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Laboratory simulations of earthquakes show that at high slip rates, faults can weaken significantly, aiding rupture [1–3] . Various mechanisms, such as thermal pressurization and flash heating, have been proposed to cause this weakening during laboratory experiments [1,4–6], yet the processes that aid fault slip in nature remain unknown. Measurements of seismic radiation during an earthquake can be used to estimate the frictional work asso- ciated with fault weakening, known as an event’s fracture energy [7–9]. Here we compile new and existing [8,9] measurements of fracture energy for earthquakes globally that vary in size from borehole microseismicity to great earthquakes. We observe a distinct transition in how fracture energy scales with event size, which implies that faults weaken di�erently during small and large earthquakes, and earthquakes are not self-similar. We use an elastodynamic numerical model of earthquake rupture to explore possible mechanisms. We find that thermal pressurization of pore fluid by the rapid shear heating of fault gouge can account for the observed scaling of fracture energy in small and large earthquakes, over seven orders of fault slip magnitude. We conclude that thermal pressurization is a widespread and prominent process for fault weakening.
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LETTERS
PUBLISHED ONLINE: 5 OCTOBER 2015 | DOI: 10.1038/NGEO2554
Ubiquitous weakening of faults due to
thermal pressurization
Robert C. Viesca
1,2
*
and Dmitry I. Garagash
2
*
Laboratory simulations of earthquakes show that at high slip
rates,faults can weakensignificantly, aiding rupture
1–3
. Various
mechanisms, such as thermal pressurization and flash heating,
have been proposed to cause this weakening during laboratory
experiments
1,4–6
, yet the processes that aid fault slip in nature
remain unknown. Measurements of seismic radiation during an
earthquake can be used to estimate the frictional work asso-
ciated with fault weakening, known as an event’s fracture en-
ergy
7–9
. Here we compile new and existing
8,9
measurements of
fracture energy for earthquakes globally that vary in size from
borehole microseismicity to great earthquakes. We observe a
distinct transition in how fracture energy scales with eventsize,
which implies that faults weaken dierently during small and
large earthquakes, and earthquakes are not self-similar. We
use an elastodynamic numerical model of earthquake rupture
to explore possible mechanisms. We find that thermal pressur-
ization of pore fluid by the rapidshearheating of fault gougecan
account for the observed scaling of fracture energy in small and
large earthquakes, over seven orders of fault slip magnitude.
We conclude that thermal pressurization is a widespread and
prominent process for fault weakening.
Throughout the seismic cycle, faults experience a wide range of
slip rates. Over this range, l aboratory observations have indicated
that the frictional strength of faults may respond in a drastically
different manner. At low, interseismic slip rates, rock friction
experiments show minor strength variations
10
. In contrast, at faster
slip rates, such as those during an earthquake-generating fault
rupture, strong weakening is observed
2,3
. An inherent problem
in inferring whether the mechanisms proposed for such strong
weakening operate on faults is that the depths at which they occur
are largely inaccessible. However, occasional forays allow direct
measurements and sampling, providing further indications that
coseismic strength may be low
11,12
.
The interest in determining whether strong weakening does
occur at high fault slip rates is heightened by implications for
earthquake hazard and fault operation. The tsunami hazard of
the 2011 Tohoku event was increased by large slip in the shallow,
near-trench regions of the subduction interface
13,14
. Thes e regions,
along with creeping segments of other major faults, have long been
considered to be stable as a result of rate-strengthening behaviour
at low slip rates; however, this st ability is undermined by strong
weakening at fast slip rates, such that these regions may participate
in the catastrophic rupture of the fault
15,16
. Low coseismic strength
also implies t hat the shear stress on a fault may be much lower than
the static strength and still support rapid rupture. This affects the
manner in which a fault ruptures, as low shear stress increases the
likelihood of a rupture occurring as a propagating pulse of slip
17,18
,
a phenomena long observed seismologically
19
.
Here we use seismological observations to help constrain the
possible weakening mechanisms operating on faults. We compile
estimates of source parameters and independent estimates of
background stress over a wide range of event sizes among crustal
and subduction interplate events. From the source information,
we calculate t he average fault fracture energy, defined as the
work done on the fault in excess of a residual level of friction.
Such inferences have shown a nonlinear scaling of fracture energy
with slip
8
. Our extensive data set allows a comparison with a
dynamic rupture model that can incorporate multiple mechanisms
of strong weakening. We show that the single, dominant weakening
mechanism (thermal pressurization) may account for the observed
fracture energy scaling over many orders of magnitude of event size.
Two mechanisms for dramatic fault weakening among the most
likely to be ubiquitous are: the reduction of the effective normal
stress by t hermal pressurization of pore fluid due to shear heating
of fault gouge
4,5,20
, and the reduction of the friction coefficient by
localized flash heating of asperity contacts in the gouge
2
. Other
candidate strong-weakening mechanisms on faults include melting
and thermal decomposition
1,3,5
, and their o ccurrence depends on the
specific mineralogical composition of the gouge and, additionally,
may require significant coseismic temperature rise. The latter may
not always be attainable, particularly in a pulse-like rupture, owing
to drastic coseismic reduction of the shear heating associated with
the thermal pressurization
21
.
We determine how these two thermal weakening mechanisms
control the slip dependence of fracture energy by coupling their
influence on frictional strength to the dynamic elastic response
of the surrounding medium. Incorporating these mechanisms in
elastodynamic simulations of crustal-scale fault rupture is a current
computational challenge. The difficulties stem from the disparity of
length scales between the rupture extent and distances of strength
evolution, anticipated to be several orders of magnitude apart. We
circumvent this computational complexity by reducing the problem
to its essential elements to find evolution of slip and strength that are
consistent with elasticity and both manners of weakening. Specifi-
cally, we focus on the tip region of the r upture, where most strength
loss is expected to occur, and model it as a steadily propagating semi-
infinite crack (Methods and Supplementary Methods).
Strength evolution behind the rupture front follows the order
in which the mechanisms operate. This order is determined by
the relative size of slip s cales that characterize weakening under
each mechanism. For flash heating, the relevant quantity is t he sub-
millimetre slip scale
10
, L, associated with evolution of the friction
coefficient, implying the coefficient may drop to a new, dynamically
reduced value over short distances behind the front of a dynamic
rupture. For thermal pressurization, there are two characteristic
slip scales and each is associated with one of two limiting regimes.
