Strong seismic waves produce frequent episodes of transient oscillating dynamic stress in the shallow subsurface within seismically active areas. Dynamic stress drives slip on a network of preexisting small fracture planes and leaves residual stresses once shaking ceases. The residual stresses act as prestresses during the next strong earthquake. Local failure occurs on fractures at low dynamic stresses where the prestresses are favorably aligned, while most of the bulk rock remains intact. Hence nonlinear attenuation of strong seismic waves commences at dynamic stresses well below those for pervasive Coulomb failure. Consideration of the fractal distribution of prestresses indicates that nonlinear attenuation increases rapidly with dynamic stress. Vertically propagating shear waves provide simple scaling relationships for quick application: (1) Nonlinear attenuation is concentrated near the quarter wavelength depth. (2) The Coulomb stress ratio (dynamic/lithostatic stress) above the quarter wavelength depth scales with the sustained dynamic acceleration in g's. Rate and state friction thus provides estimates of the maximum sustained acceleration from Coulomb stress ratio at failure: ∼1 for regolith that has repeatedly failed, ∼1.5 for shallow intact sandstone, and ∼2 for shallow intact granite. A potentially observable effect occurs with shallow brittle regolith on hillslopes where gravity causes net downslope creep during strong shaking. Analogous crack network behavior with prestress occurs in the damage zone surrounding the rupture zone of major faults. Pulverized rock forms at shallow depths when rock is repeatedly deformed in strong shaking to just beyond the elastic limit, but strain does not localize into a crack network.
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4. Geomorphic studies indicate that damaged easily eroded regolith exists near major seismically active faults [Wechsler et al., 2009]. Shattered rock and smooth topography occur in the hanging wall of major thrust faults [Brune, 2001].
5. Changes in the permeability of shallow rocks and of groundwater pressure in shallow wells in the aftermath of strong shaking are conceivably related to rock damage on fractures [Chao and Peng, 2009]. This effect, however, is usually attributed to shaking disrupting debris that clog choke points in groundwater flow paths and to transient groundwater pressure variations due to shaking [e.g., Elkhoury et al., 2006]. The time-dependent behavior of postquake clogging and crack closure should differ in principle. Available data do not provide an obvious solution for the extent to which each process operates. Resolving this issue is beyond the scope of this paper.
It is advantageous for engineering purposes to indirectly observe nonlinear attenuation (the first process) by studying the remaining four processes in addition to directly studying it with available instruments. Moreover, the existence of shallow rock damage indicates past strong motions and the lack of such damage provides an exceedance criterion [Brune, 2001]. In regard to attenuation, Sleep and Hagin  reviewed the literature on the first three processes and noted that their onset occurs at dynamic shear traction to normal traction ratios (Coulomb ratios) as low as 0.1, much lower than the expected failure criterion at a coefficient of friction >0.7. They proposed that repeated failure along cracks during strong shaking leaves randomly prestressed domains that fail again when they are aligned favorably with the dynamic stress. Semantically, these stresses are residual stresses after the shaking ceases and prestresses when new shaking commences. I use whichever term is most appropriate in each case for brevity with the understanding that the terms represent different stages of the same cycle.
The purpose of this paper is to extend Sleep and Hagin's  analysis to include the stresses at microscopic levels and interseismic relaxation of microscopic and macroscopic stresses. These modifications allow modeling the long-term effects of the interaction of oscillating stresses with static stresses. Pulverized rock provides an example application. Rock that is repeatedly damaged by strong motions near the rupture tips of major earthquakes is a deep analogy. Note that nonlinear attenuation and damage occur in shallow soil at accelerations as low as 2%–3% of ambient gravity and hence Coulomb stress ratios of ∼2%–3% [Rubinstein, 2010; Wu et al., 2010]. Failure of such clay-rich soils, as well as liquefaction, is mostly beyond the scope of the paper, which is limited to macroscopically brittle rocks.
I begin by reviewing the energy within strong seismic waves to obtain simple scaling relationships for subsequent use. I then present a domain model for failure of prestressed rock followed by applications to nonlinear attenuation. I then use examples of seismically near-fault shallow pulverized rock to put the discussion into context. The Parkfield California 2004 main shock provides the bases for quantitative example calculations.
2. Scaling Relationships for Seismic Wave Energy
I summarize well-known dimensional relationships so that I can apply them throughout this paper. Specifically, the available energy per volume in linear seismic waves yields simple well-known scaling relationships relevant for the effects of nonlinear attenuation. The kinetic energy in seismic waves is equal to the elastic strain energy. The principle applies exactly to a plane wave in a whole space at typical strong motion frequencies where gravity does not provide a significant restoring force. It applies overall and dimensionally near the free surface where kinetic energy resides at shallower depths than elastic strain energy. Mathematically, the energy per volume averaged over a wavelength in an S wave is
where ρ is density, U0 is the maximum particle velocity, τ0 is the maximum shear stress, and G is the macroscopic shear modulus [e.g., Timoshenko and Goodier, 1970, p. 491]. The S wave propagation velocity is
Equations (1) and (2) yield very useful well-known scaling relationships for maximum dynamic strain ɛD = U0/cS and dynamic stress τ0 = ρcSU0.
Body waves refract into nearly vertical paths in the low-velocity rocks near the free surface. The dynamic stress in a shear wave then goes to zero at the free surface. The above relationships for stress and strain apply with the understanding that the shear stress is doubled at the quarter wavelength depth where it causes rock damage and velocity is doubled at the free surface where it is commonly measured. The ratio of dynamic stress to lithostatic stress then is the dynamic acceleration normalized to the ambient acceleration of gravity. In general, the Coulomb stress (ratio of dynamic shear traction to normal traction from lithostatic load) scales with the dynamic acceleration for shallow body waves. These relationships apply approximately down to somewhat greater than the quarter wavelength depth for the dominant frequency on a velocity seismogram, that is, the wavelength that carries most of the seismic energy. Most of the elastic strain energy resides around that depth when the wave passes through the near surface. It is thus convenient in dimensional calculations to consider that Coulomb failure occurs near the quarter wavelength depth or dimensionally near the scale depth 1/k where k is the wave number from a velocity seismogram.
Some caveats are in order. The scaling relationship between acceleration and Coulomb stress assumes a basically sinusoidal pulse on a velocity seismogram and vertically propagating waves. In the case that particle velocity suddenly rises to its peak amplitude as in the Brune  pulse, both acceleration and Coulomb stress are unbounded at very shallow depths. Body waves are not refracted toward vertical if intact rock with a seismic velocity similar to deep rock exists in the shallow subsurface. The relationship still applies dimensionally in both cases between the stress at the quarter wavelength depth and the sustained acceleration of the dominant frequency on a velocity seismogram.
Moreover, it is inappropriate to pigeonhole seismic energy into wave types in the near field of a surface rupture. Rugged topography introduces similar complications. The body wave relationships apply dimensionally in both cases. One can model topographic amplification to predict velocity and acceleration amplitudes [e.g., Anooshehpoor and Brune, 1989; Sepúlveda et al., 2005] and evaluate the dynamic stress history at depth. This latter daunting task is numerically feasible following the work of Ma and Andrews , but well beyond the scope of this paper.
Scaling relationships involving dynamic stress and dynamic strain to particle velocity apply dimensionally to surface waves. There are not, however, simple near-surface limits involving acceleration, scale depth, and Coulomb stress ratio. Rather, there are nonzero shear tractions greatest on vertical and 45° planes at the surface for Love and Rayleigh waves, respectively, and hence imply a surface limit of infinite Coulomb stress ratios. This paper mainly discusses S waves and shear failure.
3. Fractal Elastic Strain Energy and Nonlinear Attenuation
I obtain a microscopic and macroscopic formalism for nonlinear seismic attenuation of strong seismic waves in a cracked brittle medium. Macroscopically, I model crack failure during strong shaking as Coulomb failure on numerous small slip surfaces or fault planes. I ignore cohesion (finite shear strength at zero normal traction and finite tensile strength) on the grounds that it is small on those fractures that actually fail and that the strength at tens of meters depth where there is significant lithostatic stress and dynamic elastic strain energy is more relevant than strength right at the surface. Clearly, small pieces of intact rock have finite tensile strength.
Repeated episodes of strong shaking load the small fault planes so that some are always near failure and some have residual stresses well away from failure (Figure 1). The tensor orientation of the prestressed domains is considered random at the start of strong shaking, for simplicity, as failure during oscillating dynamic stresses does not have an obvious long-term preferred orientation. Each strong seismic event releases the stresses in some domains and loads others. Over time, the distribution of residual stresses approaches a statistical steady state.
