Here one postulates first-order cause-and-effect
relationships and uses these to exclude observations that are deemed to be irrelevant.
The third step involves the selection of the
branch of continuum mechanics that provides
the general boundary conditions for modeling.
Here the scaling arguments introduced in
Chapter 4 should prove helpful, and the concepts
of strain, rate of deformation, traction, and stress
described in Chapters 5 and 6 must be understood.
Although some sub-disciplines of continuum
mechanics are spelled out on the third stepping
stone, this list is not meant to be exclusive or to
imply that coupled problems, for example of fluid
flow and solid deformation, are unimportant.
Indeed, the level of understanding of coupled
problems among structural geologists and the
modeling resources now available in terms of
both hardware and software suggest that this will
be a very fruitful area of investigation in the
twenty-first century.
After selecting the theoretical context for an
investigation one should focus on the fundamental relationships, two of which are described in
Chapter 7 as the conservation laws of mass and
momentum. This fourth step along the path commonly is overlooked, but it provides a touchstone
that makes clear how the Laws of Motion have
been formulated and how the constitutive laws
are employed to reduce these to a specific set of
governing equations identified with the fifth step
along the path. At this point one has made an
explicit commitment to a material behavior and
perhaps simplified the equations of motion to
exclude temporal or spatial variations in material
properties. In this textbook we give considerable
attention to the linear elastic material as
described in Chapter 8 and the linear viscous
material as described in Chapter 10. One of the
most commonly employed governing equations in
the elastic context is the biharmonic equation for
the stress function in two dimensions. In the
viscous context the Navier–Stokes equations are
the benchmark.
In the sixth step one chooses a particular
model geometry, which may reduce the governing equations to two or even one spatial dimension. Here one may eliminate time altogether, as
in a quasi-static elastic problem or steady-state
viscous flow problem. Then the specific boundary
and/or initial conditions are selected to provide
the constants of integration necessary to solve the
governing equations. Sometimes, the specific
boundary conditions may be constrained by field
measurements, thereby establishing a direct link
with the stepping stone at the top of the diagram.
In other cases the specific boundary conditions
are chosen arbitrarily, just to see what the
outcome might be, and thereby learn about the
behavior of the model structure. We have illustrated both of these approaches to modeling with
examples throughout the text. In Chapters 8, 9,
10, and 11 we have employed methods that
achieve analytical solutions to particular problems in elasticity theory or fluid mechanics. In the
simpler cases these solutions are derived, but
others are taken from the vast literature on these
subjects without derivation to illustrate a particular point or concept in the context of the
seventh stepping stone. In a few cases solutions
were obtained by numerical methods but these
methods are not even introduced here, much less
developed from first principles. The finite-difference method (FDM), finite-element method (FEM),
and boundary element method (BEM) have been
exploited by engineers and scientists to find
numerical solutions to boundary value problems
(Timoshenko and Goodier, 1970; Crouch and
Starfield, 1983; Hughes, 1987).
Whether found using analytical or numerical
methods, the solutions themselves are the eighth
step in this methodology and they provide the
field quantities such as stress, displacement,
velocity, and temperature as functions of the
spatial coordinates and time. Only in rare
instances are the distributions in space and time
of these quantities so simple as to be understandable from an analytical equation. Therefore, one
employs visualization techniques, the ninth stepping stone, that range from two-dimensional
graphs and contour plots to three-dimensional
projections and animations. Throughout this text
we have used Matlab to aid in visualization.
Because the scripts that produce these illustrations are available, visualization is a dynamic
process in which the reader can adjust the boundary conditions or values of the parameters and see
the response.
476
MODEL DEVELOPMENT AND METHODOLOGY
relationships and uses these to exclude observations that are deemed to be irrelevant.
The third step involves the selection of the
branch of continuum mechanics that provides
the general boundary conditions for modeling.
Here the scaling arguments introduced in
Chapter 4 should prove helpful, and the concepts
of strain, rate of deformation, traction, and stress
described in Chapters 5 and 6 must be understood.
Although some sub-disciplines of continuum
mechanics are spelled out on the third stepping
stone, this list is not meant to be exclusive or to
imply that coupled problems, for example of fluid
flow and solid deformation, are unimportant.
Indeed, the level of understanding of coupled
problems among structural geologists and the
modeling resources now available in terms of
both hardware and software suggest that this will
be a very fruitful area of investigation in the
twenty-first century.
After selecting the theoretical context for an
investigation one should focus on the fundamental relationships, two of which are described in
Chapter 7 as the conservation laws of mass and
momentum. This fourth step along the path commonly is overlooked, but it provides a touchstone
that makes clear how the Laws of Motion have
been formulated and how the constitutive laws
are employed to reduce these to a specific set of
governing equations identified with the fifth step
along the path. At this point one has made an
explicit commitment to a material behavior and
perhaps simplified the equations of motion to
exclude temporal or spatial variations in material
properties. In this textbook we give considerable
attention to the linear elastic material as
described in Chapter 8 and the linear viscous
material as described in Chapter 10. One of the
most commonly employed governing equations in
the elastic context is the biharmonic equation for
the stress function in two dimensions. In the
viscous context the Navier–Stokes equations are
the benchmark.
In the sixth step one chooses a particular
model geometry, which may reduce the governing equations to two or even one spatial dimension. Here one may eliminate time altogether, as
in a quasi-static elastic problem or steady-state
viscous flow problem. Then the specific boundary
and/or initial conditions are selected to provide
the constants of integration necessary to solve the
governing equations. Sometimes, the specific
boundary conditions may be constrained by field
measurements, thereby establishing a direct link
with the stepping stone at the top of the diagram.
In other cases the specific boundary conditions
are chosen arbitrarily, just to see what the
outcome might be, and thereby learn about the
behavior of the model structure. We have illustrated both of these approaches to modeling with
examples throughout the text. In Chapters 8, 9,
10, and 11 we have employed methods that
achieve analytical solutions to particular problems in elasticity theory or fluid mechanics. In the
simpler cases these solutions are derived, but
others are taken from the vast literature on these
subjects without derivation to illustrate a particular point or concept in the context of the
seventh stepping stone. In a few cases solutions
were obtained by numerical methods but these
methods are not even introduced here, much less
developed from first principles. The finite-difference method (FDM), finite-element method (FEM),
and boundary element method (BEM) have been
exploited by engineers and scientists to find
numerical solutions to boundary value problems
(Timoshenko and Goodier, 1970; Crouch and
Starfield, 1983; Hughes, 1987).
Whether found using analytical or numerical
methods, the solutions themselves are the eighth
step in this methodology and they provide the
field quantities such as stress, displacement,
velocity, and temperature as functions of the
spatial coordinates and time. Only in rare
instances are the distributions in space and time
of these quantities so simple as to be understandable from an analytical equation. Therefore, one
employs visualization techniques, the ninth stepping stone, that range from two-dimensional
graphs and contour plots to three-dimensional
projections and animations. Throughout this text
we have used Matlab to aid in visualization.
Because the scripts that produce these illustrations are available, visualization is a dynamic
process in which the reader can adjust the boundary conditions or values of the parameters and see
the response.
476
MODEL DEVELOPMENT AND METHODOLOGY