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Theory of the Earth
Don L. Anderson
Chapter 14. Anelasticity
Boston: Blackwell Scientific Publications, c1989
Copyright transferred to the author September 2, 1998.
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Recommended citation:
Anderson, Don L. Theory of the Earth.
Boston: Blackwell Scientific Publications,
1989.
http://resolver.calt
ech.edu/CaltechBOOK:1989.001
A scanned image of the entire book may
be found at the following persistent
URL:
http://resolver.caltech.edu/CaltechBook:1989.001
Abstract:
Real materials are not perfectly elastic.
Stress and strain are not in phase, and
strain is not a single-valued function of
stress. Solids creep when a sufficiently
high stress is applied, and the strain is
a function of time. These phenomena are
manifestations of anelasticity. The attenuation of seismic waves with distance
and of normal modes with time are ex
amples of anelastic behavior, as is
postglacial rebound. Generally, the response of a solid to a stress can be split into
an elastic or instantaneous part and an
anelastic or time-dependent part. The
anelastic part contains information ab
out temperature, stress and the defect
nature of the solid. In principle, the
attenuation of seismic waves can tell us
about such things as dislocation density
and defect mobility. These, in turn, are
controlled by temperature, pressure, stress an
d the nature of the lattice defects. If
these parameters can be estimated from seis
mology, they in turn can be used to
estimate other anelastic properties such as
viscosity. For example, the dislocation
density of a crystalline solid is a functio
n of the nonhydrostatic stress. These
dislocations respond to an applied oscilla
tory stress, such as a seismic wave, but
they are out of phase because of the fini
te diffusion time of the atoms around the
dislocation. The dependence of attenuat
ion on frequency can yield information
about the dislocations. The longer-term mo
tions of these same dislocations in
response to a higher tectonic stress
gives rise to a solid-state viscosity.
Anelastici
ty
'2s
when the massy substance
of
the
Earth
quivers.
"
-
CHRISTOPHER
MARLOWE,
TAMBURLAINE
THE
GREAT
R
eal materials are
not perfectly
elastic.
Stress
and
strain are
not in
phase,
and
strain
is
not a
single-
valued
function
of
stress. Solids creep
when
a sufficiently
high
stress is applied,
and the
strain
is a function
of
time.
These
phenomena
are manifestations
of
anelasticity. The at
-
tenuation
of
seismic
waves with
distance
and
of
normal
modes
with
time are examples
of
anelastic behavior,
as
is
postglacial rebound. Generally, the response
of
a solid
to1
a
stress
can be
split into
an
elastic or instantaneous part
and
an
anelastic or time
-
dependent part.
The
anelastic part con
-
tains
information
about
temperature, stress
and
the defect
nature
of
the solid.
In
principle, the attenuation
of
seismic
waves
can
tell
us about such
things as dislocation density
and
defect
mobility.
These,
in
turn, are controlled
by
tem
-
perature, pressure, stress
and
the nature
of
the lattice de
-
fects.
If
these
parameters can
be
estimated from seis
-
mology, they
in
turn can
be used to
estimate other anelastic
properties
such
as
viscosity. For
example, the dislocation
density
of
a crystalline
solid
is a function
of
the
nonhydro-
static stress. These dislocations respond to
an applied
oscil
-
latory stress,
such
as
a seismic wave,
but they
are out
of
phase because
of
the
finite diffusion
time
of
the atoms
around the dislocation. The dependence
of
attenuation
on
frequency
can yield
information
about
the dislocations.
The
longer
-
term motions
of these
same dislocations
in
re
-
sponse
to
a higher tectonic stress
gives
rise
to
a solid
-
state
viscosity.
SEISMIC
WAVE
ATTENUATION
Seismic
waves
attenuate
or
decay
as
they
propagate. The
rate
of
attenuation contains information about
the
anelastic
properties
of
the propagation medium.
A
propagating
wave
can
be
written
A
=
A,
exp
i(wt
-
kx)
where
A
is the
amplitude,
o
the frequency, k
the
wave
num
-
ber,
t
is travel time, x is distance
and
c
=
wlk
the phase
velocity.
