The dynamic energy balance in earthquakes expressed by fault surface morphology
TThe dynamic energy balance in earthquakes expressed by faultsurface morphology
Xin Wang a, ∗ , Juan Liu a , Feng Gao a , Zhizhen Zhang a a State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Miningand Technology, Xuzhou 221116, China
Abstract
The dynamic energy balance is essential for earthquake studies. The energy balanceapproach is one of the most famous developments in fracture mechanics. To interpretseismological data, crack models and sliding on a frictional surface (fault) models are widelyused. The macroscopically observable energy budget and the microscopic processes can berelated through the fracture energy G c . The fault surface morphology is the direct resultof the microscopic processes near the crack tip or on the frictional interface. Here weshow that the dynamic energy balance in earthquakes can be expressed by fault surfacemorphology, and that they are quantitatively linked. The direct shear experiments provesthe predictions of the theoretical discussions, and show that the strain rate has crucialinfluence on the dynamic energy balance. Keywords:
Earthquakes; Dynamic energy balance; Surface morphology; Strain rate
1. Introduction
An earthquake may be considered to be a dynamically running shear crack (Scholz,2019), in which the dynamic energy balance is essential for earthquake studies. In thefield of fracture mechanics, the energy balance approach employed by Griffith has becomeone of the most famous developments in materials science (Collins, 1993). To interpretseismological data, crack models are often used in part because the theories on cracks havebeen developed well. On the other hand, seismic faulting may be more intuitively viewed ∗ Corresponding author
Email address: [email protected] (Xin Wang) a r X i v : . [ phy s i c s . g e o - ph ] J a n s sliding on a frictional surface (fault) where the physics of friction, especially stick slip,plays a key role.The fracture energy, G c , in crack theory is the energy needed to create new crack surfacesnear the crack tip. Thus, the system must expend the threshold fracture energy G c beforethe crack can extend. In contrast, in frictional sliding model, D c , is introduced as a criticalslip before rapid sliding begins at a constant friction. If the initial stress σ of the systemdrops more or less linearly to the final stress σ , i.e., the final value of the frictional stress σ f , the energy spent in the system before this happens can be approximately written as ( σ − σ ) D c . Thus, if we are to link a crack model to a friction model, we can equate thisenergy to G c , i.e., G c = ( σ − σ ) D c . Then we can relate the macroscopically observableenergy budget to the microscopic processes in a surprisingly general way. Any constrainton fracture energy obtained from the energy budget will provide a strong bound on allmicroscopic rupture processes (Kanamori and Brodsky, 2004). Svetlizky and Fineberg(2014) have shown that interface rupture propagation is described by, essentially, the sameframework of the crack model. This suggests that an analogous ’Griffith’ condition mayexist for the onset of rupture propagation for a frictional interface.The fault surface morphology is the direct result of the microscopic processes near thecrack tip or on the frictional interface. Here we show that the dynamic energy balance inearthquakes can be expressed by fault surface morphology, and that they are quantitativelylinked.
2. The description of fault surface morphology and its links to the dynamicenergy balance in earthquakes
The description of fault surface morphology has been investigated in the literature(e.g., Ladanyi and Archambault, 1969; Barton and Choubey, 1977; Plesha, 1987; Saeband Amadei, 1992; Amadei et al., 1998) while trying to study the contribution of surfacemorphology to the shear strength of rock fractures. As shearing strictly depends on three-dimensional contact area location and distribution (Gentier et al., 2000), Grasselli et al.(2002) proposed a method for the quantitative three-dimensional description of a roughfracture surface, and based on this description, Grasselli and Egger (2003) proposed a2onstitutive criterion to model the shear strength of fractures. Wang et al. (2019) definedthe terms “quasi steps” and “quasi striations” to refer to morphological structures thatare created during the creation of new crack surfaces and the friction on the frictionalinterface. The same morphological structures are also causing anisotropy of the fracturesurface morphology, hence the anisotropy of the surface morphology’s contribution on itsshear strength. The terms “quasi steps” and “quasi striations” are broader definitions of thefault steps and fault striations whether they can be obviously seen or not. The parameter θ ∗ max /C was proposed by Grasselli et al. (2002) to describe the contribution of fracturesurface morphology on the shear strength. Wang et al. (2019) proposed