EAS 557
Introduction to Seismology
Robert L. Nowack
Lecture 9
Elastodynamic Equations of Motion
We have studied the properties of stress and strain and found a linearized equation of motion
where ui is the particle displacement, fi is a body force term, and is the stress tensor.
Assuming the solid follows a linear elastic constitutive relation where and , then
(1)
Note that we have ignored any initial stress and, for the time being, consider only deviatoric or incremental stress from the initial state. This is then the linear anisotropic elastodynamic equation.
Now, we investigate the isotropic, elastodynamic equation. Let where , then,
where we have assumed for simplicity that and are constants and the equation for a homogeneous medium.
Now
Since , then
or
(2a)
In vector form, this can be written
or (2b)
This is the elastodynamic equation for linear elastic waves in an isotropic homogeneous medium.
For an isotropic heterogeneous medium, then and the elastodynamic equation can be written
(3)
where is . Most of the rest of this class will involve solving either equation (2) or (3) for a given source distribution of body forces and boundary conditions.
For the anisotropic case, we need to go back to the anisotropic elastodynamic equation of Equation (1) to derive further results.
The Existence of “P”
and “S” Waves
The isotropic elastodynamic equation of motion is difficult to solve. In order to find simple solutions, we shall first show that there are two different types of wave solutions. Let us assume (i.e., there are no body forces) for the isotropic homogeneous elastodynamic equation.
1) Take the divergence or source terms of the equation of motion (2b)
where (see identities b) and c) below). Then
This is a simple “wave equation” for the propagation of the dilatation and can be written
where the “wave speed” is
This is a well studied equation in physics and appears in many physical problems. This same equation controls the propagation of sound waves, electromagnetic waves, etc. is the velocity of propagation of dilatational waves, also called longitudinal P waves which we will alternatively note as Vp.
2) Similarly, we can find an equation for the rotation by taking the curl of the isotropic elastodynamic equation (2b)
where from identities a), b) and c) below.
Several vector identities used above are
a) for any scalar function .
b) for any vector .
c) for any vector .
For example,
(from identity a))
(from identity b) with )
(from identify c))
Rewriting the above equation in terms of the rotation vector , we obtain
where . This is a set of three scalar wave equations for the propagation of the components of the rotation vector. or Vs is the velocity of propagation of rotational waves, also called transverse, shear, or S waves. Since , then . Thus, compressional waves propagate faster than rotational waves in an isotropic elastic body.
We often assume for the solid crust and mantle of the Earth a Poisson’s ratio of with . For this case,
This is sometimes called a Poisson solid.
For the inhomogeneous case, the elastodynamic equation (3) can still be approximately decomposed into divergence and rotational waves if are sufficiently small compared to the and terms, or the medium must be smoothly varying.
For a fluid then, and , , (the bulk modulus), and which equals velocity of sound.
Examples of seismic velocities and physical properties of different materials and rocks are given by Press (1966) in the “Handbook of Physical Properties” (S.P. Clark, Ed.). For example,
|
Vp (km/s) |
Vs (km/s) |
(kg/m3) |
Fresh water at 25oC |
1.509 |
0.0 |
1000 |
Granite ( |
5.1 |
2.79 |
2665 |
Eclogite
( |
7.31 |
4.26 |
3441 |
Dunite
( |
7.0 |
4.01 |
3264 |
But, it’s important to recall that different rock types exhibit a range of seismic velocities that also depend on pressure and temperature.
For the Earth model PREM (Dziewonski and Anderson, 1981), the physical properties for several depths in the Earth are,
|
Depth |
Vp (km/s) |
Vs (km/s) |
(kg/m3) |
Ocean |
3.0 |
1.45 |
0.0 |
1020 |
Upper Crust |
15.0 |
5.79 |
3.19 |
2600 |
Lower Crust |
25.0 |
6.79 |
3.89 |
2900 |
Uppermost mantle |
80 km |
8.07 |
4.38 |
3375 |
Mid-mantle |
1071 km |
11.55 |
6.41 |
4621 |
Lowermost mantle |
2891 km |
13.69 |
7.23 |
5566 |
Outer core |
3871 km |
9.38 |
0.0 |
11,191 |
Inner core |
5671 km |
11.14 |
3.54 |
12,982 |
Finally, knowing that there is a decomposition between and for the elastic wave equation, another direct decomposition of the original vector displacement can be written as
where . Substituting this into the elastic wave equation then results in simple wave equations for and which are called the scalar and vector potentials.
Then, and also solve simple wave equations
where and are the P-wave and S-wave components of . In more complicated cases involving heterogeneous elastic equation (3), it is often easier to work directly with the particle displacement .