Solid Earth, 11, 2283–2302, 2020
https://doi.org/10.5194/se-11-2283-2020
© Author(s) 2020. This work is distributed under
the Creative Commons Attribution 4.0 License.
Rupture-dependent breakdown energy in fault models with
thermo-hydro-mechanical processes
Valère Lambert
1
and Nadia Lapusta
1,2
1
Seismological Laboratory, California Institute of Technology, Pasadena, CA 91125, USA
2
Department of Mechanical and Civil Engineering, California Institute of Technology, Pasadena, CA 91125, USA
Correspondence:
Valère Lambert (vlambert@caltech.edu)
Received: 1 July 2020 – Discussion started: 21 July 2020
Revised: 5 October 2020 – Accepted: 11 October 2020 – Published: 26 November 2020
Abstract.
Substantial insight into earthquake source pro-
cesses has resulted from considering frictional ruptures anal-
ogous to cohesive-zone shear cracks from fracture mechan-
ics. This analogy holds for slip-weakening representations of
fault friction that encapsulate the resistance to rupture prop-
agation in the form of breakdown energy, analogous to frac-
ture energy, prescribed in advance as if it were a material
property of the fault interface. Here, we use numerical mod-
els of earthquake sequences with enhanced weakening due to
thermal pressurization of pore fluids to show how account-
ing for thermo-hydro-mechanical processes during dynamic
shear ruptures makes breakdown energy rupture-dependent.
We find that local breakdown energy is neither a constant
material property nor uniquely defined by the amount of
slip attained during rupture, but depends on how that slip is
achieved through the history of slip rate and dynamic stress
changes during the rupture process. As a consequence, the
frictional breakdown energy of the same location along the
fault can vary significantly in different earthquake ruptures
that pass through. These results suggest the need to reex-
amine the assumption of predetermined frictional breakdown
energy common in dynamic rupture modeling and to better
understand the factors that control rupture dynamics in the
presence of thermo-hydro-mechanical processes.
1 Introduction
Fault constitutive relations that describe the evolution of
shear resistance with fault motion are critical ingredients of
earthquake source modeling. When coupled with the elasto-
dynamic equations of motion, these relations provide insight
into the growth and ultimate arrest of ruptures. Earthquake
source processes are often considered in the framework of
dynamic fracture mechanics, where the earthquake rupture
may be considered as a dynamically propagating shear crack
or pulse (Ida, 1972; Palmer and Rice, 1973; Madariaga,
1976; Rice, 1980; Kostrov and Das, 1988; Heaton, 1990;
Freund, 1990; Kanamori and Heaton, 2000; Rice, 2000;
Kanamori and Brodsky, 2004; Rubin and Ampuero, 2005).
By analogy to cohesive-zone relations for mode I open-
ing cracks, slip-weakening laws have been commonly used
to describe the dynamic decrease in shear resistance during
sliding (Ida, 1972; Palmer and Rice, 1973; Madariaga, 1976;
Kostrov and Das, 1988; Kanamori and Brodsky, 2004; Bou-
chon, 1997; Ide and Takeo, 1997; Olsen et al., 1997; Bou-
chon et al., 1998; Cruz-Atienza et al., 2009; Kaneko et al.,
2017; Gallovic et al., 2019). Linear slip weakening is one of
the simplest and most commonly used versions, in which the
shear resistance decreases linearly with slip from a peak of
τ
peak
to a constant dynamic level
τ
dyn
achieved at a critical
slip distance
D
c
(Fig. 1).
The breakdown energy
G
is associated with the evolution
of shear resistance from the initial shear stress
τ
ini
to the peak
shear resistance
τ
peak
and then breakdown to the minimum
dynamic shear resistance
τ
min
. It is a part of the overall en-
ergy partitioning for dynamic ruptures, with the total strain
energy change throughout the ruptured region (
1W
) being
separated into the radiated energy
E
R
, the breakdown en-
ergy
G
, and other residual dissipated energy (Kanamori and
Rivera, 2006). The breakdown energy is analogous to frac-
ture energy from cohesive-zone models of fracture mechan-
ics (Palmer and Rice, 1973; Rice, 1980; Freund, 1990; Tinti
et al., 2005); hence, it is thought to be relevant to rupture dy-
Published by Copernicus Publications on behalf of the European Geosciences Union.
