Chapter
18
Elasticity
and
solid-state
geophysics
Mark
well
the
various
kinds
of
minerals,
note
their
properties
and
their
mode
of
origin.
Petrus
Severinus
(I
57
I)
The
seismic
properties
of
a
material
depend
on
composition,
crystal
structure,
temperature,
pres-
sure
and
in
some
cases
defect
and
impurity
con-
centrations.
Most
of
the
Earth
is
made
up
of
crystals.
The
elastic
properties
of
crystals
depend
on
orientation
and
frequency
. TI1us,
the
inter-
pretation
of
seismic
data,
or
the
extrapolation
of
laboratory
data,
requires
knowledge
of
crystal
or
mineral
physics,
elasticity
and
thermodynam-
ics.
But
one
cannot
directly
infer
composition
or
temperature
from
mantle
tomography
and
a
table
of
elastic
constants
derived
from
laboratory
experiments.
Seismic
velocities
are
not
unique
functions
of
temperature
and
pressure
alone
nor
are
they
linear
functions
of
them.
Tomographic
cross-sections
are
not
maps
of
temperature
. The
mantle
is
not
an
ideal
linearly
elastic
body
or
'an
ideal
harmonic
solid
'.
Elastic
'constants'
are
frequency
dependent
and
have
intrinsic,
extrinsic,
anharmonic
and
anelastic
contribu-
tions
to
the
pressure
and
temperature
dependen-
cies
.
Elastic
constants
of
isotropic
solids
The
elastic
behavior
of
an
isotropic,
ideally
elas
-
tic,
solid
-
at
infin
ite
frequency
- is
comp
l
etely
characterized
by t
h e
density
p
and
two
elastic
constants.
TI1ese
are
usually
the
bulk
modulus
K
and
the
rigidity
G
or
fL
Young's
modulus
E
and
Poisson's
ratio
cr
are
also
commonly
u sed.
TI1ere
are,
correspondingly,
two
types
of
elastic
waves;
the
compressional,
or
primary
(P),
and
the
shear,
or
secondary
(S)
,
having
velocities
derivable
from
pVi
=
K
+4G/3
=
K
+4J.L/3
pVl
=
G
=
fL
The
interrelations
between
the
elastic
constants
and
wave
velocities
are
given
in
Table
18
.1.
In
an
anisotropic
solid
there
are
two
shear
waves
and
one
compressional
wave
in
any
given
direction.
Only
gases,
fluids,
well-annealed
glasses
and
similar
noncrystalline
materials
are
strictly
isotropic.
Crystalline
material
with
random
ori-
entations
of
grains
can
approach
isotropy
,
but
rocks
are
generally
anisotropic
.
Laboratory
measurements
of
mineral
elas-
tic
properties,
and
their
temperature
and
pres-
sure
derivatives,
are
an
essential
complement
to
seismic
data
.
The
high-frequency
elastic-wave
velocities
are
known
for
hundreds
of
crystals.
Compilations
of
elastic
properties
of
rocks
and
minerals
complement
those
tabulated
here.
In
addition,
ab
initio
cal-
culations
of
elastic
properties
of
high-pressure
minerals
can
be
used
to
supplement
the
measurements
.
Elastic
properties
depend
on
both
crystal
structure
and
composition,
and
the
understand-
ing
of
these
effects,
including
the
role
of
temperature
and
pressure,
is
a
responsibility
of
a
discipline
ca
ll
ed
mineral
physics
or
solid-state
geophysics.
Most
meas
u
re
m
ents
are
made
under
234
EL
A S
TIC
ITY
AND
S
OLID
- ST
ATE
G E
OPHYSICS
Ta
ble
18
. 1
I
C
onn
ecting
id
entities
for
el
ast
ic
co
n
st
a
nts
of
isotrop
ic
bodies
K
A.+
21-L/3
2(1
+a)
1-L
3(
I
-
2a)
l+a
A.--
3a
G
=i-L
3(K
-
A.)/2
I-
2a
A.---
2a
I
-2a
3K--
2+2a
A.
a
K
-
2~-L/3
A.
2(A.
+
!-L)
2a
A.
~-L
--
I
-2a
3K
-A.
3K-a-
3K-
2
!-L
l+a
2(3K
+
1-L)
p(V
p
2
-
4V
l
/3)
pV
s2
p(V
p2
-2V
s2
)
A.,
1-L
=Lame
constants
G
=Rigidity
or
shear
modulus=
pV
5
2
=
1-L
K
=
Bulk
modulus
a
=
Poisson's
ratio
E
=Young's
modulus
p
=density
Vp.
