Transcript Slide 1

Localised Folding
&
Axial Plane Structures
Alison Ord1 and Bruce Hobbs1,2
1Centre
for Exploration and Targeting, Earth and Environment,
The University of Western Australia
2CSIRO
Earth Science and Resource Engineering,
Australia.
The structures developed in deformed
metamorphic rocks are commonly quasi-periodic
although localised and this suggests a control
arising from non-linear matrix response is
important in many instances.
As such these structures add yet another layer
of richness to the complexity of the folding
process observed in non-linear layered materials.
Experimental deformation of layers of cloth
Field sketches of folded quartz veins
(Hall, 1815). Model approximately 1m across
(Fletcher and Sherwin, 1978).
Examples of non-periodic folding.
Localised folding with axial plane foliation,
Localised folding, Kangaroo Island, South Bermagui, NSW, Australia. Photo: Mike
Rubenach. Outcrop approximately 0.5m across.
Australia. Outcrop about 1m across.
Foliations developed by coupled
metamorphic/deformation processes.
Metamorphic
layering
(full
line)
developed obliquely to initial bedding
(dashed line) in deformed rocks from
Anglesey, UK.
Outcrop approximately 1m across.
Folded quartz/feldspar layers (white)
crossed by a metamorphic layering (Sfull line) from the Jotun Nappe, Norway.
Photo: Haakon Fossen
The development of folds in layered rocks is
commonly analysed using Biot's theory of folding.
This theory expresses the deflection, w, of a single
layer embedded in a weaker medium in terms of the
equation
4w
2w
 P 2  F  w, x   0
4
x
x
where x is the distance measured along the layer,
P is a function of the mechanical properties of the
layer, and
F(x) is a function that represents the reaction
force exerted by the embedding medium on the
layer, arising from the deflection of the layer.
F
Biot
F = kw
w
F
w
Biot’s theory of folding involves a linear response of the embedding
medium to the deflection.
This results in strictly sinusoidal folding even at high strains with no
localised deformation in the embedding medium and hence no
development of axial plane structures.
Folding in visco-elastic Maxwell materials with no softening
in the embedding material.
In both cases the viscosity ratio between layer(s) and matrix is 100,
layer viscosity 1021 Pa s, shortening 36%,
initial strain-rate 5.7x10-13 s-1; constant velocity boundary conditions.
Bulk modulus of matrix 2.3 GPa; shear modulus of matrix 1.4 GPa. Bulk
modulus of layers 30 GPa; shear modulus of layers 20 GPa.
(a)
(a) Single layer system.
(b) Three layer system. Top and bottom
layers 3 units thick; central layer 2 units thick.
(b)
Since there is no non-linear behaviour and the viscosity ratio is relatively
small, sinusoidal fold trains develop despite a wide range in the
wavelengths of initial perturbations.
However, if the embedding medium
weakens as the layer deflects or
shows other more complicated
deformation behaviour
then the function F is
no longer linear.
F
Biot
F = kw
n=1
n=2
n=3
w
F
k
F = sinh-1  nw 
n
Localised
w
Folding with softening in the embedding material
for a single layer system.
Weakening of
elastic moduli
with
weakening of
viscosity
according to
the velocity in
the y direction
with log (viscosity)
plotted along the line
marked parallel to x1.
For a simple weakening response of the embedding layer to deflection,
the initiation of folds follows Biot's theory and the initial folding
response is sinusoidal.
Units of length are arbitrary.
Folding with softening in the embedding material
for a single layer system.
Folding response to
softening of elastic
moduli and viscosity
according to
A  Ao / n 2 w22  1
with log (viscosity)
plotted along the line
marked parallel to x1.
As the folds grow and weakening develops in the embedding
material the fold profile ceases to be sinusoidal and the folds
localise to form packets of folding along the layer.
Units of length are arbitrary.
Folding with softening in the embedding material
for a three layer system.
This model also
involves softening
according to
A  Ao / n 2 w22  1
for both elastic
moduli &
viscosity.
The variation of the
logarithm of the
viscosity along the
line shown.
Units of distance are
arbitrary.
This softening behaviour is
reflected in the embedding
medium as a series of
localised deformation zones
parallel to the deflection
direction, w, that is, parallel
to the axial plane of the folds.
These zones constitute
micro-lithons, or in an
initially finely layered
material, crenulation
cleavages.
Examples of metamorphic layering
Quartz plus muscovite (Q) layers
alternating with layers comprised
almost exclusively by muscovite (Mu).
(Photo: Ron Vernon. Picuris Range, New
Mexico, USA.). Image is 1cm across.
Metamorphic layering crossing
bedding (Anglesey)
Examples of metamorphic layering
and a model for the resulting mechanical response.
The reaction force system of the matrix
against the folding layer represented by a
Winkler model for a homogeneous matrix
(Hunt, 2006).
Each reaction unit has an elastic spring
with modulus, k, and a viscous dashpot
with viscosity, h.
The modified Winkler model for a matrix
with metamorphic layering. Planar
regions with elastic modulus and
viscosity, k1 and h1, alternate with regions
with properties k2 and h2.
We now explore the situation where a
layering arising from metamorphic
differentiation forms oblique to folding
multi-layers early in the folding history.
Localised folds with metamorphic layering in single fold
systems as outlined by the distribution of viscosity.
A  Ao  2  sin  mx1   / 5
A  Ao / n 2 w22  1
Weakening of
constitutive
parameters
Imposition of
metamorphic
layering
m=1
n=3
(a)
27% shortening.
n=3
m=3
82-y
(b)
27% shortening.
Localised folding of three layers with
metamorphic layering outlined by the
distribution of viscosity.
n=1 , m=1/p, 36% shortening.
n=1, m=1, 27% shortening.
The metamorphic layering is distorted on the limbs of folds so that it
remains approximately normal to the folded layer.
This effect is seen in natural fold systems associated with metamorphic
layering.
A simplified mechanical model of the multi-layer folding
problem with axial plane metamorphic layering.
The individual layers have their own intrinsic buckling
modes but are connected to each other by Maxwell
units that themselves weaken with respect to spring
constants and viscosity as each layer deflects (a).
The introduction of a metamorphic layering introduces
additional spatial periodicity as shown in (b).
Although the influence of initial geometrical
perturbations of large wavelength undoubtedly
have an influence in promoting the formation
of some localised folds at large viscosity ratios
an additional mechanism involves the
response to weakening in the matrix between
the buckling layers.
Non-linearities arising from large deflections
and thick beam theory using the linear nonsoftening matrix response of the Biot theory
have not so far emerged as a mechanism for
localisation.
Stress
Periodic fold form
A
B
Localised fold form
Strain
The Biot theory
appears as the analysis
that describes the initial
sinusoidal deflections in a
layered system before
softening of the
embedding matrix
appears. After the
embryonic sinusoidal fold
system forms, localisation
of the fold system can
occur and chaotic
systems develop if
localised packets of folds
interfere.
The growth of metamorphic layering
parallel or approximately parallel to the
axial planes of folds introduces a
periodicity in matrix response along the
length of the folding layer introducing
the opportunity for even greater fold
localisation.
None of the weakening behaviours proposed
here produce shear instabilities in the matrix so
that differentiated crenulation cleavages do not
develop in the models studied.
Future work needs to concentrate on matrix
constitutive relations that enable such
instabilities to develop.
It is hoped that the numerical exploration
reported here will encourage theoretical
investigations of the influence of periodic or
quasi-periodic weakening behaviour in the
matrix on fold localisation in both single and
multi-layered systems.
Thank you