2284
V. Lambert and N. Lapusta: Breakdown energy as a process quantity
Figure 1. (a)
Standard linear slip-weakening diagram where the average shear stress is assumed to increase from an initial to peak stress with
no slip and then linearly decrease to a dynamic resistance level over a critical slip distance
D
c
. The difference between the average initial
and final shear stress levels is called the static stress drop. The average stress vs. slip diagram is used to represent the energy partitioning of
the total strain energy change per unit rupture area (dashed red trapezoid) into the breakdown energy (dark gray triangle), residual dissipated
energy per unit area (light gray rectangle), and radiated energy per unit area (blue region). The additional dissipation associated with the
initial strengthening outside of the red trapezoid comes at the expense of the radiated energy (white triangle inside the dashed red trapezoid).
(b)
The case of the initial stress equal to the peak stress. Note that this diagram is an approximation even if the local behavior is governed by
linear slip-weakening friction, as different points of the rupture would have different slip, including near-zero slip close to the rupture edges,
and averaging over the dynamic rupture would produce a different curve from the local behavior (Noda and Lapusta, 2012).
namics, e.g., rupture speed. For linear slip-weakening fric-
tion, it is given by
G
=
(τ
peak
−
τ
dyn
)D
c
/
2. The term “frac-
ture energy”, while initially associated with the creation of
free surfaces during tensile fracture, has been routinely used
to refer broadly to inelastic dissipation relevant to the crack-
tip motion for both tensile and shear cracks, including con-
tributions from off-fault damage creation, plastic work, and
frictional heat (e.g., Rice, 1980; Freund, 1990; Rice, 2006).
However, here we follow the work of Tinti et al. (2005) in re-
ferring to this quantity as the “breakdown” work (or energy)
to further emphasize that
G
can incorporate various physical
sources of energy dissipation.
More involved fault constitutive laws are generally re-
quired to explain a number of aspects of faulting behavior,
most notably the restrengthening of faults between earth-
quakes. Laboratory experiments have provided significant in-
sight into the rich behavior of shear resistance, with the fric-
tional response at slip rates between 10
−
9
and 10
−
3
m/s be-
ing well described by rate-and-state friction laws (Dieterich,
2007). A number of previous studies have used models on
rate-and-state faults to provide insight into a number of
earthquake and slow slip observations, such as sequences of
earthquakes on an actual fault segment and repeating earth-
quakes (Chen and Lapusta, 2009; Barbot et al., 2012; Di-
eterich, 2007, and references therein). While incorporating
a more involved dependence of shear resistance on long-
term healing, standard Dieterich–Ruina rate-and-state fric-
tion has been shown to resemble linear slip weakening during
dynamic rupture (Okubo, 1989; Cocco and Bizzarri, 2002;
Lapusta and Liu, 2009), providing further reinforcement of
the notion that the breakdown of shear resistance during dy-
namic rupture may be adequately described by linear slip-
weakening behavior.
Many studies have attempted to infer parameters of the
slip-weakening shear resistance from the strong-motion data
resulting from natural earthquakes (Bouchon, 1997; Ide and
Takeo, 1997; Olsen et al., 1997; Bouchon et al., 1998; Cruz-
Atienza et al., 2009; Kaneko et al., 2017; Gallovic et al.,
2019). Such studies have noted substantial trade-offs in the
inferred parameters during such inversions, such as between
the slip-weakening distance
D
c
and strength excess
τ
peak
−
τ
ini
, where
τ
ini
is the initial stress (Fig. 1). It has been pre-
sumed that the spatial distribution of the static stress drop
and breakdown energy may be the most reliably determined
features, as the stress drop can be inferred from the spatial
distribution of slip, and the remaining variations in rupture
speed are largely controlled by the breakdown energy in such
linear slip-weakening representations (Guatteri and Spudich,
2000).
