Earth and Planetary Science Letters 280 (2009) 159–166
Contents lists available at ScienceDirect
Earth and Planetary Science Letters
j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e p s l
Transformation-induced jointing as a gauge for interfacial slip and rock strength
Karthik Iyer a,b,⁎, Yuri Y. Podladchikov a
a
b
Physics of Geological Processes (PGP), University of Oslo, P.O. Box 1048, Blindern, N-0316, Oslo, Norway
IFM-Geomar, Wischhofstrasse 1-3, D-24148 Kiel, Germany
a r t i c l e
i n f o
Article history:
Received 1 April 2008
Received in revised form 10 October 2008
Accepted 15 January 2009
Available online 15 February 2009
Editor: T. Spohn
Keywords:
brittle rheology
joint formation
fracture spacing to layer thickness ratio
interfacial slip
transformation-induced strain
a b s t r a c t
Various theoretical and numerical models have been proposed in order to explain joint formation and spacing in
layered rock series. However, most of these models assume that the interfaces between the rock layers are
perfectly welded, i.e. no slip occurs, and that all the layers are subjected to the same remote strain due to various
processes (e.g. tectonic processes). Other factors may also induce extensional strain in rocks, e.g. phase
transformations. However, such processes may induce different amounts of strain on the layers in a rock series
leading to a strain mismatch between these layers. In this paper, we present a 1-D finite difference linear elastic
model which allows joint formation within the middle layer in a three-layer rock series and is induced by a strain
mismatch between the fractured, central layer and the surrounding matrix. Furthermore, the central layer in our
model is not necessarily welded to the matrix layers and is allowed to slip along the interfaces between these
layers if the shear strength of the material at the interface is reached. We find that the final fracture spacing to
layer thickness ratio (S/Tf) in such layered systems is directly proportional to the ratio of the tensile and shear
strength of the material. Changes in the material properties such as the shear modulus or Young's modulus do not
affect these results. A natural analog of joint formation driven by phase transformations is found in the
orthopyroxenite dykes of the Leka Ophiolite Complex (LOC), Norway. Joint formation in orthopyroxenite dykes
results from serpentinization-driven expansion of the surrounding dunite matrix. Detailed field studies and
measurements (583 sample points) yield S/Tf ratios between 0.1 and 1.0 with a mean value of 0.45± 0.20. We
demonstrate that the strain mismatch-driven joint formation associated with interfacial slip explains the low S/Tf
ratios obtained from field measurements and may also help us constrain rock strength.
© 2009 Elsevier B.V. All rights reserved.
1. Introduction
Brittle deformation features in rocks such as joints are ubiquitous
within the Earth's crust. Brittle deformation accompanying phase
transformations in rocks are, however, poorly understood. Thermal
and volumetric changes occurring in rocks during phase transformations alter rock rheology (Karato, 1997) and complicate structural
reconstructions based on cross-sectional volume/area-balancing
methods. Additionally, joints play an important role in controlling
various physical properties such as the effective strength and
permeability of rocks. Tensile fractures are often observed in layered
rocks and are oriented parallel to the maximum compressive stress
direction. Fractures of a given set are usually parallel or sub-parallel to
each other (Price, 1966; Pollard and Aydin, 1988; Bai and Pollard,
2000a; Mandl, 2005). The open fractures are commonly confined to
the fractured layer and do not propagate into the surrounding rock due
to termination at a strength anisotropy, e.g. the interface between the
fractured layer and the surrounding rock (Gross, 1993; Gillespie et al.,
⁎ Corresponding author. IFM-Geomar, Wischhofstrasse 1-3, D-24148 Kiel, Germany.
Tel.: +49 431 6002866; fax: +49 431 600 2941.
E-mail address: kiyer@ifm-geomar.de (K. Iyer).
