R. C. Fletcher
PGP, University of Oslo
GEOMOD2010, Lisbon, Portugal
9/27/2010
with acknowledgements to:
Ezra Pound, poet
George Polya, mathematician
Bill Chapple, structural geologist
Ophelia, cat
Laccoliths of the Henry Mountains, Utah
& Gilbert’s model (1875-1880)
 pM
R 
  gZ   R  2  R Z   0
2
2 Z
p M   gZ
Analytical modeling in structural geology (AMSG)
entire, iterative process connecting two elements:
(1) sets of observations of a naturally deformed rock
mass
(2) description of the physics of deformation provided
by continuum mechanics.
Connected through the formulation,
implementation and application of a model.
Following Pappus of Alexandria (290-350), as discussed in the mathematician
G. Polya’s book “How to Solve It,” a distinction may be made between analysis,
or solution backward, or regressive reasoning, and synthesis, or constructive
solution, or progressive reasoning. Going from (1) to (2) is chiefly analysis;
going from (2) to (1) is chiefly synthesis.
1
2
3
4
5
6
FAULTS
“FOLDS”
FABRIC
EXP’TS – RHEOLOGY & ANALOG MODELS
KINEMATIC MODELS
AMSG (?)
25%
13%
25%
10%
11%
16%
N = 111 papers in JSG
Theory
 “Theory” is the underlying physics. There is no
“theory of folding”; no “theory for metamorphic
differentiation”
 A theory is closed.
 An application of continuum mechanics is to model
folding or metamorphic differentiation. Continuum
mechanics is the theory.
Exhibit: AMSG in the study of rock folding
Ramberg (1960): Ld/H = 2/(2/3R)1/3; a figure of two folds gives L/H ~ 6.2.
Using Ld/H ~ L/H, R = 1.4. Equation not used; folds used
to illustrate “contact strain” within a zone of thickness ~ Ld.
Biot (1961) : Ld/H = 2/(6/R)1/3 : R = 6. Thick-plate solution: R = 1.
Biot, Ode and Roever (1961) demonstrated Biot results in experiments, so the geologists
would “believe” them. Folds in a pitch- corn syrup assembly for which R ~ 2200,
yielded L/H = Ld/H = 45; a value yet to be seen in nature.
Currie, Patnode & Trump (1962) report L/H ~ 27, but where this value comes from
remains a mystery.
Sherwin & Chapple (1968) test Ld/H ~ L/H. Chapple surprised with small L/H
measured by Sherwin; he had seen many single-layer folds. Their AMSG application required
incorporation of layer thickening during wavelength selection. R ~ 14 – 35.
Chapple (1968), Dieterich & Onat (1968), Dietrich & Carter (1969) & Parrish (1973:
power-law) : FEM; relationships between folding and cleavage; distributions of stress and
strain. [Re: “folding mechanisms”]
Fletcher (1974) & Smith (1977) single-layer folding in
power-law materials, extended interpretation of small values L/H ~ 4 – 6.
Biot (1961) treated nonlinearity in thin-plate approximation.
Fletcher got 3 < n < 12 for limestone layer in shale.
Fletcher & Sherwin (1978): distribution of L/H provides another fitting parameter.
Hudleston & Holst (1984) estimate rheological parameters for single-layer folds of
limestone in shale; n was poorly-constrained. Petrographic study of deformation
mechanisms: twinning, stylolites, veins & of strain distribution.
Field observations [1]
Cylindrical microfolds
Castile Formation, Permian Basin
west Texas & New Mexico
Stacked SLUs from “infinite”
multilayer
THEORY [2]
  ij
x j
v j
x j
0
0
 vi
1  v j

D ij  

x j
2   xi
  n  1  / 2 
D ij  B J 2
J2 
1
2
s ij
s kl s kl
s ij   ij 
1
3
 kk  ij




