of strength and brittleness for hard rocks and fan-mechanism efficiency
The new testing machines will make it possible to study the fan mechanism and obtain complete stress–deformation dependencies of hard rocks in the entire range of confining pressures σ3 corresponding to seismogenic depths. These experimental data will allow plotting of complete strength (Figure 15a) and brittleness (Figure 15d) profiles for hard rocks, which is very important for practical purposes such as predicting earthquakes and shear rupture rockbursts in deep mines.
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Figure 15a shows schematically a model of a complete strength profile that fully characterizes the strength properties of hard rocks at different levels of σ3. It includes the conventional fracture τu and frictional τf strength profiles and the fan strength τfan profile (red curve). The variable efficiency of the fan mechanism with σ3 (as shown in Figure 8) causes corresponding variations in the fan strength and fan brittleness of hard rocks. To plot the complete profiles, it is necessary to obtain experimentally a set of complete stress–displacement curves. Figure 15b,c shows two stress–displacement curves reflecting complete postpeak rock properties (shown in red). These curves explain the meaning of the fan strength profile τfan and its relation to the conventional strength profiles τu and τf, respectively. Figure 15b shows the result at σ3 = σ3fan(opt), where the fan mechanism shows the maximum efficiency. At the peak stress, the material strength is τu. After completion of the fan structure, the specimen strength decreases to the level τfan, representing the fan strength. This means that the shear fracture governed by the fan mechanism can propagate through the material at any level of shear stress above τfan. After the fan has crossed the specimen body, the specimen strength is determined by the frictional strength τf of the new fault. For confining pressures σ3 < σ3fan(opt) or σ3 > σ3fan(opt), the fan-mechanism efficiency is lower, resulting in higher levels of τfan as shown in Figure 15c.
The complete strength profile demonstrates that the fan mechanism operates in the pressure range σ3fan(min) < σ3 < σ3fan(max), causing a decrease in rock strength from the level τu to the level τfan. This means that in this pressure range (after the formation of the initial fan structure), intact rock can fail at any level of shear stress above the red curve. Further, it will be shown that the complete strength profile makes it possible to estimate the depth of the upper and lower cutoffs of earthquakes, the nature of the distribution of earthquake frequency with depth, the most dangerous depth of seismic activity, and also the critical depth for mines below which shear rupture rockbursts can occur. In addition to the complete strength profile, the obtained stress–displacement curves also allow the construction of complete brittleness profiles as shown in Figure 14b.
THE FAN MECHANISM AS THE MAIN AND MOST DANGEROUS MECHANISM OF EARTHQUAKES IN THE EARTH'S CRUST
Depth distribution of rock strength and brittleness in the earth's crust and earthquake frequency
The complete strength and brittleness profiles of hard rocks in Figure 15 show that in a special range of confining pressure σ3, rocks show extraordinary properties induced by the fan mechanism. Thus, it is assumed that in the corresponding depth range of the earth's crust, the fan mechanism should also operate with the creation of identical rock properties. Figure 16 demonstrates the depth distribution of the following characteristics: (a) fan-mechanism efficiency; (b) profiles for the fracture strength τu, frictional strength τf and fan strength τfan; (c) the brittleness profile; and (d) earthquake frequency (Tarasov, 2013; Tarasov & Randolph, 2016).
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According to this approach, the lithospheric strength at low depths corresponding to σ3 < σ3fan(min) and at great depths corresponding to σ3 > σ3fan(max) is determined solely by frictional strength τf. Within the depth range corresponding to the fan-mechanism activity σ3fan(min) < σ3 < σ3fan(max), the situation is specific. The frictional strength τf represents the long-term lithospheric strength. However, if the fan mechanism is activated somewhere, causing new rupture development in intact rock, the transient lithospheric strength in that region decreases to the level τfan. It should be emphasized that the low transient strength arises only in the fan zone of the propagating rupture. After the completion of the failure process, the lithospheric strength reverts to the frictional strength.
It should be noted that the new concept of lithospheric strength incorporates all three types of rock strengths determining the instability in the seismogenic layer: fracture strength τu, frictional strength τf, and fan strength τfan. Fracture strength determines the level of local stress at which the initial fan structure can be generated. This issue will be discussed later. The shaded area between the frictional τf and the fan transient τfan profiles determines the level of field stress under which an initiated fan structure can propagate to create an earthquake. Importantly, the fan mechanism can cause earthquakes at any level of field stress τ within the shaded zone. Due to this, the highest probability of events is at a depth corresponding to the depth of optimal efficiency of the fan mechanism, where the rock mass is characterized by the minimum transient strength τfan and maximum brittleness. At lower and greater depths, the probability decreases. This feature determines the typical depth–frequency distribution of earthquake hypocenters. The explanation for the depth distribution of earthquake frequency on the basis of the fan mechanism differs fundamentally from the conventional explanations shown in Figure 6. In Figure 16, the upper and lower earthquake cutoffs are shown as the boundaries of the zone of the fan-mechanism activity. However, it should be noted that at shallower and greater depths, earthquakes can be generated based on the stick–slip mechanism along pre-existing faults. The map of earthquakes shown in Figure 4b supports this idea: shallow earthquakes up to a depth of about 30 km are widely dispersed outside global boundaries and can form earthquake swarms, but at greater depths, hypocenters are mainly located at global plate boundaries.