0012-821X/$ – see front matter © 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.epsl.2009.01.028
1999; Tindall and Davis, 2003; Fischer and Polansky, 2006). Field
relations of fracture spacing in layered rocks have been previously
studied with reported fracture spacing to layer thickness ratios (S/Tf)
ranging from 0.1 to greater than 10 (Bogdanov, 1947; Novikova, 1947;
Kirillova, 1949; Price, 1966; McQuillan, 1973; Narr and Lerche, 1984;
Huang and Angelier, 1989; Narr and Suppe, 1991; Gross, 1993; Gross
et al., 1995; Wu and Pollard, 1995; Becker and Gross, 1996; Ji and
Saruwatari, 1998; Bai and Pollard, 2000a) where fracture spacing is the
distance between adjacent fractures along a line perpendicular to the
average orientation of the fracture set (Narr and Suppe, 1991; Gross,
1993). Previous studies of fracture formation within layered materials
have shown that fracture infilling occurs with increasing applied strain
up to a critical fracture spacing to layer thickness ratio (e.g. Garrett and
Bailey, 1977; Parvizi and Bailey, 1978; Thouless et al., 1992; Wu
and Pollard, 1995). Fracture saturation is a phenomenon where no
further infilling of fractures takes place once this critical ratio is
reached and the additional strain is accommodated by the opening of
existing fractures (Aveston et al., 1971; Wu and Pollard, 1995).
The stress-transfer model proposed by Hobbs (1967) has been
commonly cited as an explanation for the linear relationship between
fracture spacing and rock layer thickness where fracture spacing
decreases as long as the applied strain is increased. Bai and Pollard
(2000a) pointed out that the Hobbs' model does not account for
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K. Iyer, Y.Y. Podladchikov / Earth and Planetary Science Letters 280 (2009) 159–166
fracture saturation as fracture infill occurs ad infinitum without any
critical threshold for fracture spacing being reached. Another approach
to explain fracture spacing in layered rocks is the stress-shadow model
caused by the opening of a fracture (Lachenbruch, 1961; Nur, 1982;
Pollard and Segall, 1987; Gross et al., 1995; Germanovich and Astakhov,
2004). The fracture-normal stress at the fracture faces is zero in the
absence of fluid pressures within the fractures. This stress gradually
increases away from the fracture and eventually reaches the remote
stress value. There, therefore, exists a region of reduced stresses near a
fracture where no other fractures may form as the stress is below the
strength of the rock. This region would subsequently constrain the
fracture spacing in a rock layer for a given strain. Recently, Bai and
Pollard (2000a) have shown that for a three-layer system with
perfectly welded interfaces under extension, the critical S/Tf ratio is
approximately 1.0. The normal horizontal stress component within the
layer and perpendicular to the fractures changes from tensile to
compressive when the critical ratio is reached. Bai and Pollard (2000b)
have shown that propagation of a pre-existing flaw at the interface of
the fractured layer and subsequent propagation through the layer may
occur for fracture spacing to layer thickness ratios below 1.0. The
propagation of the fracture depends on the initial size and location of
flaws present in the layer. They found that the minimum S/Tf ratio for a
system with similar elastic constants for the fractured and neighboring
layers is 0.273 and ratios below this minimum value may be possible
due to factors other than pure extension.
The above mentioned models incorporate two important assumptions. Firstly, the models assume that all the layers in the series are
subject to the same remote strains. Additionally, the models also assume
that the interfaces between the fractured layer and the surrounding
layers are perfectly welded together. Nevertheless, slip will occur at the
interface between the layers if the interfacial shear stress reaches the
interfacial slip strength (Li and Yang, 2007). In this paper, we investigate
fracture formation in a layer embedded in incompetent rocks incorporating and driven by a strain mismatch between the fractured, central
layer and the surrounding matrix. We present a 1-D finite difference
linear elastic model that accommodates the strain mismatch by joint
formation within the middle layer. In addition, the central layer in the
model is not necessarily welded to the bounding layers and is allowed to
slip along the interfaces when the interfacial shear strength is reached.
We find that the resulting fracture spacing in the central layer depends
not only on the thickness of the layer but also on the material strength.