1
Department of Civil and Environmental Engineering, Tufts University, Medford, Massachusetts 02155, USA.
2
Department of Civil and Resource
Engineering, Dalhousie University, Halifax, Nova Scotia B3H 4R2, Canada.
*
e-mail: robert.viesca@tufts.edu; garagash@dal.ca
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LETTERS
NATURE GEOSCIENCE DOI: 10.1038/NGEO2554
0.1
1
10
0.1
1
5
f
b
a
x
SlippingLocked Re-locked
p
τ
d
τ
f
τ
v
r
i
τ
τ
τ
10
−4
10
−3
10
−1
10
−2
10
1
10
2
10
3
10
4
10
5
10
−5
10
0
Normalized slip rate (V/V
)
Near-tip region
of width
Normalized shear stress ( /f
w0
)
and friction coecient (f/f
w
)
τσ
Normalized distance behind rupture tip (x/
)
Figure 1 | Cascade of weakening at the tip of a dynamic shear rupture.
a, Schematic of a slip-pulse mode of rupture: re-locking follows fault
rupture during rupture propagation. b, Evolution of the fault slip rate and
strength at the rupture tip (for example, shaded region in a), with the
parameter set in Supplementary Table 2. Initial weakening occurs through
reduction of the friction coecient by flash weakening. Subsequently,
thermal pressurization of fault gouge pore fluid reduces the eective
normal stress. Two endmembers are considered: negligible (cyan lines) and
non-negligible (red lines) flux of heat and pore fluid. Broken lines follow the
evolution under flash weakening of friction without thermal pressurization
(dashed line) or thermal pressurization under a flash-weakened coecient
of friction (dotted and dash-dotted lines).
The first regime characterizes early moments of rupture, in which
pressurization occurs under undrained, adiabatic conditions, with
the slip sc ale
4
δ
c
=
ρch
f 3
where ρc is the heat capacity, h is the thickness of the fault
gouge, f is the prevailing friction coefficient, and 3 is the thermal
pressurization coefficient relating increments of undrained pore
fluid pressure increase to increments in temperature. The second
regime occurs after sufficiently long time (slip) allows large-scale
diffusion to obscure the details of shearing of the finitely thick gouge
zone. In this regime, heating is effectively reduced to that of slip on
a plane, and weakening occurs with a characteristic slip
5,22
L
=
ρc
f 3
2
4α
V
where α is a lumped hydrothermal diffusivity
5
, and V
is a
characteristic elastodynamic slip rate
21
. Among the propert ies
0.01 0.1 1 10 100
0.0
0.2
0.4
0.6
0.8
1.0
0.1
1 10 100
0.01
0.1
1
10
Normalized slip ( /
c
)
b
a
Normalized fracture energy (G/f
w0c
)
δ
σ
Normalized shear stress ( /f
w0
)
τσ
δ
2/3
δ
2
δ
δ
Normalized slip ( /
c
)
δδ
Figure 2 | Fault fracture energy and gradual fault weakening with slip.
a, Inferred dependence of fault shear strength on slip at the tip of a
propagating rupture. Weakening here occurs by thermal pressurization
under a flash-weakened or an otherwise (approximately) constant
coecient of friction. b, Implications of the strength–slip relation for the
slip dependence of fault fracture energy. Increasing darkness of the
greyscale lines corresponds to increasing eciency of heat and pore fluid
drainage. Dashed lines assume weakening under either
undrained-adiabatic (cyan) or ‘drained’, slip-on-a-plane (red) conditions.
determining δ
c
and L
with the potential for l argest variation
are the gouge hydrothermal diffusivity and the width of the
actively shear ing gouge. However, for sub-millimetre to millimetres-
thick shear zones
23–25
and hydraulic diffusivities within an order
of magnitude or above 1 mm
2
s
1
(ref. 26), a centimetre-to-
decimetre δ
c
and millimetre-to-metre L
can be reasonably expected
(Supplementary Table 1).
We efficiently resolve the details of the compounded weakening
at the rupture tip (Fig. 1). That the evolution of the friction
coefficient is associated with such a short slip scale L—relative
to plausible values of δ
c
and L
—implies that flash heating, if
active, reduces the friction coefficient before any substantial thermal
pressurization. The subsequent manner in which effective stress is
reduced depends on the ratio L
c
, which can vary from 0.0001
to 100. For values of L
c
above unity, the majority of the increase
in gouge pore fluid pressure in large slip events occurs in a manner
consistent with slip-on-a-plane pressurization. In t his fashion, the
strength drop in large-slip, high-L
c
events occurs in a less efficient
manner than the exp onential slip weakening of undrained-adiabatic
shear in the small-slip, low-L
c
events.
Our dynamic rupture solutions provide a theoretical slip
dependence of fracture energy, def ine d as
7
G) =
Z
δ
0
τ
0
) τ)
dδ
0
(1)
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NATURE GEOSCIENCE DOI: 10.1038/NGEO2554
LETTERS
b
a
c
G'—empty symbols
G
max
—solid symbols
Chile '85
Chile '85
(aftershocks)
Subduction interplate
Pagai Is.
Peru '07
Tohoku
Sumatra'04
Nias
Maule
Aleutian
Chile'95
Sanriku
Miyagi
Mentawai
Peru Feb'96
Peru Nov'96
Benkulu
Tokachi'03
10
−1
10
0
10
1
10
3
10
4
10
5
10
6
10
7
10
8
10
9
G'—empty symbols
G—solid symbols
Big crustal
Denali
Landers
Hector Mine
Northridge
Kobe
Fukuoka
Tottori
Imp. Valley
Morgan Hill
Colfiorito
Parkfield
Subduction interplate
Chile '85 Sequence, ref. 37
Ref. 36
Ref. 35
La No Ko IV HM
Fu De MH Pa To
Colf
KTB, refs 84,85
Groß Schönebeck, ref. 81
W. Nagano, ref. 86;
W. Nagano, ref. 82
Cajon Pass, ref. 8
Long Valley, ref. 83
Korabid Aftershocks, ref. 77
Friuli, ref. 79
Au Sable Aftershocks, ref. 78
Northridge Aftershocks, ref. 87
Loma Prieta Aftershocks, ref. 80
Big crustal, refs 9,75 (G):
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
10
1
10
2
10
−2
10
−1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
Slip (m)
Slip (m)
10
−1
10
0
10
1
Slip (m)
Fracture energy (J m
−2
)
Fracture energy (J m
−2
)
10
3
10
4
10
5
10
6
10
7
10
8
10
9
Fracture energy (J m
−2
)
Peru'01
Small crustal (G'):
(G
max
= G' +
f
):
δτ
2
δ
2/3
δ
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
Figure 3 | Variation of inferred event fracture energy with event slip. a, Inferred fracture energy (G) and slip for events within continental crust and along
subduction interplate interfaces. b, For subduction events, comparison of fracture energy based on seismological observations alone (G
0
) with those
incorporating pre-event shear stress estimates and assuming near-complete coseismic strength drop (G
max
). c, For large crustal events, comparison of G
0
with G inferred from kinematic inversions of fault slip history. In ac the curves are three predictions for ruptures whose fracture energy are dominated by
thermal pressurization. Dashed curves represent scenarios of negligible (cyan) and significant (red) diusion of pore fluid and heat. The black curve
represents an intermediate scenario.