One of the most notable features of seismologically in-
ferred breakdown energies from natural earthquakes is that
the average breakdown energy from the rupture process has
been inferred to increase with the earthquake size (Aber-
crombie and Rice, 2005; Rice, 2006; Cocco and Tinti, 2008;
Viesca and Garagash, 2015; Brantut and Viesca, 2017). In-
crease in breakdown energy with slip has also been observed
in high-speed friction experiments (Nielsen et al., 2016; Sel-
vadurai, 2019), although in some experiments the increase
Solid Earth, 11, 2283–2302, 2020
https://doi.org/10.5194/se-11-2283-2020
V. Lambert and N. Lapusta: Breakdown energy as a process quantity
2285
saturates after a given amount of weakening (Nielsen et al.,
2016). Such findings are inconsistent with the breakdown en-
ergy being a fixed fault property as often assumed in linear
slip-weakening laws and as approximately follows from stan-
dard rate-and-state friction with uniform characteristic slip-
weakening distance (Perry et al., 2020), unless strong and
very special heterogeneity is assumed in fault properties. For
example, some modeling studies have assigned strongly het-
erogeneous
D
c
and hence
G
values to the fault, as if they are
properties of the interface, with larger patches having sig-
nificantly larger values of
D
c
and hence
G
, and these studies
considered sequences of events over such interfaces (e.g., Ide
and Aochi, 2005; Aochi and Ide, 2011).
Several theoretical and numerical studies have demon-
strated that enhanced dynamic weakening, as widely ob-
served at relatively high slip rates (
>
10
−
3
m/s) in laboratory
experiments (Tullis, 2007; Di Toro et al., 2011), may explain
the inferred increase in breakdown energy with slip (Rice,
2006; Viesca and Garagash, 2015; Brantut and Viesca, 2017;
Perry et al., 2020). A number of different mechanisms have
been proposed for such enhanced weakening, with many of
them due to shear heating. For example, thermal pressuriza-
tion may occur due to the rapid shear heating of pore fluids
during slip (Sibson, 1973; Andrews, 2002; Rice, 2006); if
pore fluids are heated fast enough and not allowed to dif-
fuse away, they pressurize and reduce the effective normal
stress on the fault. Flash heating is another thermally in-
duced weakening mechanism, where the effective friction
coefficient is rapidly reduced due to local melting of highly
stressed micro-contacts along the fault (Rice, 1999; Goldsby
and Tullis, 2011; Passelegue et al., 2014). Considerations of
heat production during dynamic shear ruptures provide a sub-
stantial constraint for potential fault models, as field studies
show no correlation between faulting and heat flow signa-
tures and rarely suggest the presence of melt (Sibson, 1975;
Lachenbruch and Sass, 1980). Models with enhanced weak-
ening have been successful in producing fault operation at
low overall prestress and low heat production (Rice, 2006;
Noda et al., 2009; Lambert et al., 2020), as supported by sev-
eral observations (Brune et al., 1969; Zoback et al., 1987;
Hickman and Zoback, 2004; Williams et al., 2004).
Numerical models have shown that the incorporation of
thermally activated enhanced weakening mechanisms during
dynamic rupture can have profound effects on the evolution
of individual ruptures, as well as the long-term behavior of
fault segments, with the potential to make seemingly stable
creeping regions fail violently during earthquakes (Noda and
Lapusta, 2013), and for the potential deeper penetration of
large ruptures, which may explain the seismic quiescence
of mature faults that have historically hosted large earth-
quakes (Jiang and Lapusta, 2016). Despite evolving dynamic
resistance in such models, they can also be consistent with
magnitude-invariant static stress drops (Perry et al., 2020).
At the same time, accounting for thermo-hydro-
mechanical processes during dynamic rupture can clearly
weaken or even remove the analogy between frictional shear
ruptures and idealized shear cracks of fracture mechanics.
The analogy is based on two key assumptions: (1) that
the breakdown of shear resistance is concentrated in a
small region near the rupture front, referred to as small-
scale yielding, and (2) that a constant residual stress level
τ
dyn
=
τ
min
exists throughout the ruptured region during
sliding (Palmer and Rice, 1973; Freund, 1990). For example,
the relationship between rupture speed and fracture energy
of linear elastic fracture mechanics is only valid under these
assumptions. Clearly, these assumptions can become invalid
when thermo-hydro-mechanical processes are considered.