The main paradoxes of earthquakes generated by the fan mechanism
The map of earthquake hypocenters shown in Figure 4b shows that the vast majority of earthquakes in the earth's crust nucleate outside of pre-existing general faults and can form quite wide zones of seismic activity similar to swarms. Figure 17a shows profiles characterizing the distribution of rock strength versus depths (or σ3) in the earth's crust as shown in Figure 16. The gray area here indicates the combination of stress states (σ3 and τ) at which a new fault in the intact rock mass can be generated by the fan mechanism. This section analyzes the situation in the rock mass under conditions corresponding to point R characterized by stresses σ3(R) and τR.
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Figure 17b shows a fragment of a rock mass with a pre-existing fault at four consecutive stages of new fault development (stages 0–3). Graphs under the rock fragment at each stage show the characteristics of stress and strength corresponding to point R: τu(R) denotes the fracture strength of the intact rock mass; τf(R) is the frictional strength of the pre-existing fault; τR is the field shear stress applied; and τfan(R) denotes the fan strength. At stage 0, the level of the field stress is τ0(R). The level of field stress τ0(R) is less than the frictional strength τf(R), which indicates the stable condition on the pre-existing fault. However, the dynamic rupture process can occur in intact rock adjoining the pre-existing fault, since any fault is a local stress concentrator, creating starting conditions for fan-structure formation.
Since the pre-existing fault has a step, it causes a stress concentration in the local adjacent zone of an intact rock shown by a red dotted circle. As the stress concentration approaches the fracture strength τu(R), tensile cracking begins, leading to the formation of the initial fan structure (or double fan structures resulting in a bilateral rupture). After the completion of the fan structure (stage 1), its shear resistance decreases to the level τfan(R), which is significantly lower than the applied shear stress τ0(R). As such, a new bilateral fracture (I) will spontaneously propagate through the intact rock mass at very low applied shear stress τ0(R) (for more details on the principle of bilateral rupture, see the companion Paper 1). It should be emphasized that the low rock strength arises only in the propagating fan zone during the rupture propagation.
After the rupture process, the field stress decreases to the level τ1(R), causing the stress drop Δτ = τ0(R) – τ1(R). Since any new fault is also a local stress concentrator, the formation of the following faults (II) and (III) can occur (stages 2 and 3). This mechanism of successive fault formation can cause aftershocks and swarms. It is important to note that each rupture process decreases the field stress as shown in Figure 17b. When the field stress becomes too low to create a high local stress equal to the rock strength (necessary for the initial fan-structure formation), the rock mass returns to a stable condition determined by the long-term lithospheric strength τf(R). Figure 17c shows a set of faults that formed in an intact rock mass, causing severe earthquakes in New Zealand (Temblor, 2017). Stick–slip instability was not observed on pre-existing faults.
The analysis performed shows that the fan mechanism creates a number of paradoxes from the conventional point of view: (1) it generates new faults in intact hard rocks at extremely low field stresses, which can be an order of magnitude lower than the frictional strength; (2) rupture of intact rock is accompanied by an abnormally low stress drop, which contradicts the conclusion arrived at by Brace and Byerlee (1966); and (3) despite the fact that pre-existing faults represent the weakest elements of the rock mass, the most dangerous medium in relation to earthquakes at seismogenic depths is the intact rock mass.
The energy budget of earthquake ruptures generated by the fan mechanism
This section shows that the energy budget of earthquake ruptures generated by the fan mechanism is more powerful compared to that of the stick–slip mechanism. Figure 18a shows profiles characterizing the distribution of rock strength versus depths (or σ3) in the earth's crust as shown in Figure 16. The shaded area indicates the combination of stress states (σ3 and τ) at which a new fault in the intact rock mass can be generated by the fan mechanism. What follows is the analysis of the energy budget of two earthquake ruptures generated by the fan mechanism under stress conditions corresponding to the same depth characterized by stress σ3* and two different shear stresses τ0a and τ0b. Here, τ0a represents the field stress close to the fan strength τfan and τ0b denotes the field stress close to the friction strength τfs.
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Seismic inversions have indicated that most earthquake ruptures propagate in the pulse-like mode (Heaton, 1990; Noda et al., 2009; Perrin et al., 1995; Zheng & Rice, 1998). Due to this, the energy budget of earthquakes is considered here to be pulse-like ruptures. A detailed analysis of the energy budget of extreme ruptures generated by the fan mechanism has been carried out and presented in the companion Paper 1. Here, the most important features of this budget are briefly recalled. Figure 18b,c shows the energy budget per unit rupture area for two pulse-like ruptures propagating at relatively low τ0a and large τ0b shear stresses. The point is that the energy budget is fundamentally different when the applied shear stress is less than the dynamic frictional strength τ0a < τfd (Figure 18b) and higher than the dynamic frictional strength τ0b > τfd (Figure 18c). These two energy budgets per unit rupture area are compared by using the average diagrams of shear stress (shown in black) and shear resistance (shown in red) versus slip within the slip zone. It should be emphasized that in this paper, for clarity, a conditional joint diagram is used to analyze both the distribution of shear stresses and shear resistance along the rupture and to analyze the energy budget. When analyzing the energy budget the segment OS on the horizontal axis is considered as the total slip (displacement) between the rupture faces occuring inside the slip zone. The segment ODc is the breakdown displacement and the segment DcS is the slip displacement.