where τ is the fault shear and δ is the average distance the fault
slipped in an event. B ecause the slip scale L is comparatively small,
the initial weakening due to flash heating has a negligible con-
tribution to t he total fault fracture energy, which is consequently
controlled by thermal pressurization
27
. The slip dependence is then
determined solely by the ratio L
c
(Fig. 2). However, for any value
of the parameter, there are two endmember scalings of the fracture
energy: for small slip, the early time undrained-adiabatic deforma-
tion results in fracture energy scaling as G δ
2
, and for large slip,
where shear heating resembles slip on a plane, we find that Gδ
2/3
.
This latter large-slip scaling is in contrast to the finite fracture
energy (ρch ¯σ /3) reached in the limit of large slip under undrained-
adiabatic conditions. Our dynamic analysis, which couples evo-
lution of the slip rate to that of the strength, supports previous
kinematic models for evolution of the coseismic strength
5,22
. In the
latter a constant slip rate of coseismic magnitude is proscribed,
resulting in behaviour qualitatively comparable to, but quantita-
tively different from (with the exception of the small-slip limit)
our model.
The definition of the fracture energy, equation (1), presumes
that the dissipation of energy is dominated by work done on
the fault surface—yet, inelastic deformation of the adjoining rock
is expected to occur. Fault rupture models allowing for such
deformation
28
suggest that off-fault dissipation, although scal-
ing with event size, remains at most comparable to the on-fault
dissipation (Methods).
We estimate the average fault fracture energy G for a wide range
of event sizes using previously published s eismological inferences
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LETTERS
NATURE GEOSCIENCE DOI: 10.1038/NGEO2554
of source parameters (Methods and Supplementary Tables 3–5 and
Supplementary Figs 1–3), namely the total moment and seismic
energy released and a characteristic source radius. On the basis of
seismic source spectra, for each event, we estimate an intermediate
energetic quantity G
0
=(1τ/2 τ
a
(where is the stat ic stress
drop and τ
a
is the apparent stress). By conservation of energy
G=G
0
+
f
τ
d
, w here τ
d
is the minimum coseismic strength,
and τ
f
is the final (after-event) fault stress
8
. For crack-like ruptures,
reasonably characterized by minimal dynamic over/undershoot
(τ
f
τ
d
), G G
0
. For large pulse-like ruptures, consistent with the
hypothesis of thermal pressurization that the coseismic strength
drop at large slip is near-complete (τ
d
0), G G
max
=G
0
+τ
f
δ,
the estimate of which requires inferences of the absolute fault
stress (Methods).
In Fig. 3a we compile our frac ture energy estimates and estimates
made previously
8,9
. Estimates of G are available for the largest events:
for those occurring on a subduction interplate interface, we assume
G =G
max
and make use of published inferences of fault stress; for
those occurring within the crust, we rely on available estimates of G
from kinematic inversions
9
, independent of our estimates of G
0
from
source spectra. In both cases, we find that G
0
largely underestimates
the fracture energy (Fig. 3b,c), which may indicate that the strong-
weakening slip-pulse scenario represents rupture behaviour of large
subduction and crustal events
19
. However, for the smallest events
(borehole seismicity and crustal aftershock sequences, Fig. 3a),
studies of the stress field and dynamic rupture models imply that
rupture may frequently be crack-like, in which case we may take
G G
0
. Inferences of the stress field in regions surrounding the
KTB and Cajon Pass boreholes indicate a critically stressed crust,
in which high values of the shear-to-normal-stress ratio on opti-
mally oriented faults
29,30
favour crack-like rupture propagation
17,18
.
Alternatively, these small events may be limited by a fault extent
such that a rupture nucleated in a crack-like mode reaches fault
boundaries and arrests before a transition to a pulse-like mode
ever occurs.
The estimates of fracture energy are in general agreement with
the prediction of the dynamic rupture tip solution with thermal
pressurization over more than six decades of event slip. For small
event sizes, we observe that the scaling of fracture energy with slip is
similar to δ
2
, consistent with linear slip weakening under undrained-
adiabatic pressurization for event slip smaller than δ
c
. At large event
sizes, the apparent quadratic scaling of the fracture energy breaks
down and is replaced by a sublinear scaling characteristic of slip-on-
a-plane (or drained’) pressurization. Remarkably, the t he oretically
predicted evolution of fracture energy with slip (Fig. 3, curves) is
made using a single value of δ
c
(0.1 m) and a single value of the
nominal fault strength f ¯σ
0
(110 MPa). The former may imply similar
coseismic shear zone thicknesses on different faults; the latter may
indicate either approximately constant values of f and ¯σ
0
in the crust,
or, alternatively, a variable high or low (flash-weakened) value of
friction, inversely related to a variable effective normal stress level
from event to event.
Although multiple weakening mechanisms may operate on
faults, the results suggest that thermal pressurization may be the
dominant contributor to fault fracture energy. Furthermore, that
fracture energy seems to follow a scaling attributable to thermal
pressurization, considering a large number of events occurring on a
wide variety of fault zones, also suggests the potential prevalence of
this mechanism. Understanding how faults lose their strength dur-
ing the rapid slip of an earthquake-generating rupture is important,
for instance, to constrain the minimum level of stress a fault requires
to rupture catastrophically.
Methods
Methods and any associated references are available in the online
version of the paper.