For example, shear heating can raise the pore fluid pressure
in regions away from the rupture front and weaken the fault
there, contributing to the breakdown of fault resistance away
from the rupture tip and varying the dynamic resistance
level. Furthermore, the shear heating itself would depend on
the overall dissipated energy, making the fault weakening
behavior, and hence the “breakdown”, depend on the abso-
lute stress levels, and not just the stress changes, as typically
considered by analogy with traditional fracture mechanics.
Moreover, studies that infer dynamic parameters from
natural earthquakes using dynamically inspired kinematic
models suggest more complicated evolutions of shear stress
with slip, including heterogeneous dynamic resistance levels
(Ide and Takeo, 1997; Bouchon et al., 1998; Tinti et al.,
2005; Causse et al., 2013).
In this study, we use numerical models of earthquake
sequences with enhanced weakening due to thermal pres-
surization to illustrate how the inclusion of thermo-hydro-
mechanical processes during dynamic shear ruptures makes
breakdown energy rupture-dependent, in that the values of
both local and average breakdown energy vary among rup-
tures on the same fault, even with spatially uniform and time-
independent constitutive properties. As such, the breakdown
energy is not an intrinsic fault property, but develops differ-
ent values at a given location, depending on the details of the
rupture process, which in part depend on the prestress before
the dynamic rupture achieved as a consequence of prior fault
slip history. Moreover, the local breakdown energy is not
uniquely defined by the amount of slip attained during rup-
ture, but depends on how that slip was achieved through the
complicated history of slip rate and dynamic stress changes
throughout the rupture process. Additional fault characteris-
tics that we do not consider here, such as heterogeneity in
fault properties and dynamically induced, evolving, inelas-
tic off-fault damage (Dunham et al., 2011a, b; Roten et al.,
2017; Withers et al., 2018), should result in qualitatively sim-
ilar effects and add even more variability to the breakdown
energy.
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V. Lambert and N. Lapusta: Breakdown energy as a process quantity
2 Description of numerical models
We conduct numerical simulations of spontaneous sequences
of earthquakes and aseismic slip (SEAS) utilizing the spec-
tral boundary integral method (BIE) to solve the elastody-
namic equations of motion coupled with friction boundary
conditions, including the evolution of pore fluid pressure and
temperature on the fault coupled with off-fault diffusion (La-
pusta et al., 2000; Noda and Lapusta, 2010). Our simulations
consider mode III slip on a 1-D fault embedded into a 2-
D uniform, isotropic, elastic medium slowly loaded with a
long-term slip rate
V
pl
(Fig. 2). The simulations resolve the
full slip behavior throughout earthquake sequences, includ-
ing the nucleation process, the propagation of individual dy-
namic ruptures, as well as periods of post-seismic and the
interseismic slip between events that can last from months to
hundreds of years.
Our fault models adopt the laboratory-derived Dieterich–
Ruina rate-and-state friction law with the state evolution gov-
erned by the aging law (Dieterich, 1979; Ruina, 1983):
τ
=
σf(V,θ)
=
(σ
−
p)
[
f
∗
+
a
log
V
V
∗
+
b
log
θV
∗
D
RS
]
,
(1)
̇
θ
=
1
−
Vθ
D
RS
,
(2)
where
σ
is the effective normal stress,
σ
is the normal stress,
p
is the pore fluid pressure,
f
∗
is the reference steady-state
friction coefficient at reference sliding rate
V
∗
,
D
RS
is the
characteristic slip distance, and
a
and
b
are the direct effect
and evolution effect parameters, respectively. Other formu-
lations for the evolution of the state variable exist, such as
the slip law (Ruina, 1983) as well as various composite laws,
and the formulation that best describes various laboratory ex-
periments remains a topic of ongoing research (Bhattacharya
et al., 2015, 2017; Shreedharan et al., 2019). However, the
choice of the state evolution law should not substantially in-
fluence the results of this study, as the evolution of shear
resistance during dynamic rupture within our simulations is
dominated by the presence of enhanced weakening mecha-
nisms. We use the version of the expressions (1) and (2) reg-
ularized for zero and negative slip rates (Noda and Lapusta,
2010).