a theoretical modelthat describes the contribution of quasi steps and quasi striations on the shear strength.And by fitting this theoretical model to θ ∗ max /C from each slip direction on the fracturesurface, the amount of quasi steps R G and quasi striations R H can be estimated. Combinedwith the method proposed by Wang et al. (2017) for outcrop fracture surface extraction frompoint cloud data, the estimated quasi steps and quasi striations data had good applicationsin tectonics (e.g., Wang and Gao, 2020).Let’s consider the formation of quasi steps during the crack growth. The voids (Fig. 1as suggested by Bouchaud and Paun (1999)) are nucleated under the influence of the stressfield adjacent to the tip, but not at the tip, due to the existence of the plastic zone thatcuts off the purely linear-elastic (unphysical) crack-tip singularities. The crack grows bycoalescing the voids with the tip, creating a new stress field which induces the nucleationof new voids (Bouchbinder et al., 2004). The morphological structures of quasi steps arethen formed during the crack growth under shear load as iillustrated in Fig. 1. A crucialaspect of this picture is the existence of a typical scale, ξ c , which is roughly the distancebetween the crack tip and the first void, at the time of the nucleation of the latter. In thispicture there exists a “energy dissipation zone ” in front of the crack tip in which plasticyield is accompanied by the evolution of damage cavities. From this picture, it can also beseen that the typical scale ξ c is positively related with the estimated amount of quasi steps R G since the quasi steps are more developed on the fracture surface with larger value of ξ c .A simple model for ξ c was developed (Bouchbinder et al., 2004; Afek et al., 2005) byassuming the energy dissipation zone to be properly described by the Huber–von Mises plas-3icity theory. The material yields as the distortional energy exceeds a material-dependentthreshold σ Y and the typical distance ξ c scales as (Bouchbinder et al., 2004) ξ c ∼ K II σ Y , (1)where K II is the stress intensity factor for mode II (in-plane shear) cracks. On the otherhand, the linear-elastic solution is still valid outside the energy dissipation zone, and theenergy release rate G ∗ can be expressed as G ∗ = K II E (cid:48) , (2)where E (cid:48) is related to Young’s modulus E and Poisson’s ratio ν (for plane strain): E (cid:48) = E − ν . The definition of the energy release rate G ∗ is G ∗ = ∂ Ω ∂s , (3)where Ω is the strain energy and s is the crack growth area. Then Eq. 1 can be rewrittenas ξ c ∼ G ∗ / σ Y E (cid:48) = ∂ Ω ∂s / σ Y E (cid:48) . (4)The amount of quasi steps R G is a description of the degree of quasi steps developmentover the fracture surface, so it is an average quantity over the fracture surface. R G shouldscale with ξ c , the average of ξ c over the fracture surface S : R G ∼ ξ c = 1 S (cid:90) s ξ c ∼ Ω γ G , (5)where γ G = (cid:90) S σ Y E (cid:48) . (6)4 rack ξ Energy dissipation zoneξ: Damage correlation length
Figure 1: The model of the tip growth of mode II (in-plane shear) cracks. The formation of voids aredocumented in Bouchaud and Paun (1999).
Let’s consider the formation of quasi striations during the friction on the frictionalinterface. As suggested by Svetlizky and Fineberg (2014), the interface rupture frictionis described by, essentially, the same framework of the crack model. The main differenceis that the crack model consider the crack growth in the rock mass, while the frictionmodel consider the crack growth through the local protruding obstacles on the frictionalinterface, which are then removed or deformed during the friction. As the slip goes on, moreprotruding obstacles that face the slip direction with high angles are removed or deformedwhile protruding obstacles that are high angle perpendicular to the slip direction are kept.This anisotropy results in the macroscopic quasi striations structures. The amount of quasistriations R H is a description of the degree of quasi striations development over the fracturesurface, so it is an average quantity over the fracture surface. R H should scale with E f ,the frictional energy like this: R H ∼ E f ω , (7)where ω is the average strain energy needed to remove or deform a protruding obstacle.Suppose n protruding obstacles are removed or deformed, according to Eq. 5 and Eq. 6,the strain energy scale as ω ∼ (cid:90) S ∗ σ Y E (cid:48) , ω ∼ (cid:90) S ∗ σ Y E (cid:48) , ..., ω n ∼ (cid:90) S ∗ n σ Y E (cid:48) , S ∗ i is the crack area in the i -th protruding obstacle. The average strain energy ω canbe written as ω = 1 n n (cid:88) ω i ∼ (cid:90) S ∗ (cid:18) σ Y E (cid:48) (cid:19) , (8)where S ∗ = 1 n n (cid:88) S ∗ i , and (cid:18) σ Y E (cid:48) (cid:19) = n (cid:88) (cid:90) S ∗ i σ Y E (cid:48) (cid:30) n (cid:88) S ∗ i . Finally, we have the link between the fault surface morphology and the dynamic energybalance in earthquakes: R G R H ∼ ωγ G Ω E f . (9)
3. The experimental results and discussions