In Figure 18b, the total slip OS is conditionally equal to the length of the fan structure ℓfan on the diagram. The dissipated energy represented by the breakdown energy Wr (dark gray thin triangle) and the friction energy Wf (light gray area) are very low for the fan mechanism. The breakdown energy Wr is associated with the constantly recurring creation of the frontal tensile crack forming the frontal slab. The breakdown strength τr decreases sharply from τu (fracture strength) to τfan (fan strength) on the distance ODc, which is determined by the width of the frontal tensile crack. The following slip DcS along the fan structure is characterized by a very low shear resistance τfan.
The driving power consists of two types of energies. The yellow area denotes the radiated (or seismic) energy Ws representing a portion of the energy released from the rock mass due to the rupture motion accompanied by the stress drop Δτ = τ0 – τ1. The pink area denotes the amplified energy Wamp generated by the fan mechanism. Due to the much higher driving power Ws + Wamp compared to the dissipated energy Wr + Wf, the rupture velocity increases considerably (up to a supersonic level) and causes a violent rupture process.
The amplified energy Wamp ensures constant readiness and continuity of the rupture process in intact rocks. Due to this energy, compared to the conventional frictional mechanisms, the fan mechanism provides a different physics of energy supply to the rupture tip. According to classic theories, spontaneously propagating ruptures are driven by elastodynamic waves, where the energy released by the rupture motion (yellow area on the model in Figure 18d) is transferred through the medium to the rupture tip region at the maximum speed equal to the pressure wave speed vp. This mechanism of the energy supply restricts the maximum rupture velocity by vp (Freund, 1998; Gori et al., 2018; Needleman, 1999; Rice, 2001; Rosakis et al., 2007). Unlike classic theories, the source of energy represented by the amplified energy Wamp surrounds the rupture tip, providing energy directly. In this case, there is no need to deliver energy from elsewhere to provide conditions to achieve supersonic rupture velocity. The seismic energy per unit area Ws (yellow area) represents the energy released from the rock mass due to the rupture and slipping processes accompanied by the stress drop Δτa = τ0a – τ1. The stress drop Δτa at failure under natural conditions is small compared to the stress drop obtained on the same rock in the laboratory Δτlab = τu – τfs (shown on the diagram).
Figure 18c shows the energy budget for the earthquake shear rupture generated at the field shear stress τ0b, which exceeds the dynamic frictional strength τfd and is close to the static frictional strength τfs. Since the initial stress is higher and close to the static frictional strength τfs, the energy budget of this rupture will have the following differences compared to that of the rupture in Figure 18b. (1) The length of the fan structure ℓfan is longer (for an explanation, see the companion Paper 1). (2) Slipping continues behind the fan, since the applied stress τ1 exceeds the dynamic frictional strength τfd, and the slipping will not cease until the applied stress decreases to the level of dynamic frictional strength τ1 = τfd. Dynamic friction is a velocity-dependent parameter, and currently, there is no consensus on the actual frictional resistance during earthquakes. For rupture processes controlled by the fan mechanism, the level of dynamic friction does not play a decisive role, as in the stick–slip mechanism. In Figure 18c, τfd is about 30% lower than static friction τfs. (3) The pulse-like head OS of the rupture here is complex and consists of two parts (fan and frictional), which is longer than that in Figure 18b (for a discussion of this question on the basis of experimental results, see the companion Paper 1). (4) The frictional energy Wr (light gray area) at the fan stage is low, but is high at the frictional stage. (5) Despite this, the seismic energy Ws (yellow area) is higher compared to Figure 18b). The stress drop Δτb = τ0b – τ1 is also higher. All these differences indicate that the rupture process under this stress condition in Figure 18c is much more powerful than that in Figure 18b.
Figure 18d shows a diagram representing the conventional energy budget of an earthquake rupture generated in a pre-existing fault by the frictional stick–slip mechanism (analog of the typical diagram in Figure 18e from Abercrombie & Rice, 2005). As is discussed earlier, such diagrams do not include stress conditions for dynamic spontaneous rupture development at the initial stage of failure between points A and B. The zone within the dashed green trapezoid represents the total strain energy change per unit rupture area, which is determined by the average stress versus slip diagram τ0 ÷ τ1. This energy is partitioned into the rupture energy Wr (including the part above point B), the frictional energy Wf, and the radiated free energy Ws. To cause dynamic failure just beyond the rupture tip (point A), the energy released by the rupture motion is transferred from the region shown in yellow through the medium to the rupture tip region. The dissipated rupture and frictional energy Wr and Wf are relatively high, which makes the rupture process governed by the stick–slip mechanism less powerful than that by the fan mechanism.
Das and Scholz (1983) pointed out that although, by definition, tectonic earthquakes may nucleate anywhere within the seismogenic layer, in almost all cases, earthquakes that are severe nucleate near the base of the seismogenic layer. According to Das and Scholz (1983), this is because frictional strength and stress drop should increase with depth, and in situ stress measurements suggest that the ambient stress increases with depth. The diagram of the lithospheric strength in Figure 18a shows that near the base of the seismogenic layer (lower cutoff), shear stresses that can activate the fan mechanism are close to the frictional static strength τfs. This means that under these stress conditions, the energy budget of earthquake ruptures will be similar to that shown in Figure 18c, which is characterized by higher release of seismic energy, larger stress drop, and larger slipping rupture head.