I use fractals to represent the scalar distribution of residual stress that exists between earthquakes. My scalar treatment involves the probability distribution of static stresses and does not specifically utilize the scale invariance of a fractal network of cracks. Scale invariance as discussed in section 3.1 is inappropriate in the sense that the intrinsic microscopic strength of crystals is orders of magnitude larger that the macroscopic Coulomb strength of shallow fractures. Sections 4 and 5 and Appendix B qualitatively discuss the effects on large corestones (that is, islands) of intact rock within a fractured matrix (Figure 1).
Mathematically, I follow Marsan [2005, Figure 12] and use his stress distribution exponent ∼–2 for high stresses in the near field of faults on the grounds that the near surface is extensively damaged in seismically active areas, rather than the far field value of −1.5. It is necessary in my applications to concentrate on high residual stresses that are likely to cause interseismic creep and to trigger dynamic failure during shaking, rather than the lowest stresses. “Near field” implies that a spot in the rock is within a distance comparable to the length of a fracture that slipped. Islands of intact rock that are much further away from any fracture exist as noted in the previous paragraph (Figure 1).
Considerable notational simplification accrues if the fractal exponent is exactly −2. The number of domains or the fraction of the volume occupied by domains where the stress is less than τ is
where τnorm is a normalizing constant with the dimensions of stress and σ is a dummy variable for stress. The lower limit of the integral τmin is mathematically necessary for the integral to be bounded; it can be set so that the integral over the full range of stress is 1, yielding volume fractions. Mathematically,
where τmax is the maximum stress present. Physically, the lower limit represents that the entire volume is near fracture planes, so places where the stress is essentially zero are quite rare and the lower truncation of the distribution is an acceptable approximation. Additionally, τmax ≫ τmin when a truncated fractal distribution is a relevant approximation. Then τmin ≈ τnorm.
The total elastic strain energy per volume where stress is less than τ is
where the effective value of the shear modulus G may in general depend on the local scale and hence stress. The upper limit on stress τmax is mathematically necessary for the integral (4) to be bounded by a finite energy. The final approximate equality assumes constant G where there is equal strain energy in each stress bin and the lower limit is set to zero. It applies at high stresses where τ ≫ τmin. The total strain energy per volume is then that at the geometric mean stress .
The situation differs from simple nonlinear elasticity. The microscopic stresses and shear moduli are much greater than macroscopic ones in the shallow subsurface. I consider processes at the micron scale of real contacts with stresses of a few GPa and Coulomb stress processes on the 1–10 m scale of crack and joints that one sees at typical outcrops separately. I do not specifically consider the intermediate “mesoscopic” ∼1 mm length scale of full grains and microcracks although treatments are available [e.g., Paliwal and Ramesh, 2008]. That is, I avoid bulky relationships by modeling microscopic and microscopic behavior separately, each different constant values of material parameters.
3.1. Microscopic Real Stresses and Their Dissipation
I first apply (4) to the energy balance of microscopic stresses. Modern real contact theory represents friction as ductile thermally activated creep on the scale of real contracts where the stress is a few GPa [e.g., Berthoud et al., 1999; Baumberger et al., 1999; Rice et al., 2001; Nakatani, 2001; Nakatani and Scholz, 2004; Beeler, 2004; Sleep, 2005, 2006a]. Slip on these surfaces transfers stress to other parts of the lattice (Figure 1). I apply the fractal distribution (4) with the understanding that microscopic stresses exist only within solid grains. I ignore factors of the fraction of the volume occupied by grains 1 − f, where f is porosity, to avoid cluttering scaling relationships that do not retain precision to this factor.
Creep relaxes stresses both during rapid sliding and the interseismic interval. I apply the theory of exponential creep at very high stress to constrain the level of residual stress immediately after shaking and the prestress after a significant postseismic interval. The microscopic strain rate tensor in isotropic materials is formally
where γ′ is the strain rate tensor, ij are tensor indices, γ′0 is a constant with dimensions of strain rate, τij is the microscopic deviatoric stress tensor, s is a material constant with dimensions of stress, and the second invariant of deviatoric stress r is normalized without loss of generality so that it is the shear traction in simple shear. Calibration of (5) by the macroscopic behavior of rate and state friction indicates that the value of r/s at the yield stress r = τY where creep occurs at an observable rate is ∼100 [Sleep, 2006a].
In this formalism, the microscopic strength of a contact represented by τY is a mineral property that does not depend on history. The macroscopic strength does depend on history. Compaction creep on microscopic contacts increases the surface area available to support macroscopic stress, thereby increasing the macroscopic stress needed to bring the microscopic stress to failure. In addition as discussed in the next paragraph, creep relaxes contact stresses, increasing the stress change needed for sudden failure.
Returning to elastic strain energy in (4), the yield stress is ∼4 GPa and the intrinsic mineral shear modulus ∼40 GPa applies. The stress scale s is ∼0.04 MPa for r/s = 100. Postseismic behavior provides a calibration of the constant τnorm. Creep during strong motion occurs quickly over a time scale of 1/ω where ω is angular frequency. I retain the value of ∼0.1 s for the 2004 Parkfield California main shock from the work of Sleep and Hagin  in examples. Interseismic relaxation of the microscopic stress occurs over a time scale of ∼30 years (109 s) for Parkfield in examples. So the seismic and interseismic creep rates that result in significant strains differ by a factor of ∼1010. The peak stress at the end of the interseismic interval τint is thus sln(1010) = 0.92 GPa less than the 4 GPa yield stress τmax = τY, that is, keeping extra digits, 3.08 GPa. This quick result involves a logarithm and hence is not sensitive to the assumed ratio of creep rates. The change in elastic strain energy per volume in (5) from the relaxation of the larger stress τY to τint is
Stresses lower than τint do not relax in this approximation.
The energy in (6) provides a minimum estimate of the microscopic elastic strain energy supplied and then dissipated during each cycle. The fraction of the ambient elastic strain energy that dissipates is
which is here 0.053.
The actual coseismic fractional dissipation of energy within a failed domain in the rock may be near this minimum or considerably larger as (7) includes only the strain energy at the end of shaking and not strain energy imposed and then relaxed during shaking. In the former case, about 5% of the elastic strain energy dissipates in each cycle and ∼20 seismic cycles are required for the microscopic elastic strain energy to build up to quasi-steady state. See Appendix A for more on energy partitioning.
Conversely, significant microscopic strain energy can persist in intact rock over geological time if exponential creep is the rate-limiting processes. For a numerical example, I assume stresses have relaxed over 3 Ma (1014 s) so the fast and slow strain rates differ by a factor of 1015. The maximum microscopic stress τint is hence sln(1015) = 1.38 MPa less than peak stress of τY = 4 MPa or 2.62 MPa. The ratio is (7) is 0.12, so 88% of the initial microscopic strain energy remains.
Still the application of (7) is not straightforward to repeated damage as the fraction of the total rock volume that fails by microscopically during each episode of strong shaking and thus has significant changes in the microscopic stress in not self-evident. Intuitively, macroscopic slip concentrates within many small fault zones rather than evenly throughout the rock mass and the microscopic strain energy changes with these zones (Figure 1). The rest of macroscopically intact rock rarely fails microscopically. That is, the intact rock away from the macroscopic fractures is in the far field on microscopic contact failures and near-field microscopic stress distribution within fractures does not apply.
For an example calculation with total failure, I dimensionally balance macroscopic energy per volume in the seismic wave with microscopic energy dissipation to dimensionally calibrate τnorm for the case that microscopic stresses change throughout the entire rock mass. For the energy per volume available for dissipation, I assume a generic strong wave with velocity amplitude of 1 m s−1 in broken regolith with a density of 2000 kg m−3. The total per volume energy in (1) is 1000 J m−3. This energy per volume is dimensionally available to restore the microscopic elastic energy that dissipated since the last earthquake. Solving for τnorm yields 0.4 MPa, which satisfies the fractal criterion that τnorm ≈ τmin ≪ τY. The total ambient microscopic strain energy per volume is 20 times the dissipated energy, 2 × 104 J m−3.
The behavior of macroscopic porosity change in section 3.3 and hillslope creep induced by strong shaking in Appendix B are consistent with localized failure. For comparison, section 4 considers pulverized rock where the entire volume eventually fails.
