G141 Earthquakes and Volcanoes
STRESS, STRAIN, AND THE PHYSICS OF EARTHQUAKE GENERATION,
by Dr. Michael Hamburger, Indiana University
Introduction
In science, the simplest questions often prove to be among the hardest to answer. The same is true
of seismology. The first question that comes to the student of seismology - Why do earthquakes
occur? - usually calls forth a long-winded, circuitous response. This much we do know: that
earthquakes are related to fracturing of the Earth’s crust, and that this fracturing results from forces
applied to the rocks in the crust. Thus, we must first understand something about the nature of the
forces acting on the Earth; then we must explore the behavior of Earth materials when subjected to
these forces; and finally, we must examine the processes that lead to catastrophic failure of the rock.
In the previous chapter, we examined some of the deep-seated processes of plate tectonics that lead to
movement of the Earth’s lithospheric plates. In this chapter, we examine the manner in which these
forces can act on rocks in the Earth’s crust, and in turn, to trigger earthquakes.
Because earthquakes occur some 10-20 km below the Earth’s surface, it is often difficult for
geologists to infer the exact conditions that lead to earthquake-related fracturing. As an alternative,
geologists have tried to examine the behavior of rocks under controlled conditions in the laboratory.
Geologists can study how rocks behave by subjecting them to large forces, under a wide variety of
conditions, such as ambient temperature, pressure, water content, or chemical environment. The
discipline of rock mechanics involves characterizing the physical behavior of earth materials in the
laboratory. A more complete understanding of what controls the fracture of rock is also crucial to
understanding the fundamental causes of earthquakes, and ultimately, how to predict them.
Stress, Strain, and Deformation
In order to understand the forces that generate earthquakes, we first need to acquaint ourselves with
some of the specialized terminology of rock mechanics.
What measure do we use to describe the forces that act on a body of rock? We must begin with a
simple definition of force - going back to Isaac Newton’s classic 17th century tract on physics,
Principia. Anyone remember Newton’s Second Law of Motion? "The change of motion of a body is
proportional to the force impressed upon it." Newton expressed this relationship in his famous
equation
F = m a,
where F represents the force applied to a body, m is its mass, and a is the acceleration (the rate of change
of its velocity) induced by the force. Thus, a force can be calculated by measuring the mass of an
object (expressed in units of kilograms, or kg), and the rate at which it accelerates (units of meters per
second per second, or m/sec2) when subjected to the force. [Note that although we commonly
interchange the concepts, what we refer to as the weight of an object is not its mass, but the force
acting on that object by the Earth’s gravity.] Thus, force can be expressed in units of kg m/sec2, or,
for convenience, Newtons (or N). [Just to confuse beginning students of physics, in the English
system, we usually use the same unit for both mass and force--pounds.]
Figure 3.1. Schematic diagram of a
rock press, with the cylindrical rock
specimen placed beneath a powerful
piston, encased in a specially designed
pressure vessel to mimic the conditions
of high confining pressure that exist
deep within the Earth. From Paterson,
Experimental Rock Deformation, NY,
Springer-Verlag, 1978.
Now, what force does it take to break a piece of rock? It depends entirely on the manner in which
the force is applied. Consider the slab of rock shown in Figure 3.2. In one case (Fig. 3.2a), the force
is applied over a broad area of the rock’s surface. In another configuration (Fig. 3.2b), the force is
concentrated on a small area of the rock’s face. The rock might be able, in the first case, to
withstand the force, whereas in the second, the rock might fracture. Physicists describe this degree
of concentration of forces as the stress to which the rock is subjected. Formally, stress is described as
the force applied per unit area of the rock, in units of Newtons per square meter (or N/m2). Stress can be
expressed in units of pascals (or Pa), where 1n/m2 = 10 Pa. (In the English system, the units are pounds
per square inch, or psi). When stresses are applied equally on all sides of a body, as when a rock is
surrounded by water or air, it is referred to as pressure, using the same units as stress.
Figure 3.2. The same force, applied to a rock over a broad surface area results in a low applied
stress, whereas the same force, applied over a smaller area produces a much higher stress.
All materials respond to the application of moderate stresses by changing their form, or size, or
both. This change is referred to as strain or deformation. The nature of the strain depends on the
manner in which the stress is applied (Figure 3.3). For instance, strain can result from acting in
compressional stresses, where the material is pushed together, or from tensional stresses, where the rock is
pulled apart. Stress can be applied equally on all sides of a body (i.e., pressure), or in a single
direction ( "uniaxial stress"). Stress can act to bend or twist a body ("shear stress"). Because strain
can be expressed as a percentage change - in length, in volume, or in shape - it is commonly
expressed as a percent, without any metric units attached.