The results show that the final fracture spacing to layer thickness ratio
in such layered systems is directly proportional to the tensile strength
and inversely proportional to the shear strength of the material. We
also show that changes in other material properties such as the shear
modulus or Young's modulus do not affect these results. The model
results are compared to a natural analog found in the fractured
orthopyroxenite dykes of the Leka Ophiolite Complex (LOC), Norway.
Fracture formation in the orthopyroxenite dykes occurs as a result of a
differential volume increase in the orthopyroxenite dykes and the
surrounding dunites during serpentinization. Detailed field studies and
measurements yield S/Tf ratios between 0.1 and 1.0. These values cannot
be explained adequately by welded interface models due to the fracture
saturation effects mentioned above. Independent of this theoretical
challenge, field observations also suggest a strong strain mismatch
between the orthopyroxenite layers and the surrounding dunite matrix.
We demonstrate that the strain mismatch-driven joint formation
associated with interfacial slip may explain the field observations. If
the model holds true for the processes occurring in the LOC, it may help
us to better evaluate the strength of natural interfaces undergoing
metamorphism at geological time scales and settings.
2. Numerical modeling
In order to understand the effects of shear stress along the
interfaces of the middle rock layer on fracture spacing, we use a quasi
Fig. 1. 2-D sketch of an elastic layer (M) with length L and height H sandwiched
between two rock layers (S). The middle layer, M, is fixed on the left side with respect to
horizontal movement (black diamond) and extension is driven by a background strain,
εbg. A weak material heterogeneity is introduced in the middle of the layer (white circle)
to initiate fracturing. The horizontal normal stress is calculated along the centre of
the middle layer (dashed line). Initially, the layer (M) and surrounding matrix (S) are
perfectly bonded at the interfaces (gray arrows). Interfacial slip is allowed to occur only
when the interfacial shear stress reaches the interfacial shear strength.
1-D, linear elastic, finite difference model of a layer under extension
which incorporates fracture formation. The model is based on a threelayer rock formation where the horizontal normal stress within the
middle layer, M, is calculated (Fig. 1). The length and height of the
modeled layer can be varied. The layer is fixed on the left boundary
and extension of the layer is incorporated in the model by the use of a
constant remote background strain (Fig. 1). Tensile strength along the
middle layer is constant except in the middle where a heterogeneity
(80% of tensile strength) is introduced to initiate fracturing. Open
fractures are allowed to develop within the middle layer when the
normal horizontal stress component, σxx, reaches the prescribed
tensile strength of the material, σt. The horizontal stress component is
set to zero where fractures occur. The layer interface is decoupled from
the bounding layers once the shear stress at the interface reaches the
shear strength of the layer.
Consider an elastic layer of length L and height H under extension
(Fig. 1). By integrating the horizontal force balance equation over the
layer thickness, we have:
bot
τ top
Ahσ xx i
xy − τ xy
+
=0
Ax
H
ð1Þ
where
⟨σxx⟩
top
τxy
bot
τxy
is the thickness averaged horizontal stress component
is the shear stress acting on the top interface of the layer
is the shear stress acting on the bottom interface of the layer
The average relative horizontal displacement, ⟨ur⟩, is related to the
average displacements in the layer, ⟨ul⟩, and surrounding matrix, ⟨ubg⟩,
by:
hur i = hul i − hubg i
ð2Þ
Henceforth, quantities that are averaged over thickness are
denoted with ⟨ ⟩.
K. Iyer, Y.Y. Podladchikov / Earth and Planetary Science Letters 280 (2009) 159–166
The thickness averaged horizontal strain in the layer, ⟨εxx⟩, is given
by:
hexx i =
Ahul i
Ax
ð3Þ
Substituting Eq. (2) in Eq. (3):
hexx i =
Ahubg i
Ahur i
+
Ax
Ax
ð4Þ
The average mismatch transformation strain between the matrix
and the layer, ⟨Δεxx⟩, can be written as:
bg
l
hΔexx i = hexx i − hexx i
ð5Þ
l
where ⟨εbg
xx ⟩ and ⟨εxx⟩ are the average transformation strains in the matrix
and layer, respectively.
