negligible. In this section we describe a variety of
simplification procedures, and the methods that
are employed to make and to justify these choices.
The procedure for idealization, described in the
previous section, and the selection and simplification of general boundary conditions are not
independent. The former is described as a field procedure and it should, for the most part, be undertaken during mapping or data collection, when the
decisions can be evaluated by direct observations.
The latter is described as an office procedure, and it
is likely to be undertaken with a continuum
mechanics textbook in hand. In fact most structural geologists, including Gilbert, intertwine
these two procedures. Apparently Gilbert carried
an engineering handbook into the field and presumably used it to help conceptualize and set up
problems (Pyne, 1980). An idealization may be
designed specifically to take advantage of a known
solution, for example in elasticity theory or fluid
mechanics. Later, having found a solution for a
more realistic set of parameters, certain constraints imposed in the former idealization may be
relaxed. In this way an analysis may progress, for
example, from one employing a linear and isotropic viscous fluid (Chapter 10), to an anisotropic
viscous fluid, to a fluid with more complex rheology
(Chapter 11). This give-and-take relationship
between the idealization of field observations and
the selection of general boundary conditions for
modeling is vital to the development of effective
models and the analysis of structural data.
12.2.1 Plate theory: what controls the
shape of the domed strata over a
laccolith?
It is clear that Gilbert understood the strata over
laccoliths in the Henry Mountains did not behave
as rigid pistons (Fig. 1.18) because his conceptual
model that preceded the piston laccolith shows
the strata bent into a domed shape (Fig. 12.5). He
noted that resistance to uplift for the piston
model is proportional to the overburden thickness and speculated in his report of 1877 that
resistance to bending might increase with some
greater power of the overburden thickness.
I am led by the analogy of allied problems in mechanics
to assume that the resistance of the body of strata varies
with some power of its depth, but I am unable to say
what power. So far as I am aware, neither mathematical
analysis nor experimentation has been directed to the
problem in question. According to Rankine “the resistances of flexure of similar cross-sections (of elastic
beams) are as their breadths and as the squares of their
depths” (“Applied Mechanics”, page 316), and it is possible that the same law applies to the resistances which
continuous strata oppose to the uplifts of domes. But it
appears more probable that the greater complexity of
the strains developed in the formation of domes causes
the depth to enter into the formula with a higher power
than second (Gilbert, 1877).
To address questions about the shape of the
domed strata and their resistance to bending a
number of models have been proposed since
Gilbert’s report based on a theory for the
deflection of thin plates (Johnson, 1970; Pollard
and Johnson, 1973; Koch et al., 1981; Kerr and
Pollard, 1998).
The context, or general boundary conditions,
for the analysis of thin plates is the engineering
discipline called strength of materials or plate theory.
Reference textbooks are available on this subject
from the engineering literature (Timoshenko,
1958; Timoshenko and Woinowsky-Krieger, 1959).
In Section 4.3.1 the dimensionless group for plate
bending was derived, (4.45), and showed that displacements scale with the fourth power of the
length and inversely with the elastic stiffness and
the third power of the thickness. Thus, bending
under distributed loads is most sensitive to the
geometry of the plate.
To apply plate theory to the laccolith problem
the sedimentary strata overlying the laccolith are
idealized as a stack of n thin plates of thickness, h i
(Fig. 12.6a). Particular plates may or may not correspond to stratigraphic units, but each behaves
as an independent mechanical unit, capable of
slipping relative to adjacent mechanical units
along bedding-plane faults. Unlike Gilbert’s rigid
piston, these plates deform according to linear
elastic relationships among the stress and infinitesimal strain components defined by Hooke’s
Law (Chapter 8). For plates that are isotropic and
homogeneous with respect to elastic properties,
the two relevant material constants for each plate
are Young’s modulus, E i , and Poisson’s ratio, i .
The plates are bent by upward directed forces due
12.2 SELECTION OF GENERAL BOUNDARY CONDITIONS
463
simplification procedures, and the methods that
are employed to make and to justify these choices.