A striking example of compressional strain is shown in Figure 3.4. The fossils shown in this
photograph were originally nearly circular. Prolonged exposure to compressional stress (during the
collision between Africa and North America about 300 million years ago) led to deformation of the
rock, and the fossils changed their form from circular to elliptical. Geologists can use such deformed
fossils to quantify the stresses that acted on the rock and the degree of compressional strain that
resulted from this applied stress.
Figure 3.3. Types of stress
application and their resultant
strain.
What is the relation between stress and strain? Many materials, when subjected to stress, will deform
in an amount that is directly proportional to the stress, just as a spring will shorten or lengthen in
proportion to the force applied (Figure 3.5). Thus, if we make a plot of stress vs. strain (or applied
force vs. length change for the spring), the graph will form a straight line. And, as with a spring, the
material commonly returns to its original shape when the applied stress is removed. In this case, we
can describe the strain as reversible. These two properties - a linear relation between stress and strain,
and reversibility of deformation - are what define elastic deformation. Any solid that displays this style
Figure 3.4. An example of compressional strain. The undeformed fossils shown in the left-hand
photograph are nearly spherical, whereas those in the right-hand photograph were subjected to
intense compressional stress, and have been deformed into elliptical shapes. From Twiss & Moores,
Structural Geology, NY, W.H. Freeman, 1992.
of deformation can be called an elastic solid. These properties of elastic materials were first described
by the British physicist Sir Robert Hooke, and Hooke’s Laws, which describe deformation of elastic
solids, bear his name.
Figure 3.5. Deformation of elastic
materials, according to Hooke’s Laws.
In (a) the change of length of the
spring is proportional to the applied
stress. Similarly, in (b), a cylindrical
rock specimen subjected to
compressional stress has its volume
reduced in proportion to the applied
stress. In both cases, a plot of stress
vs. strain results in a straight line.
From Skinner & Porter, The Dynamic
Earth, NY, J. Wiley, 1989.
Most rocks, when subjected to small stresses, at pressures and temperatures typical of those near the
Earth’s surface, can be described as elastic solids: their deformation remains proportional to the
applied stress, and the strain disappears after the stress is removed (Figure 3.6a). However, the key
to this statement is small stresses: all materials have a yield strength, beyond which elastic deformation no
longer takes place. At this point, one of two things may happen. For some materials, the stress
applied to the rock may exceed its breaking strength, and the rock may respond by brittle fracture
(Figure 3.6a). When a rock is fractured, its deformation is irreversible, and the strain can no longer
be described as elastic. For other materials, another kind of irreversible strain can occur: ductile
deformation. Ductile deformation occurs by plastic-like flow of rock. As any kid knows (even if he
can’t pronounce the term ‘ductile deformation’!), a piece of bubble gum, when pulled, deforms
ductilely. In ductile deformation, the strain need not be proportional to the applied stress (Figure
3.6b), arid the strain typically is not reversible.
Figure 3.6. Examples of brittle (left) and ductile (right) deformation.
Stress and Strain in the Earth
In the Earth, we can see examples of both brittle and ductile deformation. Brittle deformation
(Figure 3.7a) is commonly expressed by the occurrence of faults (fractures along which relative
movement has taken place) and joints (fractures without relative movement). Ductile deformation
(Figure 3.7b) is commonly observed by folding of rock layers. Deep in the earth, ductile deformation
can take place by plastic flow of hot, weak rock (Figure 3.7c).
Figure 3.7. Examples of brittle deformation, resulting in faults (left), and ductile deformation,
resulting in folding of rock strata (center), or in plastic flow of rock (right). Photographs are from
Press & Siever, Understanding Earth, NY, W.H. Freeman, 1994, and from Suppe, Principles of Structural
Geology, Englewood Cliffs, Prentice-Hall, 1985.
Why are faults observed in some locales, and folds in others? Certainly, the material itself is partly
responsible. A slab of wood, for instance, will respond very differently to applied stress than will a
slab of rock. Similarly, different rock materials - for instance, shale vs. granite - have very different
elastic limits and rupture strengths. In fact, some common materials (such as steel) may exhibit all
three styles of deformation (Figure 3.8). Under low applied stress, the strain is elastic, and is
proportional to the applied stress. As the stresses increase, the material may reach its elastic limit,
beyond which the material begins to behave in a ductile manner. And continued stress eventually
will lead to brittle fracture.
Figure 3.8. Material properties of
steel. When subjected to moderate
stresses, the strain is proportional
to stress, and the steel deforms
elastically. At higher stresses, where
the elastic limit is exceeded, the
steel deforms plastically (ductile
behavior). At even higher stresses,
the yield strength is exceeded, and
the steel breaks by brittle failure.
To make matters even more complicated, the same material can have different elastic properties
depending on its physical state. To illustrate this phenomenon, consider what would happen if you
took that same piece of bubble gum, and left it in the freezer for a few hours, and then pulled on it?