However, as ⟨εlxx⟩ = 0,
bg
hΔexx i = hexx i
ð6Þ
The thickness averaged horizontal strain in the layer, ⟨εxx⟩, and the
relative horizontal displacement, ur, are related by:
hexx i =
Ahur i
+ hΔexx i
Ax
ð7Þ
Hooke's Law for linear elasticity states that:
hσ xx i = Ehexx i
ð8Þ
where E is an appropriately chosen elastic constant to represent threedimensional straining in a one dimensional model (i.e. a combination
of plane strain in the X–Y plane and constant stress in the Y-direction
assumptions).
Using Eqs. (7) and (8), we get the interlayer normal horizontal
stress by:
Ahur i
hσ xx i = E
+ hΔexx i
ð9Þ
Ax
The shear stress acting on the layer can be expressed as a function
of the shear modulus (G) and the shear strain (γ):
Auy
Aux
+
τxy = Gγ = G
Ay
Ax
ð10Þ
where ux and uy are components of the elastic displacement field.
Although more elaborate solutions for elastic shear stresses are
available (e.g. Ji and Saruwatari, 1998; Li and Yang, 2007), for simplicity
reasons, we neglect the variation of the shear stress along the interface
and approximate it, similar to the classical Hobbs model (1967), by:
top
bot
τxy = − τ xy ≈G
hur i
H
ð11Þ
bot
where τtop
xy and τxy are the shear stresses at the interface.
Two-dimensional Finite Element modeling verified that neither
the choice of shear modulus, G, nor the severe oversimplification of
this approximation of elastic shear stresses along the interface have
any noticeable results for the joint spacing at saturation.
In order to accommodate plastic failure at the interfaces, a
maximum cut-off for the shear stress (τxy) is imposed and is equal
to the shear strength (τs) of the material:
161
this paper, we use the mechanical engineering definition of plastic
deformation where plastic yield at the interface occurs when the shear
stresses exceeds the elastic limit (Hill, 1950; Zhang 1998). Plastic
yielding in this context does not relieve the stress present at the
interface but imposes a limit on it as opposed to joint formation within
the layer which results in a complete stress drop.
We solve for horizontal stress (σxx) and horizontal relative
displacement (ur) using Eqs. (1) and (9). In order to calculate shear
stress in the layer, we use Eq. (11) in the elastic regime and Eq. (12) in
the plastic regime. Therefore, we obtain:
8
u
>
< −2G r2
Ahσ xx i
H
=
>
Ax
: 2 τs
H
if
if
ur
b τs
H
u
G r N τs
H
G
ð13Þ
Eqs. (9) and (13) form a closed system of equations for the two
unknown profiles, σxx and ur.
Sliding of the layer across the frictional interface occurs once the
shear strength of the material is reached and the shear stress is kept
constant with a value equal to the shear strength. Experiments in rock
mechanics show that the unconfined shear strength of a rock may be
between three to five times the uniaxial tensile strength (e.g. Afrouz,
1992; Zhao, 2000). Runs were conducted on the numerical model
using these values as limits for the shear strength of the layer. In
addition to variations in shear strength, the thickness of the modeled
layer was also systematically varied to obtain values for the critical
fracture spacing to layer thickness ratio.
3. Results
3.1. Modeling
The development of the horizontal stress component (σxx) with
increasing strain mismatch for a run is shown in Fig. 2. The shear
strength of the layer (τs) is kept fixed at three times the tensile
strength (σt) for the example run. The initial horizontal stress
distribution within the layer is uniform and set to the tensile strength
of the heterogeneity in the middle of the layer. Joint formation is
initiated at the heterogeneity after which σxx at that point is set to
zero (Fig. 2a). The horizontal stress component increases with
increasing strain mismatch and results in the formation of joints
where σxx reaches σt (Fig. 2b–e). Frictional sliding at the interface
occurs when the shear stress (τxy) reaches the shear strength value
(τs). The shear stress value at the interface where slip occurs is kept
constant and equal to the value of shear strength. The horizontal stress
component stops evolving with increasing strain mismatch once the
shear resistance at the interface reaches its maximum value and no
more joints are formed (Fig. 2e). The resulting fracture spacing to layer
thickness ratio for the run where τs = 3σt is 0.166.