The procedure for idealization, described in the
previous section, and the selection and simplification of general boundary conditions are not
independent. The former is described as a field procedure and it should, for the most part, be undertaken during mapping or data collection, when the
decisions can be evaluated by direct observations.
The latter is described as an office procedure, and it
is likely to be undertaken with a continuum
mechanics textbook in hand. In fact most structural geologists, including Gilbert, intertwine
these two procedures. Apparently Gilbert carried
an engineering handbook into the field and presumably used it to help conceptualize and set up
problems (Pyne, 1980). An idealization may be
designed specifically to take advantage of a known
solution, for example in elasticity theory or fluid
mechanics. Later, having found a solution for a
more realistic set of parameters, certain constraints imposed in the former idealization may be
relaxed. In this way an analysis may progress, for
example, from one employing a linear and isotropic viscous fluid (Chapter 10), to an anisotropic
viscous fluid, to a fluid with more complex rheology
(Chapter 11). This give-and-take relationship
between the idealization of field observations and
the selection of general boundary conditions for
modeling is vital to the development of effective
models and the analysis of structural data.
12.2.1 Plate theory: what controls the
shape of the domed strata over a
laccolith?
It is clear that Gilbert understood the strata over
laccoliths in the Henry Mountains did not behave
as rigid pistons (Fig. 1.18) because his conceptual
model that preceded the piston laccolith shows
the strata bent into a domed shape (Fig. 12.5). He
noted that resistance to uplift for the piston
model is proportional to the overburden thickness and speculated in his report of 1877 that
resistance to bending might increase with some
greater power of the overburden thickness.
I am led by the analogy of allied problems in mechanics
to assume that the resistance of the body of strata varies
with some power of its depth, but I am unable to say
what power. So far as I am aware, neither mathematical
analysis nor experimentation has been directed to the
problem in question. According to Rankine “the resistances of flexure of similar cross-sections (of elastic
beams) are as their breadths and as the squares of their
depths” (“Applied Mechanics”, page 316), and it is possible that the same law applies to the resistances which
continuous strata oppose to the uplifts of domes. But it
appears more probable that the greater complexity of
the strains developed in the formation of domes causes
the depth to enter into the formula with a higher power
than second (Gilbert, 1877).
To address questions about the shape of the
domed strata and their resistance to bending a
number of models have been proposed since
Gilbert’s report based on a theory for the
deflection of thin plates (Johnson, 1970; Pollard
and Johnson, 1973; Koch et al., 1981; Kerr and
Pollard, 1998).
The context, or general boundary conditions,
for the analysis of thin plates is the engineering
discipline called strength of materials or plate theory.
Reference textbooks are available on this subject
from the engineering literature (Timoshenko,
1958; Timoshenko and Woinowsky-Krieger, 1959).
In Section 4.3.1 the dimensionless group for plate
bending was derived, (4.45), and showed that displacements scale with the fourth power of the
length and inversely with the elastic stiffness and
the third power of the thickness. Thus, bending
under distributed loads is most sensitive to the
geometry of the plate.
To apply plate theory to the laccolith problem
the sedimentary strata overlying the laccolith are
idealized as a stack of n thin plates of thickness, h i
(Fig. 12.6a). Particular plates may or may not correspond to stratigraphic units, but each behaves
as an independent mechanical unit, capable of
slipping relative to adjacent mechanical units
along bedding-plane faults. Unlike Gilbert’s rigid
piston, these plates deform according to linear
elastic relationships among the stress and infinitesimal strain components defined by Hooke’s
Law (Chapter 8). For plates that are isotropic and
homogeneous with respect to elastic properties,
the two relevant material constants for each plate
are Young’s modulus, E i , and Poisson’s ratio, i .
The plates are bent by upward directed forces due
12.2 SELECTION OF GENERAL BOUNDARY CONDITIONS
463