If spatial attenuation occurs,
then
k is complex and
A
=
A,
exp
i
(ot
-
k)
.
exp
-
k*x
where k
and
k*
are
now
the real
and
imaginary parts
of
the
wave
number. k* is
called
the spatial attenuation co
-
efficient.
The elastic moduli,
say
M,
are
now
also complex:
M
=
M
+
iM*
The specific quality factor, a convenient dimensionless
measure
of
dissipation, is
QM'
=
M*IM
This
is related to the energy dissipated per cycle
A
E
:
where
(
E
)
is
the
average stored energy
(O'Connell
and
Budiansky,
1978).
This is commonly approximated as
where
E,,
is the
maximum
stored
energy.
Since phase
ve
-
locity,
c,
is
it follows that
In
general
c(w),
M(o),
k(w)
and
Q(w)
are all functions
of
frequency.
For
standing
waves,
or free oscillations,
we
write a
complex
frequency,
w
+
iw*,
where
w*
is
the
temporal
attenuation
coefficient and
In general, all
the
elastic
moduli
are complex,
and
each
wave
type
has
its
own
Q
and
velocity.
For
an
isotropic solid the
imaginary parts
of
the
bulk modulus and
rigidity are de
-
noted
as
K*
and
G*.
Most
mechanisms
of
seismic
-
wave
absorption
affect
G,
the rigidity,
more
than
K, and usually
QK
>>
QG
Important exceptions over certain frequency ranges
have to
do
with
thermoelastic mechanisms
and
composite systems
such
as
fluid
-
filled
rocks.
Frequency
Dependence
of
Attenuation
In
a perfectly elastic homogeneous
body,
the elastic
-
wave
velocities are independent
of
frequency.
In
an
imperfectly
.
elastic,
or
anelastic,
body
the velocities are dispersive, that
is,
they
depend
on
frequency. This is important
when
com
-
paring seismic data taken at
different
frequencies
or
when
comparing seismic
and
laboratory
data.
A
variety
of physical processes contribute to attenu
-
ation in a crystalline
material:
motions
of
point defects, dis
-
locations, grain boundaries
and
so on. These processes all
involve a high
-
frequency, or unrelaxed,
modulus and
a
low-
frequency, or relaxed, modulus.
At
sufficiently high
fre
-
quencies
the
defects,
which
are characterized
by
a time con
-
stant,
do
not
have
time to contribute,
and
the
body behaves
as a perfectly elastic
body.
Attenuation is
low and
Q is
high
in
the
high
-
frequency limit.
At
very low
frequencies the de
-
fects
have
plenty
of
time to respond to the applied force
and
they
contribute
an
additional strain. Because the stress
cycle time
is
long
compared to
the response time
of
the
defect, stress
and
strain are in phase
and
again Q
is high.
Because
of
the additional
relaxed
strain,
however,
the
mod
-
ulus
is low
and
the
relaxed velocity
is
low.
When
the fre
-
quency is comparable to the characteristic time
of
the de
-
fect, attenuation reaches a maximum,
and
the
wave
velocity
changes rapidly
with
frequency.
These characteristics are embodied in the
standard
lin
-
ear solid,
which
is composed
of
a spring
and
a
dashpot
(or
viscous
element)
arranged
in a parallel circuit,
which
is then
attached to another spring.
At
high frequencies the second,
or series, spring responds to
the
load,
and this
spring con
-
stant is the effective
modulus and
controls the total exten
-
sion.
At
low
frequencies the other spring
and
dashpot
both
extend,
with
a time constant
r
characteristic
of
the
dashpot,
the total extension is greater,
and
the effective modulus is
therefore lower. This
system
is sometimes described as a
viscoelastic solid.
The
Q-l
of
such
a system
is
where
k2
and
k,
are, respectively,
the
spring constants (or
moduli)
of
the series spring
and
the parallel spring
and
r
is
the
relaxation time,
7
=
r)
/
k2
where
77
is
the viscosity.
Clearly,
Q-l(w)
is a maximum,
Q;&,
at
WT
=
1,
and
and
Q-'(w)
-+
0
as
wr
when
w
+
0
and was
(07)-I
as
w
+
m.
The resulting absorption
peak
is shown
in
Fig
-
ure
14
-
1.
The phase
velocity
is approximately given
by
where
c,
is the zero
-
frequency velocity.