The model was also run for shear strength values (τs) of the layer
equal to four and five times the tensile strength (σt). The length of the
layer is kept constant for all runs and the normalized thickness is
varied. The minimum fracture spacing to layer thickness ratio
obtained for all runs with τs = 3σt is 0.166, for all runs with τs = 4σt
is 0.125 and for all runs with τs = 5σt is 0.10 (Fig. 3). It is found that the
approximation of shear stress in the elastic regime as shown in Eq. (6)
as well as variations of material properties (e.g. Young's Modulus, E)
does not affect the results obtained for the fracture spacing to layer
thickness ratios (Fig. 3).
ð12Þ
3.2. Fracture formation in orthopyroxenite dykes of the Leka Ophiolite
Complex
The negative sign in the above equation represents the opposing
force of friction to the relative direction of motion during sliding. In
The Leka Ophiolite Complex outcrops on the island of Leka, NordTrøndelag, Norway and is a part of the Upper Allochthon of the
top
bot
τxy = − τ xy ≈ − τ s
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K. Iyer, Y.Y. Podladchikov / Earth and Planetary Science Letters 280 (2009) 159–166
Fig. 2. Growth of the normalized horizontal stress component (σxx/σt) in the layer with an increase in strain mismatch. The shear strength of the material (τs) is three times the
tensile strength (σt) for the model run. The thickness and length of the layer are kept constant for the run. See text for details.
Scandinavian Caledonides (Furnes et al., 1988). The LOC formed as a
part of the oceanic lithosphere of the North Iapetus Ocean and was
obducted during the Caledonian orogeny between the Ordovician and
Silurian periods (Dunning and Pedersen, 1988; Titus et al., 2002)
during which the mantle section was folded into an open fold
(Maaløe, 2005).
The Leka Ophiolite Complex is one of the most completely preserved
ophiolites present in the Scandinavian Caldeonides (Prestvik, 1972) and
contains all the principle components of an ophiolite, including the
mantle section, the layered crustal sequence and the overlying
sediments (Fig. 4). The ultramafic units of the LOC are serpentinized
to variable degrees (b10 to ~90%). The mm to dm thick orthopyroxenite
dykes are of particular interest and are exposed within the layered
cumulates (dunite with minor wehrlite) near Steinstind (Fig. 4). The
dykes locally crosscut the dunites and wehrlites, and have variable
orientations. They are invariably fractured and the spacing between the
fractures is positively correlated with the thickness of the dykes (Iyer
et al., 2008a). The fractures are always sub-perpendicular to the contact
between the orthopyroxenite dukes and the dunite matrix. The process
of fracturing in the orthopyroxenite dykes has been attributed to the
serpentinization of the dunite matrix (Iyer et al., 2008a,b). One of the
important petrophysical effects of serpentinization is the density/
volume changes occurring in the serpentinized rocks. The reaction
of orthopyroxene to talc and olivine in the presence of fluids, and
subsequently to serpentine, controls a major part of the density changes
in the orthopyroxenite dykes which takes place between ~550 and
K. Iyer, Y.Y. Podladchikov / Earth and Planetary Science Letters 280 (2009) 159–166
Fig. 3. Plot of the ratios of fracture spacing to layer thickness (S/Tf) against the ratio of
tensile to shear strength (σt /τs) of the material for the 1-D FDM layer model with
frictional sliding interfaces. The diamonds and circles represent runs conducted with
different values of the shear parameters for the elastic regime (e.g. G) and show no
variations in the fracture spacing to layer thickness ratios.
