Gabriel Lamé (1795–1870) introduced another
elastic constant, , in 1852 (Fung, 1969) and this is
related to Young’s modulus and Poisson’s ratio as:
(8.27)
Poisson’s ratio is related to Lamé’s constant as:
(8.28)
The constant characterizes the spectrum of
behaviors from perfectly compressible materials,
ϭ 0, for which ϭ 0, to incompressible materials,
ϭ 1/2, for which ϭϱ. Taken together, and G
sometimes are called Lamé’s constants. Equations
(8.19) may be written in terms of Lamé’s constants
using (8.26) and (8.27) such that:
(8.29)
The relatively simple behavior of the isotropic and
linear elastic solid makes it ideally suited for
analysis in continuum mechanical models. One
can solve problems that involve quite complex
geometries and boundary conditions by restricting the material behavior in this way, and the
solutions provide important insights about the
origins of some geologic structures.
8.3 Quasi-static displacement
boundary value problems
One may formulate a problem in elasticity
theory in terms of the displacement components, or in terms of the stress components, as
the dependent variables. In this section we take
the equations of motion written in terms of the
displacement components as the dependent
variables. A specific example is provided using
the two-dimensional solution for an edge dislocation which has surprisingly broad applications in structural geology, ranging from
micrometer-scale defects in mineral grains to
plate-bounding faults at continental margins
(Weertman and Weertman, 1964; Weertman,
1996). Textbooks on elasticity theory provide
many other useful solutions for displacement
boundary value problems (Muskhelishvili, 1954;
ij ϭ 2G ij ϩ kk ␦ ij
ϭ
2(G ϩ )
ϭ
Ev
(1 ϩ )(1 Ϫ 2)
Sokolnikoff, 1956; Timoshenko and Goodier,
1970; Barber, 1992).
The mathematical structure of the theory of
elasticity makes it possible to solve boundary
value problems of elasticity using displacement,
traction, or mixed boundary conditions. This fact
led to a debate about whether boundary displacements “cause” the elastic solid to deform (Marrett
and Peacock, 1999; Tikoff and Wojtal, 1999;
Peacock and Marrett, 2000; Pollard, 2000). In the
authors’ opinion the opportunity to employ displacement boundary conditions is no more than
a consequence of the underlying mathematical
relationships among traction and displacement
components. In a Newtonian context one would
seek an “explanation” for the specified displacements in terms of forces applied exterior to the
model boundary, and these forces would be the
ultimate causative agents for the resulting deformation in the interior. On the other hand, where
tractions are the prescribed boundary conditions
one may refer to the associated forces as the
causative agents for the deformation.
8.3.1 Two-dimensional plane strain
solutions for cylindrical structures
All structures in the Earth are three dimensional,
but there are circumstances in which it is appropriate to ignore some of the components of stress,
strain, and displacement. This has practical implications because the mathematical complexity of
the boundary value problem is greatly reduced.
One circumstance involves structures that are
very long in one dimension relative to their size in
the other two dimensions. If the geometry of such
a structure does not change significantly along its
length, it may be described as a cylindrical structure.
A common geological example would be the
surface of a sedimentary layer, folded into a shape
that may be approximated by moving the fold axis
through space without changing its orientation
(Fig. 8.7a). Other common examples include the
surfaces of blade-like dikes in volcanic rift zones
and of vertical joints confined between two horizontal sedimentary layers (Fig. 8.7b).
The special case of deformation that applies to
two-dimensional cylindrical structures is called
plane strain. Here the (x, y)-plane is taken as the
plane of interest (Fig. 8.7), and we postulate that
8.3 QUASI-STATIC DISPLACEMENT BOUNDARY VALUE PROBLEMS
299
elastic constant, , in 1852 (Fung, 1969) and this is
related to Young’s modulus and Poisson’s ratio as:
(8.27)
Poisson’s ratio is related to Lamé’s constant as:
(8.28)
The constant characterizes the spectrum of
behaviors from perfectly compressible materials,
ϭ 0, for which ϭ 0, to incompressible materials,
ϭ 1/2, for which ϭϱ. Taken together, and G
sometimes are called Lamé’s constants. Equations
(8.19) may be written in terms of Lamé’s constants
using (8.26) and (8.27) such that:
(8.29)
The relatively simple behavior of the isotropic and
linear elastic solid makes it ideally suited for
analysis in continuum mechanical models. One
can solve problems that involve quite complex
geometries and boundary conditions by restricting the material behavior in this way, and the
solutions provide important insights about the
origins of some geologic structures.
8.3 Quasi-static displacement
boundary value problems
One may formulate a problem in elasticity
theory in terms of the displacement components, or in terms of the stress components, as
the dependent variables. In this section we take
the equations of motion written in terms of the
displacement components as the dependent
variables. A specific example is provided using
the two-dimensional solution for an edge dislocation which has surprisingly broad applications in structural geology, ranging from
micrometer-scale defects in mineral grains to
plate-bounding faults at continental margins
(Weertman and Weertman, 1964; Weertman,
1996). Textbooks on elasticity theory provide
many other useful solutions for displacement
boundary value problems (Muskhelishvili, 1954;
ij ϭ 2G ij ϩ kk ␦ ij
ϭ
2(G ϩ )
ϭ
Ev
(1 ϩ )(1 Ϫ 2)
Sokolnikoff, 1956; Timoshenko and Goodier,
1970; Barber, 1992).
The mathematical structure of the theory of
elasticity makes it possible to solve boundary
value problems of elasticity using displacement,
traction, or mixed boundary conditions. This fact
led to a debate about whether boundary displacements “cause” the elastic solid to deform (Marrett
and Peacock, 1999; Tikoff and Wojtal, 1999;
Peacock and Marrett, 2000; Pollard, 2000). In the
authors’ opinion the opportunity to employ displacement boundary conditions is no more than
a consequence of the underlying mathematical
relationships among traction and displacement
components. In a Newtonian context one would
seek an “explanation” for the specified displacements in terms of forces applied exterior to the
model boundary, and these forces would be the
ultimate causative agents for the resulting deformation in the interior. On the other hand, where
tractions are the prescribed boundary conditions
one may refer to the associated forces as the
causative agents for the deformation.
8.3.1 Two-dimensional plane strain
solutions for cylindrical structures
All structures in the Earth are three dimensional,
but there are circumstances in which it is appropriate to ignore some of the components of stress,
strain, and displacement. This has practical implications because the mathematical complexity of
the boundary value problem is greatly reduced.
One circumstance involves structures that are
very long in one dimension relative to their size in
the other two dimensions. If the geometry of such
a structure does not change significantly along its
length, it may be described as a cylindrical structure.
A common geological example would be the
surface of a sedimentary layer, folded into a shape
that may be approximated by moving the fold axis
through space without changing its orientation
(Fig. 8.7a). Other common examples include the
surfaces of blade-like dikes in volcanic rift zones
and of vertical joints confined between two horizontal sedimentary layers (Fig. 8.7b).
The special case of deformation that applies to
two-dimensional cylindrical structures is called
plane strain. Here the (x, y)-plane is taken as the
plane of interest (Fig. 8.7), and we postulate that
8.3 QUASI-STATIC DISPLACEMENT BOUNDARY VALUE PROBLEMS
299