I
n this chapter we describe how the elastic properties of rock are measured in the laboratory
and provide tables of numbers representing the
range of values for different rock types. However,
the need to understand and measure the resistance to deformation of rocks goes well beyond
the simple accumulation of numbers in handbooks of rock properties. To paraphrase Truesdell
and Noll (1965), the aim of structural geology is to
construct mathematical models that enable us,
from use of knowledge gathered in a few observations, to predict by logical processes the outcomes
in many other circumstances. To analyze a geologic structure one must choose the appropriate
boundary or initial value problem to serve as a
mechanical model. To formulate such a problem
one must postulate a particular mechanical
behavior. That is, one must say exactly what the
relationship is between the stress acting within a
material and some measure of the deformation,
usually strain or rate of deformation. These
relationships are called constitutive equations. For
example, researchers studying the displacement
field around an edge dislocation in silicon as
revealed by electron microscopy (Chapter 8, frontispiece) postulated an anisotropic linear elastic
constitutive law and calculated model displacements that are remarkably similar to those
observed.
We begin this chapter by describing the deformation of the Mancos Shale associated with the
emplacement of a basaltic dike near Ship Rock,
New Mexico, about 30 million years ago. In this
example the shale was compressed as magma
pressure forced the dike open, so the thickness of
the dike gives us a measure of the resistance to
deformation of the surrounding rock. Next we
introduce the formal concept of linear elasticity
as first envisioned by Robert Hooke in 1676, along
with the constitutive equations that connect
stress to strain for idealized elastic materials.
These constitutive equations are central to the
infinitesimal theory of an elastic continuum. As
Truesdell and Noll (1965) suggest in the opening
quotation for this chapter, elastic theory has
played an enormous role in the development of
science and engineering. Some examples are
reviewed to illustrate the type of contributions
this theory has made to structural geology.
Next, the measurement techniques used to
determine elastic properties in the laboratory and
at engineering field sites are described. This leads
to a discussion of how elastic properties vary with
the size of the rock mass being tested. Considerations of scale effects are closely linked to
those of heterogeneity with respect to elastic
properties. To build intuition a solution to a
boundary value problem is examined for a circular inclusion with different elastic stiffness and
compressibility than the surrounding material.
Finally, we consider how the elastic properties of
rock may vary with orientation, in other words
how anisotropic is rock with respect to elastic
properties? After reviewing Hooke’s Law for
anisotropic materials in general, and providing
some representative values of the elastic moduli
from laboratory measurements, we describe a
solution to a boundary value problem for an
orthotropic elastic material. While it is well
known that rock masses at the scale of Earth’s
crust can be heterogeneous and anisotropic to
some degree, it is noteworthy how well isotropic
and homogeneous elastic models correspond to
measured deformation. This is illustrated using
data on surface displacements during the 1999
Hector Mine earthquake.
8.1 Estimating rock properties
from geological field tests
To determine how rock masses resist deformation
one must conceive an experiment and build or
identify a testing apparatus that controls the
applied loads and facilitates measurement of the
resulting deformation. In the context of elastic
solids the experiment is designed to determine a
quantitative relationship between the stress and
the strain. Here we focus on such experiments,
conducted in the field rather than in a laboratory,
because this enables one to estimate the elastic
properties of rock at length scales of kilometers
which are relevant to the geological structures
under consideration.
288
ELASTIC DEFORMATION
n this chapter we describe how the elastic properties of rock are measured in the laboratory
and provide tables of numbers representing the
range of values for different rock types. However,
the need to understand and measure the resistance to deformation of rocks goes well beyond
the simple accumulation of numbers in handbooks of rock properties. To paraphrase Truesdell
and Noll (1965), the aim of structural geology is to
construct mathematical models that enable us,
from use of knowledge gathered in a few observations, to predict by logical processes the outcomes
in many other circumstances. To analyze a geologic structure one must choose the appropriate
boundary or initial value problem to serve as a
mechanical model. To formulate such a problem
one must postulate a particular mechanical
behavior. That is, one must say exactly what the
relationship is between the stress acting within a
material and some measure of the deformation,
usually strain or rate of deformation. These
relationships are called constitutive equations. For
example, researchers studying the displacement
field around an edge dislocation in silicon as
revealed by electron microscopy (Chapter 8, frontispiece) postulated an anisotropic linear elastic
constitutive law and calculated model displacements that are remarkably similar to those
observed.
We begin this chapter by describing the deformation of the Mancos Shale associated with the
emplacement of a basaltic dike near Ship Rock,
New Mexico, about 30 million years ago. In this
example the shale was compressed as magma
pressure forced the dike open, so the thickness of
the dike gives us a measure of the resistance to
deformation of the surrounding rock. Next we
introduce the formal concept of linear elasticity
as first envisioned by Robert Hooke in 1676, along
with the constitutive equations that connect
stress to strain for idealized elastic materials.
These constitutive equations are central to the
infinitesimal theory of an elastic continuum. As
Truesdell and Noll (1965) suggest in the opening
quotation for this chapter, elastic theory has
played an enormous role in the development of
science and engineering. Some examples are
reviewed to illustrate the type of contributions
this theory has made to structural geology.
Next, the measurement techniques used to
determine elastic properties in the laboratory and
at engineering field sites are described. This leads
to a discussion of how elastic properties vary with
the size of the rock mass being tested. Considerations of scale effects are closely linked to
those of heterogeneity with respect to elastic
properties. To build intuition a solution to a
boundary value problem is examined for a circular inclusion with different elastic stiffness and
compressibility than the surrounding material.
Finally, we consider how the elastic properties of
rock may vary with orientation, in other words
how anisotropic is rock with respect to elastic
properties? After reviewing Hooke’s Law for
anisotropic materials in general, and providing
some representative values of the elastic moduli
from laboratory measurements, we describe a
solution to a boundary value problem for an
orthotropic elastic material. While it is well
known that rock masses at the scale of Earth’s
crust can be heterogeneous and anisotropic to
some degree, it is noteworthy how well isotropic
and homogeneous elastic models correspond to
measured deformation. This is illustrated using
data on surface displacements during the 1999
Hector Mine earthquake.
8.1 Estimating rock properties
from geological field tests
To determine how rock masses resist deformation
one must conceive an experiment and build or
identify a testing apparatus that controls the
applied loads and facilitates measurement of the
resulting deformation. In the context of elastic
solids the experiment is designed to determine a
quantitative relationship between the stress and
the strain. Here we focus on such experiments,
conducted in the field rather than in a laboratory,
because this enables one to estimate the elastic
properties of rock at length scales of kilometers
which are relevant to the geological structures
under consideration.
288
ELASTIC DEFORMATION