650 °C at 1 kbar. The density changes in the dunites during
serpentinization are, on the other hand, dominated by the reaction of
olivine to serpentine and brucite at temperatures lower than 400 °C at
1 kbar. The density and volume changes in the ultramafic rocks occur at
163
different temperatures and are thus temporally separated which plays
an important role in the deformation of the orthopyroxenite dykes.
The volume changes occurring in the system, at lower temperatures, will therefore be dominated by the serpentinization of dunite
during which the orthopyroxenite dykes are mostly unreactive.
Approximately 50–60% serpentinization of the surrounding dunite
causes a volume increase of 25% and is also reflected in the bulk volume
change calculated from the phase transformations (Iyer et al., 2008b).
The expansion occurring in the dunite during serpentinization
will impose an extensional strain via the initially welded interface on
the orthopyroxenite dykes and cause them to fracture. The surfacenormal direction of the fractures, independent of the dyke orientation,
suggests that the dykes are subject to least compressive stresses
parallel to the dyke-dunite contact. This is consistent with a stress-field
generated by dunite expansion during serpentinization. The geometrical and statistical characteristics of the 2-D fracture networks in the
LOC are typical of patterns generated during hierarchical fracturing
where the layer is progressively broken up in to smaller domains (Iyer
et al., 2008a).
Field measurements across cross-sections through the dykes show
a linear correlation between the fracture spacing and the layer
thickness with a slope of approximately 0.45 ± 0.2 (R2 = 0.67) for 583
data points as shown in Fig. 5. The fracture spacing in the field was
measured as the average distance between two adjacent fractures.
Care was taken to exclude fractures that did not completely cut
through the dykes. The ratio of fracture spacing to layer thickness for
Fig. 4. Simplified geological map of the Leka Ophiolite Complex. The orthopyroxenite dykes are embedded in a dunite matrix and are exposed at Steinsind (filled orange circle). The
orthopyroxenite dykes are extensively fractured with the fractures oriented sub-perpendicular to the dyke-dunite contact.
164
K. Iyer, Y.Y. Podladchikov / Earth and Planetary Science Letters 280 (2009) 159–166
the fracture spacing to layer thickness ratios at saturation presented
above do not depend on elastic material properties. We have
considered the optimal scenario that would produce the highest
possible fracture density. Any transient or strain reversal effects that
deviate from this scenario would produce higher fracture spacing to
layer thickness ratios. The aim of the model is to quantify the spacing of
fractures driven by phase transformations and does not intend to
explain the modes of fracture propagation and initiation.
4.1. Activation of a frictional sliding interface
Fig. 5. Scatter plot of layer thickness against fracture spacing (583 data points). Data
points plot below the line with a slope of 1 (dashed line) which is expected from
extension models (e.g. Bai and Pollard, 2000a, Bai et al., 2000). A linear fit to the data
(solid line) gives a slope of 0.45 ± 0.2 (R2 = 0.67).
the field data ranges from 0.1 to 1. Two-dimensional observations
(plane view) of the fracture networks verify the above narrow fracture
spacing observed along the one-dimensional profiles (Iyer et al.,
2008a). This slope is considerably less than a critical ratio between 0.8
and 1.2 predicted from 2-D extension models for layered rocks with
welded interfaces (e.g. Bai and Pollard, 2000a, Bai et al., 2000; Fig. 5).
4. Discussion
The minimum fracture spacing to layer thickness ratios (S/Tf)
obtained from the 1-D FDM layer model with interfacial slip is
considerably lower than the S/Tf obtained from the previously
mentioned 2-D models with perfectly welded layers along the top
and bottom boundaries of the interface. From force balance across the
layer of height Tf and length S, with a normal horizontal stress
component within the layer (σxx) and a shear stress component acting
along the interface (τxy):
σ xx Tf = 2τxy S
ð14Þ
σ xx
S
=
Tf
2τ xy
ð15Þ
If we consider the fracture spacing in the layer with interfacial slip
at saturation, and substituting in Eq. (15):
σt
S
=
Tf
2τ s
Previous 2-D numerical studies of an elastic, layered system with
perfectly adhered interfaces under extension predict that further
infilling of fractures within the layer is not possible if the fracture
spacing to layer thickness ratio (S/Tf) falls below a critical value (0.8 to
1.2) and is independent of the average strain of the fractured layer. A
transition in stress-state occurs within the layer when S/Tf ratio is
below the critical value and inhibits further formation and propagation of fractures.
