Purdue University



Purdue University

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

[pic]

Let the particle velocity be [pic] and let dm be the mass of a particle at P, then the integrated momentum is [pic]. This is the total momentum summed over all the particles of mass that build up the body.

Conservation of linear momentum requires

[pic]

where [pic] 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 [pic].

b) Body forces

These are forces applied in the interior of the body by external force fields. For gravity,

[pic]

where [pic] is body force and [pic] 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

[pic]

Using conservation of mass (see below), we may write

[pic]

In component form, the linear momentum equation then can be written as

[pic]

We would like to have the surface integral transformed into an equivalent volume integral. This may be done using

[pic]

From Cauchy’s formula [pic], we can write the boundary tractions in terms of the stress tensor. Then,

[pic]

This surface integral may be transformed into a volume integral using [pic] as

[pic]

Putting this into the momentum equation gives

[pic]

Define the density as [pic], then,

[pic]

If the volume under consideration is arbitrary, we can equate the integrands to find

[pic]

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 [pic] 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

[pic]

where [pic] is the total derivative, [pic] is the local derivative, and [pic] 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, [pic], the particle will have moved [pic]. Then, the change of an arbitrary function g will be

[pic]

Expanding in a Taylor series,

[pic]

Then,

[pic]

Finally, the total derivative is

[pic]

In the linear approximation, we ignore the advection term and make the approximation

[pic]

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

[pic]

then

[pic]

Conservation of Mass

To show conservation of mass, let the mass M = [pic]. Then,

[pic]

where the last term is the material flux through the boundary. This can be changed to a volume integral to obtain

[pic]

where [pic] is the flux of the material. Now for conservation of mass,

[pic]

Since volume is arbitrary, then [pic], which can be written

[pic]

In terms of the total derivative [pic], then

[pic]

This is called the continuity equation.

In terms of the dilatation,

[pic] and [pic]

Thus, the density change is proportional to the negative of the dilatation change. The negative sign implies that density decreases when [pic] increases or volume dilates.

This can be used for a function

[pic]

where [pic] is the density.

The total derivative of G is

[pic]

[pic]

[pic]

Because the last two terms in the integrand are zero from the continuity equation, then

[pic]

For example, for the total momentum equation

[pic]

from conservation of linear momentum,

[pic]

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, [pic] 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.

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