In conclusion, as it is, the fan mechanism represents the most dangerous rupture mechanism in the earth's crust in relation to earthquakes. According to the frictional stick–slip mechanism, instability can be generated in dry rocks at stresses exceeding the frictional strength τfs, indicated by the line “frictional” in the diagram in Figure 18a. In contrast, the fan mechanism can generate earthquakes in dry rocks under a wide range of stress conditions, represented by the gray shaded area located between the curve ‘frictional’ and the curve ‘fan’ in the diagram in Figure 18a. The fan mechanism can generate earthquakes even at very low shear stresses, which can be a decimal order of magnitude lower than the frictional strength. The failure process generated by the fan mechanism under any stress conditions is accompanied by the release of much higher seismic energy compared to the stick–slip mechanism (see diagrams of the energy budget in Figure 18). The outstanding energy budget and special physics of the energy supply to the rupture tip provide conditions for very high rupture velocity including supersonic levels (for the experimental proof, see the companion Paper 1).
New concept of the interrelation between volcanoes and earthquakes
Known sources of heat for rock melting in the earth's crust
For a more complete characterization of the fan mechanism as the most dangerous at seismogenic depths, it is also necessary to note its role in the creation of stress-release shocks in the earth's crust, accompanied by the formation of large volumes of molten rocks that can serve as a source of magma for volcanoes. Currently, two sources of heat caused by dynamic processes are considered to be responsible for rock melting in the earth's crust: friction-induced heat along a fault surface during seismic slip (Fukuyama, 2009; Lin, 2008; McKenzie & Brune, 1972; Sibson, 1975; Spray, 1987) and meteorite impact (French, 2003; Langenhorst & Hornemann, 2005; Melosh, 1989; Zel'dovich & Raizer, 2002). Pseudotachylytes are widely accepted as the most accurate rock indicator of seismicity on ancient faults and meteorite collisions. The basic features of rock melting due to these two processes are as follows:
- 1.
Quantitative models and laboratory experiments have demonstrated that frictional melting could occur during coseismic slip if the slipping zone is of the order of a few millimeters thick, but once a thin film of melt is formed on a seismic fault plane, the marked decrease in fault friction is assumed to suppress further melting. It has been recognized that the generation zones where pseudotachylytes are formed during seismic slip represent overlapping tips of faults (jogs) in an echelon array (Grocott, 1981; Sibson, 1975; Swanson, 1992) as shown in Figure 19 (modified from Curewitz & Karson, 1999). Reservoir zones in Figure 19 are large, dike-like pseudotachylyte bodies that are commonly >10 m wide and occupy extensional voids in fault zones (Sibson, 1975). The following mechanism associated with the specific geometry of segmented faults was proposed. Localized decompression at dilational jogs can create significant fluid pressure gradients, leading to rapid migration of the melt to sites of low pressure, thereby re-establishing frictional contact across the fault surface and promoting further melt generation (Curewitz & Karson, 1999). Actually, pseudotachylytes can involve not only melted but also crushed rocks. This mechanism, however, cannot explain the following fact: in some cases, shear displacement along the fault is too small for the creation of observed large volumes of molten material (Lin, 2008).
- 2.
In the case of meteorite impact, the huge amount of energy causing crush, melting, and vaporization of rocks is derived from the kinetic energy of impacting objects flying at speeds of tens of kilometers per second, exceeding the velocity of pressure elastic waves Vp in the impacted target rocks (French, 2003; Melosh, 1989; Zel'dovich & Raizer, 2002); Langenhorst & Hornemann, 2005). In the case of hard rocks, Vp ≈ 5–8 km/s. Melting of rocks during impact is the result of shock compression and subsequent release. Impact shock waves are characterized by an instantaneous onset of extreme pressures (tens of GPa). Shock compression is an irreversible process where plastic work done on the target material remains in the rock as heat after subsequent pressure release and can increase the temperature of the target above the melt temperature. If the shock pressure near the impact point is about 40 GPa, it may increase the postshock temperature sufficiently up to melting of the rock material. At shock pressures ≥60 GPa, the postshock temperature would exceed 2000°C and thus cause immediate large-scale melting after the shock wave has passed (Langenhorst & Hornemann, 2005).
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Rock melting due to “meteorite-like” collision of extreme ruptures (supersonic) during earthquakes governed by the fan mechanism
This paper proposes a previously unknown source of heat capable of producing large volumes of molten rock during earthquakes based on the fan mechanism. As shown in Figure 18, the fan mechanism generates an outstanding energy budget and offers new physics for applying energy to the rupture tip, which provides the conditions for the rupture propagation in intact hard rocks at supersonic speeds. According to the experimental results obtained at high confining pressures by Otsuki and Dilov (2005), segmented faults propagate due to advanced triggering of a new bilaterally propagating shear fracture (new segment) in front of the propagating current shear fracture (current segment). This is shown in Figure 20a, which shows four stages of segmented fault propagation. All ruptures are driven by the fan mechanism, where the fan structure represents the rupture head. Red asterisks mark the points of origin of new segments. Two segments propagating toward each other to make contact form an overlapping zone (jog). As Figure 20a shows, compressional-type jogs are similar to those observed in the experiments by Otsuki and Dilov (2005) and in ultra-deep South African mines by Ortlepp (1997).