As discussed above in Section 3.2, equilibrium thermodynamic
calculations show that serpentinization of the dunites generates a
bulk volume increase of approximately 25%. This expansion translates
to 8% strain along any given direction imposed on the embedded
orthopyroxenite dykes, for the simple case where expansion of the
dunites is isotropic. Previous models with welded interfaces assume
that the remote strain imposed on all the layers in the model is equal
(e.g. Bai and Pollard, 2000a, Bai et al., 2000). However, the large strain
generated by the serpentinization of dunites is not equally imposed on
the embedded orthopyroxenite dykes and the surrounding dunites
themselves. The strain generated by the expansion of the dunites
drives the extension of the orthopyroxenite layers in our model. The
misfit in strain between the layer and the surrounding matrix is too
large and, cannot be accommodated by elastic deformation alone.
Therefore, the interface between the layer and the matrix could be
decoupled in order to accommodate the strain. The strain may also be
accommodated by ductile deformation. Some of the orthopyroxenite
dykes display pinch-swell structures in addition to tensile fractures
(Fig. 6). The mechanism and stress pattern involved in the formation
of pinch-swell structures is similar to those observed in boudin
formation. Boudin formation requires a competent rock layer
sandwiched between incompetent rocks (Wegmann, 1932). Stretching parallel to the layering of such a multi-layered rock series results
in the rupture of the competent layer while the incompetent layers
yield by plastic flow (Ramberg, 1955). Elongation of the rock layers
can be due to (a) stretching under tensile stress, (b) compression in a
direction perpendicular to the layering or (c) a stress couple at about a
ð16Þ
The above dependence of fracture spacing on the ratio of tensile
and shear strength of the material is similar to the model proposed by
Price (1966) and that used by Cox (1952) for fiber pullout, and is much
simpler than that developed by Ji et al. (1998) for interbed slip.
The result is the same as found by Kelly and Tyson (1965) and Lloyd
et al. (1982).
Using the values for shear and tensile strength obtained from rock
mechanics experiments (Afrouz, 1992; Zhao, 2000), the minimum
fracture spacing to layer thickness ratios for τs = 3σt, τs = 4σt and τs =
5σt are 0.166, 0.125 and 0.10, respectively. These minimum ratios are
also predicted by the 1-D elastic layer model with interfacial slip and
can further constrain the S/Tf ratios observed in the orthopyroxenite
dykes. As mentioned in Section 3.1, in the numerical model, elastic
effects do not affect the outcome of fracture spacing to layer thickness
ratios at saturation. We also find that force balance considerations for
Fig. 6. Fractured orthopyroxenite dyke (arrow) with a pinch-swell structure embedded
in a dunite matrix.
K. Iyer, Y.Y. Podladchikov / Earth and Planetary Science Letters 280 (2009) 159–166
45° angle to the elongation. The expansion of the dunite surrounding
the orthopyroxenite dykes fulfils conditions (a) and (b) required for
the formation of pinch-swell structures. It is, therefore, possible that
some of the strain accompanying the expansion of dunites is
associated with the plastic necking of the orthopyroxenite dykes. It
could also be possible that the plastic flow resulting in the pinch-swell
structures are related to the emplacement of the orthopyroxenite
dykes within the surrounding matrix rocks. Furthermore, only a few of
the observed orthopyroxenite dykes display pinch-swell structures
while others do not show any evidence for ductile flow but are
nevertheless similarly fractured.
