I
n the context of structural geology we can construct a working model of mountain building
from those small number of physical laws and
then leap backward in time and understand the
development of geological structures such as those
depicted in the structural block diagram (Chapter
7, frontispiece) of a part of the Penninic Alps of
Switzerland constructed by Emile Argand and published in 1911 (Argand, 1911). This is one of the earliest published block diagrams in the literature of
structural geology (McIntyre and Weiss, 1956;
Howarth, 1999) and it illustrates what was known
in the early part of the twentieth century about
one of the most interesting and complex regions of
folding and faulting in that mountain chain.
Among the small number of physical laws that
can be employed to understand tectonic processes
and their structural products are those of mass,
momentum, and energy conservation. Newton’s
Second Law of Motion, for example, relating force,
mass, and linear acceleration is embodied in a
generalized statement of momentum conservation. Kinetics is the branch of mechanics that considers the action of forces and torques on particles
and rigid bodies, and their resulting accelerations. This should be familiar ground for students
recently exposed to the mechanics section of a
college physics course. However, we offer a short
review for those who would benefit from a second
encounter with these topics. This review serves
another important purpose. Textbooks in structural geology typically fail to make a clear connection between the material taught in the
mechanics section of an introductory college
physics course and the mechanical concepts
employed in analyzing geological structures. By
making that connection explicit students are prepared to use the mechanics effectively and with a
confidence that stems from understanding the
fundamentals of the discipline.
We begin with the concepts of linear and
angular momentum as treated in particle dynamics
and generalize this to rigid-body dynamics. While
the methods of classical dynamics have application in many familiar human endeavors (from
tracking satellites to playing snooker) where accelerations are key to understanding, the rock masses
that comprise Earth’s crust do not experience
appreciable accelerations, apart from those brief
moments during rare events such as earthquakes.
In the absence of appreciable accelerations, conservation of momentum requires that the resultant forces and torques acting on a rock mass are
negligible. This condition ensures that the linear
and angular momenta are nearly constant with
respect to time. These requirements are fundamental and should be examined at an early stage
of any program of modeling geologic structures.
Our review of concepts introduced in the
typical college physics class leaves us short of
having all of the necessary tools of mechanics to
analyze tectonic processes. Given our human time
scale and the poor resolution of our eyes for discerning small changes in the shapes of objects,
rock does seem quite rigid. One might think that
the dynamics of rigid bodies would be as far as we
have to go to tackle problems of rock deformation
in the Earth. Indeed, the early practitioners of
plate tectonics conceptualized plates of the lithosphere as thin rigid masses slowing moving over a
mobile aesthenosphere. Similarly, the typical midtwentieth century structural geology textbook
introduced students to faulting with diagrams
showing rigid blocks moving relative to one
another (Billings, 1972), and laboratory exercises
utilized painted wooden blocks to illustrate the
patterns of offset strata (Fig. 7.1). While these
figures and blocks may be instructive guides to
understanding the map patterns of faulted strata,
the perceptive students might ask: what happens
near the end of a fault? Clearly a rigid block model
could not provide a satisfactory answer, because
the “model fault” has no end.
Moving beyond the rigid block models, it is necessary to consider a continuous and deformable
body of rock and to broaden our perspective to constrain explicitly how mass and momentum are
conserved throughout such a body. Here we postulate that temperature changes and chemical
changes within the body are negligible, so the
model is isothermal and isochemical. If heat flow and
chemical reactions play an important role, than
conservation of energy and conservation of chemical species must be included. With these limitations in mind, conservation of mass leads to the
equation of continuity and the conservation of
momentum leads to the equations of motion for the
material continuum. In turn, the equations of
244
CONSERVATION OF MASS AND MOMENTUM
n the context of structural geology we can construct a working model of mountain building
from those small number of physical laws and
then leap backward in time and understand the
development of geological structures such as those
depicted in the structural block diagram (Chapter
7, frontispiece) of a part of the Penninic Alps of
Switzerland constructed by Emile Argand and published in 1911 (Argand, 1911). This is one of the earliest published block diagrams in the literature of
structural geology (McIntyre and Weiss, 1956;
Howarth, 1999) and it illustrates what was known
in the early part of the twentieth century about
one of the most interesting and complex regions of
folding and faulting in that mountain chain.
Among the small number of physical laws that
can be employed to understand tectonic processes
and their structural products are those of mass,
momentum, and energy conservation. Newton’s
Second Law of Motion, for example, relating force,
mass, and linear acceleration is embodied in a
generalized statement of momentum conservation. Kinetics is the branch of mechanics that considers the action of forces and torques on particles
and rigid bodies, and their resulting accelerations. This should be familiar ground for students
recently exposed to the mechanics section of a
college physics course. However, we offer a short
review for those who would benefit from a second
encounter with these topics. This review serves
another important purpose. Textbooks in structural geology typically fail to make a clear connection between the material taught in the
mechanics section of an introductory college
physics course and the mechanical concepts
employed in analyzing geological structures. By
making that connection explicit students are prepared to use the mechanics effectively and with a
confidence that stems from understanding the
fundamentals of the discipline.
We begin with the concepts of linear and
angular momentum as treated in particle dynamics
and generalize this to rigid-body dynamics. While
the methods of classical dynamics have application in many familiar human endeavors (from
tracking satellites to playing snooker) where accelerations are key to understanding, the rock masses
that comprise Earth’s crust do not experience
appreciable accelerations, apart from those brief
moments during rare events such as earthquakes.
In the absence of appreciable accelerations, conservation of momentum requires that the resultant forces and torques acting on a rock mass are
negligible. This condition ensures that the linear
and angular momenta are nearly constant with
respect to time. These requirements are fundamental and should be examined at an early stage
of any program of modeling geologic structures.
Our review of concepts introduced in the
typical college physics class leaves us short of
having all of the necessary tools of mechanics to
analyze tectonic processes. Given our human time
scale and the poor resolution of our eyes for discerning small changes in the shapes of objects,
rock does seem quite rigid. One might think that
the dynamics of rigid bodies would be as far as we
have to go to tackle problems of rock deformation
in the Earth. Indeed, the early practitioners of
plate tectonics conceptualized plates of the lithosphere as thin rigid masses slowing moving over a
mobile aesthenosphere. Similarly, the typical midtwentieth century structural geology textbook
introduced students to faulting with diagrams
showing rigid blocks moving relative to one
another (Billings, 1972), and laboratory exercises
utilized painted wooden blocks to illustrate the
patterns of offset strata (Fig. 7.1). While these
figures and blocks may be instructive guides to
understanding the map patterns of faulted strata,
the perceptive students might ask: what happens
near the end of a fault? Clearly a rigid block model
could not provide a satisfactory answer, because
the “model fault” has no end.
Moving beyond the rigid block models, it is necessary to consider a continuous and deformable
body of rock and to broaden our perspective to constrain explicitly how mass and momentum are
conserved throughout such a body. Here we postulate that temperature changes and chemical
changes within the body are negligible, so the
model is isothermal and isochemical. If heat flow and
chemical reactions play an important role, than
conservation of energy and conservation of chemical species must be included. With these limitations in mind, conservation of mass leads to the
equation of continuity and the conservation of
momentum leads to the equations of motion for the
material continuum. In turn, the equations of
244
CONSERVATION OF MASS AND MOMENTUM