abstraction or idealization of nature and we must
address its limitations. As John Wheeler reminds
us in his forward to The Continuum: A Critical
Examination of the Foundation of Analysis by
Hermann Weyl:
For the advancing army of physics, battling for many a
decade with heat and sound, fields and particles, gravitation and space–time geometry, the cavalry of mathematics, galloping out ahead, provided what it thought
to be the rationale for the real number system. Encounter with the quantum has taught us, however,
that we acquire our knowledge in bits; that the continuum is forever beyond our reach. Yet for daily work the
concept of the continuum has been and will continue
to be as indispensable for physics as it is for mathematics. In either field of endeavor, in any given enterprise,
we can adopt the continuum and give up absolute
rigor, or adopt rigor and give up the continuum, but
we can’t pursue both approaches at the same time in
the same application (Weyl, 1987).
We refer to mathematical points at which density
is defined in a continuum mechanical model of a
faulted rock mass. Those “points” and the
motions or material properties attributed to
them, must be thought of as representative of a
finite piece of rock, perhaps several cubic centimeters in volume.
4.2 Physical dimensions and
dimensional analysis
One can express physical quantities in terms of
many different units of measure and a particular
quantity can take on very different numerical
values under the different systems of units. For
example, 1 m Ϸ 2.85 ϫ 10
3 printer points Ϸ 3.94 ϫ
10
1 inches Ϸ 3.28 ϫ 10
0 feet Ϸ 4.97 ϫ 10
Ϫ3 furlongs
Ϸ 6.21 ϫ 10
Ϫ4 miles. However, the underlying
physics must be independent of the choice of
units: it can’t depend on the length of the King’s
Foot! This leads us to the concept that there is
something more fundamental than the units
attached to a physical quantity and this is the physical dimension of that quantity. Regardless of the
units chosen, for example, for mechanical work
(newton и meter, pound force и foot, or dyne и centimeter) this physical quantity has the dimensions
of force times displacement. Thus, when analyzing the relationships among various quantities it
is instructive to consider the dimensions of those
quantities. In this section we introduce the
dimensions commonly encountered in mechanical processes.
Dimensional analysis is a useful tool for
working with and understanding theoretical constructs in all of science and engineering. The
paper by M. K. Hubbert (1937) puts the use of
dimensional analysis in a geological context and
relies on the methods put forward in the book by
P. W. Bridgman (1931). In this section we use
dimensional analysis to understand whether a
given equation, which reportedly describes some
aspect of rock deformation, is consistent from a
dimensional point of view. If not, the equation is
invalid and should be discarded. Then we introduce the technique for plotting dimensionless
graphs and illustrate why this is the preferred
method to present scientific results.
4.2.1 Dimensionally homogeneous
equations
The dimensions of the fundamental mechanical
quantities (length, mass, time, and temperature)
are given as: L, M, T, and ⌰ respectively. The
dimensions of derived quantities are composed of
products and powers of these fundamental
dimensions. Reading {ϭ} “has dimensions of” we
have, for example:
(4.10)
(4.11)
(4.12)
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
(4.18)
(4.19)
(4.20)
Both the stretch and the angle are dimensionless
physical quantities, but we use the symbol l
3.141 592 65, . . . , {ϭ}1
stretch, S{ϭ}L L Ϫ1 ϭ L 0 ϭ 1
thermal expansion, ␣{ϭ}⌰ Ϫ1
stress, {ϭ}M L Ϫ1 T Ϫ2
force, F{ϭ}M L T Ϫ2
mass density, {ϭ}M L Ϫ3
acceleration, a{ϭ}L T Ϫ2
velocity, v{ϭ}L T Ϫ1
displacement, u{ϭ}L
volume, V{ϭ}L 3
area, A{ϭ}L 2
4.2 PHYSICAL DIMENSIONS AND DIMENSIONAL ANALYSIS
127
address its limitations. As John Wheeler reminds
us in his forward to The Continuum: A Critical
Examination of the Foundation of Analysis by
Hermann Weyl:
For the advancing army of physics, battling for many a
decade with heat and sound, fields and particles, gravitation and space–time geometry, the cavalry of mathematics, galloping out ahead, provided what it thought
to be the rationale for the real number system. Encounter with the quantum has taught us, however,
that we acquire our knowledge in bits; that the continuum is forever beyond our reach. Yet for daily work the
concept of the continuum has been and will continue
to be as indispensable for physics as it is for mathematics. In either field of endeavor, in any given enterprise,
we can adopt the continuum and give up absolute
rigor, or adopt rigor and give up the continuum, but
we can’t pursue both approaches at the same time in
the same application (Weyl, 1987).
We refer to mathematical points at which density
is defined in a continuum mechanical model of a
faulted rock mass. Those “points” and the
motions or material properties attributed to
them, must be thought of as representative of a
finite piece of rock, perhaps several cubic centimeters in volume.
4.2 Physical dimensions and
dimensional analysis
One can express physical quantities in terms of
many different units of measure and a particular
quantity can take on very different numerical
values under the different systems of units. For
example, 1 m Ϸ 2.85 ϫ 10
3 printer points Ϸ 3.94 ϫ
10
1 inches Ϸ 3.28 ϫ 10
0 feet Ϸ 4.97 ϫ 10
Ϫ3 furlongs
Ϸ 6.21 ϫ 10
Ϫ4 miles. However, the underlying
physics must be independent of the choice of
units: it can’t depend on the length of the King’s
Foot! This leads us to the concept that there is
something more fundamental than the units
attached to a physical quantity and this is the physical dimension of that quantity. Regardless of the
units chosen, for example, for mechanical work
(newton и meter, pound force и foot, or dyne и centimeter) this physical quantity has the dimensions
of force times displacement. Thus, when analyzing the relationships among various quantities it
is instructive to consider the dimensions of those
quantities. In this section we introduce the
dimensions commonly encountered in mechanical processes.
Dimensional analysis is a useful tool for
working with and understanding theoretical constructs in all of science and engineering. The
paper by M. K. Hubbert (1937) puts the use of
dimensional analysis in a geological context and
relies on the methods put forward in the book by
P. W. Bridgman (1931). In this section we use
dimensional analysis to understand whether a
given equation, which reportedly describes some
aspect of rock deformation, is consistent from a
dimensional point of view. If not, the equation is
invalid and should be discarded. Then we introduce the technique for plotting dimensionless
graphs and illustrate why this is the preferred
method to present scientific results.
4.2.1 Dimensionally homogeneous
equations
The dimensions of the fundamental mechanical
quantities (length, mass, time, and temperature)
are given as: L, M, T, and ⌰ respectively. The
dimensions of derived quantities are composed of
products and powers of these fundamental
dimensions. Reading {ϭ} “has dimensions of” we
have, for example:
(4.10)
(4.11)
(4.12)
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
(4.18)
(4.19)
(4.20)
Both the stretch and the angle are dimensionless
physical quantities, but we use the symbol l
3.141 592 65, . . . , {ϭ}1
stretch, S{ϭ}L L Ϫ1 ϭ L 0 ϭ 1
thermal expansion, ␣{ϭ}⌰ Ϫ1
stress, {ϭ}M L Ϫ1 T Ϫ2
force, F{ϭ}M L T Ϫ2
mass density, {ϭ}M L Ϫ3
acceleration, a{ϭ}L T Ϫ2
velocity, v{ϭ}L T Ϫ1
displacement, u{ϭ}L
volume, V{ϭ}L 3
area, A{ϭ}L 2
4.2 PHYSICAL DIMENSIONS AND DIMENSIONAL ANALYSIS
127