The
high
-
frequency
or elastic
velocity
is
Far
away
from
the
absorption peak,
the velocity can be
written
and
the
Q
effect
is only
second
order.
In
these limits,
ve
-
locity is
nearly
independent
of
frequency, but Q
varies
as
o
or
o-l.
In
a dissipative system, Q
and
c cannot
both
be
independent
of
frequency, and the
velocity
depends on the
attenuation.
When
Q is constant, or
nearly
so,
the fractional
change in phase
velocity
is proportional
to
Q-I
rather
than
Q-2
and becomes a
first
-
order
effect.
By
measuring the variation
of
velocity
or Q in the
laboratory
as
a function
of
frequency or temperature,
the
nature
of
the attenuation
and
its characteristic or relaxation
time,
r,
can often
be
elucidated.
For
activated processes,
r
=
ro
exp
E*IRT
(4)
where
E*
is an
activation
energy. Most
defect mechanisms
at seismic frequencies
and
mantle
temperatures
can be
de
-
scribed as activated relaxation effects. These include
stress-
induced diffusion
of
point
and line
(dislocation) defects.
At
very
high
frequencies
and low
temperatures, other
mecha
-
nisms come into
play, such
as
resonances of defects,
and
these cannot
be
so
described.
For
activated processes, then,
w.r,
exp
E*
IRT
Q-l
(w)
=
2Q;k
(1
+
(wrJ2
exp
2E*/RT)
(5)
an~d
the relaxation
peak
can
be defined
either
by
changing
w
or
changing
T.
Activation energies
typically
lie
in
the range
of
10
to
100
kcal/mole.
At
high temperatures, or
low
frequencies,
Q-l
(o)
=
2&&or0
exp
E*IRT
46)
This is contrary
to the general intuition
in
the seismological
literature
that
attenuation increases
with
temperature.
How
-
ever,
if
7
differs
greatly from seismic periods,
it is possible
that
we
may
be
on
the
low
-
temperature or high
-
frequency
portion
of
the
absorption peak,
and
Q-l
(o)
=
2Qi&
/
(or0
exp
E*IRT),
w7
>>
1
(7)
In
that case
Q
does
decrease
with
an
increase
in
T,
and
in
that regime
Q
increases
with
frequency.
This appears to
be
the case for short
-
period
waves
in the mantle. It is also gen
-
erally
observed that
low
-
Q
and
low
-
velocity regions
of
the
upper mantle
are
in
tectonically active
and high heat
-
flow
areas. Thus, seismic frequencies appear
to
be
near the
high-
frequency,
low
-
temperature side
of
the absorption peak
in
the
Earth's
upper mantle. Melting, in fact,
may
occur
before
the condition
o~
=
or,
exp
E*IRT
=
1
is satisfied. More generally,
FIGURE
14
-
1
Specific attenuation function
Q
-l(w)
and
phase velocity
c(o)
as
a
function
of
frequency
for
a linear viscoelastic
solid
with
a single relaxation
time,
7
(after Kanamori and Anderson,
1977).
where
V*,
the activation
volume,
controls the
effect
of
pressure
on
r
and
Q.
Because
of
the pressure
and
tempera
-
ture gradient in the mantle,
we
can
expect
r
and
therefore
Q(o)
to vary
with
depth. Increasing temperature drives the
absorption peak
to
higher frequencies (characteristic fre
-
quencies increase
with
temperature). Increasing pressure
drives the
peak to lower
frequencies.
Absorption
in
a medium with
a single characteristic
frequency
gives
rise
to
a bell
-
shaped
Debye
peak
centered
at a frequency
o
=
7-
l,
as
shown
in Figure
14
-
1.
The
specific dissipation function,
Q
-I,
and
phase
velocity
sat
-
isfy
the differential equation for the standard linear solid
and
can
be
written
Q-'(o)
=
2Q;&or/(1
+
w2r2)
+
~w~T~Q&]
The
high
-
frequency
(c,)
and
low
-
frequency velocities are
related
by
so that the total dispersion depends on the magnitude
of
the
peak dissipation.
For
a
Q
of
200,
a typical
value
for the
upper mantle, the
total velocity
dispersion is
2
percent.