Vs
=compressional
and
shear
velocities
I
(V
p
/V
s)
2
- 2
l(V
p
/V
5
)2-
I
4!-L-
E
pVi
=A.+
2!-L
=
3K
-
2A.
=
K
+
4~-L
/3
=
~-L---
3t.L-
E
3k
+
E
1
-
a
2 -
2a
1
-
a
=3K
=A.--=t.L--=3K--
9K
-
E
a
1
-
2a
1
+a
conditions
far
from
the
pressure
and
temper-
ature
conditions
in
the
deep
crust
or
mantle.
The
frequency
of
laboratory
waves
is
usually
far
from
the
frequency
content
of
seismic
waves
.
The
measurements
themselves,
therefore,
are
just
the
first
step
in
any
program
to
predict
or
interpret
seismic
velocities.
Some
information
is
now
available
on
the
high-frequency
elastic
properties
of
all
major
rock-forming
minerals
in
the
mantle.
On
the
other
hand,
there
are
insufficient
data
on
any
mineral
to
make
assumption-free
comparisons
with
seismic
data
below
some
100
km
depth
. It
is
essential
to
have
a
good
theoretical
understand-
ing
of
the
effects
of
frequency,
temperature,
com-
position
and
pressure
on
the
elastic
and
thermal
properties
of
minerals
so
that
laboratory
mea-
surements
can
be
extrapolated
to
mantle
con-
ditions.
Laboratory
results
are
generally
given
in
terms
of
a
linear
dependence
of
the
elastic
moduli
on
temperature
and
pressure.
The
actual
variation
of
the
moduli
with
temperature
and
pressure
is
more
complex.
It
is
often
not
justi-
fied
to
assume
that
all
derivatives
are
linear
and
independent
of
temperature
and
pressure;
it
is
necessary
to
use
physically
based
equations
of
state
.
Unfortunately,
many
discussions
of
upper-
mantle
mineralogy
and
interpretations
of
tomog-
raphy
ignore
the
most
elementary
considerations
of
solid-state
and
atomic
physics
.
The
functiona
l
form
of
a(T,
P),
the
coeffi
-
cient
of
thermal
expansion,
is
closely
related
to
the
specific-heat
function,
and
the
neces-
sary
theory
was
developed
long
ago
by
Debye
,
Griineisen
and
Einstein.
Yet
a(T,
P)
is
some-
times
assumed
to
be
independent
of
pressure
and
temperature,
or
linearly
dependent
on
tem-
perature.
Likewise
,
interatomic-potential
theory
shows
that
the
pressure
derivative
of
the
bulk
modulus
dK/dP
must
decrease
with
compression,
yet
the
moduli
are
often
assumed
to
increase
linearly
with
pressure.
There
are
also
various
thermodynamic
relationships
that
must
be
sat-
isfied
by
any
self-consistent
equation
of
state,
ELA
ST I C
CON
ST
ANTS
OF
I S
OTROP
IC
SOL
I
DS
235
and
certain
inequalities
regarding
the
strain
dependence
of
anharmonic
properties.
Processes
within
the
Earth
are
not
expected
to
give
random
orientations
of
the
constituent
anisotropic
min-
erals.
On
the
other
hand
the
full
elastic
tensor
is
difficult
to
determine
from
seismic
data.
Seismic
data
usually
provide
some
sort
of
average
of
the
velocities
in
a
given
region
and,
in
some
cases,
estimates
of
the
anisotropy
.
The
best-quality
labo
-
ratory
data
are
obtained
from
high-quality
single
crystals.
The
full
elastic
tensor
can
be
obtained
in
these
cases,
and
methods
are
available
for
com-
puting
average
properties
from
these
data.
It
is
simpler
to
tabulate
and
discuss
average
properties,
as
I
do
in
this
section.
It
should
be
kept
in
mind,
however,
that
mantle
minera
ls
are
anisotropic
and
they tend
to
be
readily
oriented
by
mantle
processes
.
Certain
seismic
observations
in
subducting
slabs,
for
example,
are
best
inter-
preted
in
terms
of
oriented
crystals
and
a
result-
ing
seismic
anisotropy
.
If
all
seismic
observations
are
interpreted
in
terms
of
isotropy,
it
is
possible
to
arrive
at
erroneous
conclusions.
The
debates
about
the
thickness
of
the
lithosphere,
the
deep
structure
of
continents,
the
depth
of
slab
pene-
tration,
and
the
scale
of
mant
le
convection
are.
to
some
extent,
debates
about
the
anisotropy
and
mineral
physics
of
the
mantle
and
the
interpreta-
tion
of
seismic
data
.