Unknowns
Equations
6
3
3
1
6
6
0
6-1
SLU VII
Ld/H = 4.2
Hi: 1, 1, 1
Ai: 0.4, 1, 0.6
SLU VI
SLU VII
Link (1) and (2): The model
The sciences do not try to explain, they hardly even try to interpret,
they mainly make models. By a model is meant a mathematical
construct which, with the addition of certain verbal interpretations,
describes observed phenomena. The justification of such a
mathematical construct is solely and precisely that it is expected to
work.
John von Neumann
What is the question? Rheological parameters for a fit & are they
plausible?
Two alternate models for SLU VII
3 layers, equal thickness,
bounded by
rigid, frictionless platens
slip parameter Q*=h*H/h*
stress exponent n
3 layers, equal thickness,
bounded by
viscous ½-space
R = h/h2
Note: actual thicknesses & amplitudes shown;
models are symmetric equivalents
SLU VII
3 equal layers
L/H = 4.2
Bounded by rigid,
frictionless platens
Interfacial viscous slip
Q = (h*H)/(h*)
Stress exponent
n
Contours of Ld/H (black)
& qd (red)
Fit: n > 50
Q ~ 0.1
n > 12
Consistency:
R ~ (1/Q)1/2
Webster, Massachusetts
Nipmuc Native American:
“You fish on your side. I fish on my side. Nobody
fishes in the middle.”
The descendents of the fishing Nipmucs now run the biggest
gambling casino in Connecticut
The Four Tribes –
warlike & independent
fought for “research wampum”
Buffalo Hunters: Field-based structural geologists
Lived in teepees
Meat eaters
Outcrop scalpers
Mound Builders: Model builders
Urban dwellers
Ate pumpkins & corn meal
Build large mounds of ??? For burial of the dead?
To escape floods? On which to erect fancy homes?
Coastal Indians: Rock rheologists
Built totem poles
Ate oysters & lobsters, the local cheeses
Drank expensive alcoholic beverages
Paleo-Indians: Petrofabricators
Where they lived now covered with tiny rock chips
Left untranslatable manuscripts on moose hide
Analytical solutions
1. Infinite space
internal folding
2. Layers
3. Wedges
4. Inclusions – in infinite space
SCA in polycrystals & other composites
5. Cracks, anticracks
Joints, faults, stylolites
6. Pipes
magma flow/wall erosion
Should kinematic modeling go the way of the Ptolemy’s
Epicycles? [YES]
Is it based on theory?
NO – No closure
Does it address field observations?
Over-constrains form
Does it provide insight into processes operative
in development of structure & fabric?
NO – Excludes quantities containing dimension of
mass
TRISHEAR: deformation associated with
propagating basement fault
vr
Governing
equation:
r

vr
r

1  v
r 
0
v r  v r    v r 
KINEMATIC
 v

0
VISCOUS
What does pursuit of a model that can be treated by an analytical
solution, as a part of AMSG, do for you?
All modeling, including AMSG, centers on posing interesting questions
Write out the equations = physics
Sketch the field example
Assemble a set of measurements
Simplify
Pose a tractable boundary-value problem
Intuitive ability closely resembles artistic talent (M. A. Biot, 1963)
“Though I haven't applied your work in exactly the way you did,
what you describe for the "effective aspect ratio" is a perfectly sound approach
to me as well, but is in a way an a posteriori method. The core challenge for
an analytical solution in my mind remains an a priori approach, that is to say
to look at a distribution of hard and soft phases and predict/know/easily measure
what the effective aspect ratio is. For a conglomerate or similar, it's straightforward,
but I am not sure how to know for situations such as anastomosing shear zones
(with various degrees of connectedness).”
Postmodernism
a tendency in contemporary culture characterized by rejection of objective
truth. Emphasizes roles of language, power relations, and motivations;
attacks use of sharp classifications such as white versus black.
Postmodernist thought is an intentional departure from modernist
approaches previously dominant. Term "postmodernism" comes from its
critique of the "modernist" scientific mentality of objectivity and progress
associated with the Enlightenment.
Modernism and postmodernism are understood as cultural projects."Postmodernism"
in critical theory refers to a point of departure for works of literature, …, and design,
as well as in marketing and business and in the interpretation of law, culture, and
religion. Indeed, postmodernism … can be understood as a reaction to modernism….
Whereas modernism was primarily concerned with principles such as identity, unity,
authority, and certainty, postmodernism is often associated with difference, plurality,
textuality, and skepticism.
Four worldviews are the Postmodern-ironist, which sees truth as socially
constructed; the scientific-rational, in which truth is found through methodical,
disciplined inquiry; the social-traditional, in which truth is found in the heritage of
American and Western civilization; and the neo-romantic, in which truth is found
through attaining harmony with nature and/or spiritual exploration of the inner self.