During conditions of steady-state sliding (
̇
θ
=
0), the fric-
tion coefficient is expressed as
f
ss
(V)
=
f
∗
+
(a
−
b)
log
V
V
∗
.
(3)
The combination of frictional properties
(a
−
b) >
0 results
in steady-state velocity-strengthening (VS) behavior, where
stable slip is expected, and properties resulting in
(a
−
b) <
0 lead to steady-state velocity-weakening (VW) behavior,
where accelerating slip and, hence, stick slip occur for suffi-
ciently large regions (Rice and Ruina, 1983; Rice et al., 2001;
Rubin and Ampuero, 2005).
An important, yet often underappreciated, implication of
the rate- and state-dependent effects observed in labora-
tory experiments is that notions of static and dynamic fric-
tion coefficients, as well as the slip-weakening distance, are
not well-defined and fixed quantities, as would be consid-
ered by standard linear slip-weakening laws (Cocco and Biz-
zarri, 2002; Rubin and Ampuero, 2005; Ampuero and Ru-
bin, 2008; Lapusta and Liu, 2009; Barras et al., 2019; Perry
et al., 2020). Instead, they depend on the history and current
style of motion. For example, the dynamic friction, compa-
rable to the steady-state friction at dynamic slip rates, de-
pends on the slip rate (Eq. 3), which can vary substantially
throughout rupture and between different ruptures. More-
over, the peak friction and effective slip-weakening distance
under standard rate-and-state friction depend on the history
of motion through the state variable
θ
as well as the sliding
rate during fast slip (Fig. 3). Let us consider a point with the
same initial friction but different periods of inter-event heal-
ing, captured by increasingly larger values of the pre-rupture
state variable. If the point is now driven to slide at a fixed
sliding rate, the peak friction and slip-weakening distance
would be larger for points that (i) have a higher pre-rupture
value of the state variable, representing better healed inter-
faces, and/or (ii) sliding at faster slip rates (Fig. 3). For stan-
dard rate-and-state friction, these effects typically translate
into generally mild variations in dynamic and static stress
drop and breakdown energy, due to the logarithmic depen-
dence of the shear stress evolution on slip rate, resulting in
both the static stress drop and breakdown energy being effec-
tively rupture-independent (Cocco and Bizzarri, 2002; Rubin
and Ampuero, 2005; Ampuero and Rubin, 2008; Lapusta and
Liu, 2009; Perry et al., 2020), at least compared to the large
variations in breakdown energy with slip inferred from natu-
ral earthquakes as discussed in Sect. 1. However, such vari-
ations in stress evolution become more substantial with en-
hanced dynamic weakening mechanisms that lead to stronger
rate-dependent weakening.
Laboratory experiments indicate that the standard rate-
and-state laws (Eqs. 1–2) provide good descriptions of
frictional behavior at relatively slow slip rates (10
−
9
to
10
−
3
m/s). However, at higher sliding rates, including aver-
age seismic slip rates of
∼
1 m/s, additional enhanced weak-
ening mechanisms can occur, such as the thermal pressur-
ization of pore fluids. Thermal pressurization is governed in
our simulations by the following coupled differential equa-
tions for the evolution of temperature and pore fluid pressure
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V. Lambert and N. Lapusta: Breakdown energy as a process quantity
2287
Figure 2. (a)
The fault model incorporates a velocity-weakening (VW) seismogenic region surrounded by two velocity-strengthening (VS)
sections. A fixed plate rate is prescribed outside of these regions.
(b)
We incorporate enhanced dynamic weakening due to the thermal
pressurization of pore fluids by calculating the evolution of temperature and pore fluid pressure due to shear heating and off-fault diffusion
throughout our simulations.
(c)
The beginning of the accumulated slip history for simulated sequences of crack-like earthquake ruptures and
aseismic slip. Seismic events are illustrated by red lines with slip contours plotted every 0.5 s and interseismic slip plotted in black every
10 years. The total simulated slip history spans 2675 years corresponding to cumulative slip of 84 m and contains 200 seismic events.