Although
it
is
important
to
understand
the
effects
of
temperature
and
pres-
sure
on
physical
properties.
it
is
also
important
to
realize
that
changes
in
crystal
structure
(solid-
solid
phase
changes)
and
orientation
have
large
effects
on
the
seismic
velocities.
Tomographic
images
are
often
interpreted
in
terms
of
a
single
variable
,
temperature.
Thus
blue
regions
on
tomo-
graphic
cross-sections
are
often
called
cold
slabs
and
red
regions
are
often
called
hot
plum
es.
Many
of
the
current
controversies
in
mantle
dynam-
ics
and
geochemistry,
such
as
deep
slab
penetra-
tion,
whole
mantle
convection
and
the
presence
of
plumes
can
be
traced
to
over-simplified
or
erro-
neous
scaling
relations
between
seismic
veloci-
ties
and
density.
temperature
and
physical
state
.
Table
18.2
is
a
compilation
of
the
elastic
prop-
erties,
measured
or
estimated
,
of
most
of
the
important
mantle
minerals,
plus
pressure
and
temperat
u
re
derivatives.
Anharmonicity
There
are
var
i
ous
routes
whereby
temperature
affects
the
elastic
moduli
and
seismic
velocity.
The
main
ones
are
ane
l
asticity
and
anharmonic-
ity.
The
first
one
does
not
depend,
to
first
order,
on
vol
u
me
or
density
and
therefore
many
geo-
dynamic
scaling
re
l
at
i
ons
are
invalid
if
ane
las-
ticity
is
importa
n t.
Elast
ic
mod
u li
also
depend
on
parameters
other
than
temperature,
such
as
composition.
The
visua
l
interpretations
of
tomo-
graphic
color
cross-sections
assume
a one-to-one
correspondence
between
seismic
velocity,
density
and
temperat
u re.
The
thermal
oscillation
of
atoms
in
their
(asymmetric)
potential
well
is
anharmonic
or
nonsinusoidal.
Thermal
oscillation
of
an
atom
causes
the
mean
position
to
be
displaced
,
and
thermal
expansion
resu
lts.
(In
a
symmetric,
or
parabo
lic,
potential
well
the
mean
positions
are
unchanged,
atomic
vibrations
are
harmonic,
and
no
thermal
expansion
results.)
Anharmonici
ty
ca
u ses
atoms
to
take
up
new
average
positions
of
eq
u
ilibrium,
dependent
on
the
amplitude
of
the
vibrations
and
hence
on
the
temperature,
but
the
new
positions
of
dynamic
equilibrium
remain
near
ly
harmonic.
At
any
given
volume
the
har-
monic
approximation
can
be
made
so
that
the
characteristic
temperature
and
frequency
are
not
explicit
functions
of
temperature.
This
is
called
thequ
asi-harmonic
approximation.ffit~
assumed
that
the
frequency
of
each
normal
mode
of
vibration
is
changed
in
simple
proportion
as
the
volume
is
changed.
There
is
a
close
relation-
ship
between
lattice
thermal
conductivity
,
ther-
mal
expansion
and
other
properties
that
depend
intrinsically
on
anharmonici
ty
of
the
interatomic
potential.
The
atoms
in
a
crystal
vibrate
about
equilibrium
positions,
but
the
normal
modes
are
not
independent
except
in
the
idealized
case
of
a
harmonic
solid.
The
vibrations
of
a
crystal
lattice
can
be
resolved
into
interacting
traveling
waves
that
interchange
energy
due
to
anharmonic
,
non-
linear
coupling.
In
a
harmonic
solid:
there
is
no
thermal
expansion;
adiabatic
and
isothermal
elastic
constants
are
equal;
Table
18
.2
I
Elastic
properties
of
mantle
minerals
(Duffy
and
Anderson,
1988)
formula
Densit
y
Ks
G
-K
s
-G
(structure)
(g/cm
3
)
(GPo)
(GPa)
K'
s
G'
(GPa/K)
(GPa/K)
(Mg
,F
e)2Si04
3.222+
129
82-
5.1
1.8
0.016
0.013
(olivine)
1.182XF
e
31
XFe
(Mg,Fe)2Si04
3.472+
174
114-
4.9
1.8
0.018
0.0
14
(.8
-spinel)
I
.24XF
e
41XF
e
(Mg.
