Earthquakes and faults

The causes of earthquakes have interested man since antiquity. As was mentioned in section 1.1, various ideas have been proposed from the time of the ancient Greek natural philosophers to our days. During the 19th century systematic field studies after earthquakes were started and the first attempts to relate them to tectonic processes were made by Mallet (Naples, Italy, 1857), Koto (Neo, Japan, 1891), and Oldham (Assan, India, 1897) among others. With the increase in number of field observations and in precision of localization of epicenters, the correlation between earthquakes and faults became clearer. Authors such as Suess, Koto, Montessus de Ballore, and Sieberg assigned the cause of earthquakes to stresses accumulated in the Earth's crust by tectonic processes and their release by its fracture. The first mechanical model was presented by Reid (1911) in order to explain the origin of the San Francisco earthquake of 1906. His theory, known as elastic rebound, proposes that earthquakes take place by fracturing of the Earth's crust with the total or partial release of the elastic strain accumulated in a region owing to tectonic stress. According to plate tectonics, which was developed in 1960, tectonic stresses are ultimately related to the relative motion of lithospheric plates.

An earthquake can be considered to be produced by rupturing of part of the Earth's crust with a relative displacement of its two sides and the release of the accumulated elastic strain that had been produced by tectonic processes.

The representation of the source. Kinematic and dynamic models

We saw in Chapter 15 that earthquakes are produced by fractures in the Earth's crust. In Reid's model of elastic rebound, faulting is caused by the sudden release of accumulated elastic strain when the strength of the material is overcome. In seismology the problem of the source mechanism consists in relating observed seismic waves to the parameters that describe the source. In the direct problem, theoretical seismic wave displacements are determined from source models, whereas in the inverse problem, the parameters of source models are derived from observed wave displacements. The first step in both problems is to define the seismic source in terms of a mechanical model that represents the physical fracture. These models or representations of the source are defined by parameters whose number depends on their complexity. Simple models are defined by a few parameters whereas more complex ones require a larger number of parameters (Madariaga, 1983; Udías, 1991; Koyama, 1997).

Fracturing can be approached in two different ways, kinematic and dynamic. Kinematic models of the source consider the slip of the fault without relating it to the stresses that cause it. Fracturing is described purely in terms of the slip vector as a function of the coordinates on the fault plane and time. From models of this type, it is relatively simple to determine the corresponding elastic displacement field.

The historical evolution of seismographs

The oldest instrument used to detect the occurrence of an earthquake was probably constructed in China during the second century AD and is attributed to the philosopher Chian-hen. This instrument consisted in a bronze figure of a dragon with eight heads in whose mouths there were eight balls. Inside the figure there was some kind of pendular device that pushed the balls and made them fall when it was shaken by an earthquake. The figure was oriented in the geographic directions so that, upon the arrival of seismic waves, the corresponding ball will fall and show the occurrence and orientation of a shock. In Europe, the first instrument was a mercury seismoscope designed by De Haute-Feuille in 1703, consisting in a vessel with mercury connected by eight channels to eight cavities. Earthquakes will make the mercury flow into one or several of the cavities, indicating their orientations and sizes (quantities of mercury spilled). It is not certain that the instrument was actually built, although we have a description of it, but similar instruments were built in 1784 by Cavalli and in 1818 by Cacciatori (Ferrari, 1992). Vertical and horizontal pendulums started to be used around 1750. These instruments have an alarm to indicate the occurrence of an earthquake or a stylus attached to the mass that left a mark on sand or smoked plate of glass in which case they are called seismoscopes.

This textbook has been developed from 25 years of experience teaching seismology at the universities of Madrid and Barcelona. The text is at an introductory level for students in the last years of European licentiate or American upper-division undergraduate courses and at similar levels in other countries. As a first book, no previous knowledge of seismology, as such, is assumed of the student. The book's emphasis is on fundamental concepts and basic developments and for this reason a selection of topics has been made. It has been noticed that sometimes even graduate students lack a true grasp of the very fundamental ideas underlying some aspects of seismology. The most fundamental concepts are developed in detail. Simple cases such as one-dimensional problems and those in liquid media are used as introductory topics. Mathematical developments are worked out in complete detail for the most fundamental problems. Sometimes more difficult subjects are introduced, but not fully developed. In these cases references to more advanced books are given.

The book presupposes a certain amount of knowledge of mathematics and physics. Knowledge of mathematics at the level of calculus and ordinary and partial differential equations as well as a certain facility for vector and tensor analysis are assumed. Cartesian, spherical, and cylindrical coordinates and some functions such as Legendre and Bessel functions are used. Tensor index notation is used preferentially throughout the book. Fundamental ideas about certain mathematical subjects are given briefly in Appendixes 1–4.

There are many important systems that exhibit nonequilibrium or noncontinuum behavior. This final chapter examines some important examples of such systems. In doing so, we have two objectives. The first is to understand how, and under what conditions, the system behavior may deviate from the idealizations embodied in equilibrium theory or continuum theory. The second is to demonstrate theories and methods that are commonly used to model nonequilibrium and noncontinuum systems. Because they are commonly used to analyze such systems, kinetic theory and the Boltzmann transport equation are introduced. Nonequilibrium and noncontinuum phenomena associated with multiphase systems and electron transport in solids are examined in detail. The final section of Chapter 10 uses results from previous chapters to examine length scales and time scales at which classical and continuum theories become suspect. Doing so defines the range of conditions for which we expect classical and continuum theories to be accurate models of real physical systems. Although limited in its coverage, this chapter provides an introduction to microscale aspects of nonequilibrium and noncontinuum phenomena and serves to illustrate how they relate to the theoretical framework developed in the preceding chapters.

Basic Kinetic Theory

With increasing frequency engineers are dealing with microscale systems in which the applicability of classical macroscopic equilibrium thermodynamics becomes questionable. Generally, the applicability of classical equilibrium theory breaks down because the system is far from equilibrium and/or the system behavior deviates from a continuum model.