EAS 557
Introduction to Seismology
Robert L. Nowack
Lecture 7
Conservation Principles
The theorems of conservation are the physical laws that govern the deformation of the continuum and are derived from physical principles. The physical laws we are concerned with are:
Conservation of linear momentum
Conservation of mass
Conservation of angular momentum
Conservation of energy
Conservation of Linear Momentum – global derivation
Let the
particle velocity be 
and let dm be the mass of a particle at P, then
the integrated momentum is 
. This is the total
momentum summed over all the particles of mass that build up the body.
Conservation of linear momentum requires
where 
is the total force
applied on the body. Applied forces are
of two types:
a) Boundary forces
These
are external forces applied on the boundary such as 
.
b) Body forces
These are forces applied in the interior of the body by external force fields. For gravity,
where 
is body force and 
is acceleration of
gravity. Gravitational body forces are
proportional to the mass of the particles P.
We can then write the linear momentum equation as
Using conservation of mass (see below), we may write
In component form, the linear momentum equation then can be written as
We would like to have the surface integral transformed into an equivalent volume integral. This may be done using
From Cauchy’s formula 
, we can write the boundary tractions in terms of the stress
tensor. Then,
This surface integral may be transformed into a volume
integral using 
as
Putting this into the momentum equation gives
Define the density
as 
, then,
If the volume under consideration is arbitrary, we can equate the integrands to find
This is the general equation of motion in continuum mechanics. This equation is valid for elastic, viscoelastic, liquid, and plastic materials, since no assumptions about the behavior of the material were involved in its derivation.
Linearization of the Equation of Motion
The
derivative 
is usually called a
total derivative since it includes both the time variability and the
variability due to any flow of the material.
In elasticity, we prefer to solve for the velocity at a given spatial position. That is, instead of following a particle, we solve for the velocity of the particle that at time t is at position x.
The total derivative (or material derivative) is
where 
is the total
derivative, 
is the local
derivative, and 
is the advection
term. Note that the total derivative of
the particle velocity is generally nonlinear because of the advection terms.
To derive this for a small time
increment, 
, the particle will have moved 
. Then, the change of
an arbitrary function g will be
Expanding in a
Then,
Finally, the total derivative is
In the linear approximation, we ignore the advection term and make the approximation
In this approximation, the particles don’t move very much from their initial position. This is true for the propagation of a seismic wave that we are interested in here. The linearized equation of motion is
then
Conservation of Mass
To show conservation
of mass, let the mass M = 
. Then,
where the last term is the material flux through the boundary. This can be changed to a volume integral to obtain
where 
is the flux of the material. Now for conservation of mass,
Since volume is arbitrary, then 
, which can be written
In terms of the total derivative 
, then
This is called the continuity equation.
In terms of the dilatation,
and 
Thus, the density change is proportional to the negative of
the dilatation change. The negative sign
implies that density decreases when 
increases or volume
dilates.
This can be used for a function
where 
is the density.
The total derivative of G is
Because the last two terms in the integrand are zero from the continuity equation, then
For example, for the total momentum equation
from conservation of linear momentum,
This was used in derivation of linear momentum equation above.
Conservation of
Angular Momentum
We showed
in an earlier lecture that conservation of angular momentum results in the
symmetry of the stress tensor. Thus, 
which reduces the
stress matrix to six independent numbers.
Conservation of
Energy
We will typically assume that heat transfer is small in elastic wave propagation problems (it is an adiabatic process). To first order, work done on system is stored as strain energy and is completely reversible upon unloading since the strain is on the order of 10-4 in seismic wave propagation problems. In general, however, there will be some heat dissipation which will result in attenuation of the seismic wave.