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A new possible mechanism is proposed for the melting of rocks caused by earthquake ruptures, controlled by the fan mechanism. This mechanism can work both in compressional and in dilational jogs. The general idea of this mechanism is shown in Figure 20b–d. Figure 20b shows two supersonic ruptures propagating toward each other in three stages. Since the pressure wave speed for hard rocks is about Vp ≈ 5–8 km/s, a supersonic rupture speed of about 10 km/s can be assumed. The relative speed of the two ruptures is 20 km/s. Mach waves generated by these ruptures represent pressure shock fronts. Figure 20c shows images of Mach shear and pressure waves obtained experimentally by Gori et al. (2018). When the shock fronts meet, they create a huge instantaneous compression in the meeting zone and then an instantaneous release of pressure, that is, conditions similar to a meteorite impact causing melting and vaporization of rocks. Figure 20d shows a plane view of the ruptures. This mechanism can create large reservoir zones and large dike-like pseudotachylyte bodies that are commonly >10 m wide and occupy extensional voids in fault zones as mentioned by Sibson (1975). This mechanism explains how large volumes of molten material can be created at very small fault displacement reported by Lin (2008). Figure 20e shows the typical configuration of reservoir zones of molten rocks in ancient seismic faults. Another version of the “meteorite-like” impact leading to rock melting can occur when a supersonic rupture enconunters an obstacle that stops the rupture propagation. An obstacle can be, for example, a pre-existing fault located in the perpendicular direction. As shown in Section 4.5 of Paper 1, when encountering an obstacle, the fan stucture mobilises its maximum capacity to stress amplification, which is applied as a shock. The combination of two forms of impact (pressure shock wave and amplified stress shock) can cause rocks to melt.
The fan mechanism of volcanic earthquakes as a generator of magma
The maps in Figure 21 show the typical earthquake activity in volcanic zones. The red dots on the map in Figure 21a correspond to the earthquake hypocenters in the Yellowstone volcano zone. Figure 21b shows the depth distribution of earthquake hypocenters within the seismogenic layer along the line AA′ in Figure 21a (from Smith et al., 2009). Magnitudes for some of the larger events are labeled. Dots of different colors on other maps also indicate earthquake hypocenters. Figure 21c shows earthquake activities in Kilauea Volcano during May 1–14, 2018 (from Hawaiian Volcano Observatory October 11, 2018). On May 4, 2018, a powerful magnitude-6.9 earthquake on the south flank of Kīlauea Volcano shook the Island of Hawaii. The inferred rupture area (white dashed line) with its foreshocks and the aftershocks in the first 10 days spanned an area of about 800 km2. Circle size indicates earthquake magnitude; color indicates earthquake depth. Magnitudes for some of the larger events are labeled. The depth range of these earthquakes is 5–13 km. The earthquake map in Figure 21d evidences over 9000 earthquakes at Mt Thorbjorn, Iceland, during a month in 2022 (Flis, 2022).
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The close relationship between earthquakes and volcanic outbursts is evident from the maps depicting the locations prone to both these phenomena. This is because the main modern theory behind both these natural calamities includes plate tectonics (Anderson & Natland, 2014; Clague & Dalrymple, 1987; Fouch, 2012; Leonard & Liu, 2016). Earthquakes can trigger volcanic eruptions through severe movement of tectonic plates. Similarly, volcanoes can trigger earthquakes through the movement of magma within a volcano. This theory is supported by the fact that most of the Earth's volcanoes are formed at the edges of tectonic plate boundaries. But some of the Earth's most well-known volcanoes—such as those of the Hawaiian Islands and Yellowstone—arise in the middle of a plate, and are fueled by isolated “hot spots” of magma rising from deep within the mantle. Scientists are still trying to understand why and how these hot spots come into being. Some argue that a deep mantle plume sourced at the base of the mantle supplies the heat beneath Yellowstone, whereas others claim that shallower subduction or lithospheric-related processes can explain the anomalous magmatism.
The fan theory proposes a new mechanism of magma supply into volcanoes. According to the fan theory, the vast majority of earthquakes occur as a result of the formation of new dynamic faults in intact rocks. Natural faults usually have a multisegment structure (similar to that shown in Figure 20a). Jog zones between segments serve as magma generators for volcanoes due to the “meteorite-like” collision of extreme (supersonic) ruptures (illustrated in Figure 20). The numerous earthquakes that occur around each volcano can produce enough magma to sustain volcanic activity. In this case, the volcanic zone is a self-sustaining organism in which high-temperature gradients cause numerous earthquakes (due to local stress concentrations), and earthquakes generate new magma for the volcano. The new volcanic concept does not require the delivery of magma from deep within the mantle.
Thus, it can be concluded that the fan mechanism is the most dangerous rupture mechanism in the earth's crust, both in relation to natural disasters in the form of earthquakes and volcanic eruptions.