(Noda and Lapusta, 2010):
∂T(y,z,t)
∂t
=
α
th
∂
2
T(y,z,t)
∂y
2
+
τ(z
;
t)V(z,t)
ρc
exp
(
−
y
2
/
2
w
2
)
√
2
πw
,
(4)
∂p(y,z,t)
∂t
=
α
hy
∂
2
p(y,z,t)
∂y
2
+
3
∂T(y,z
;
t)
∂t
,
(5)
where
T
is the pore fluid temperature,
α
th
is the thermal dif-
fusivity,
τV
is the shear heating source which is distributed
over a Gaussian shear layer of half-width
w
,
ρc
is the specific
heat,
y
is the fault-normal distance,
α
hy
is the hydraulic dif-
fusivity, and
3
is the coupling coefficient that provides the
change in pore pressure per unit temperature change under
undrained conditions.
The total fault domain of size
λ
is partitioned into a fric-
tional region of size
λ
fr
where we solve for the balance of
shear stress and frictional resistance, as well as loading re-
gions at the edges where the fault is prescribed to slip at a
tectonic plate rate (Fig. 2a). The frictional interface is com-
posed of a 24 km region with VW frictional properties of size
λ
VW
, surrounded by a VS domain. The majority of the seis-
mic events arrest within the VW region, which we refer to as
“partial ruptures”; however, some events span the entire VW
region, which we refer to as “complete ruptures” (Fig. 2c).
Weakening due to thermal pressurization is confined to the
region with the VW properties. The parameter values used
for the simulations presented in this work are motivated by
prior studies (Rice, 2006; Noda and Lapusta, 2010; Perry
et al., 2020) and are provided in Table 1.
3 Energy partitioning and the notion of breakdown
energy
G
In the earthquake energy budget, the total strain energy
change per unit source area
1W/A
is partitioned into the
dissipated energy per unit area,
E
Diss
/A
, and the radiated en-
ergy per unit area,
E
R
/A
:
1W/A
=
E
Diss
/A
+
E
R
/A.
(6)
The total strain energy released per unit area
1W/A
is given
by
1W/A
=
1
2
(
τ
ini
+
τ
fin
)
δ,
(7)
where
δ
is the average final slip for the event, and
τ
ini
and
τ
fin
are the average initial and final shear stress weighted by
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V. Lambert and N. Lapusta: Breakdown energy as a process quantity
Figure 3.
Illustration of the rate- and state-dependence of the peak and dynamic friction coefficients,
f
peak
and
f
dyn
, respectively, as well
as the effective slip-weakening distance
D
c
.
(a–c)
Evolution of the friction coefficient with slip for points with the same initial friction
coefficient of 0.58 but different values of the initial state variable
θ
ini
, corresponding to different histories of previous motion. The initially
locked point slips at an imposed slip rate of
V
=
1 cm/s (black) or
V
=
1 m/s (red), to approximately reproduce transition from the locked
state to dynamic sliding as the rupture propagates through. For a given slip rate, the friction evolves to a new steady-state level,
f
dyn
=
0
.
54
and
f
dyn
=
0
.
56 for
V
=
1 m/s and
V
=
1 cm/s, respectively. These levels are similar, as expected from the logarithmic dependence on the
slip rate and a narrow range of dynamic slip rates. The peak friction coefficient and effective slip-weakening distance vary more significantly
with
θ
ini
, where the peak friction coefficient increases for higher
θ
ini
associated with longer inter-event healing times. The example uses
typical laboratory values of
(a
−
b)
=
0
.
004,
f
∗
=
0
.
6,
D
RS
=
1 μm, and
V
∗
=
10
−
6
m/s.
Table 1.
Model parameters used in simulations of earthquakes and aseismic slip.
