Purdue University

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 (Barre, VT)
(at 10 bars)

5.1

2.79

2665

Eclogite (Healdsburg, CA)
(at 10 bars)

7.31

4.26

3441

Dunite (Webster, NC)
(at 10 bars)

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 .