THE FAN MECHANISM AS A GENERATOR OF INDUCED EARTHQUAKES
Shear rupture rockbursts generated by the fan mechanism
As hard rock mining progresses to depth, rockburst problems increase. Below a certain depth (different for different rocks), a special form of rockburst—shear rupture rockburst—comes into play. It is established (Gay & Ortlepp, 1979; McGarr et al., 1979) that shear rupture rockbursts are caused by the dynamic formation of new faults in intact hard rock. These mine tremors are seismically indistinguishable from natural earthquakes and share the apparent paradox of failure under low shear stress (McGarr et al., 1979). Some additional features of shear rupture rockbursts were described by David Ortlepp (Ortlepp, 1997; Ortlepp et al., 2005): “Shear rupture rockbursts nucleate in pristine rocks within the rock space at a point some considerable distance away from the surface of an opening … after which a very large amount of energy is suddenly and violently emitted from somewhere in the semi-infinite rock-space surrounding a mine, to express itself as a large rockburst. Large incidents cause damage of such intense violence that it seems that our knowledge of the mechanism of damage is completely inadequate.”
Thus, the outstanding features of shear rupture rockbursts include great depths, generation of new faults of extreme dynamics in intact rocks, fault nucleation at a point considerably away from the opening surface, fault propagation at low shear stresses, and extreme volatility. Indeed, it is difficult to explain the entire combination of the features of this phenomenon on the basis of the traditional understanding of rock failure mechanisms. This section demonstrates that all these features of shear rupture rockbursts represent manifestations of the intrinsic properties of the fan mechanism (Tarasov, 2010, 2014b, 2018).
Figure 22 shows the condition around a deep mine. Figure 22a shows an example of a complete strength profile for rocks surrounding the mine. This strength profile indicates the range of confining pressure σ3fan(min) / σ3fan(max), within which the fan mechanism can be activated. The red curve shows the variation of the fan strength τfan within this pressure range and reflects the variation of the fan-mechanism efficiency ψ = τf/τfan. The complete profile allows analysis of the possibility of generation of extreme ruptures governed by the fan mechanism around the mine. Figure 22b shows a schematic cross-section of a fragment of the earth's crust involving the opening. The black graph illustrates the depth distribution of minor stress σ3 in this area. The fan mechanism can be activated below a critical depth (upper cutoff) corresponding to the critical level of minor stress σ3fan(min). The green curve symbolically illustrates the depth distribution of the fan-mechanism efficiency ψ = τf/τfan. The graphs show that around the opening, the minor stress is below the critical level σ3 < σ3fan(min). Hatched areas on the picture above the upper cutoff and around the opening indicate zones where the fan mechanism cannot be generated. The zone of the fan-mechanism activity is shown by the gray area.
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Figure 22c shows the variation in rupture mechanisms with increasing confining pressure σ3 and, consequently, with distance from the mine surface. Different rupture mechanisms operating at different distances from the mine surface determine the corresponding behavior of the rock mass surrounding the mine. Behind the boundary of the unloaded zone characterized by σ3 = σ3fan(min), the rock mass is exposed to stress conditions corresponding to the fan-mechanism activity. The mine alters the stress state in the surrounding rock mass and can cause a local stress concentration on the basis of a pre-existing defect, leading to the tensile cracking process and formation of the initial fan structure. This can only occur at a considerable distance away from the opening surface (Figure 22d), where the stress conditions correspond to the fan-mechanism activity. The completed fan will cause extreme rupture, propagating even at low shear stress. Since the energy budget of rockburst ruptures is similar to that for earthquake ruptures as shown in Figure 18, the failure process will be very violent. Thus, all characteristic features of shear rupture rockbursts mentioned above are caused by the fan mechanism. This knowledge of the actual mechanism of shear rupture rockbursts can aid in the development of methods for prevention of disasters.
The fan-hinged shear as a possible mechanism of induced seismicity
The shear rupture rockburst discussed above is an example of induced seismicity. Induced seismicity is typically earthquakes and tremors that are caused by human activity that changes the stress state in the earth's crust. Induced seismicity is associated with different industrial processes including geothermal energy extraction, mining, dam building, construction, hydraulic fracturing, and underground gas storage (Gaucher et al., 2015; Gibowicz, 2009; Kisslinger, 1976; McGarr et al., 2002; Suckale, 2009). According to the existing concepts, induced seismicity can be caused by various mechanisms, such as pore pressure variations, geochemical reactions, temperature effects, and reactivation of pre-existing faults. Following classical rock mechanics, it is assumed that in all cases, the materials must be pre-stressed to a substantial fraction of their breaking strength in order for seismicity to be induced. The source mechanism of induced seismicity remains a topic of active research. As industrial projects expand, and new technologies become available, both the frequency and the scale of human-induced earthquakes will increase considerably.