Fe)2Si04
3.548+
184
119
-
4.8
1.8
0.017
0.014
(y -spinel)
1.30XF
e
41XF
e
(Mg,Fe)Si0
3
3.204+
104
77-
5.0
2.0
0.012
0.011
(orthopyroxene)
0.799XF
e
24XFe
CaMgSi206
3.277
113
67
4.5
1.7
0.013
0.010
(clinopyroxene)
NaAISi
20 6
3.32
143
84
4.5
1.7
0.016
0.013
(clinopyroxene)
(Mg.Fe)O
3.583+
163
-
131
4.2
2.5
0 .016
0 .024
(magnesiowustite)
2.28XFe
8XFe
77XFe
AI203
3.988
251
162
4.3
1.8
0 .014
0.019
(corundum)
Si0
2
4.289
316
220
4.0
1.8
0.027
0 .018
( stishovite)
(Mg.Fe)3AI2Si3012
3.562+
175+
90+
4.9
1.4
0.021
0.010
(garnet)
0.758XFe
I
XFe
8XF
e
Ca3(AI,Fe)2Si3012
3.595+
169
-
140-
4.9
1.
6
0.016
0 .015
(garnet)
0.265X
Fe
I I
XFe
14XF
e
(Mg.
Fe)Si03
3.810+
212
132-
4.3
1.7
0.017
0 .017
(ilmenite)
I.IO
XFe
41
XFe
(Mg.Fe)Si03
4 . 104+
266
153
3.9
2.0
0.031
0.028
(perovskite)
1.07XF
e
CaSi0
3
4 .
13
227
125
3.9
1.9
0.027
0.023
(perovskite)
(Mg
,F
e)4Si4012
3.
518+
175+
90+
4 .9
1.4
0.021
0.010
(majorite)
0.973XFe
I
XFe
8XFe
Ca2Mg2Si4012
3.53
165
10
4
4.9
1.6
0.0
16
0.015
(majorite)
Na
2
AI
2S
i401
2
4.00
200
127
4.9
1.6
0 .016
0.015
(majorite)
XFe
is
molar
fraction
of
Fe
end
member.
Alternative
va
l
ues
for
parameters
in
Table
18.2
and
data
for
other
minerals
are
given
in
the
followi
n g
references.
Anderson,
0.
L.
and
I
saak,
D.
G.
(1995)
Elastic
constants
of
mant
le
minerals
at
high
temperature,
in
Mineral
Physics
and
Crystallography:
A Handbook
of
Pl1ysical
Constants,
pp.
64-97,
ed.
T.
J.
A11rens,
American
Geophysica
l
Union,
Washington,
DC.
Bass,
J.
D.
(1995)
Elasticity
of
minera
ls,
glasses,
and
melts,
in
Mineral
Physics
and
Crystallography:
A
Handbook
of
Physical
Constants,
pp.
45-63,
ed.
T.
J.
A11ren
s.
American
Geophysica
l
Union,
Washington,
DC.
Duffy,
T.
and
Anderson
, D.
L.
(1989)
Seismic
velocit
ies
in
mant
le
minerals
and
the
mineralogy
of
the
upper
mantle
.
].
Geophys.
Res.,
94
(B2),
1895-912.
Li,
B.
and
Zhang,
J.
(2005)
Pressure
and
temperat
u
re
dependence
of
e l
astic
wave
velocity
of
MgSi03
perovskite
and
the
compos
i
tion
of
the
lower
mantle,
Pl1ys.
Eart/1
Planet.
Inter,
1 51 ,
143-54.
Mattern,
E.,
Matas,
J.,
Ricard,
Y.
and
Bass,
J.
D.
(2005)
Lower
mant
le
composit
i
on
and
temperature
fTom
mineral
physics
and
thermodynamic
modelling
,
Geophys.].
Int.
,
160 ,
973-90.
CORRECTING
ELA
S
TIC
PROPERT
IES
F
OR
TEM
P
ERATURE
237
the
elastic
constants
are
independent
of
pres-
sure
and
temperature;
the
heat
capacity
is
constant
at
high
temper-
ature
(T
>
0);
and
the
lattice
thermal
conductivity
is
infinite.
TI1ese
are
the
res
u
lt
of
the
neglect
of
anharmo
n ic-
ity
(higher
than
quadratic
terms
in
t h e
inter-
atomic
displacements
in
the
potential
energy).
In
a
real
crystal
the
presence
of
lattice
vibra-
tion
causes
a
periodic
elastic
strain
that,
throug
h
anharmonic
interaction,
modulates
the
elastic
constants
of
a
crystal.
