Geoscience Reference
In-Depth Information
Table 18.1
Connecting identities for elastic constants of isotropic bodies
K
G
=
μ
λ
σ
λ
2
3
3(
K
)
2
K
2
3
λ
+
μ/
−
λ
/
−
μ/
2(
)
λ
+
μ
2(1
)
1
2
2
+
σ
−
σ
σ
λ
μ
λ
μ
3(1
2
)
2
1
2
3
K
−
σ
σ
−
σ
−
λ
1
1
2
3
K
2
+
σ
3
−
σ
σ
−
μ
3
K
3
K
λ
2
2
1
2(3
K
)
σ
+
σ
+
σ
+
μ
ρ
(
V
p
−
4
V
S
/
3)
ρ
V
s
ρ
(
V
p
−
2
V
s
)
—
V
s
)
2
1
2
(
V
p
/
2
−
—
—
—
(
V
p
/
V
s
)
2
1
−
λ, μ
=
Lamé constants
V
s
G
=
Rigidity or shear modulus
=
ρ
=
μ
=
K
Bulk modulus
σ
=
Poisson's ratio
E
=
Young's modulus
ρ
=
density
V
p
,
V
s
=
compressional and shear velocities
μ
−
4
E
V
p
ρ
=
λ
+
2
μ
=
3
K
−
2
λ
=
K
+
4
μ/
3
=
μ
3
μ
−
E
3
k
+
E
1
−
σ
σ
2
−
2
σ
1
−
σ
=
3
K
E
=
λ
=
μ
=
3
K
9
K
−
1
−
2
σ
1
+
σ
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
(
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,
Gruneisen
functional
form
of
α
(
T
,
P
)issome-
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 d
K
/d
P
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,
and
Einstein.
Yet
α