Received 13 January 2015; accepted 3 September 2015;
published online 5 October 2015
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20. Mase, C. W. & Smith, L. Effects of frictional heating on the thermal,
hydrologic, and mechanical response of a fault. J. Geophys. Res. 92,
6249–6272 (1987).
21. Garagash, D. I. Seismic and aseismic slip pulses driven by thermal
pressurization of pore fluid. J. Geophys. Res. 117, B04314 (2012).
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of the Punchbowl fault, San Andreas system, California. Tectonophysics 295,
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24. Boullier, A.-M., Yeh, E.-C., Boutareaud, S., Song, S.-R. & Tsai, C.-H.
Microscale anatomy of the 1999 Chi-Chi earthquake fault zone. Geochem.
Geophys. Geosyst. 10, Q03016 (2009).
25. Kuo, L.-W., Hsiao, H.-C., Song, S.-R., Sheu, H.-S. & Suppe, J. Coseismic
thickness of principal slip zone from the Taiwan Chelungpu fault Drilling
Project-A (TCDP-A) and correlated fracture energy. Tectonophysics 619–620,
29–35 (2014).
26. Wibberley, C. A. J. Hydraulic diffusivity of fault gouge zones and implications
for thermal pressurization during seismic slip. Earth Planet. Space 54,
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27. Brantut, N. & Rice, J. R. How pore fluid pressurization influences
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NATURE GEOSCIENCE DOI: 10.1038/NGEO2554
LETTERS
29. Zoback, M. D. & Healy, J. H. In situ stress measurements to 3.5 km depth in
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Acknowledgements
This study was supported by the Natural Science and Engineering R esearch Council
of Canada (Discovery grant 371606), the National Science Foundation (grant
EAR-1344993), and the Southern California Earthquake Center (SCEC; contribution
No. 6004). SCEC is funded by NSF Cooperative Agreement EAR-1033462 and
USGS Cooperative Agreement G12AC20038.
Author contributions
Both authors contributed to developing the main ideas, interpreting the results and
producing the manuscript.
Additional information
Supplementary information is available in the online version of the paper. Reprints and
permissions information is available online at www.nature.com/reprints.
Correspondence and requests for materials should be addressed to R.C.V. or D.I.G.
Competing financial interests
The authors declare no competing financial interests.
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LETTERS
NATURE GEOSCIENCE DOI: 10.1038/NGEO2554
Methods
The dependence of fracture energy with event slip is theoretically estimated by
examining the dependence of strength w ith slip at the tip of a propagating rupture.
The region near the tip is modelled as a steadily propagating semi-infinite crack
rupturing a planar fault dividing two linearly elastic half-spaces. The fault strength
is determined by an evolving friction coefficient and effective normal stress. The
evolution of the former is slip-rate- and state-dependent, with strong weakening at
high slip rates
2,18
. The evolution of fault pore fluid pressure alters the effective
normal stress and depends on the history of shear heating and hydrothermal
diffusion
4,5,18
. We formulate a sharp-tip rupture approximation that requires a
regularization of the frictional e volution to allow for a slip velocity that is
vanishingly small at the rupture front. The resulting coupled nonlinear integral
equations are solved by amending a pseudospectral boundary integral equation
method. Laboratory measurements of frictional and hydrothermal properties of
fault material indicate that evolution of pore pressure and friction occur over
disparate fault length and slip scales. Our method of solution allows the resolution
of the compounded weakening with large-scale separation. When calcu lating the
fracture energy, the details of frict ional evolution are reasonably neglected, and the
model reduces to one with a single parameter that captures the relative efficiency of
hydrothermal diffusion
22
. We compile estimates of fracture energ y for crustal and
subduction interplate ruptures using published seismological data and, where
available, estimates of pre-event shear stress for large subduction events. Primarily,
we infer key source properties (seismic moment, radiated energy, and characteristic
event radius) on the basis of the recorded source spectra, which we use in turn to
estimate the average event f racture energy and slip
8
.
Seismological estimates of fracture energy. In this work we compiled
seismological estimates of the fracture energy of a large number of events (from
borehole seismicity to great subduction megathrust eart hquakes) which are based
on two distinct approaches. The first approach
9,31
relies on the finite-fault
kinematic inversion for the slip-history, and corresponding calculation of the
elastic stress change over the fault to calculate G directly via its definition
7
G) =
Z
δ
0
[τ
0
) τ )]dδ
0
(2)
The other method
8
relies on estimating an energetic quantity G
0
(see (3) below)
from the source spectra only—that is, from the s eismically inferred moment M
0
,
radiated energy E
s
, and e quivalent source radius r (inferred for the conner
frequency, for example, by way of a circular rupture model
32,33
). As we show below,
G
0
can be used to approximate the fracture energy G for expanding crack-like
ruptures, whereas a measure we call G
max
(see (5) below) additionally requires an
independent estimate of t he fault stress (either b efore or after the event) and is a
suitable approximation of the frac ture energy of large, pulse-like ruptures.
G
0
and G
max
approximations of the fracture energy. Potential energy release in an
earthquake can be written as
7
1W /A=
i
+τ
f
)δ/2 =E
s
/A+G +F
where E
s
/A= radiated energy, G = fracture energy and F =τ
d
δ = frictional heat
per unit slipped fault surface. The energetic quantity
8
G
0
=
i
τ
f
)δ/2 τ
a
δ (3)
can be estimated from source spectra by using the following expressions for the
stress drop τ
i
τ
f
=7M
0
/16r
3
, the apparent stress τ
a
=E
s
/πr
2
δ (A=πr
2
) and the
average slip δ =M
0
πr
2
. Fracture energy can then be expressed as
G=G
0
+
(
τ
f
τ
d
)
δ (4)
Modelling of crack-like events with arbitrary slip show smal l dynamic stress
overshoot
32
(that is, minimum coseismic stress τ
d
is only slightly larger than the
final static stress τ
f
), potentially allowing it to be neglected in the expression (4) for
the fracture energy. That is,
crack-like events,τ
f
τ
d
: GG
0
On the other hand, we expect near-complete coseismic strength loss, τ
d
0, for
large slip events on faults with strong dynamic weakening, as well as a finite final
static stress τ
f
for pulse-like events (for which stress is expected to partially recover
in the wake of the pulse from its minimum, coseismic value τ
d
). Thus,
large pulse-like events,τ
f
τ
d
: GG
max
=G
0
+τ
f
δ (5)
Fracture energy estimates of big crustal and subduction events. The relevant
source data and corresponding estimates of fracture energy G (for large crustal
events
9,34
), as well as our estimates of energetic quantities G
0
(for large
subduction
35–51
and crustal events
8,9,52–72
) and G
max
(for great subduction events for
which independent estimates of t he fault prestress, τ
i
15 MPa, are available
73,74
)
are shown in Supplementary Tables 3 and 4 for selected earthquakes.