3.2. Macroscopic Prestresses Stresses and Their Dissipation
Macroscopically frictional failure on a tabular crack relaxes stresses in (6) within a volume around the crack. I apply the well-known formalism of rate and state friction to represent these macroscopic stresses in rock damaged by repeated dynamic stress for application to regolith.
The difference between starting friction before shaking and friction during dynamic sliding is relevant to the energy balance and the onset of dynamic failure. I use the strain rate form of a unified theory of rate and state friction compiled by Sleep [1997, 2006a] and Sleep et al. . The instantaneous shear traction τ as a function of normal traction P is then
where the dominant term μ0 ≈ τ/P represents the approximation that the coefficient of friction has a constant value for a given surface (Amonton's law), a and b are small ∼0.01 dimensionless constants, ɛ′ ≡ V/W is the strain rate, V is sliding velocity, W is the width of the sliding zone, ɛ′ref is a reference strain rate that may set to a convenient value, and ψ is the state variable that includes the effects of healing and damage, and the ψnorm is the normalizing value of the state variable which I constrain in the next paragraph.
I use the “aging” evolution law to represent the state variable ψ in terms of the past history of the fault to illustrate well-known implications of (8). This law represents has kinematically explicit terms [Dieterich, 1979]
where the first term represents healing and the second damage. The variable t is time, ɛint is the strain to significantly change the properties of the sliding surface, α is a dimensionless parameter that represents the behavior of the surface after a change in normal traction from the work of Linker and Dieterich , and Pref is a reference normal traction. The steady state value of the state variable in (9) is
If the coefficient of friction at a given strain rate is independent of normal traction, the normalized value of the state variable is
For simplicity for now, I consider only changes in shear traction. I set the pressure and the reference pressure to the ambient lithostatic pressure and ψnorm = 1. The evolution equation then has the form
The state variable after significant sliding at the dynamic sliding strain rate ɛ′D is ɛ′ref/ɛ′D. Sliding ceases after the strong shaking and the state variable increases to ɛ′reftpost/ɛint. The logarithm of the ratio of these values of the state variable gives the “starting” friction at the end of the postseismic interval in relation to the dynamic steady state friction
where μD is the coefficient of friction during dynamic slip. An alternative way to view the processes is that dynamic stress needs to increase the instantaneous strain rate by the ratio the postseismic time 109 s to the duration of slip tseis = ∼0.1 s to cause significant slip. This formulation yields
Both formulations involve logarithms and yield similar results as a ≈ b in the shallow subsurface.
I supply a generic example for dry regolith with density 2000 kg m−2 at 50 m depth with Parkfield California in mind. The ambient lithostatic pressure is 1 MPa. I let the coefficient of friction at a dynamic sliding velocity be μD = 0.8 and a = b = 0.007. The S wave velocity is 300 m s−1, so the shear modulus is 0.18 GPa. The starting coefficient of friction in domains where the state variable is 1010 of that during rapid sliding is 0.961 at the end of the interseismic interval.
Just after sliding ceases, the peak residual stress in unbroken domains is the lithostatic pressure times the starting coefficient of friction 0.961, yielding 0.961 MPa. The microscopic value of τnorm of 0.4 × 106 J m−3 from section 3.1 is nearly half of the yield stress and thus is inconsistent with a fractal distribution for macroscopic stress. I assume that the macroscopic τnorm is 0.1 the yield stress to provide a numerical example. Then from (4), the energy per volume is 257 J m−2 as expected is a modest fraction of seismic wave energy per volume of 1000 J m−3 in the strong generic seismic wave that causes failure. Stresses relax over 109 s until the next earthquake, so the peak prestress at the end of that time is 0.8 times the confining pressure or 0.8 MPa. From (7), 5% of the ambient strain energy relaxes interseismically, the same as with the microscopic example. Note that microscopic failure of real contacts in section 3.1 leads to macroscopic rate and state behavior. I selected my generic value b = 0.007 with forethought, so that my macroscopic and microscopic example rheologies are consistent.
To reiterate, the ambient microscopic and macroscopic strain energies per volume in domains that fail are a factor of ∼20 greater than the change of these strain energies over each cycle. The ambient macroscopic strain energy per volume is bounded by that at the macroscopic stress at failure. It also cannot have a value greater than maintained over earthquake cycles by the energy per volume supplied by strong seismic waves. Specifically, the ambient microscopic strain energy per volume is limited the need to restore it from the energy within strong seismic waves; it was ∼20 times the available energy per volume in the seismic wave in the example. As already noted, only a small fraction of the volume may fail in cracking and have its microscopic energy reset during each event. For example, if 1% of the volume fails the overall change in ambient microscopic energy per volume is 200 J m−3, comparable to the estimate of ambient macroscopic strain energy.
3.3. Work Against Confining Pressure
I show that failure that occurs on isolated fault planes as suggested at the end of section 3.2 yields acceptable energetics [Sleep and Hagin, 2008]. Conversely, a situation where the entire volume fails each time microscopically does not.
Specifically, the decrease in the state variable during strong shaking implies an increase in porosity. The state variable is empirically
where ϕ is a reference porosity, f is porosity, and Cɛ is a dimensionless material property [Segall and Rice, 1995]. Sleep et al.  obtained an estimate of the parameter Cɛ = 3.4 × 10−3 by considering the effects of strain localization. A decrease in the state variable by a factor of 1010 during failure implies an increase of porosity of 7.8%. The energy per volume work against 1 MPa of lithostatic pressure is 78,000 J m−3, which is much greater than the energy per volume in a strong seismic wave. Moreover, this porosity increase even over a 10 m depth range would cause 0.78 m of coseismic uplift, which is grossly excessive. Rather, the state variable changes significantly only within the cracks that actually fail (Figure 1).
For an example with forethought, I assume that failed crack zones occupy ∼0.25% of the volume. This assumption yields an energy per volume estimate for porosity change against lithostatic pressure of 250 J m−3, which is modest fraction of 1000 J m−3 of available dynamic energy per volume and by intent similar to the ambient microscopic and macroscopic elastic strain energy estimates per volume.
Note that the changes in microscopic strain energy and porosity represent different kinematic processes if strain localization within crack zones occurs. A strain of less than ∼ɛint is needed to cause slip on grain-grain contacts that resets microscopic elastic strain energy. The value of ɛint is 0.06–0.12 in experiments where strain localization in gouge is considered [Sleep et al., 2000]. The grain scale (or equivalently gouge scale) elastic strain at microscopic failure is ∼0.1. Microscopic stress and strain localizes on contact asperities so some microscopic anelastic strains of this magnitude are expected well before the anelastic gouge strain reaches 0.1, so some resetting of microscopic prestresses occurs at low gouge strains.
With regard to resetting porosity during rapid slip, the healing term of the evolution equation (12) is negligible until steady state is approached. That is, a strain of
is needed to change the porosity to its steady state value for dynamic sliding. As ln(1010) ≈ 23, only highly strained parts of crack zones significantly contribute to porosity change. It is thus reasonable the fraction of the volume that strongly dilates is less than the fraction of volume with microscopic strain by a factor of a few as assumed in the examples.
Sleep and Hagin  estimated the energy per volume from expanding porosity during strong shaking from changes in low-amplitude seismic velocity before and after strong shaking. These concluded that this energy sink was a significant source of the macroscopic nonlinear dissipation, but did not obtain precise estimates. I do not explicitly represent this effect in my model for nonlinear attenuation of strong seismic waves. Rather, I include it in the macroscopic friction on cracks that fail in my model of dissipation in section 3.4.
For completeness, the observed postseismic changes in seismic velocity indicate that the shear modulus varies throughout the seismic cycle. This change should strictly be included in the energy integrals including (6). Postseismic deformation makes the rock less compliant, but mainly where creep relaxes stresses. Some care is needed not to count the dissipation of the same elastic energy twice. A self-consistent treatment of this second-order effect is beyond the scope of this paper that mainly seeks dimensional expressions.
3.4. Nonlinear Dissipation of Strong Shallow Seismic Waves
The previous part of the paper prepares the groundwork to represent the societal relevant process of nonlinear attenuation of strong seismic waves in the shallow subsurface. I extend the model of Sleep and Hagin  by explicitly considering that the highest residual stresses relax during the interseismic period and by using the implied near-field macroscopic fractal stress distribution to obtain prestresses.