Now, it might behave quite differently; it might deform by brittle fracture rather than by ductile
deformation. In other words, its elastic properties have changed as a result of the change in ambient
conditions. Similarly, earth materials may respond differently under different conditions. There are
many factors that control the style of deformation in the Earth. In the following section, we briefly
discuss a few of the principal ones.
Rock composition. The minerals that make up a rock have a primary control over its behavior. For
instance, minerals with a sheet-like structure (such as mica, clay, or gypsum) tend to have low elastic
limits, and permit a rock containing them to deform in a ductile fashion. Rocks containing salt, gypsum, or calcite also tend to be relatively ductile. The common rock types that contain these minerals
are limestone, marble, shale, and schist. Rocks dominated by stronger minerals (such as quartz, olivine, or garnet) tend to be stronger, and deform elastically up to a high yield strength. These include
sandstone, granite, basalt, diorite, and gneiss. The nature of the materials in the rocks pore spaces
and fractures also plays a significant role. The presence of interstitial water tends to weaken chemical
bonds and reduce friction between neighboring rock grains. Thus, relative to rocks with low water
content, "wet" rocks tend to have lower strength, and are more liable to ductile deformation.
Temperature. As we warm up our piece of frozen bubble gum, its behavior changes from brittle to
ductile. Similarly, at higher temperatures, rocks become more ductile. Thus, for a given material, as
the temperature is increased, the elastic limit is reached more quickly, and ductile behavior is favored
(Figure 3.9a). Because the Earth’s temperature rapidly increases with depth (see chapter 2) rocks are
expected to change from brittle to ductile at depths of 12-20 km below the Earth’s surface. When
rocks that were buried deep in the Earth’s crust are exposed at the surface, we can see textures that
look a little like toothpaste, suggesting that the rocks deformed by plastic flow (Figure 3.7c). This
may explain why in most areas of the world, earthquakes occur only in the upper 20 km of the crust.
[The exception to this rule is in subduction zones, where earthquakes may extend to depths as great as
700 km. Why should earthquakes occur at these unusual depths?]
Figure 3.9. Influence of
temperature and pressure on the
elastic properties of rock.
Pressure. With increasing depth in the Earth, the pressure on the rock rapidly increases, due to the
enormous weight of the overlying rock, just as the water pressure increases with depth in the oceans.
This increasing pressure hinders the formation of fractures, and enhances mechanisms of ductile
flow; thus the effect is to reduce the elastic limit and expand the domain of ductile deformation
(Figure 3.9b). The effect of pressure is dramatically illustrated by the effects of compression on a
cylindrical rock sample at various confining pressures (Figure 3.10).
Figure 3.10. The effect of
confining pressure on the style of
deformation in marble. All of the
samples of marble began with the
same original length. From left to
right, the samples were subjected
to progressively greater confining
pressure, and the mode of
deformation changed from brittle
to ductile. From Twiss & Moores,
Structural Geology, NY, W.H.
Freeman, 1992.
Thus pressure acts in concert with temperature increases to ensure that brittle deformation is
normally limited to the Earth’s outermost layers. Seismologist Arch Johnston of the University of
Memphis has suggested that the extra pressure generated by large ice sheets in Antarctica and
Greenland is sufficient to inhibit brittle deformation - thus explaining the absence of earthquakes in
the world’s polar regions.
Time and Strain Rate. Ever played with a piece of silly putty? When pulled slowly, the putty stretches
into long rubbery strands, a perfect example of ductile deformation. Yet when stretched suddenly,
the putty snaps like a piece of hard resin, i.e., it behaves elastically. Believe it or not, over much
longer time periods, ordinary window glass displays the same brittle/ductile behavior. We all know
that when bent quickly, the glass will shatter, but have you ever seen what happens to a sheet of
glass in an antique cabin? The window glass, just left to stand in a window for a hundred years or so,
begins to slowly flow, resulting in an irregular glass sheet, thin at the top and thick at the bottom,
with flow lines distorting the view out the window. Time has a similar effect on rocks in the Earth’s
crust - the longer they are subjected to a given stress, the more apt they are to behave in a ductile
manner. In other words, the rate at which a rock changes its shape or volume-its strain rate - governs
its behavior: like the silly putty or glass, at lower strain rates, rocks tend to exhibit ductile behavior,
while at high strain rates, the rocks become brittle (Figure 3.8c).
Measuring Deformation in the Earth
As a result of the slow, deep-seated movements of the Earth’s lithospheric plates, the rocks at the
Earth’s surface are subjected to deforming forces - acting in compression, tension, or shear. These
forces in turn deform the Earth’s surface. How do we know that this deformation is taking place?
