1. Identify a natural experiment run under the
conditions and at the scale of interest.
2. Use mapping techniques in the field to characterize the structures, identify the lithologies,
and measure the relevant geometric parameters.
3. Infer the sequence of deformation and appropriate boundary conditions for loading and
displacement (or other relevant physical quantities) from the field data.
4. Set up and solve, or borrow from the literature,
the appropriate mechanical problem (usually a
boundary or initial value problem from continuum mechanics).
5. Derive from the solution to this problem an
equation for the physical property of interest in
terms of the measured and/or inferred quantities.
6. Use the derived equation and available data to
estimate the physical property.
This method is not limited to rocks with elastic
properties, but is generally applicable to any material behavior described by a well-defined constitutive law. Nor is it limited to dikes, but is
generally applicable to any geological structure
that can be modeled using continuum mechanics.
8.2 The idealized elastic material
The general concept of an elastic material is one in
which the current configuration depends only on
the initial (unstressed) configuration and the
current state of stress, and not on the history of
deformation from the initial to the current state
(Truesdell and Noll, 1965). In most applications of
elasticity theory one does not study the most
general elastic material, but rather one in which
the stress is linearly related to the infinitesimal
strain. Because the infinitesimal strain is an approximation these elastic models must be considered an
approximation. Never-the-less linear elastic theory
has provided a wide variety of useful solutions for
mechanical problems in structural geology.
8.2.1 The elastic solid
One common experience with the mechanical
behavior of rock comes at the moment the geologist’s rock hammer hits an exposure. If the exposure is fresh granite, the hammer springs back
quickly with a high-pitched ringing and vibrates
vigorously for a second or two. The rock also gives
off a sharp audible report. Perhaps a few small
chips of rock or metal shoot out from the point of
impact if the blow is particularly aggressive, but
lighter blows permanently deform neither the
hammer nor the rock. Both rock and hammer
return nearly to their shapes just before the
impact. If you held your hand on the exposure
near the point of the hammer blow you would feel
vibrations in the granite, a result of waves propagating out from the point of impact. Other types
of rock respond in a similar fashion, but with
some quantitative differences. For example, sandstone might respond with a duller sound and the
hammer will not seem to jump back as quickly.
When materials return essentially to their original shape after the applied loading is removed, we
say they are elastic.
A linear relationship between force and extension is credited to the English natural philosopher
Robert Hooke (1635–1703) who published a statement on the subject in 1676 in the following
remarkable form (Gordon, 1976):
c e i i i n o s s s t t u u
Hooke apparently was intent on laying claim to
this area of research, without providing the
specifics, before the appearance of a lengthier
treatise. The anagram has the following solution
published in 1679 by Hooke:
Ut tensio sic vis
J. E. Gordon provides the following translation
(Gordon, 1976):
As the extension, so the force
In other words, the extension is proportional to
the force.
Despite his obfuscation in presenting this discovery, Robert Hooke is honored by having this
relationship, and a generalized version that
extends the simple one-dimensional concept to
three dimensions, referred to as Hooke’s Law of
linear elasticity. The book The Abyss of Time by
Claude C. Albritton (1980) contains an informative
account of Hooke’s life and his scientific contribu292
ELASTIC DEFORMATION
conditions and at the scale of interest.
2. Use mapping techniques in the field to characterize the structures, identify the lithologies,
and measure the relevant geometric parameters.
3. Infer the sequence of deformation and appropriate boundary conditions for loading and
displacement (or other relevant physical quantities) from the field data.
4. Set up and solve, or borrow from the literature,
the appropriate mechanical problem (usually a
boundary or initial value problem from continuum mechanics).
5. Derive from the solution to this problem an
equation for the physical property of interest in
terms of the measured and/or inferred quantities.
6. Use the derived equation and available data to
estimate the physical property.
This method is not limited to rocks with elastic
properties, but is generally applicable to any material behavior described by a well-defined constitutive law. Nor is it limited to dikes, but is
generally applicable to any geological structure
that can be modeled using continuum mechanics.
8.2 The idealized elastic material
The general concept of an elastic material is one in
which the current configuration depends only on
the initial (unstressed) configuration and the
current state of stress, and not on the history of
deformation from the initial to the current state
(Truesdell and Noll, 1965). In most applications of
elasticity theory one does not study the most
general elastic material, but rather one in which
the stress is linearly related to the infinitesimal
strain. Because the infinitesimal strain is an approximation these elastic models must be considered an
approximation. Never-the-less linear elastic theory
has provided a wide variety of useful solutions for
mechanical problems in structural geology.
8.2.1 The elastic solid
One common experience with the mechanical
behavior of rock comes at the moment the geologist’s rock hammer hits an exposure. If the exposure is fresh granite, the hammer springs back
quickly with a high-pitched ringing and vibrates
vigorously for a second or two. The rock also gives
off a sharp audible report. Perhaps a few small
chips of rock or metal shoot out from the point of
impact if the blow is particularly aggressive, but
lighter blows permanently deform neither the
hammer nor the rock. Both rock and hammer
return nearly to their shapes just before the
impact. If you held your hand on the exposure
near the point of the hammer blow you would feel
vibrations in the granite, a result of waves propagating out from the point of impact. Other types
of rock respond in a similar fashion, but with
some quantitative differences. For example, sandstone might respond with a duller sound and the
hammer will not seem to jump back as quickly.
When materials return essentially to their original shape after the applied loading is removed, we
say they are elastic.
A linear relationship between force and extension is credited to the English natural philosopher
Robert Hooke (1635–1703) who published a statement on the subject in 1676 in the following
remarkable form (Gordon, 1976):
c e i i i n o s s s t t u u
Hooke apparently was intent on laying claim to
this area of research, without providing the
specifics, before the appearance of a lengthier
treatise. The anagram has the following solution
published in 1679 by Hooke:
Ut tensio sic vis
J. E. Gordon provides the following translation
(Gordon, 1976):
As the extension, so the force
In other words, the extension is proportional to
the force.
Despite his obfuscation in presenting this discovery, Robert Hooke is honored by having this
relationship, and a generalized version that
extends the simple one-dimensional concept to
three dimensions, referred to as Hooke’s Law of
linear elasticity. The book The Abyss of Time by
Claude C. Albritton (1980) contains an informative
account of Hooke’s life and his scientific contribu292
ELASTIC DEFORMATION