Elasticity of the Earth - Quantitative Plate Tectonics - page 241

where is the dilatation (Eq. 7.38 ) and›,which
is called bulk modulus , is a measure of the in-
compressibility of the material. In fact, by ( 8.2 )
we see that this quantity represents the ratio
of hydrostatic pressure to the resulting volume
change. Values of the adiabatic bulk modulus for
the main crustal and mantle minerals are listed
in Table 1.1 . Two other common parameters,
especially in engineering and applied geology,
are the Young modulus , Y ,and Poisson ' s ratio
. The Young modulus is defined as the ratio
between extensional stress and resulting exten-
sional strain for a cylinder that is pulled by both
ends. It can be shown (e.g., Ranalli 1995 )thatis
given by:

Therefore,

œ.1 /
2

© ij n j D n i (8.7)

As a consequence, a principal axis of stress, n ,
with eigenvalue œ is also a principal axis of strain
with eigenvalue:

œ.1 /
2

œ 0 D

(8.8)

8.2 Equations of Motion for

Elastic Media

.3œ C 2/
œ C

Now we are going to search a solution to
the homogeneous version ( 7.51 ) of Cauchy's
momentum equation, which links the second
time derivatives of the displacement field (inertial
term) to the spatial variations of the stress tensor
(surface forces field). To this end, we use Hooke's
law ( 8.1 ) to write the components of the stress
tensor in terms of displacement field. Substituting
the expression ( 7.32 ) for the strain tensor into
Hooke's law gives:
£ ij D œ• ij r u C @ u i

Y D (8.3)

Finally, the non-dimensional Poisson's ratio
is the ratio between the lateral contraction of
a cylinder that is pulled by both ends and its
longitudinal extension. It is given by:

œ
2.œ C /

D (8.4)

This parameter varies between -1 and a maxi-
mum value D 0.5 in the case of a liquid ( D 0).
A Poisson solid is a material such that œ D ,
so that D 0.25. In seismology, crustal rocks are
often approximated as Poisson solids in the esti-
mation of seismic velocities. In general, 0.25
0.30 for most crustal rocks. An interesting
property of isotropic media is that in this instance
the principal axes of stress coincide with the
principal axes of strain. To prove this assertion,
let us assume that:
£ ij n j D œn i
© ij n 0 j D œ 0 n 0 i

@ u j
@x i

@x j C

(8.9)

This equation is a version of Hooke's law that
makes explicit the dependence of the stress field
from the displacements. It is important to note
that in this relation the variables are evaluated
at a given time t and position r , and the small
displacement u i ( r , t ) associated with the elastic
response instantaneously accompanies the varia-
tions of stress in time. However, the momentum
equation tells us that it is the spatial variability
of the stress tensor to drive changes in the dis-
placement field, which occur with velocity and
acceleration that are given by the first and second
time derivatives of u i . Therefore, while ( 8.9 ) links
the local components of stress at r to the spatial
variations of displacement in a neighbor of r ,the
momentum Eq. ( 7.51 ) relates the spatial variabil-
ity of the stress field to the temporal changes in
the local displacement.

(8.5)

where œ an œ 0 are eigenvalues.
Substituting Hooke's law ( 8.1 ) in the first of
these equations gives:
£ ij n j D œ• ij C 2© ij n j

D œn i C 2© ij n j D œn i (8.6)