The uncertainty in G-estimates for large events based on finite-fault kinematic
inversions may stem from non-uniqueness of finite source inversion for a given
event, as well as from different s ets of assumptions involved w ith modelling
coseismic stress changes b ased on a g iven fault slip kinematics. For example,
Tinti et al.
9,75
estimate the average fracture energy over fault area with significant
slip for the Landers event to be 51 MJ m
2
and 42 MJ m
2
for the two respective
kinematic slip models
52,76
. A different study
31
of the same two kinematic models for
Landers yielded 90 MJ m
2
and 32 MJ m
2
, respectively. This is a fairly
representative cas e, which suggests that the uncertainties of G-estimates can b e
roughly constrained to within a factor of two.
Fracture energy estimates of borehole seismicity and crustal aftershock sequences.
We use source parameters from a number of recent studies
8,77–87
of aftershock
sequences and borehole-recorded microseismicity (Supplementar y Table 5) to
infer values of the energetic quantity G
0
and average event slip. In doing so, we
recalculate the source parameters from the reported individual spectral
measurements using a consistent set of assumptions. For all but t wo event
sequences (Chile 1985 and Northridge 1994 aftershock sequences), we use sp ectral
relations
32,33
between the corner frequencies (f
cS
, f
cP
) and the source radius r of a
circular rupture to estimate
r =geometric mean
h
k
S
i
c
s
f
cS
,
h
k
P
i
c
s
f
cP
(6)
where k
S
and k
P
are tabulated functions
32
of rupture velocity and azimuth θ (the ray
take-off angle with the fault normal), and the
h
k
i
are the corresponding average
values over the representative azimuth range for an assumed representative value of
the rupture velocity. (When one corner frequency is reported but not the
other—that is, either f
cS
or f
cP
—we modify (6) accordingly). We assume v
r
=0.9c
s
(ref. 8), and select average values of k as either
h
k
S
i
=0.30,
h
k
P
i
=0.39 (average over 0 θ 90
) (7)
or
h
k
S
i
=0.21,
h
k
P
i
=0.32 (average over 45
θ 90
) (8)
depending on the assumed azimuthal coverage of events. Specifically, we have
assumed the entire azimuthal coverage, (7), for all borehole-recorded events and
the locally recorded Au Sable Forks 2002 aftershock sequence; and the upper-half
of that range, (8), for all other crustal aftershock sequences that we analysed. The
latter choice can be understood on an example of strike-slip (vertical) fault with
receiving stations located on the earth surface at or near the fault (with the distance
to t he fault not exceeding a typical event depth). When both corner frequencies are
reported, we apply a filter 0.75 f
cP
/f
cS
2.25 to remove the events with
unrealistically small/large values of the frequency ratio (this is motivated by
elastodynamic solutions
32
with v
r
=0.9c
s
, which suggest the corner frequency ratio
values bet ween 1 and 1.5, depending on the azimuth). These removed events
normally account for a small percentage of the total number of events in a sequence
(for example, 3% of KTB borehole events, 6% of Loma Prieta aftershocks) with the
exception of the Western Nagano events analysed in ref. 86, of which about 40%
had unrealistic corner frequency ratios. The static stress drop, average event slip,
and energetic quantity G
0
are estimated from source values M
0
, r, and E
s
, as
explained earlier
8
.
In the case of Northridge 1994 aftershock sequence, reported signal durations
t
signal
(ref. 87) allow an estimation of the equivalent circular rupture radius
88
r =v
r
t
signal
1+max
v
r
c
s
sinθ,
v
r
c
p

1
(9)
where, as before, we assume v
r
=0.9c
s
, and average
h
sinθ
i
=0.9 over the azimuthal
range 45
θ 90
. The latter estimate of
h
sinθ
i
is justified for the Northridge
aftershocks, for which we can calculate
h
sinθ
i
0.92 using the stations locations
and hypocentre/fault-plane solutions
89
. We note that our values of the source radii
for Northridge aftershocks are on average a half of the values originally reported
87
.
This discrepancy can be tracked to the assumption that a pulse duration t
pulse
(used
in previous source radius estimates
88
) is half of the signal duration t
signal
. On the
basis of the preceding estimates
88
(equations (9) and (11) therein), we can obtain
t
pulse
0.265
t
signal
for the studied Northridge aftershocks
87
, which suggests that
the pulse durations and corresponding source radii were possibly overestimated in
their study. We also note that the original range of stress-drop values is 2–66 MPa
and that our values range from 3 to 100 MPa.
Assessing off-fault contributions to fr acture energy. Dynamic rupture models
indicate that stress concentrations at the tip of a dynamically propagating rupture
may be sufficient to cause significant off-fault inelastic deformation
28,90
, which may
be reflected in the observed damage zones of surrounding faults
91,92
. This
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NATURE GEOSCIENCE DOI: 10.1038/NGEO2554
LETTERS
coseismic, off-fault inelastic deformation is part of the total fracture energy of an
event. Constraints on the magnitude of that contribution (relative to on-fault
processes, such as those studied here) may be deduced from dynamic rupture
models in which inelastic deformation is allowed to occur
28,93
. In these models, the
extent of inelastic deformation is proportional to the rupture propagation
distance, which implies a linear scaling of the off-fault dissipation with event
slip for crack-like and unsteady (g rowing) slip-pulse rupture modes. We seek an
order-of-magnitude constraint on the off-fault contribution to the fracture energy
by determining the prefactor of that linear scaling. To do so we use detailed
reports on the extent and intensity of off-fault deformation in dynamic rupture
models
94
, along with a relation between rupture size and mean slip. We find that
the off-fault contribution to the fracture energy G
off
(0.1–10) MJ m
2
(δ/1 m),
where the lower limit roughly corresponds to the case of low fault shear stress
inferred on major plate-bounding faults
95
. This estimate is approximately an
order of magnitude below the fracture energy estimates from seismological
observations for larger events (Fig. 3, δ & 0.1 m). Although we caution that this is a
preliminary estimate, this suggests that on-fault contributions to fracture energy
dominate when fault strength weakens slowly with slip, as for weakening by
thermal pressurization.
