Download lecture 5.5 (Alon Ziv)

(Introduction to) Earthquake Energy Balance
• Mechanical energy, surface energy and the Griffith criteria
• Seismic energy and seismic efficiency
• The heat flow paradox
• Apparent stress drop
Earthquake energy balance: related questions
• Are faults weaker or stronger than the surrounding crust?
• Do earthquakes
release most, or just a
small fraction of the
strain energy that is
stored in the crust?
stress
stress
stress
drop
recurrence
time
time
time
Earthquake energy balance: A crack within an elastic medium
We first consider the energy balance of a crack embedded within
an elastic medium.
Why cracks?
• Because “there’s a crack in
everything” (Anthem, Leonard
Cohen).
• At any given moment the rupture
may be envisioned as a shear
crack with well defined crack tips,
beyond which the slip velocity is
equal to zero.
Earthquake energy balance: Griffith criteria
The static frictionless case:
U  U M  U S  (U Mpotential  U Minternal)  U S
• UM is the mechanical energy.
• UMpotential is the potential energy of

the external
load applied on the
system boundary.
• UMinternal is the internal elastic strain
energy stored in the medium
• US is the surface energy.
dU dc  0
crack extends if:
crack at equilibrium if dU dc  0
dU dc  0
crack heals if:
Earthquake energy balance: dynamic shear crack
Dynamic shear crack with non-zero friction:
U  U Minternal  U Mpotential  U S  U K  U F
Here, in addition to UMinternal, UMpotential and US:
• UK is the kinetic energy.
• UF isthe work done against friction.
During an earthquake, the partition of energy (after less before) is
as follows:
E S  U K  U Mpotential  U Minternal  U S  U F ,
where ES is the radiated seismic energy.

Earthquake energy balance: dynamic shear crack
Since earthquake duration is so small compared to the inter
seismic interval, the motion of the plate boundaries far from the
fault is negligible, and UMpotential=0. Thus, the expression for the
radiated energy simplifies to:
E S  U Minternal  U S  U F .
Question: what are the signs of UMinternal, US and UF?

U Minternal  0
U S  0
U F > 0
Let us now write expressions for UMinternal, US and UF .

Earthquake energy balance: elastic strain energy
To get a physical sense of what UMinternal is, it is useful to
consider the spring-slider analog.
The reduction in the elastic strain
energy stored in the spring during
a slip episode is just the area
under the force versus slip curve.
For the spring-slider system, UMinternal is equal to:
U
internal
M
F1  F2

u.
2
Earthquake energy balance: elastic strain energy
Similarly, for a crack embedded within an elastic medium,
UMinternal is equal to:
U
internal
M

1   2
2
uA,
where 1 and 2 are initial and final stresses, respectively, and the
minus sign indicates a decrease in elastic strain energy.

Earthquake energy balance: frictional dissipation and surface
energy
The frictional dissipation:
UF (t)    F (t)Vdt,
where F is the friction, V is sliding speed, and t is time. Frictional
work is converted mainly to heat.

The surface energy:
US  2A,
where  is the energy per unit area required to break the atomic
bonds, and A is the rapture dimensions. Experimental studies
show that  is very
small, and thus surface energy is very small
compared to the radiated energy (but not everyone agrees with
this argument).
Earthquake energy balance: the simplest model
Consider the simplest model, in
which the friction drops
instantaneously from 1 to 2.
In such case: F=2, and we
get:
 2
ES  1
uA.
2

Earthquake energy balance: seismic efficiency
We define seismic efficiency, , as the ratio between the seismic
energy and the negative of the elastic strain energy change, often
referred to as the faulting energy.
ES

,
internal
UM
which leads to:

1   2



,
1   2 1   2
with  being the static stress drop. While the stress drop may be
determined from seismic data, absolute stresses may not.

Earthquake energy balance: seismic efficiency
u
  CG ,
L˜
where G is the shear modulus, C is a geometrical constant, and
(the tilde) L is the rupture characteristic length.
The static stress drop is equal to:

The characteristic rupture
length scale is different for
small and large earthquakes.
For small earthquakes, L˜  r and C  7 16. Combining this with the
expression for seismic moment we get:
16 3
M  r  .

7

Both M and r may be inferred from seismic data.
Earthquake energy balance: seismic efficiency
Stress drops vary between 0.1
and 10 MPa over a range of
seismic moments between 1018
and 1027 dyn cm.
Figure from: Schlische et al., 1996
Earthquake energy balance: seismic efficiency
constraints on absolute stresses: In a hydrostatic state of stress,
the friction stress increases with depth according to:
 F (z)  (c  w )gz,
where  is the coefficient of friction, g is the acceleration of gravity,
and c and w are the densities of crustal rocks and water,
respectively.