Parameter
Symbol
Value
Loading slip rate
V
pl
10
−
9
m/s
Shear wave speed
c
s
3299 m/s
Shear modulus
μ
36 GPa
Rate-and-state parameters
Reference slip velocity
V
∗
10
−
6
m/s
Reference friction coefficient
f
∗
0.6
Characteristic slip
D
RS
1 mm
Rate-and-state direct effect (VW)
a
0.010
Rate-and-state evolution effect (VW)
b
0.015
Rate-and-state direct effect (VS)
a
0.050
Rate-and-state evolution effect (VS)
b
0.003
Thermal pressurization parameters
Interseismic effective normal stress
σ
=
(σ
−
p)
25 MPa
Coupling coefficient (when TP present)
3
0.34 MPa/K
Thermal diffusivity
α
th
10
−
6
m
2
/s
Hydraulic diffusivity
α
hy
10
−
3
m
2
/s
Specific heat
ρc
2.7 MPa/K
Shear zone half-width
w
10 mm
Length scales
Fault length
λ
96 km
Frictional domain
λ
fr
72 km
Velocity-weakening region
λ
VW
24 km
Cell size
1z
3.3 m
Quasi-static cohesive zone
3
0
75 m
Nucleation size (Rice and Ruina, 1983)
h
∗
RR
200 m
Nucleation size (Rubin and Ampuero, 2005)
h
∗
RA
490 m
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V. Lambert and N. Lapusta: Breakdown energy as a process quantity
2289
the final slip (Noda and Lapusta, 2012), respectively,
τ
ini
=
∫
τ
ini
(z)δ
fin
(z)
d
z
∫
δ
fin
(z)
d
z
,
(8)
τ
fin
=
∫
τ
fin
(z)δ
fin
(z)
d
z
∫
δ
fin
(z)
d
z
.
(9)
Here,
represents the ruptured domain. The static stress
drop is a measure of the difference in average stress before
and after the rupture. The relevant definition of the average
static stress drop for energy considerations is the energy-
based or slip-weighted stress drop (Noda et al., 2013):
1τ
=
τ
ini
−
τ
fin
=
∫
[
τ
ini
(z)
−
τ
fin
(z)
]
δ
fin
(z)
d
z
∫
δ
fin
(z)
d
z
.
(10)
The dissipated energy per unit rupture area can be computed
from the evolution of shear resistance with slip:
E
Diss
/A
=
∫
[
∫
δ
fin
(z)
0
τ(δ
′
)
d
δ
′
]
d
z
∫
d
z
.
(11)
The dissipated energy
E
Diss
/A
is often further partitioned
into the average breakdown energy
G
(Palmer and Rice,
1973; Rice, 1980; Tinti et al., 2005) and the residual dissi-
pated energy (dark gray triangle and light gray rectangle in
Fig. 1, respectively). The average breakdown energy repre-
sents the spatial average of the local breakdown energy
G
loc
within the source region,
G
=
∫
G
loc
(z)
d
z
∫
d
z
,
(12)
where the local breakdown energy is defined as
G
loc
(z)
=
D
c
(z)
∫
0
[
τ(δ
′
)
−
τ
min
(z)
]
d
δ
′
,
(13)
and
τ
min
(z)
is the minimum local shear resistance during
seismic slip after the initial strengthening from the initial to
peak shear resistance via the direct effect.
D
c
is defined as the
critical slip distance during the rupture such that
τ(D
c
(z))
=
τ
min
(z)
.
Seismological studies have attempted to estimate the av-
erage breakdown energy for natural earthquakes based on
the standard energy partitioning diagram (Fig. 1) as follows
(Abercrombie and Rice, 2005; Rice, 2006):
G
′
=
δ
2
(
1τ
−
2
μE
R
M
0
)
,
(14)
where
G
′
is the approximation for the average breakdown
energy
G
,
δ
is the average slip during the rupture,
1τ
is
the seismologically inferred average static stress drop,
μ
is
the shear modulus,
E
R
is the radiated energy, and
M
0
is
the seismic moment. The definition of
G
′
assumes that the
rupture area exhibits negligible stress overshoot/undershoot
or that the average level of dynamic resistance during slid-
ing is the same as the final average shear stress. Numeri-
cal studies have shown that
G
′
may indeed provide a rea-
sonable estimate of the average breakdown energy (within a
factor of 2) for crack-like ruptures, which exhibit mild over-
shoot/undershoot compared with the static stress drop (Perry
et al., 2020); however, such estimates can dramatically differ
from the true values for ruptures that experience a consider-
able stress undershoot, as is the case of self-healing pulse-
like ruptures (Lambert et al., 2020).
Note that the energy balance shown in Eq. (6) reflects
the energy partitioning over the rupture process as a whole.