I use a yield stress representation to dynamic failure as I seek the volume-averaged macroscopic effect, rather than the fate of a particular crack. Domains where the macroscopic dynamic stress exceeds yield stress fail and the stress drops to ∼0. This is an implication of the hypothesis that the domains that fail producing small earthquakes contain partly opening mode cracks with near total stress drop [Fischer and Sammis, 2009]. Larger shallow events may involve transient opening mode cracks [Aoi et al., 2008] or shear cracks [Sleep and Ma, 2008]. Failed domains have finite residual stress after the seismic wave passes and dynamic stress returns to zero.
The dynamic energy per volume of τD2/2G is then dissipated when failure occurs. The stress in a given domain is the prestress σ plus the macroscopic dynamic stress τD when these stresses are favorably oriented. There are both domains that have relaxed to stress τint and domains that have not relaxed because they already had a lower stress after the previous earthquake. The contribution from the relaxed domains is
The contribution from the unrelaxed domains is
The total energy dissipation per volume is the sum of these quantities
The second bracket in the final equality is the dynamic energy per volume present in the seismic wave. Thus the first bracket is proportional to the attenuation Q−1. The limits of the integral in (17) and (18) require attention. If τY − τD > τint, the dynamic stress is too low to cause failure at the yield stress and the nonlinear attenuation is zero. If τY − τD ≤ τint, all the relaxed domains with stress τint fail. Thus attenuation Q−1 increases from 0 to a term proportional to the first bracket in (17) at τD = τY − τint. If τY − τD < τmin, the dynamic stress causes failure in all the domains and the available seismic energy is dissipated. More precisely, strong attenuation diminishes the amplitude of a propagating wave, so this definition implies that a full cycle of stress peaking at τD would dissipate the energy per volume in the wave given by (19).
Three parameters arise in the treatment. The yield stress τY represents the starting strength of fractures in the rock and the peak residual stress after shaking. It is measurable in the laboratory and given by (13) and (14). Intact rock and fractures that have not slipped for a geologically significant time will be stronger than regolith fractures the repeatedly slip during strong shaking as discussed in section 4. The peak prestress τint at the start of strong shaking depends on aging in (12) and the parameter b in the shear traction equation (8). The parameters τnorm and τmin are measures of the overall level of prestress; the requirement that (3a) integrates to 1 determines τmin in (3b). The elastic strain energy per volume is linearly proportional to this quantity in (4). The change in microscopic strain energy per volume and the total macroscopic strain energy per volume scale with the energy per volume in a strong seismic wave that causes frictional failure, which provides dimensional calibration.
The difference τY − τint implies that there is a threshold dynamic stress for nonlinear attenuation (Figure 2). The attenuation in (19) above the threshold is independent of τint. Thus the curves as in Figure 2 can be calibrated by measurements of attenuation at two levels of shaking. The magnitude of damage (from low-amplitude S wave velocity changes) at two shaking levels would provide significant information.
With regard to low dynamic stresses, the threshold is sharp in the model, but likely to be gradual in a real rock with heterogeneous properties and heterogeneous local normal traction prestress. In addition, creep in some domains may load adjacent ones, maintaining a finite number of domains near failure. Nonlinear attenuation is weak at the threshold in Figure 2; the continuous attenuation curve in (19) could be used for numerical convenience in a dynamic calculation. This threshold needs to be distinguished from the dynamic stress where nonlinear attenuation is comparable to ordinary linear attenuation and hence easily observable. To this point, the model assumes that many the cracks that can fail were reset in the last strong seismic event. It also assumes that these cracks form a throughgoing network that can anelastically accommodate the dynamic strain. I examine induced hillslope creep in Appendix B and pulverized rock in section 4 to address these issues. I finally apply the formalism to deep near-fault damage in section 5.
4. Application to Pulverized Shallow Near-Fault Rock
Pulverized sandstone and granite crop out near major fault zones in southern California [e.g., Dor et al., 2009; Rockwell et al., 2009]. The exposed material appears to be fresh rock when casually observed on a smooth surface. However, the bulk rock and its grains can be fragmented by hand along existing cracks into dust. The material weathers to badlands morphology where it is freshly exposed. I examine the physics of its formation to constrain the behavior of the shallow subsurface during strong shaking.
Shattered rock in the hanging wall of major thrust faults [Brune, 2001] is analogous to pulverized rock in that it exhibits no obvious shear deformation. It differs in that failure occurred on a network of fractures rather than at a subgrain scale. My discussion of pulverized rock applies to shattered rock with the difference that preexisting flaws allowed a fracture network to develop.
Some reluctance is warranted in applying rate and state friction to macroscopically intact rock as the formalism arose from studies of shear on planar surfaces. Intuitively, real intact rock is unlikely to be free of microscopic defects that lead to stress concentration and failure. Empirically, rate and state friction with the Linker and Dieterich  relationship (10) and (B3) does give an acceptable representation of internal shear failure of intact rocks. These include granite [Sleep, 1999], sandstone [Sleep and Hagin, 2008], and the Coulomb failure of cylindrical samples of numerous porous rocks in end cap experiments [Sleep, 2010a]. Moreover, the physics of exponential creep at crack tips in similar to shear creep at asperities. An acceptable thermodynamic formulation of rate and state friction is obtained considering crack tips as the rate-limiting step [Beeler, 2004]. It is thus justified to explore the implications of rate and state friction to macroscopically intact rocks.
4.1. Scaling Relationships and Depth to Damage
I obtain formalism for pulverized rock starting with strain in a vertically propagating shear wave. Anelastic strain for body waves is most likely to occur dimensionally near the scale depth 1/k = cS/ω where the Coulomb stress ratio and the absolute dynamic stress are high. The lithostatic stress at that depth is
where ρ is density and ω is the angular frequency. The material fails at an internal friction coefficient of μS so the shear traction is approximately from (B3)
The particle velocity at failure is
which is independent of density and seismic velocity. The dynamic strain at failure is
Pulverized rock extends over at least tens of meters depth and laterally to hundreds of meters, requiring a significant wavelength and period of the damaging wave. The dominant angular frequency of near-fault body wave velocity seismograms from large earthquakes is typically comparable to or less than 10 s−1 as observed with the Parkfield 2004 main shock. Equations (22) and (23) are singular at the low-frequency limit, which implies that feasible very long period waves do not cause failure at their quarter wavelength depth.
4.2. Generic Example of Pulverized Granite
I provide a numerical example relative to pulverized granite with (B3). The starting material should have properties between intact granite and the pulverized material. For intact rock, I assume the density ρ is 2600 kg m−3; the shear modulus is 40 GPa; and the S wave velocity cS is 3922 m s−1. I use angular frequency of 10 s−1 for Parkfield events. The scale depth is thus 392 m and the lithostatic pressure is 10 MPa. Doan and Gary  state that the elastic modulus of pulverized rock is less than 1/5 of that of intact granite in their analysis. I use this modulus as a proxy for the still lower modulus of pulverized rock in examples. The density does not change much. The seismic velocity, scale depth, and lithostatic pressure for pulverized granite are thus, 1754 m s−1, 175 m, and 4.56 MPa. I use both pristine and damaged rocks to provide a range of estimates. The quarter wavelength depth moves upward over geological time as pulverization and/or shattering reduce the seismic velocity of the rock. Obviously erosion gradually exhumes the damaged rock.
Proceeding to obtain yield criteria, Sleep  gave 2 eyeball fits to (B3) for the starting friction of intact granite using results by calculated by Lockner  from data obtained by Byerlee . I use both to provide a measure of uncertainty. The value of Pref was set at 1400 MPa where the coefficient of friction of gouge is the same as the coefficient of friction of intact rock. In one model, μref was 0.76 and α is 0.30. In the other model, these parameters were 0.8 and 0.21. These models yield a coefficient of friction of 2.24 and 1.84 at 10 MPa. They yield 2.48 and 2.01 at 4.56 MPa. Failure of intact rock at 1 MPa is relevant to the survival of islands of intact rock in granite regolith at ∼40 m depth. The coefficient of friction from (B3) is 2.93 to 2.32, compared with 1.5 for intact sandstone. With regard to crustal faults, the coefficient of friction of intact granite at 100 MPa normal traction is 1.55 to 1.35.
Intuitively, pulverized granite is likely to be weaker than intact granite and possibly intact sandstone. Sleep and Hagin [2008, Figure 16] estimated the coefficient of friction for sandstone at the scale depth from percolation theory as a function of S wave velocity. This treatment is sensitive to crack-like pore space that aids both elastic compliance and frictional failure; it applies to tight sandstone (as well as porous sandstone) and hence to granite. It attempts to empirically represent the effects of microcracks [e.g., Paliwal and Ramesh, 2008]. Sleep and Hagin's  computed model used μref = 0.7 and α = 0.21. Adjusting their curve for my value of μref = 0.8, the starting friction (for S wave velocity of 1754 m s−1 at the scale depth) is 1.25, well below that of intact granite and intact sandstone.
