Geoscience Reference
In-Depth Information
Chapter 18
Elasticity and solid-state geophysics
Mark well the various kinds of
minerals, note their properties and
their mode of origin.
constants. These are usually the bulk modulus
K
and the rigidity
G
or
μ
Young's modulus E and
Poisson's ratio
are also commonly used. There
are, correspondingly, two types of elastic waves;
the compressional, or primary (P), and the shear,
or secondary (S), having velocities derivable from
σ
Petrus Severinus (1571)
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. Thus, 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.
ρ
V
P
=
K
+
4
G
/
3
=
K
+
4
μ/
3
ρ
V
S
=
G
=
μ
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 called
mineral physics
or
solid-state
geophysics
. Most measurements are made under
Elastic constants of isotropic solids
The elastic behavior of an isotropic, ideally elas-
tic, solid -- at infinite frequency -- is completely
characterized by the density
ρ
and two elastic