Since early in this century, geologists have recognized that modern surveying methods could be used
to measure the rate and style at which the Earth’s surface is deforming. Beginning early in the last
century, surveyors marked the territory of the United States with a network of survey benchmarks;
these survey marks are the official basis for all of the country’s boundary lines - including political
borders, property boundaries, mineral rights - and thus have to be done pretty precisely.
Surveyors have been able to use an array of fancy surveying tools, including optical theodolites and
transits to measure angles, laser reflectors to measure distances, and more recently, satellite-based
Global Positioning System (GPS) receivers to obtain precise relative positions of their survey
benchmarks. With these measurements in hand, surveyors began noting that their benchmarks
weren’t staying in the same place; rather they were moving around in systematic ways. Geologists
working near the San Andreas fault, for instance, noted that some of their geodetic lines were
systematically growing longer, while others were growing systematically shorter (see Bolt, Figure
6.2). The pattern could be explained simply by the accumulation of shear strain across the San
Andreas fault. In other words, the areas east of the fault are gradually moving to the southeast, at
rates of 2-3 cm/yr. with respect to areas west of the fault (See Figure 3.11). This gradual movement
is entirely consistent with the theory of plate tectonics, which contends that points to the east of the
San Andreas must be attached to the North American plate, while those to the west of the fault, lie
on the Pacific Plate and are moving, together with Hawaii and Los Angeles, to the northwest. These
gradual motions, amounting to a few cm/yr, translate to a few thousand kilometers over the course of
100 million years - enough to explain the global-scale movement of continents that Wegener first
documented at the turn of the century. On a more political level, that has enormous implications as
well: we can calculate that in about 20 million years, San Francisco will become just another eastern
suburb of L.A!
Strain Leads to Stress
Two or three centimeters per year may not sound like much, but over the course of a century, the
total offset amounts to 2-3 meters-equivalent to the slip that has been observed to take place in a
large earthquake. It was just this observation that pointed geologist H.F. Reid in 1906 to his
revolutionary theory of elastic rebound. Reid made careful measurements of the offsets of the Earth’s
crust that took place during the 1906 earthquake, which varied along the fault - anywhere from 1 to
6 meters of slip was observed. He then compared these observations with geodetic measurement of
strain over the fifty years prior to the 1906 quake. He calculated that 3.2 meters of relative movement had taken place during those 50 years, and as much as 6-lOm of slip might have accumulated
since the last large earthquake on the San Andreas. Reid hypothesized that this slow accumulation of
Figure 3.11. Measurement of gradual
deformation of the Earth’s crust
surrounding the San Andreas Fault,
using Global Positioning System
(GPS) satellite measurements. The
arrows in this map represent motion
of points on the Earth’s surface, with
respect to the North American Plate.
Note 10 mm/yr (1 cm/yr) reference
arrow in upper right. Thus, the points
near the Pacific coast are moving at
rates approaching 5 cm/yr. The San
Andreas (represented by the light line
running diagonally across the map)
accommodates about ½ this total
movement. From Bennett et al,
Journal of Geophysical Research, v. 101, p.
21946.
strain, if it were not continually released, could explain the catastrophic release of energy that takes
place during an earthquake. Thus, as illustrated in Figure 3.12, the rocks on either side of a fault
must be gradually slipping past one another, whereas at the fault zone, the crust is locked, gradually
accumulating strain in the Earth’s crust.
Figure 3.12. Cartoon illustrating
the principle of elastic rebound. In
the years prior to an earthquake (a),
slow motion of the crust on either
side of a fault (black line) results in
gradual deformation of the ground
surface. When the crust is strained
beyond its yield strength, a rupture
takes place, and the rock
surrounding the fault snaps back to
its original shape, but leaving a
marked rupture along the fault.
From Bolt, Earthquakes, W.H.
Freeman, 1993.
This flexure of the crust, like the bending of a wooden stick, places a greater and greater stress on
the rock near the fault. When that stress exceeds the yield strength of the rock, the fault slips, and
the surrounding rock rapidly snaps back, or rebounds, to its original shape - and the result is an
earthquake. Expressed in more formal terms, the rock can be said to be storing strain energy over
the years preceding an earthquake, as a result of the gradual movement of the Earth’s plates. When
the rock eventually fractures, all of that strain energy is suddenly released in the form of kinetic energy
(i.e., energy of motion). Some of this kinetic energy is released in the form of the destructive seismic
waves that we associate with an earthquake, and much of it is released as heat. Once released, the
cycle begins anew, with strain gradually accumulating again, until the rock is once again brought to
its yield strength. This process, of gradual strain accumulation and sudden strain release, can explain
the cyclicity of earthquakes observed in most earthquake–prone areas. Ultimately, it is this cycle of
strain accumulation and release that offers us some hope of predicting earthquakes in the future.