‘Homogeneous’ fault model predictions for ruptures with heterogeneous slip.
Kinematic slip inversions of large events often show complex/spatially
heterogeneous slip. It is therefore a relevant question as to how much can we
learn about the physics of earthquake slip by contrasting fault-averages of
heterogeneous fields inferred from observations to predictions for the s ame
field based on ‘homogeneous fault models. Systematic studies of such
comparisons, involving predictive solutions for heterogeneous faults, are needed
to answer this question conclusively, and are outside of the scope of this
contribution. Yet, we can suggest two types of observations that may lend some
credence to the application of homogeneous fault model predictions to their
heterogeneous prototypes.
It has been observed that slip and fracture energy inferred from
finite-fault kinematic inversions, although heterogeneous, can strongly correlate
9,96
.
This suggests that the fault-area-averaged fracture energy inferred from an
events kinematic inversion
h
G
i
observ
can be meaningfully compared to the
theoretical predictions from a homogeneous fault model (that is, fault with
homogeneous properties and prestress) if the scale of the heterogeneity is larger
than the characteristic dimension of a dynamically slipping patch (making
pulse-like ruptures more viable candidates than crack-like ones); and the slip
magnitude versus fault area distribution is known from the inversion. For
example, assuming a linear distribution of area with slip in the range between 0 and
2
h
δ
i
, one can readily evaluate the fault-averaged prediction
G
theory
)
to be
contrasted to values inferred from observations—that is, to
h
G
i
observ
. Such a
comparison is shown on Fig. 3, after introducing an additional simplification by
using G
theory
(
h
δ
i
) in place of
G
theory
)
. The maximum error associated with this
last simplification is between +33% and 5% for a power-law G
theory
versus δ
relation, whose exponent may vary from 2 to 2/3, respectively, with increasing slip,
as predicted by the thermal pressurization model. This latter error is
inconsequential in the log–log scale comparisons over many orders of magnitude
of slip and G (Fig. 3).
Another observation pertains to the question how well do the ‘local’ G
versus slip data inferred f rom a kinematic inversion agree with the fault-averaged
data. We provide an illustration of such an agreement in Supplementary Fig. 1,
where we plot the local fracture energy versus slip data (cyan stars) from selected
locations in the finite-fault model of the Imperial Valley event
96
together with the
fault-average values for other large crustal earthquakes (as in Fig. 3c). Evidently, the
local fault data scales in approximately the same way as the averaged data from
different events (including the fault-average value for the Imperial Valley
event itself).
Modelling of the tip of a dynamic fault rupture. Our fault rupture model reduces
to coupled nonlinear integral equations relating the evolution of f riction and the
development of fault pore fluid pressures to the elastodynamics of a propagating
rupture. Below we summarize the governing equations and highlight key
assumptions that enable efficient, approximate solutions for the slip- and
slip-rate-weakening behaviour behind the rupture front. Knowledge of this
behaviour enables a calcul ation of the fracture energy (including its slip
dependence) as it depends on the manner of dynamic weakening.
The fault shear strength is determined by
τ =f p) (10)
where f is the frict ion coefficient, σ is the fault-normal stress, and p is the fault
pore pressure.
Numerical solutions
18
indicate that the classical friction coefficient dependence
on the rate and ‘state of frictional contacts
10,97–99
amended for high-velocity thermal
weakening (fl ash heating) results in a marked slip acceleration and corresponding
friction/stress build up to some peak value (f
p
) immediately on arrival of the
rupture front, followed by weakening towards the high-velocity (‘flash-heated’),
steady-state friction value (f
w
f
p
). This allows one to significantly simplify the
modelling by approximating the rupture front as a sharp crack with a p eak level of
friction f
p
established at the tip, whereas the decline in the friction coefficient
immediately behind the sharp front is dominated by the evolution of the state
variable over the characteristic slip scale L, which on integration yields an
exponential decay with slip
27,100
f f
w
(f
p
f
w
) exp(δ/L) (11)
To solve for simultaneous evolution of the fault pore pressure, we consider the
frictional heating of the fault and simultaneous heat flux in the fault-normal
direction. Assuming that t he fault slip velocity is accommodated by a Gaussian
distribution of the strain-rate ˙γ over a thin layer of gouge of nominal width h,
˙γ =(V /h) exp[−π(y/h)
2
], and g iven a history of shear stress and slip velocity at a
point along the fault, the current pore pressure at the gouge centre at that point can
be given by
p(t) =p
0
+
3
ρch
Z
t
0
τ (t
0
)V (t
0
)K
t t
0
t
d
;
α
hy
α
th
dt
0
(12)
where t
d
=h
2
/(4α) is a characteristic diffusion timescale, and K is a convolution
kernel whos e form depends on the ratio of hydraulic-to-thermal diffusivities
α
hy
th
(refs 5,21,101).
We consider the steady, dynamic propagation of a sharply tipped mode-II or
mode-III crack (in-plane or anti-plane rupture) with rupture velocity v
r
. We define
x as the distance behind the rupture tip such that t =x/v
r
is the time since passage
of the rupture front, and the slip velocity is given by
V =v
r
dδ
dx
(13)
where δ is the relative fault displacement. In view of (13), the shear stress
distribution along the fault, ahead and behind the rupture tip, is given by
102
τ (x)=
¯µ
2πv
r
Z
0
V (s)
sx
ds (14)
where ¯µ is a shear modulus that depends on both the mode of rupture as
well as v
r
.
Solutions to (10)–(14) are found by first discretizing the equations spatially
along two sets of singular integral quadrature points
103
with quantities defined on
the one set related to those on the other by barycentric interpolation
104
. The
resulting nonlinear equations for the discretized slip rate, friction and effective
normal stress are solved via Newton–Raphson iteration, with details given in a
subsection to follow. The solution yields a relation between shear strength and slip,
τ ) (as in Fig. 2), which allows an estimate of the slip dependence of the fracture
energy G via (2).