While the dissipated energy is a local quantity along the
fault, the radiated energy is not and can only be related to
the stress-slip behavior in the averaged sense over the en-
tire rupture process (Fig. 1). Seismological estimates of the
average breakdown energy can be made assuming the stan-
dard energy partitioning following the slip-weakening dia-
gram (Fig. 1) and using Eq. (14) with the total radiated en-
ergy, with the results dependent on the accuracy of the ra-
diated energy estimates and validity of the assumed energy
partitioning model, which has been shown to breakdown for
pulse-like ruptures (Lambert et al., 2020). Estimating the lo-
cal breakdown energy is more challenging. One approach is
to use finite-fault slip inversions to determine the stress evo-
lution during rupture and, hence, the breakdown work (e.g.,
Tinti et al., 2005), with the results dependent on the accuracy
of finite-fault inversions that are known to be nonunique and
affected by smoothing.
4 Breakdown energy in models with thermal
pressurization of pore fluids
The local slip and stress evolution are determined at every
point along the fault within our simulations at all times; thus,
we can calculate the local dissipation and breakdown en-
ergy throughout each rupture as well as study the evolution
of these quantities in different ruptures throughout the se-
quence. We can also compute the average energy quantities
and construct the average stress vs. slip curves for the total
rupture process in a manner that preserves the overall energy
partitioning (Noda and Lapusta, 2012). We define seismic
slip to occur when the local slip velocity exceeds a velocity
threshold
V
thresh
=
0
.
01 m/s. As slip rates during sliding are
typically around 1 m/s or higher and drop off rapidly during
the arrest of slip, modest changes of this velocity threshold of
an order of magnitude produce very mild differences in
D
c
and
G
(of less than 1 %).
The average breakdown energy
G
computed from our
simulations increases with average slip and matches esti-
mates of breakdown energy for natural events (Fig. 4), as ex-
pected from the simplified theoretical considerations (Rice,
https://doi.org/10.5194/se-11-2283-2020
Solid Earth, 11, 2283–2302, 2020
2290
V. Lambert and N. Lapusta: Breakdown energy as a process quantity
Figure 4. (a)
The simulations result in a sequence of mostly crack-like ruptures that, despite including dynamic weakening due to thermal
pressurization of pore fluids, are capable of reproducing nearly magnitude-invariant average static stress drops, with values between 1 and
10 MPa.
(b)
These crack-like ruptures display the overall increasing trend in the average breakdown energy with average slip, as inferred
for natural earthquakes (Abercrombie and Rice, 2005; Rice, 2006).
(c)
The simulated fault maintains reasonable temperatures and avoids
melting, due to a relatively low interseismic effective normal stress of 25 MPa (and, hence, chronic fluid overpressurization) and sufficiently
efficient enhanced weakening due to thermal pressurization of pore fluids.
2006). As demonstrated in previous numerical studies (Perry
et al., 2020), when our fault models combine moderately ef-
ficient thermal pressurization with persistently weak condi-
tions, such as from relatively low interseismic effective nor-
mal stresses (25 MPa) due to substantial chronic fluid over-
pressurization, the models produce mostly crack-like rup-
tures that reproduce all main observations about earthquakes,
including magnitude-invariant average static stress drops of
1–10 MPa, breakdown energy values that are quantitatively
comparable to estimates from natural earthquakes, and fault
temperatures well below representative equilibrium melting
temperatures near 1000
◦
C for wet granitic compositions in
the shallow crust (Rice, 2006). It is important to note that the
presence of enhanced dynamic weakening is critical for pro-
ducing reasonable values of static stress drop (
>
1 MPa) in
such fault models with chronic fluid overpressurization; oth-
erwise, the stress changes due to the standard rate-and-state
friction would be too low (as they are proportional to the
effective normal stress). As such, dynamic weakening due
to thermal pressurization still dominates the overall weak-
ening behavior during dynamic rupture. These results sug-
gest that fault models incorporating chronic fault weakness
and enhanced weakening may be plausible representations
of rupture behavior on mature faults. The work of Perry et al.
Solid Earth, 11, 2283–2302, 2020
https://doi.org/10.5194/se-11-2283-2020