This approach would be much more useful if the coefficient of starting friction had been measured on pulverized granite at some normal traction. Field measurements do exist for shattered gneiss near Pacoima Dam California [Brune, 2001]. This site experienced shaking in the 1971 San Fernando main shock [e.g., Anooshehpoor and Brune, 1989] and the 1994 Northridge main shock [Sepúlveda et al., 2005]. The peak accelerations were 1.25 and 1.58 g. The S wave velocity is ∼2000 m s−1, similar to the example in the previous paragraph. There were significant topographic effects [e.g., Anooshehpoor and Brune, 1989; Sepúlveda et al., 2005], so the relationship between acceleration and Coulomb stress ratio applies only dimensionally. The predicted maximum Coulomb ratio 1.25 from the previous paragraph is in reasonable agreement, pending a full numerical calculation as in the work of Ma and Andrews .
4.3. Energy Budget of Pulverization
The dynamic velocity at the failure stress provides an estimate of the energy in a seismic wave that is estimated from the results of section 4.2 and hence constrains on the mechanism in section 5.4. At 10 MPa lithostatic stress, it is 1.84–2.24 m s−1 and is 2.01–2.48 m s−1 at 4.56 MPa, using intact granite strength. These dynamic velocities are reasonable near major faults.
I use a dynamic velocity of 2.27 m s−1 within the estimated range to provide a numerical example of the energy balance. The energy per volume in a seismic wave is 6500 J m−3. Rockwell et al.  estimate that the average grain size in pulverized granite is 26–201 μm. The surface free energy is ∼1 J m−2. The surface energy per volume for simplicity treating the grains as cubes is 23–3 × 104 J m−3. This is 35–4.6 times the energy per volume in the strong seismic wave. In section 3.1, it was noted that ∼5% of the microscopic elastic strain energy decays per seismic cycle so that the ambient residual elastic strain energy per volume scales to ∼20 times that in the strong seismic wave. The value of 20 is in the range estimated for the ratio of surface to seismic energy so it is thus attractive that ambient surface energy and ambient microscopic strain energy have comparable values.
Neither Doan and Gary  nor Rockwell et al.  give constraints on the porosity increase in forming pulverized rock. The inelastic volumetric strain of 2% in the experiments of Doan and Gary  provides the basis for my numerical example. The work per volume against lithostatic stress is 20 × 104 J m−3 at 10 MPa and 9.12 × 104 J m−3 at 4.56 MPa. These values are comparable to the estimated elastic strain and surface energies, again supporting approximate equal partition of energy.
4.4. Dynamic Model of Pulverization
Doan and Gary  certainly obtained pulverized rock in their experiments. They proposed a single-event model for pulverization at shallow depths and dynamic stresses of ∼100 MPa. Their implied energy balance is reasonable. The computed particle velocity is 9.6 m s−1 for intact granite and ∼20.4 m s−1 for already pulverized granite. The energy per volume in the seismic wave is 11.5 × 104 J m−3 and 57.5 × 104 J m−3, which is comparable to the work per volume to create surfaces, elastic strain energy, and porosity.
However, dynamic seismic velocities become increasingly rare at high values. Velocities exceeding 10 m s−1 in the history of a shallow rock thus imply numerous events with velocities over 2 m s−1. I thus propose a mechanism where numerous strong events impose dynamic stresses that just exceed the elastic limit of the rock. Limited pervasive failure occurs each time with microcracking but not macroscopic cracking. Such behavior is well known in hard rocks and engineering materials [e.g., Tapponnier and Brace, 1976; Paliwal and Ramesh, 2008]. This sequence is essentially the model of Wechsler et al.  for near-fault shallow damage by numerous similar events. It is also a weak form of a characteristic earthquake model. The pulverized rock is near a major fault that produces virtually all of the episodes of strong shaking; in this example, most have amplitudes of ∼2 m s−1.
There are two published phenomenological objections to my hypothesis. Doan and Gary  noted correctly that sudden very high stress pulverizes rock before strain has a chance to localize and conversely cracks localize at low stress and low inelastic strain rates. However, the granite studied by Rockwell et al.  and the sandstone studied by Dor et al.  are massive and do not have lots of obvious flaws to concentrate inelastic strain. Given the relative rarity of pulverized rock compared to fractured regolith and shattered rock, it is reasonable to invoke uncommon starting conditions. Rockwell et al.  objected to failure in shear, as there is correctly no obvious shear deformation within pulverized rocks. However, the amount of shear deformation expected even from single event failure is small, ∼2% in the experiments of Doan and Gary . The elastic strain at yield is ∼0.1% for 2.27 m s−1 motion within pulverized rock with a shear wave velocity of 1754 m s−1 and ∼1/2 this for intact rock. There are no strain markers in granite or sandstone that provide such fine resolution or even resolve 2% strain. Strain compatibility precludes large local shear strains that would produce either macroscopic voids or high macroscopic stresses in a medium, which otherwise returns to low strains after each event. As with hillslopes in Appendix B, domains of strong rock kinematically preclude large systematic shear deformations in the adjacent pulverized rock.
To reiterate, it is statistically expected that moderately large dynamic stresses and motions are much more common than extremely large ones. Thus, pulverized rock has been brought just beyond its elastic limit numerous times but never very far beyond it. Failure each time is pervasive with microscopic prestress and the different elastic constants and anisotropic orientations of minerals providing stress concentrations leading to microcracks [e.g., Tapponnier and Brace, 1976]. After shaking, microscopic elastic strain energy in pulverized rock relaxes somewhat. Elastic strain energy, porosity, and surface area build up as comparable quantities until there is enough ambient microscopic strain energy that its dissipation over each seismic cycle removes much of the energy brought in by each the strong seismic wave. After that the new damage per event decreases.
The initial state of the rock is important. The pulverized rock forms at the expense of massive rock with few macroscopic flaws to nucleate cracks. Conversely, flaws did exist and a fracture network built up within shattered rock. In pulverized rock, the dynamic strain tensor differs between episodes, so there is reduced tendency for microscopic damage to nucleate macroscopic cracks. In addition, rock becomes more elastically compliant with damage. High stresses are unlikely within compliant domains driven by essentially displacement boundary conditions of the surrounding stiff domains. This effect tends to delocalize macroscopic damage when the stiff domains support the dynamic stress. Conversely, throughgoing compliant domains cushion strong domains and maintain damage localization.
5. Rupture Tip Damage Near Crustal Fault Zones
The formalism developed above for the shallow subsurface is applicable to damage near major crustal depth faults. The modern theory of seismic rupture in large earthquakes invokes brief large dynamic stresses near the rupture tip [e.g., Noda et al., 2009; E. M. Dunham et al., Earthquake ruptures with strongly rate-weakening friction and off-fault plasticity: 1. Planar faults; 2. Nonplanar faults, submitted to Bulletin of the Seismological Society of America, 2010]. The shear traction on the fault plane reaches the starting friction for fractured rock. This high stress occurs in a region around the rupture tip, causing damage. This process contributes to macroscopic friction and acts to attenuate the seismic wave. I develop the analogy of this processes to hillslope behavior during dynamic shaking (Appendix B) with the caveat that the dynamic stress history near rupture tips is systematic to some extent as discussed in section 5.2.
I provide a generic example in hard crustal rock. The density ρ is 2600 kg m−3 and the shear modulus is 40 GPa. The S wave velocity cS is thus 3922 m s−1. The assumed starting shear traction at fault failure is 100 MPa, similar to the normal traction. For comparison, the coefficient of friction of intact granite at 100 MPa normal traction in (B3) is 1.55 to 1.35.
5.1. Strain and Energy Balance
To cause near-fault damage, the particle velocity U0 at the rupture tip needs imply dynamic stresses comparable to the starting shear traction of the main fault. This dynamic stress is obtained from 1/2 the macroscopic slip velocity US of the fault as the relevant coordinate system is centered on the fault plane [Noda et al., 2009]. The scale stress for near-fault damage is thus cSUSρ/2. For the assumed strength of the damageable material of 100 MPa, this relationship yields a slip velocity of 20 m s−1.