Endmembers of the fracture energy of thermal pressurization. When the
slip-weakening scales for t he flash heating (L) and thermal pressurization
(δ
c
or L
) separate—that is, Lmax
c
,L
)—a cascade of weakening is
observed in our numerical solutions, which show that the early flash heating
weakening acts to set up the nominal level of friction (f f
w
) on a
dynamically slipping fault, although its contribution to the event fracture
energy is negligible compared to that of the ther mal pressurization ac ting
over larger slip and spatial scales. Although the numerical solution allows
one to recover the complete dependence of the fracture energy on slip, it is
instructive to provide the analytical endmembers of this dependence for
relatively small and large values of slip, when the dynamic response of the
sheared fault gouge can b e described as essentially undrained and
drained, respectively.
The fault gouge response to the frictional heating and pressurization is
undrained (drained) when the slip time t is much smaller (larger) than the
hydrothermal diffusion timescale t
d
defined in the above. Elastodynamics (14) and
the characteristic scale of the strength reduction by the thermal pressurization
(τ
=f
w
¯σ
0
) suggest the characteristic slip velocity V
=(f
w
¯σ
0
/ ¯µ)v
r
near the rupture
front. Then, the fault response is expected to be undrained (drained) when the
dynamically accrued slip is much smaller (larger) than the characteristic diffusion
slip δ
d
=V
t
d
, which interestingly is not an independent slip scale (that is, it can
be expressed in terms of the two previously introduced TP slip-weakening
scales, δ
d
=δ
2
c
/L
).
Undrained fracture energy. When δ δ
d
(or t t
d
) and f f
w
, the constitutive
equations (10) with (12) and t =x/v
r
reduce to the slip-law
4
, τ =f
w
¯σ
0
exp(δ/δ
c
),
and the corresponding f racture energy dep endence on slip is found from (2),
G=f
w
¯σ
0
δ
c
1(1 +δ/δ
c
) exp(δ/δ
c
)
. The small-slip limit, δ δ
c
, corresponds to
the quadratic slip dependence
undrained, small slip: G f
w
¯σ
0
δ
2
/2δ
c
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LETTERS
NATURE GEOSCIENCE DOI: 10.1038/NGEO2554
Drained fracture energy. In the large, drained slip limit δ δ
d
(or t t
d
), (10)
with (12) and t =x/v
r
reduce to the slip-on-a-plane constitutive equation
5
,
τ (x)f
w
¯σ
0
=
3
ρc
v
r
Z
x
0
τ (s)V (s)ds
4πα(x s)
(15)
We look for the large-slip, far-field (large distance x from the rupture front)
asymptotic solution of (14) and (15) among the candidate monomial solutions
105
of
(14), V /V
=Bx
λ
and τ
=(B/2)cot(πλ)x
λ
, where the values of the exponent
and the prefactor, λ =1/4 and B =
2(x
/π)
1/4
, with x
=( ¯µ/τ
)L
, are recovered
by substitution into (15) and balancing the dominant terms in the limit x .
Integrating for slip and relating it to the strength, we can recover the corresponding,
drained large-slip law τ
(2/3π)
1/3
(δ/L
)
1/3
. The corresponding expression for
the fracture energy slip dependence follows by integrating (2),
drained, large slip: G(12π)
1/3
f
w
¯σ
0
L
1/3
δ
2/3
Numerical method of solution for slip rate near the rupture tip. When the total
slip incurred in dynamic rupture is much larger than the characteristic
slip-weakening distance, most of the strength reduction is accommodated in the
near-tip region of the rupture. This region can be modelled as a semi-infinite shear
crack steadily propagating at the instantaneous rupture velocity v
r
(ref. 21). The
corresponding elastodynamic equation in the coordinate system x moving with the
crack tip (x =0) is given by
τ (x)=τ
+
¯µ
2π
Z
0
dδ
ds
ds
sx
(16)
where ¯µ =µ
1(v
r
/c
s
)
2
is the apparent shear modulus for a mode-III crack, c
s
is
the shear wave speed of the surrounding medium, µ is the shear modulus, and τ
is the far-field driving stress. Requiring no stress intensity at the tip and a constant
stress drop far b ehind determines the asymptotic behaviour of the slip gradient
towards the crack tip and far from it
dδ
dx
x0
x,
dδ
dx
x→∞
1
x
(17)
We use t he following coordinate transform to map x [0, ) to z [−1, 1]
x =
1+z
1z
and arrive at
τ [x(z)]=τ
+
¯µ
2π
Z
1
1
(1z)(1u)
2
dδ
dt
du
uz
which takes the familiar form of the Hilbert transform when expressed as
g(z) =
¯µ
2π
Z
1
1
h(u)
uz
du (18)
with
h(z) =(1z)
dδ
dz
g(z) =2
τ [x(z)]τ
1z
(19)
Reflecting the asymptotics of (17), h(z) behaves as
h(z 1)
1+z h(z 1)
1
1z
With this anticipated we may define a function φ (z) such that
h(z) =
r
1+z
1z
φ(z) (20)
Substituting this expression in (18) we find that the integral takes a form
approximable by a Gauss–Chebyshev quadrature of the third kind
Z
1
1
r
1+u
1u
F(u)du
n
X
j=1
w
j
F(u
j
) (21)
with u
j
=cos[π(2j1)/(2n +1)] and w
j
=2π(1+t
j
)/(2n+1). When F(u) is the
singular kernel 1/(u z), the approximation continues to hold provided z is
evaluated at points z
i
=cos[2πi/(2n+1)] for i =1, 2,. .. , n (ref. 103). This reduces
the evaluation of (16) to that of
g(z
i
) =
¯µ
2π
n
X
j=1
w
j
φ(u
j
)
u
j
z
i
(22)
To evaluate the left-hand side of (22), we calc ulate f and ¯σ at points z
i
and note
that for ruptures including undrained thermal pressurization, τ
=0. Considering
friction, we may integrate a slip-law formulation for friction evolution (refs 18,98),
neglecting the direct effect as discussed and making use of the Galilean
transformation to the crack-tip coordinate x =v
r
t,
f (x) f
p
=
1
L
Z
x
0
dδ
ds
f f
ss
(V )
ds
We transform the coordinates via (17), replacing dδ/dz with its representation by
φ(z), and include a Heaviside step function H(z) to extend the domain of
integration to one consistent with the quadrature (21)
f [x(z)]f
p
=
1
L
Z
1
1
H(z u(u)
r
1+u
1u
f f
ss
(V )
1u
du
such that
f [x(z
i
)]f
p
=
1
L
n
X
j=1
S
ij
w
j
φ(u
j
)
f [x(u
j
)]f
ss
(V )
1u
j
(23)
where S
ij
=1 if u
j
< z
i
, and 0 otherwise, and f
ss
(V ) may be evaluated by recognizing
that V =v
r
dδ/dx with dδ/dx =φ[z(x)]
1z
2
/2.