The fan-hinged shear rupture mechanism, which is the most dangerous rupture mechanism in the earth's crust, has the following outstanding feature: it can create new ruptures in intact rocks when the field shear stresses are significantly lower than the frictional strength. For this reason, it can be one of the most active mechanisms of induced seismicity, operating at great depths in hard rocks. Figure 23 shows schematically how the fan mechanism, in the presence of a source of additional stress, located on the earth's surface (dam), can cause an earthquake at a great depth at very low level of field stress. Figure 23 shows a fragment of the earth's crust with a pre-existing fault (black solid line), located at a great depth in the zone of the fan-mechanism activity (diagram on the right). The graphs below indicate that before the dam construction, the initial field stress τ0 is much lower than the static frictional strength τfs of the fault (green dotted lines). The local stress τloc amplified on the step is also lower than the fracture strength τu of the intact rock adjoining the fault. During the dam construction, the applied stresses τ0 and τloc increase. It is important to note that the stress τ0 increases slightly because the additional stress from the remote load is distributed over the entire fault area (the conditional stress increment in the graph Δτ0 = τ1 – τ0), while the local stress τloc accumulated on the step increases significantly (the conditional stress increment Δτloc = τu – τloc). When the local stress reaches the level of rock fracture strength τu, the tensile cracking process creates the initial fan structure. After the completion of the fan structure, its shear resistance τfan becomes lower than the field stress τ1 and the new rupture propagates spontaneously through the intact rock mass, causing the earthquake. As shown in Figure 17, each new fault can serve as a local stress concentrator for the next rupture. Suckale (2009) noted that nearly all studies of induced seismicity in hydrocarbon fields have confirmed that earthquakes have a strong tendency to form clusters or swarms. Accordingly, it is supposed that some of this seismicity could be generated by the fan mechanism.
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The fan-hinged shear instead of hydraulic fracturing
The fault structure generated by the fan mechanism has another very important property: high permeability. Void space in the form of channels in the row of slabs results due to slab rotation. Figure 24a shows schematically how slab rotation changes the cross-sectional area of the channels. At the initial stage, tensile cracks and slabs are oriented at an angle α0 to the fault plane and the structure is very compact. The value of the initial angle α0 depends on the stress state and can be within the range α0 = 30° ÷ 40° (Horii & Nemat-Nasser, 1985; Reches & Lockner, 1994). During shear displacement of the fault, all slabs rotate, which leads to an increase in the void space between them. The maximum space is formed at angle αr = 90°. The ratio z between the area of the void space and the area of the entire fault structure can be calculated using the schema in Figure 24a. The schema shows that the distance between the centers of adjacent slabs is s and the slab width is w. Hence, z = (s – w)/s = 1 – sin α0. For α0 = 30° and αr = 90°, the ratio z = 0.5. This rate indicates conditions of very high permeability.
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The images in Figure 24b–d, taken from Ortlepp's (1997) monograph “Rock Fracture and Rockbursts,” demonstrate the following behavior of shear ruptures generated by the fan mechanism in intact quartzite surrounding a mine in South Africa. The photographs in Figure 24b show shear ruptures consisting of a row of slabs oriented at angle αr = 65° and 90°. The value of the angle αr depends on the magnitude of shear displacement along the fault. The image in Figure 24c shows that the orientation of a fault depends on the stress state in the rock mass and thus can be very different. The red lines in Figure 24d indicate the shear faults that have caused a series of rockbursts in the mine during mining. This means that shear ruptures can be generated as long as the starting conditions are created. Figure 24e shows that in laboratory earthquake experiments, extreme bilateral ruptures were triggered by a local pressure pulse caused by the explosion of a thin wire (Lu et al., 2007). It can be assumed that extreme shear ruptures in a rock mass can also be triggered by local explosions.
So far, there has been no experimental study of the activation and control of extreme ruptures in nature, but it is hoped that such studies will make it possible to understand this process. Due to the high permeability of fan ruptures, they can be used instead of hydraulic fracturing for a variety of purposes, including geothermal energy, oil, and gas extraction (see Figure 24f). The extreme ruptures governed by the fan mechanism have a number of advantages over hydraulic fracturing:
- 1.
The fan rupture propagates due to the energy released by the rock mass and does not require any energy from the outside.
- 2.
It does not require a massive amount of water.
- 3.
It does not require thousands of tons of frac sand (small, solid particles used to keep the fractures in the rock formation open after the pressure from injection subsides).
- 4.
It does not require the use of chemical additives.
FURTHER DEVELOPMENT OF METHODS FOR EXPERIMENTAL STUDY OF CLASS III ROCKS
Compared with our current understanding, the fan mechanism provides fundamentally different properties of hard rocks under conditions of seismogenic depths. Experimental results described in Section 4 demonstrate that the unloading rate of the used modified testing machine (25 000 GPa/s) was not sufficient to prevent the spontaneous rupture of hard rocks at high σ3. Stress–strain and stress–time graphs in Figure 25 show the current situation. The red curves indicate the elusive postpeak properties of the tested specimen that need to be recorded experimentally. The blue stress–time curve demonstrates that the unloading rate of the testing machine used is too slow to follow the red curve. Thus, for the experimental study of the elusive class III properties of hard rocks, testing machines of a new generation are needed. This section will discuss the possible design of such a testing machine proposed in the collaboration with Professor's Manchao He and Weili Gong from China University of Mining and Technology, Beijing.
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To minimize the response time of the new testing machine, the following requirements must be fulfilled:
- 1.
Minimization of the mechanical and hydraulic compliance of all load train components with a monolithic stiff frame.
- 2.
Minimization of the inertia mass of the loading–unloading piston of the actuator.
- 3.
Minimization of fluid volume in the loading and unloading chambers of the actuator.
- 4.