The energy balance treatment for regolith in section 3.2 applies with the modification that the microscopic and macroscopic shear moduli are essentially the same. The energy per volume in a seismic wave with the assumed particle velocity is 5 × 105 J m−3. The fractional change of microscopic strain energy per cycle form (7) remains ∼5%; microscopic yield stress τY = ∼4 GPa and the change per cycle sln(1010) = 0.92 GPa do not change obviously with depth. The value of τnorm from the total strain energy being 20 times the strain energy change is 200 MPa, which is again much lower than the microscopic yield stress and too high to apply to macroscopic stresses. Macroscopic τnorm is likely to be a small fraction of the macroscopic yield stress.
As with regolith, nonlinear behavior of the near-fault region commences at dynamic stresses well below those for Coulomb failure as each rupture events leaves residual stresses in its wake. The width of the dynamically damaged zone is thus likely to be greater than that expected from a naïve Coulomb criterion. Inferring prestress from exhumed samples is beyond the scope of this paper.
5.2. Long-Term Creep in Damage Zone
The analysis for hillslope creep in Appendix B applies with the modification that the dynamic stress at rupture tips implies systematic anelastic strain accommodating the gross direction of plate motion. In addition, anelastic strain occurs in the damage zone mainly during the passage of the rupture tip, rather than throughout the seismic event. The duration of rupture tip conditions at any one place is brief and the width of the damage zone scales to the duration times the shear wave velocity. I use a generic width of 10 m on each side of the fault to provide numerical examples.
The dynamic elastic strain in the damage zone provides scaling for anelastic effects. That is, 100 MPa shear stress implies an elastic strain of 0.25% and an elastic displacement of 25 mm on one side of the fault and 50 mm in total. This amount of anelastic failure is ∼1% of the 5 m displacement in a generic large earthquake. The slip in the damaged zone thus adds slightly to the macroscopic moment of the earthquake (R. Viesca-Falguières, personal communication, 2009).
The mechanics suggested for hillslopes (Appendix B) and pulverized rocks (section 4) may well apply to damage zones. Intact rock is stronger than the damaged rock. That is, domains of intact rock lock when they jostle in the damaged zone during strong shaking and do not accommodate large net shear strains.
6. Discussion and Practical Implications
This paper provides formalism for relating rock damage to nonlinear attenuation in the shallow subsurface. Practical applications include predicting the maximum amplitude of a seismic wave that can propagate to the surface and estimating the maximum amplitude of seismic waves that have impinged on a site in the past. It is intended to apply to a wide range of macroscopically brittle rocks, including sandstone and granite.
My derivations are dimensional and thus require calibration. There is some hope of success in that the predicted nonlinear attenuation Q−1 in (19) increases rapidly with wave amplitude. Detection of its effects at modest wave amplitude thus indicates much stronger attenuation at somewhat higher amplitudes.
With the intent of improving instrument deployment and encouraging observational studies, I qualitatively discuss issues that arise from the formalism, beginning with calibration of dynamic stress versus attenuation. Intact shallow rocks are of interest as they may transmit strong waves. However, their persistence indicates that they have not yet been damaged over a geological time by such waves. Finally, one must distinguish ductile soil effects from the brittle effects considered in the paper.
6.1. Calibration of Attenuation Versus Dynamic Stress
The attenuation equation (19) requires calibration before it can be used in numerical modeling. This task is not currently straightforward as data are lacking. Nonlinear attenuation may be detected directly from paired borehole and surface stations. It may be inferred indirectly by studying shallow rock damage without having to wait for a major earthquake from transient variations in the shallow low-amplitude S wave velocity after moderate shaking.
Ambient noise studies have the potential to provide surface wave dispersion curves and hence low-amplitude seismic velocity changes before and after strong shaking. The main difficulty is that stations need to be closely spaced to measure high-frequency waves that reside in the uppermost tens of meters. Combined with measurements of the strong surface wave amplitude it is straightforward to relate damage to Coulomb stress ratios [see Sleep, 2010b].
Strong surface waves in sedimentary basins are promising as significant elastic strain energy resides in the shallow subsurface and persists over numerous period cycles. Coulomb failure and tensional failure criteria are thus exceeded at very shallow depths for even moderate basin waves. Conversely, it is attractive to use measured damage from strong body waves to constrain the onset of nonlinearity for application to surface waves.
6.2. Intact Rock
The behavior of more intact rock may be modeled by this formalism with the understanding that the Coulomb stress ratio at failure is higher and that there are few domains with high prestress. The Linker and Dieterich  relationship (B3) provides an estimate of the starting internal friction of intact rock. It predicts Coulomb stress ratios greater than 2 for intact granite. The threshold stress τY − τint from (13) and (14) is somewhat larger than for regolith, as the highest macroscopic stresses have relaxed over geological time.
However, truly intact hard rock rarely exists in the shallow subsurface, except in recently deglaciated regions. Physically, erosion slowly exposes rock so it has time to experience numerous seismic events as well as nonseismic causes of fracturing. The S wave velocities of shallow hard rocks are thus much less that ∼4000 m s−1 of truly intact rock. Brocher [2008, Figure 10] gave a range of 300–1500 m s−1 for the San Francisco area of California.
Conveniently, the effect of normal traction (depth) in (B3) on yield Coulomb ratio is minor over a broad range in fractured rock S wave velocities. The scale depth for highly fractured rock is small since it has a low S wave velocity, which increases the contribution of the logarithmic term. In contrast, the fractured material has a smaller value of the state variable which reduces the μref. I thus estimated for the Coulomb failure stress at the scale depth of 1.25 for pulverized rock to that for fractured shallow hard rocks in general. This estimate requires more calibration, but using S wave velocity to constrain nonlinear attenuation is attractive as that parameter is already used in empirical site response studies.
Islands of shallow intact rock in the shallow subsurface are a complication (Figure 1). The corestones that become precarious rocks are an example [Brune, 2001]. The starting coefficient of friction of in these islands from (B3) is much more than that of the surrounding fractured rock. Such islands thus tend to survive once they are cushioned by elastically and frictionally compliant regolith. A practical consideration is that significant anelastic strain in the regolith may lock the islands, limiting the amount of nonlinear attenuation. Consideration of long-term erosion rates indicates that such locked domains greatly retard net downhill creep during strong shaking (Appendix B). Overall, the existence of intact rock, islands of intact rock, or pervasively fractured regolith is observable at sites. It is also often known somewhat whether sites are frequently or rarely shaken by strong seismic waves.
6.3. Damage and Exceedance
The extent of petrological damage and the ease which strong shaking changes causes seismic velocity to change provide long-term and short-term exceedance parameters. The damage in frequently shaken rock concentrates near the quarter wavelength depth of typical strong S waves. Thus the observable depth of damage provides an indication of the wave period of typical strong events at a site. The presence of residual stress in the shallow subsurface can be potentially measured to provide evidence of past damage.
With regard to long-term damage, Brune  pointed out that shattered rock occurs in the hanging walls of thrusts where accelerations over 1 g are observed. This constraint applies to the geological time for the rock to be exhumed by erosion to the quarter wavelength depth to typical strong waves. Damaged regolith to erodes preferentially and hence can be sensed remotely [Brune, 2001; Wechsler et al., 2009].
Overall, the formalism leads to a somewhat unsatisfying exceedance implication to sustained dynamic S wave accelerations. A site with intact rock has not experienced strong shaking over ∼1 g that would have caused extensive fracturing, yet the rock has weak nonlinear attenuation and would transmit strong seismic waves if they impinged from below. Conversely, a site with well-developed seismic regolith and postseismic damage has experienced repeated strong shaking at approaching least ∼1 g, but the rock has strong nonlinear attenuation and would not transmit much larger waves than it typically experiences from below. Buried mildly fractured massive rock with corestones that erodes to become precarious rocks [Brune, 2001] is the best case. This material has not experienced extreme shaking that would shatter it, but has a sufficient fracture network to facilitate nonlinear attenuation.
6.4. Distinguishing Ductile Soil Effects From Brittle Effects
Nonlinear effects in regolith need to be separated from those in shallow soil. With regard to engineering practice, nonlinear attenuation in very shallow soil is often not a concern for substantial structures when building interests remove soil before laying foundations. It is, however, a direct concern for family houses and indirectly a concern to the extent that the behavior of this layer affects the recorded motion at surface seismic stations and hence empirical site responses.