We may find an expression for the effective stress σ p via (12) in a similar
manner. Following the shift to the crack-tip coordinate,
¯σ (x) ¯σ
0
=
3
ρch
Z
x
0
τ (s)
dδ
ds
K
x s
t
d
v
r
;
α
hy
α
th
ds (24)
Subsequently, the coordinate transformation from x and s to z and u, similar use of
the Heaviside function, and the quadrature approximation of the integral, enable
the approximation of ¯σ (x) as
¯σ [x(z
i
)] ¯σ
0
=
3
ρch
n
X
j=1
S
ij
w
j
φ(u
j
)K
ij
f [x(u
j
)]¯σ [x(u
j
)]
1u
j
(25)
where K
ij
=K
(x(z
i
) x(u
j
))/(v
r
t
d
)
.
In (23) and (25), f and ¯σ are evaluated at two sets of points at x(z
i
) and x(u
j
).
To reduce the set of solution variables, we interpolate the quantity p (that is, f or ¯σ )
from x(u
j
) to x(z
i
) by
p[x(z
i
)]=
n
X
j=1
M
ij
p[x(u
j
)] (26)
We make convenient use of the points u
j
lying at the zeros of a Chebyshev
polynomial and use the barycentric form of Lagrange interpolation
104
, for which
M
ij
=
W
j
z
i
u
j

n
X
m=1
W
m
z
i
u
m
!
and, as u
j
are the zeros of a Chebyshev polynomial of the third kind, the weights are
W
j
=(1)
j
cos
θ
j
2
sinθ
j
θ
j
=π
2j1
2n+1
Using the shorthand notation w here indices i and j stand for the evaluation at
the z
i
and u
j
nodes, respectively, (for example, f
i
=f [x(z
i
)], f
j
=f [x(u
j
)], and,
according to (26), f
i
=
P
n
j=1
M
ij
f
j
), and defining m
j
=1/(1u
j
), we arrive at the
system of 3n equations
f
i
=f
p
1
L
n
X
j=1
S
ij
w
j
φ
j
m
j
f
j
f
ss
(V
j
)
¯σ
i
= ¯σ
0
3
ρch
n
X
j=1
S
ij
w
j
φ
j
m
j
K
ij
f
j
¯σ
j
(27)
2f
i
¯σ
i
1z
i
=
¯µ
2π
n
X
j=1
w
j
φ
j
u
j
z
i
for 3n unknowns in the order of f
j
, ¯σ
j
, and φ
j
. On evaluating the Jacobian of (27), its
solution is amenable to the Newton–Raphson iteration method.
The quadrature (22) can also be used in the solution to the problem of the
exponentially slip-weakening semi-infinite crack. Such a manner of slip weakening
occurs, for example, for the case of undrained-adiabatic thermal pressurization
4
.
For this case, when evaluating g (z), we use
τ τ
=f ¯σ
0
exp(δ/δ
)
in which slip is evaluated at points z by applying the coordinate transform to the
integral δ(x) =
R
x
0
(dδ/dx
0
)dx
0
to ar rive at
δ[x(z)]=
Z
1
1
H(z u)
r
1+u
1u
φ(u)
1u
du (28)
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© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved
NATURE GEOSCIENCE DOI: 10.1038/NGEO2554
LETTERS
which is approximated at z
i
nodes as
δ
i
=S
ij
w
j
φ
j
m
j
with terms as previously defined.
Amendment to account for non-classical asymptotic behaviour. The above
method may be amended to account for an exponent of the far-field slip gradient
other than one-half, that is,
dδ
dx
x0
x,
dδ
dx
x→∞
=Ax
λ
1
2
< λ < 0
where the latter asymptote is assumed to be known explicitly. For example, in a
preceding section concerning endmember cases, we discuss a case where the
weakening in the far field occ urs by slip-on-a-plane thermal pressurization, for
which λ =1/4 and A =[(4/π)(f ¯σ
0
/ ¯µ)
3
L
]
1/4
. We use the behaviour of h(z)
(equation (19)) in the far field, h(z) (1 z)
1λ
, to amend the definition of the
function φ(z) (equation (20)) by introducing an additional term (φ
asy
(z)) that
reflects the asymptotics
h(z) =
r
1+z
1z
(
φ
asy
(z) +(z)
)
, φ
asy
(z) A
2
1z
1/2+λ
Here (z) is f inite and continuously differentiable at the interval’s ends allowing
one, as previously, to use Gauss–Chebyshev quadrature methods to evaluate
related integrals.
Denote the contributions from the φ
asy
-term in h(z) to the elastic shear stress
(18) with (19), the effective stress (24), and slip (28) as
τ
asy
[x(z)]=
¯µ
2π
1z
2
Z
1
1
r
1+u
1u
φ
asy
(u)
uz
du
¯σ
asy
[x(z)]=
3
ρch
Z
z
1
r
1+u
1u
φ
asy
(u
asy
[x(u)]
1u
K
x(z) x(u)
v
r
t
d
du
δ
asy
[x(z)]=
1
2π
Z
z
1
r
1+u
1u
φ
asy
(u)
1u
du
We can write the amended set (27) of 3n algebraic equations as
f
i
=f
p
1
L
n
X
j=1
S
ij
w
j
m
j
asy
j
+
j
)
f
j
f
ss
(V
j
)
¯σ
i
¯σ
asy
i
= ¯σ
0
3
ρch
n
X
j=1
S
ij
w
j
m
j
K
ij
asy
j
+
j
)f
j
¯σ
j
τ )
asy
j
2(f
i
¯σ
i
τ
asy
i
)
1z
i
=
¯µ
2π
n
X
j=1
w
j
j
u
j
z
i
in 3n unknowns chosen here in the order of f
j
, 1 ¯σ
j
= ¯σ
j
τ
asy
j
/f
, and
j
,
where, as previously, indices i and j correspond to the values of a variable
evaluated at nodes z
i
and u
j
, respectively, and related by the interpolation
expression (26).
Code avai lability. The approach used to produce the theoretical curves in
Figs 1–3 is detailed in the Methods. The resulting algebraic equations are easily
solved using a Newton–Raphson procedure, making inclusion of the underlying
code unnecessary.
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