Laboratory experiments show:
  0.6.

Byerlee, 1978
Earthquake energy balance: seismic efficiency
Using:
, the coefficient of friction = 0.6
c, rock density = 2600 Kg m-3
w, water density = 1000 Kg m-3
g, the acceleration of gravity = 9.8 m s-2
D, the depth of the seismogenic zone, say 12x103 m
We get an average friction of:
(c  w )gD
2
 56MPa,
and the inferred seismic efficiency is:



 0.1.
1   2
Earthquake energy balance: seismic efficiency
So, the radiated energy makes only a small fraction of the energy
that is available for faulting.
Based on this conclusion a strong heat-flow anomaly is expected
at the surface right above seismic faults.
Earthquake energy balance: the heat flow paradox
At least in the case of the San-Andreas fault in California, the
expected heat anomaly is not observed.
A section perpendicular
to the SAF plane:
Figure from: Scholz, 1990
The disagreement between the expected and observed heat-flow
profiles is often referred to as the HEAT FLOW PARADOX.
Earthquake energy balance: the heat flow paradox
A section parallel to the SAF plane:
Figure from: Scholz, 1990
Earthquake energy balance: the heat flow paradox
• A possible way out of the heat-flow paradox is to question the
validity of Byerlee law for geologic faults.
• What can get wrong with Byerlee law?
• It turned out that at
any given moment
during rupture, the slip
occurs over a small
portion of the total
rupture area.
The 1992 Landers earthquake:
Wald and Heaton, 1994
Earthquake energy balance: the heat flow paradox
• To many seismologists,
this “pulse-like” slip reminds
a moving carpet wrinkle.
• The reason the carpet
wrinkle can slide under very
low shear stress is because
the normal stress is locally
(i.e., under the wrinkle)
much smaller than the
average normal stress.
Earthquake energy balance
The assumptions underlying the ''simple model'' are:
• Instantaneous drop from static to kinetic friction, and constant
friction during slip.
• Uniform distribution of slip and stresses.
• Zero overshoot.
• Constant sliding velocity.
• No off fault deformation.
The first point means that continuity is violated...
Earthquake energy balance
Other conceptual models:
Static-kinetic friction
slip weakening
quasi-static
• The (simple) static-kinetic model.
• The slip-weakening model. Significant amount of energy is
dissipated in the process of fracturing the contact surface. In the
literature this energy is interchangeably referred to as the breakdown energy, fracture energy or surface energy.
• A silent (or slow) earthquake - no energy is radiated.
Earthquake energy balance
In reality, things are probably more complex than that.
We now know that the distribution of slip and stresses is highly
heterogeneous, and that the source time function is quite
complex.
Earthquake energy balance: radiated energy versus seismic
moment and the apparent stress drop
Radiated energy and seismic moment of a large number of
earthquakes have been independently estimated. It is interesting
to examine the radiated energy and seismic moment ratio.
Remarkably, the
ratio of radiated
energy to seismic
moment is fairly
constant over a wide
range of earthquake
magnitudes.
M=4.3
Figure from: Kanamori, Annu. Rev. Earth Planet. Sci., 1994
M=7.3
Earthquake energy balance: radiated energy versus seismic
moment and the apparent stress drop
What is the physical interpretation of the ratio ES to M0? Recall
that the seismic moment is:
M0  GuA,
and the radiated energy for constant friction (i.e., F = 2):

ES 
1   2
2
uA.
Thus, ES/M0 multiplied by the shear modulus, G, is simply:

E S 1   2
G

.
M0
2
This is often referred to as the 'apparent stress drop'.
Earthquake energy balance: radiated energy versus seismic
moment and the apparent stress drop
• Thanks to the improvement
of seismic data, it is now
possible to directly measure
the radiated energy of small
earthquakes.
• In recent years, the
constancy of the apparent
stress drop is a matter of
debate.
Figure from: Figure from Kanamori and
Brodsky, Rep. Prog. Phys., 2004
Further reading
• Scholz, C. H., The mechanics of earthquakes and faulting, NewYork: Cambridge Univ. Press., 439 p., 1990.
• Kanamori, H., Mechanics of Earthquakes: Ann. Rev. Earth and
Planetary Sciences, v. 22, p. 207-237, 1994.