Recent results indicate that separation of soil from rock effects may be feasible with before and after low-amplitude seismic velocity data. Rubinstein  used resonance frequency changes to measure nonlinearity near Parkfield. Wu et al. [2009b, 2010] used surface and borehole stations in Japan. Both studies are most sensitive to changes within the uppermost resonating layer, for example, 2.3 m thick with a S wave velocity of 135 m s−1 at one Parkfield site [Rubinstein, 2010]. Nonlinearity commenced at accelerations as low as 2%–3% of ambient gravity and hence Coulomb stress ratios of that magnitude. The resonant period change persisted briefly, with much of the recovery in the first 20 s. Hence this effect is unlikely to be present in data sets, such as those from repeating earthquakes, which are observed hours to days after strong shaking.
This paper considered shear failure during strong seismic shaking in macroscopically brittle rocks with direct application to shallow S waves. The mechanics are also applicable to strong surface waves and dynamic failure near crustal fault rupture tips. Practical implications involve nonlinear attenuation in the shallow subsurface.
The formalism represents the shallow subsurface as fractured regolith that repeatedly fails in numerous strong earthquakes. The material self-organizes so that a range of prestresses exists. Favorably oriented fractures fail attenuating strong seismic waves. This process leaves residual stresses that become the prestresses in the next strong event. The highest microscopic and macroscopic stresses thus relax before shaking recommences. Considering energy budgets indicate that most regolith fails along cracks where the porosity and state variable change significantly (Figure 1). The rest >99% of the rock is not damaged in a given event.
Modeling the stress as a scalar fractal distribution over a finite range yields a simple scaling expression for nonlinear attenuation (19). Rate and state friction provides macroscopic quantification. Fractured domains heal and strengthen between seismic events. They can support starting Coulomb stress ratios of ∼1 (0.961 in the example model). The maximum stresses immediately after shaking approach this criterion. The highest stresses relax interseismically, so that most fractured domains are modestly away from failure at the start of the next earthquake. Only the fractured domains with high prestresses fail in modest shaking. The material cannot sustain a yield stress, above which all the fractures fail.
Analogous crack network situations exist. Earthquake triggering involves gradual external loading occurs in the interseismic interval or equivalently systematic ambient tectonic stress [Hill, 2008]. Shallow hillslopes (Appendix B) are a case in which gravity provides a continuous driving force and ambient stress field. Their net downslope creep during strong shaking, but the amount predicted by a simple model is excessive. Studies to resolve coseismic downslope creep are warranted as the provide information on nonlinear brittle processes with regolith.
Pulverized rock provides a counterexample to crack networks. Failure occurs pervasively throughout the rock mass, just beyond the elastic limit. Shear deformation is not visible in samples because the strain at the elastic limit ≪1% cannot be observed by petrological markers. Numerous seismic events ∼20 are needed for microscopic residual stresses, surface free energy, and porosity to build up to quasisteady values.
Finally, the strong motions of a vertically propagating S wave provide simple scaling relationships using the dominant frequency on a velocity seismogram for quick calculations. Behavior of the material at the scale depth 1/k of the seismic wave (or equivalently the quarter wavelength depth) is relevant to nonlinear attenuation and damage. The sustained dynamic acceleration normalized to ambient gravity is approximately the Coulomb stress ratio on horizontal planes at shallow depths. This acceleration scaling applies to compressional P waves, but porous rock fails in crushing not shear [Andrews et al., 2007; Sleep, 2010a].
Ory Dor presented experiments related to the mechanism proposed for forming pulverized rock in section 4 at the 2010 SCEC meeting.
Appendix A:: Energy Partitioning During Failure
The discussions of macroscopic stress concluded following Sleep and Hagin  that comparable energies per volume go into creating residual stress, dilating cracks against confining pressure, and sliding cracks during dynamic friction. Analogously, the microscopic energy budget discussion in section 4.3 concluded that comparable energies go into creating new grain surfaces, anelastic deformation of grains, and opening cracks with pulverized rock. I briefly discuss the mechanics of these processes and the inference of frequent near-total local stress drop in this appendix, beginning with kinematic macroscopic behavior.
Macroscopic kinematic behavior in the shallow subsurface differs from that at great depths because many domains of the rock have some tensile strength while others have none. Geometrically, the triggered small seismic events discussed by Fischer and Sammis  have source dimensions of submeter to several meters in the size range of rocks that can be lifted by the top (and the size range of overhangs) and thus support transient absolute tension.
First, the failure of a granular substance in shear results in kinematic dilatancy. For example, consider the two-dimensional rolling of square blocks supporting both sides of a shear crack. The initial opening and shear velocities are equal. Once the blocks have rolled so that the corners support the surface, the shear displacement is times the maximum possible opening displacement. Further rolling, however, leads to no additional dilatancy. Rolling friction is most likely at low normal tractions [Anthony and Marone, 2005].
Moreover, the Coulomb stress ratio at failure can exceed 1. As an example, consider a vertically propagating S wave that produces shear traction greater than the lithostatic stress on a horizontal plane. This wave also produces absolute tension on dipping planes. Once tensional failure begins on a dipping plane, near total shear stress drop can occur [Fischer and Sammis, 2009]. Prestress also causes the local normal traction to be much less than lithostatic along favored places in shallowly dipping planes. The cracks fail relieving the dynamic stresses with comparable shear and opening displacements. Residual opening remains, as displaced cracks do not fit together.
Third, once dynamic shaking has ceased, the stress along a crack that failed with near-total stress drop is at the failure stress as assumed in the fractal treatment. The ratio of shear to normal stresses can exceed 1. There is thus opportunity to open adjacent cracks producing dilatancy.
The microscopic shear failure stress ∼4 GPa greatly exceeds lithostatic stress throughout the crust. Microscopically, material has strength in tension until opening at crack tips occurs. Total stress drop occurs both at opening crack tips and when sliding asperities lose contact. Large strains as in sliding asperities and small strains in pulverized rock behave differently. Contact asperities accumulate anelastic strains many times the yield strain [Sleep, 2006b]. The residual stresses once a contact is abandoned cannot exceed the yield stress. The asperity leaves residual stresses in the surface over which it passed that scale as a fraction ∼3/8 of the yield strain or equivalently as ∼3/8 of the ratio of shear yield stress to shear modulus.
In contrast, an asperity in pulverized rock does not accumulate large strains. It behaves instead like a crack with total stress drop in regolith. Once dynamic stress ceases, the asperity has residual stresses near the yield stress. These stresses do work to open pore space and to create new grain surfaces.
Obtaining an equivalent rheology that keeps track of porosity, grain size, and residual strain energy for material that is episodically stressed is a reasonable goal. Nguyen and Einav  discuss such a model for cataclastic deformation. Paliwal and Ramesh  discuss microcracks. I do not attempt here to obtain treatment for rock pulverized by transient events.
Appendix B:: Interaction of Oscillating Dynamic Stresses With Static Stresses on Hillslopes
Hillslopes near major seismically active faults provide information on the behavior of regolith. The static gravitational stresses, as with a block on a ramp, interact with the dynamic stresses during shaking (Figure B1). It is thus straightforward to construct a trial model of a hillslope with regolith. The ratio of the hillslope stress to the dynamic stress at failure in strong seismic waves is reasonably constrained for an example calculation. This process is distinct from the effect where repeatedly damaged rock near faults is more easily eroded by rain and runoff than distal intact rock [Brune, 2001; Wechsler et al., 2009] and from shallow seismically triggered massive landslides [e.g., Sepúlveda et al., 2005]. It differs dynamically driven shallow soil creep [Rubinstein, 2010; Wu et al., 2009b, 2010] by involving Coulomb failure.
B1. Kinematic Equivalent Rheology of Regolith
For strong nonlinear attenuation to occur, anelastic strain needs to relax the available seismic energy in (1) or equivalently the anelastic strain scales with the dynamic strain ɛD = U0/cS, where U0 is the dynamic particle velocity and cS is the shear wave velocity. The downslope stress scales with the slope. This strain and this displacement occur dimensionally once every earthquake cycle lasting tpost.
The process is analogous to the well-known effect where an object gradually slides down a vibrating ramp along a slight slope. In analogy, regolith domains may fail either when the dynamic and slope stress are favorably aligned or when they are misaligned. Continuing to assume near-total stress drop, the slip is proportional to the sum of dynamic and slope stresses τdyn + τslope in the first case and τdyn − τslope in the second case. The long-term strain rate scales with the dynamic strain U0/cS, inversely to the duration of the seismic cycle tpost, and proportionally to the ratio of slope stress to dynamic stress:
This strain acts over a depth range scaling to the inverse of the wave number k of the seismic wave on a velocity seismogram. The creep velocity is thus
where U0/cSk is the dynamic displacement, which by assumption occurs each cycle.