Other
phonons
are
scat
-
tered
by
these
modulations.
TI1is
is
a
nonlinear
process
that
does
not
occur
in
the
absence
of
anharmonic
terms.
The
concept
of
a
strictly
harmonic
crysta
l
is
highly
artificial.
It
implies
that
neig
h
boring
atoms
attract
one
another
with
forces
propor-
tional
to
the
distance
between
them,
but
such
a
crystal
would
collapse.
We
must
disting
u
ish
between
a
harmonic
solid
in
which
each
atom
executes
harmonic
motions
about
its
equilibrium
position
and
a
solid
in
which
the
forces
between
individual
atoms
obey
Hooke's
law.
In
the
for-
mer
case,
as
a
solid
is
heated
up,
the
atomic
vibrations
increase
in
amplitude
but
the
mean
position
of
each
atom
is
unchanged.
In
a
two-
or
three-dimens
i
onal
lattice,
the
net
restoring
force
on
an
individual
atom,
when
all
the
near-
est
neighbors
are
considered,
is
not
Hookean.
An
atom
oscillating
on
a
line
between
two
adjacent
atoms
will
attract
the
atoms
on
perpendicular
lines,
thereby
contracting
the
lattice.
Such
a solid
is
not
harmonic;
in
fact
it
has
negative
thermal
expansion.
TI1e
quasi-harmonic
approximation
takes
into
account
that
the
equilibrium
positions
of
atoms
depend
on
the
amplitude
of
vibrations,
and
hence
temperature,
b u t
that
the
vibrations
about
the
new
positions
of
dynamic
equilib-
rium
remain
closely
harmonic.
One
can
then
assume
that
at
any
given
volume
V
the
harmonic
approximation
is
adequate
.
In
the
simplest
quasi-
harmonic
theories
it
is
assumed
that
the
frequen-
cies
of
vibration
of
each
normal
mode
of
lattice
vibration
and,
hence,
the
vibrational
spectra,
the
maximum
frequency
and
the
characteristic
tem
-
peratures
are
functions
of
volume
alone.
In
this
approximation
y
is
independent
of
temperature
at
constant
vo
l
ume,
and
a
has
approximately
the
same
temperature
dependence
as
molar
specific
heat
.
Correcting
elastic
properties
for
temperature
The
elastic
properties
of
solids
depend
primarily
o n
static
latt
ice
forces.
b
ut
vibrational
or
ther-
ma
l
motions
become
increasingly
important
at
high
temperature.
The
resistance
of
a
crystal
to
deformation
is
partially
due
to
interionic
forces
and
partially
due
to
the
radiation
pressure
of
high-frequency
acoustic
waves,
which
increase
in
intensity
as
the
temperature
is
raised.
If
the
increase
in
vo
l
ume
associated
with
this
radiation
press
u
re
is
compensated
by
the
application
of
a
suitable
external
pressure,
there
still
remains
an
intrinsic
temperature
effect.
Thus,
these
equa-
tions
provide
a
convenient
way
to
estimate
the
properties
of
the
static
lattice.
that
is,
K(V,
0)
and
G(V.
0).
and
to
correct
measured
values
to
differ-
ent
temperatures
at
constant
volume.
The
static
lattice
values
should
be
used
when
searching
for
velocity-density
or
modu
l
us-volume
systematics
or
when
attempting
to
estimate
the
properties
of
unmeasured
phases.
The
first
step
in
forward
modeling
of
the
seis-
mic
properties
of
the
mantle
is
to
compile
a
table
of
the
ambient
or
zero-temperature
proper-
ties,
including
temperature
and
pressure
deriva-
tives,
of
all
relevant
minerals.
The
fully
normal-
ized
extrinsic
and
intrinsic
derivatives
are
then
formed
and
,
in
the
absence
of
contrary
infor-
mation.
are
assumed
to
be
independent
of
tem-
perature.
The
coefficient
of
thermal
expansion
can
be
used
to
correct
the
density
to
the
tem-
perature
of
interest
at
zero
pressure.
It
is
impor-
tant
to
take
the
temperature
dependence
of
a(T)
into
account
properly
since
it
increases
rapidly
from
room
temperature
but
levels
out
at
high
T.
TI1e
use
of
the
ambient
a
will
underestimate
the
effect
of
temperature;
the
use
of
a
plus
the
initial
slope
will
overestimate
the
volume
change
at
high
temperature.
Fortunately.
the
shape
of
a(T)
is
well
known
theoretically
(a
De
bye
f u
nction)
and
has
been
meas
u
red
for
many