An experiment was designed to test the above theoretical discussion. The dimensionsand experiment settings are illustrated in Fig. 2. 14 red sandstone rock samples are tested inthis simple direct shear experiment with the strain rate ranging from 0.01/min to 0.083/min.The final fracture surfaces after the crack growth and the interface friction processes arescanned and analyzed, the amount of quasi steps R G and quasi striations R H are estimated.On the other hand, the stress-strain curves are recorded during the tests. As shown in Fig. 3,the strain energy is mainly accumulated before the formation of the macroscopic crackthrough the rock sample, while the frictional energy is mainly spent after the formationof the macroscopic crack. Thus the strain energy Ω and the frictional energy E f can beroughly estimated. 6 ixed F N F S mm mm
150 mm mm xyz zy F N
150 mm
Figure 2: Experiment settings of direct shear test.Figure 3: Examples of the stress-strain curve. Roughly the green regions represent the strain energy tobe released during the crack growth, and the red regions represent the frictional energy spent during thefriction on the frictional interface. R G /R H ) is plotted against log (Ω) / log ( E f ). The logarithm of the strain energy and thefrictional energy are taken here in order to compare their values with R G and R H , althoughthe logarithm relationship is not explicitly shown in above discussions (e.g., Eq. 5 and Eq. 7).Results of rock samples are scatter ploted in Fig. 4 with their sample numbers labeled ontheir corresponding scatter points for reference. Table 1: Rock sample numbers and their strain rate during the test.
Sample N.O. 0 1 2 3 4 5 6Strain rate (%/min) 0.08 0.077 0.07 0.063 0.037 0.03 0.01Sample N.O. 7 8 9 10 11 12 13Strain rate (%/min) 0.017 0.023 0.033 0.043 0.057 0.067 0.083 E f )2.02.53.03.54.04.55.05.5 R G / R H
01 23 45 678 910 1112 13
Figure 4: The experimental results. The ratio between the amount of quasi steps and the amount of quasistriations ( R G /R H ) is plotted against log (Ω) / log ( E f ). Each labeled point is the result of the correspondingrock sample, and its strain rate can be found in Table 1. ω/γ G (e.g., sample number12, 1, 10, 5, 2, 6, 11 and 9) because ω and γ G are both integrals of σ Y /E (cid:48) over some area,whose average tend to be the same for the same rock sample, although it varies locally.Thus the link predicted by Eq. 9 has the property of material-independent, at least for drybrittle materials discussed here.Note that sample number 3, 13, 4 and 0 seem have slightly larger value of ω/γ G , andthey seem have larger strain rate. On the contrary, sample number 7 and 8 have slightlysmaller value of ω/γ G and they have smaller strain rate. If we take the rupture speed V into account, Eq. 9 becomes R G R H ∼ ω (1 − V ( Bβ ) ) γ G Ω E f , (10)where B is a constant of the order of 1 and β is the shear-wave speed (Kanamori andBrodsky, 2004), because G = G ∗ g ( V ) = ∂ Ω ∂s (1 − V ( Bβ ) ) . But this doesn’t explain the data and the rupture speeds V aren’t significantly different inthose tests, so the rupture speeds V doesn’t have a significant effect here. One explanation isthat for real materials, σ Y depends on the state of deformation and its history. Experimentsdiscussed here show that high strain rate results in larger value of ω , i.e., on average, itneeds more energy to remove or deform a protruding obstacle on the fracture surface duringthe friction. In the formation of the macroscopic crack and the protruding obstacles on thecrack surface, high strain rate may make less plastic deformation inside the protrudingobstacles, and hence more energy is needed to remove or deform them in the followingdeformation.
4. Conclusions
The dynamic energy balance is essential for earthquake studies. To interpret seismolog-ical data, crack models and sliding on a frictional surface (fault) models are widely used.9rom these two types of models, the macroscopically observable energy budget and themicroscopic processes can be related through the fracture energy G c .The fault surface morphology is the direct result of the microscopic processes near thecrack tip or on the frictional interface. Here we show that the dynamic energy balance inearthquakes can be expressed by fault surface morphology, and that they are quantitativelylinked.The direct shear experiments proves the predictions of the theoretical discussions, andshow that the strain rate has crucial influence on the dynamic energy balance. The linkpredicted by the theoretical discussions has the property of material-independent, at leastfor dry brittle materials discussed here. Acknowledgments
This research was funded by the Fundamental Research Funds for the Central Uni-versities (grant no. 2020QN29), the China Postdoctoral Science Foundation (Grant no.2020M681759), and the State Key Program of National Natural Science Foundation ofChina (Grant no. 51934007).
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