Elasticity of the Earth - Quantitative Plate Tectonics - page 240

8

Elasticity of the Earth

Abstract
An important source of data in plate tectonics comes from seismology.
This is the first of three chapters devoted to the fundamental laws of
propagation of seismic waves. In this chapter, I describe the elastic
response of Earth's rocks to deformation, which is quantified by Hooke's
law, the seismic wave equation, and the concept of seismic energy.

The scalar fields œ D œ( r )and D ( r )are
called Lamé parameters and represent two of
the three quantities determining the local elastic
properties of a rock, the third one being the
density . In the case of pure shear deformation,
the dilatation D © kk is zero, so that ( 8.1 )re-
duces, component-by-component, to ( 7.53 ) with
Y D 2 . The quantity is termed shear modulus
or rigidity modulus and is a measure of the
resistance or rocks to shear deformation. Values
of for the main crustal and mantle minerals are
listed in Table 1.1 . This parameter is always zero
in the case of liquids, gases, and plasma, because
matter in the fluid states does not exhibit resis-
tance to finite strain. Conversely, the parameter
œ does not have a simple physical interpretation,
thereby it is often convenient to introduce some
additional, more descriptive, elastic parameters.
For example, we can separate in ( 8.1 ) isotropic
and deviatoric components. If we define the mean
normal stress £ 0 D - p £ kk /3, then taking the
trace in ( 8.1 )gives:

8.1 Hooke's Law

In this chapter, we consider the elastic behaviour
of Earth's rocks in response to dynamic loads at a
time scale of a few tens seconds. In the context of
plate tectonics, slip along plate boundaries is the
ultimate cause of most earthquakes, but are the
elastic properties of the rocks to determine the
mode of propagation of seismic waves through
the Earth. Therefore, these properties constitute
the “hard background” for the study of seismol-
ogy. The basic formulation of this subject as-
sumes that the elasticity of rocks does not depend
on direction, so that the material is isotropic .
In this instance, it is possible to show that the
number of independent parameters necessary to
describe the elastic behaviour of a rock body
reduces to three scalar quantities, and the three-
dimensional time-dependent constitutive equa-
tion describing the relation between stress and
strain is a tensor form of Hooke's law ( 7.53 ) :

œ C

3 © kk ›

2

ij .r;t/ D œ.r/• ij © kk .r;t/ C 2.r/© ij .r;t/
(8.1)

£ 0 D (8.2)