B2. Parkfield Example
I provide a numerical example with generic values appropriate to Parkfield California, retaining parameters from Sleep and Hagin . The shear wave velocity is 300 m s−1; the particle velocity is 1 m s−1, the seismic cycle is 30 years (109 s), and the inverse of the wave number 1/k is 30 m. This scale thickness of the creeping zone is large enough that vegetation has little effect. The dynamic displacement is 0.1 m. The angular frequency is 10 s−1. The dynamic acceleration is 10 m s−2, equal by intent to the ambient acceleration of gravity (1 g), a canonical value for strong motion. The downslope component of ambient gravity causes the block to slide preferentially in that direction. The dynamic stress at shallow depths in strong motion is then approximately the lithostatic stress. The second parentheses in (B2) is thus twice the slope, in this example ∼0.2 for a slope of ∼0.1. These parameters yield that the creep velocity is 0.67 mm a−1 or 670 m Ma−1. In terms of an individual event, downslope movements scale to the product of twice the dynamic displacement 200 mm and the slope 0.1, which is 20 mm for the parameters used. (Sleep and Hagin  showed a Parkfield example where the dynamic displacement is 130 mm.) Such systematic downslope movement in strong earthquakes would have already been detected by radar studies if it frequently occurred.
Consideration of the mass balance of material on the slope provides further quantitative comparison with this prediction. The equivalent flux in the example is dimensionally the creep rate times the scale depth ∼30 m, yielding 0.020 m2 a−1. Kinematically, the downslope flux supplies the measurable material carried away on average by streams. Studies give material fluxes per length of contour line of ∼0.001 m2 a−1 [e.g., Heimsath et al., 2005]. Most of this flux occurs within a ∼1 m thick soil layer where measurements are available. The deep flux from the previous paragraph must be less than ∼0.001 m2 a−1 (and could be much less than), rather than ∼20 times this amount as computed.
Note for context that the derivation of flux is easily modified to represent creep in a thin soil layer as studied by Rubinstein  and Wu et al. . The shear traction in an S wave that ascends vertically is approaches zero at the horizontal free surface. The shear traction increases linearly with depth when the soil thickness is much less than the scale length 1/k. The elastic displacement across the soil layer is then Zsoil2a/2c2, where Zsoil is the thickness of the soil layer (2.3 m in a site studied by Rubinstein ), a is particle acceleration at the surface (here the generic value of 1 g), and c is the S wave velocity in the soil (135 m s−1 in a site studied by Rubinstein ). The computed elastic displacement and the implied anelastic deformation scale are 1.45 mm. Assuming a recurrence interval of again 30 years and a 0.1 slope, the hillslope flux is 2.6 × 10−5 m2 a−1, which is negligible.
B3. Mechanics of Regolith on Hillslope
The excessive estimate of hillslope flux from (B2) involves a dimensional approximation with an implicit unknown multiplicative constant. As given, the computed creep flux from the expression thus may be high by a factor of a few or even an order of magnitude. First, it uses the total elastic strain as a measure of the anelastic strain necessary to strongly attenuate the wave. The component of anelastic strain parallel to topographic contours does not facilitate downslope creep. Neither does significant anelastic strain that opens porosity against lithostatic stress as discussed in section 3.3. Strong waves with such attenuation may not recur at a given site during every earthquake cycle.
Still, it is productive to seek a physical explanation for why the simple model gives an excessive result. The macroscopic mechanical material is regolith cut by numerous fractures between more intact blocks. Typically, the fractured domains fail during strong shaking, jostling the blocks. Interlocking strong domains may greatly retard large amounts of downslope creep over many cycles (Figure B1). The strong domains only rarely fail and limit the overall rate of creep.
Rate and state friction is useful for quantifying the starting friction of intact sandstone blocks. The Linker and Dieterich  relationship in (8) and (11) provides reasonable estimates of the coefficient of internal starting friction of fractured sandstone at shallow depths. A laboratory scientist measures the shear traction at sudden failure of intact samples at various normal tractions [Sleep and Hagin, 2008]. The starting state variable ψ is the same in these experiments and the strain rate is approximately the same and similar to that for coseismic failure. Letting the strain rate and the state be constant in (8) yields that the starting coefficient of friction is
where the starting coefficient of friction measured at normal traction Pref is μref. Sleep and Hagin  obtained 100 MPa and 0.81 for these parameters and 0.15 for α for sandstone. Extrapolating with this equation to 1 MPa normal traction near the base of the regolith yields that the coefficient of starting friction is 1.5 for sandstone. Equation (B3) has the net effect of cohesion in that the failure shear traction at large normal tractions does not extrapolate to zero. The equation does not include cohesion in the sense that the shear traction is zero at zero normal traction. I acknowledge that small blocks of rock do have cohesion at very low normal traction, but ignore it on the grounds that failure occurs elsewhere.
Cracking that is pervasive on a scale that is a modest fraction of the scale length 1/k of the seismic wave reduces macroscopic dynamic stress, attenuates the wave, and cushions the intact domains of rock from stresses higher than the crack yield stress. The intact domains thus rarely fail. The overall rheology is similar to that of a wooden jigsaw puzzle that locks after finite shear. The macroscopic elastic strains needed to start cracking involve the Coulomb stress ratio of 0.961 in the examples that are only modestly less than the elastic strains to produce failure at the Coulomb stress ratio of ∼1.5 within intact rock in section 5. The failed domains in a repeatedly jostled rock mass may well be able to take up the latter strain inelastically and cushion the intact rock in individual events. Equation (B2) may still apply but with an unknown ≪1 multiplicative factor that accounts for the rare instances of failure in the intact rock.
rate friction parameter, dimensionless.
state friction parameter, dimensionless.
S wave velocity, m s−1.
parameter relating to porosity to state, dimensionless.
elastic energy per volume integral, J m−3.
volume fraction, dimensionless.
shear modulus, N m−2.
tensor indices, dimensionless.
seismic wave number, m−1.
normal traction, N m−2.
lithostatic pressure, N m−2.
reference normal traction, N m−2.
seismic attenuation, dimensionless.
second invariant of microscopic deviatoric stress, N m−2.
stress scale for exponential creep, N m−2.
duration of the seismic cycle, s.
duration of strong seismic stress, s.
maximum dynamic particle velocity, m s−1.
fault sliding velocity, m s−1.
creep velocity of hillslope, m s−1.
width of fault sliding zone, m.
soil layer thickness, m.
parameter representing effect of normal traction changes on shear traction, dimensionless.
microscopic strain rate tensor, s−1.
constant with dimensions of strain rate, s−1.
energy per volume integral change over relaxed domains, J m−3.
energy per volume integral change over unrelaxed domains, J m−3.
total energy per volume change, J m−3.
hillslope strain rate, s−1.
strain to significantly change the properties of the sliding surface, dimensionless.
dynamic seismic strain, dimensionless.
strain to approach steady state, dimensionless.
strain rate, s−1.
reference strain rate, s−1.
approximate coefficient of friction, dimensionless.
coefficient of friction during dynamic slip, dimensionless.
coefficient of friction, dimensionless.
state variable, dimensionless.
normalizing value of the state variable, dimensionless.
steady state variable, dimensionless.
density, kg m−3.
dummy variable for stress, N m−2.
shear stress or traction, N m−2.
maximum shear stress, N m−2.
dynamic seismic stress, N m−2.
microscopic deviatoric stress tensor, N m−2.
lower limit of integral over stress, N m−2.
maximum stress present after dynamic damage, N m−2.
normalizing constant with the dimensions of stress, N m−2.
failure stress of intact rock, N m−2.
hillslope stress, N m−2.
yield stress, N m−2.
angular frequency, s−1.
John Anderson, Rasool Anooshehpoor, and Jim Brune quickly responded to my questions on exceedance. George Hilley discussed hillslope creep. Eric Dunham read the paper, provided preprints, and made helpful suggestions. Justin Rubinstein provided a helpful preprint on attenuation with suggestions. Zhigang Peng provided several helpful preprints. This work benefited from two anonymous reviews. This research was in part supported by NSF grants EAR-0406658 and EAR-0909319. The latter grant is funded under the American Recovery and Reinvestment Act of 2009 (ARRA) (Public Law 111-5). This research was supported by the Southern California Earthquake Center. SCEC is funded by NSF Cooperative Agreement EAR-0106924 and USGS Cooperative Agreement 02HQAG0008. This paper is SCEC publication 1449. The 2008 and 2009 SCEC meetings indicated the need to consider pulverized rock and regolith.