The use of a high-speed pressure relief valve in addition to a high-speed servo valve.
Figure 25 shows a possible configuration of the new machine. To minimize the inertial mass of the actuator piston, it is necessary to use servo-valves SV with a significantly higher operating pressure than what is commonly used at present. In the testing machine used, the operating pressure of SV was 30 MPa. With a high operating pressure (e.g., 100 MPa), it is possible for the diameter ø and height h of the actuator piston to be decreased correspondingly. To decrease the piston inertial mass further, it should be a hollow cylinder (shown schematically in yellow). The working stroke ∆ of the piston should be restricted by the axial deformation of hard rock specimens. An analysis of available data shows that ∆ = 5 mm is enough for the majority of hard rocks tested up to σ3 = 300 MPa. This very small working stroke and the reduced diameter of the actuator piston can minimize the total volume of fluid in the loading and unloading chambers, which must be released from the chamber to drop the applied load. A relief valve RV can be used in addition to the servo-valve SV. The interconnections between the SV, RV, and the closed-loop system are shown in Figure 25. If the SV cannot provide sufficiently fast unloading to follow the command signal at the initial postpeak stage, the RV will aid in decreasing the pressure dynamically in the loading chamber of the actuator piston, thus increasing the unloading rate. When the stress level reaches σfan, further failure corresponding to class III postpeak behavior will be controlled by the SV. There are other ways to increase the response rate for the postpeak control of hard rocks at high σ3.
CONCLUSION
This paper shows that the vast majority of earthquakes at seismogenic depths of the earth's crust are caused by the rupture of intact rocks, rather than by stick–slip along pre-existing faults, as has been accepted during the past 60 years. This concerns natural, induced, and volcanic earthquakes. It is shown that all the specific properties of earthquakes, which are currently associated exclusively with the friction stick–slip mechanism, are also created by the fan mechanism during the rupture of intact rocks.
The preference of the fan mechanism over the stick–slip mechanism is clear due to the outstanding properties of the fan structure, which include the ability to
- 1.
generate new faults in intact dry rocks even at shear stresses that are an order of magnitude lower than the frictional strength;
- 2.
provide shear resistance close to zero;
- 3.
cause a low stress drop;
- 4.
provide an extraordinary energy budget of extreme ruptures with huge release of seismic energy;
- 5.
use new physics of energy supply to the rupture tip, providing supersonic rupture velocity, yielding a previously unknown interrelation between earthquakes and volcanoes; and
- 6.
create conditions for “meteorite-like” melting of rocks at seismogenic depths that serve as a source of magma for volcanoes.
The fracturing of intact rocks by the fan mechanism can be used instead of hydraulic fracturing for a variety of purposes, including geothermal energy, oil, and gas extraction.
Intensive study of the fan mechanism and the anomalous properties of rocks determined by it requires a new generation of testing machines and testing methods.
ACKNOWLEDGMENTS
The author would like to thank the co-authors of some of his papers for their outstanding contributions to the study of the fan-hinged shear rupture mechanism at various stages of the author's work on this topic. Professor Mark Randolph made a great contribution to the development of a special stiff servo-controlled machine for an experimental study, to the analysis of experimental results, and to the understanding of the role of the fan mechanism in the creation of earthquakes. Professors Mikhail Guzev and Vladimir Sadovskii put a lot of effort and provided unique knowledge to create mathematical models of the fan mechanism. The author is very grateful to all co-authors for their cooperation and help during this study.
CONFLICT OF INTEREST STATEMENT
The author declares no conflict of interest.
DATA AVAILABILITY STATEMENT
Data are openly available in a public repository that issues data sets with DOIs.
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Abstract
Frictional stick–slip instability along pre‐existing faults has been accepted as the main mechanism of earthquakes for about 60 years, since it is believed that fracture of intact rocks cannot reflect such features inherent in earthquakes as low shear stresses activating instability, low stress drop, repetitive dynamic instability, and connection with pre‐existing faults. This paper demonstrates that all these features can be induced by a recently discovered shear rupture mechanism (fan‐hinged), which creates dynamic ruptures in intact rocks under stress conditions corresponding to seismogenic depths. The key element of this mechanism is the fan‐shaped structure of the head of extreme ruptures, which is formed as a result of an intense tensile cracking process, with the creation of inter‐crack slabs that act as hinges between the shearing rupture faces. The preference of the fan mechanism over the stick–slip mechanism is clear due to the extraordinary properties of the fan structure, which include the ability to generate new faults in intact dry rocks even at shear stresses that are an order of magnitude lower than the frictional strength; to provide shear resistance close to zero and abnormally large energy release; to cause a low stress drop; to use a new physics of energy supply to the rupture tip, providing supersonic rupture velocity; and to provide a previously unknown interrelation between earthquakes and volcanoes. All these properties make the fan mechanism the most dangerous rupture mechanism at the seismogenic depths of the earth's crust, generating the vast majority of earthquakes. The detailed analysis of the fan mechanism is presented in the companion paper “New physics of supersonic ruptures” published in DUSE. Further study of this subject is a major challenge for deep underground science, earthquake and fracture mechanics, volcanoes, physics, and tribology.
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1 Laboratory of Geomechanics of Highly Stressed Rock and Massives, Far Eastern Federal University, Vladivostok, Russia