We have indirect evidence, from petrological observations and
thermodynamic constraints, of the amount of strain generated during
serpentinization of the dunites. As serpentinization occurs throughout
the dunites, the accompanying strain would be the same for all regions
on the interface between the orthopyroxenite dykes and the dunites.
Although joint formation in the orthopyroxenite dykes would relieve
some of the strain, the presence of unfractured dyke domains between
two consecutive joints suggests that another mechanism, besides joint
formation within the dykes, must have released the strain at the
interface. The only other mechanism available to the system in order to
release strain would be the decoupling of the interface between the
dunites and the orthopyroxenite dykes. Hence, microstructures present
at the interface between the orthopyroxenite dykes and the dunites may
be an important indicator of interfacial slip. However, as serpentinization is an ongoing process in the studied lithologies, any structures
present at the interface that might give us direct insight into the slip
generated at the interface have been completely obliterated due to the
formation of secondary hydrous phases (Fig. 7). Therefore, the necessity
of interfacial slip to occur is a theoretical prediction as discussed above.
Another important implication of the model is the estimation of
the strength of rocks deduced from field observations of fracture
spacing to layer thickness ratios. The Leka Ophiolite Complex (LOC)
165
was a part of the oceanic lithosphere formed at a slow spreading
ridge (Maaløe, 2005). The average thickness of the oceanic crust is
approximately 7.1 ± 0.8 km except at very slow-spreading ridges
where the oceanic crust is anomalously thin (White et al., 1992). From
Byerlee's Law, the shear strength of rocks at a depth of 6 and 8 km
would be approximately 135 and 180 MPa, respectively (Byerlee, 1978;
Turcotte and Schubert, 2002). The typical tensile strength of rocks is
between 10 and 20 MPa (e.g. Afrouz, 1992; Zhao, 2000). Using these
values as constraints for the relation between the fracture spacing to
layer thickness ratio and the tensile to shear strength ratio, the model
predicts a minimum fracture spacing to layer thickness ratio in the
orthopyroxenite dykes of around 0.05. The measured field fracture
spacing to layer thickness ratios falls above this predicted minimum
ratio. However, the average value of fracture spacing to layer thickness
ratios measured in the field is 0.45 ± 0.2. This suggests that shear
strength of the rock may be significantly lowered as compared to that
predicted by Byerlee's Law which could be an effect of the phase
transformations occurring within the rock layer and/or the presence
of high fluid pressures.
5. Conclusions
The use of interfacial slip between the fractured layer and the
surrounding matrix of a layered rock series yields minimum fracture
spacing to layer thickness ratios that are significantly lower than the
ratios predicted by models with perfectly welded interfaces. We have
also shown that the minimum fracture spacing to layer thickness ratio
for our model is directly proportional to the ratio of tensile and shear
strength of the material. Lower minimum fracture spacing to layer
thickness ratios are possible if the difference between the tensile
and shear strength of the rock is greater. The fracturing of the
orthopyroxenite dykes of the Leka Ophiolite Complex driven by a
strain mismatch during serpentinization may have been deformed in a
Fig. 7. A. Photomicrograph of the interface between dunite and orthopyroxenite dyke. B. Photomicrograph of the interface (red square in 7A) shows complete serpentinization of the
original mineral assemblages. C. Photomicrograph of bastite (altered orthopyroxene) present in the orthopyroxenite dyke away from the interface. D. Photomicrograph of meshtextured serpentinization of olivine present in the dunite away from the interface.
166
K. Iyer, Y.Y. Podladchikov / Earth and Planetary Science Letters 280 (2009) 159–166
similar fashion accompanied by interfacial slip and, therefore, yielding
low fracture spacing to layer thickness ratios. The model also provides
constraints on the rock strength. Furthermore, the average value of
fracture spacing to layer thickness ratios measured in the field
suggests that phase transformations and/or fluid pressure could also
significantly affect rock rheology.
Acknowledgements
The authors are grateful to Sergei Medvedev, Dani Schmid, Marcin
Dabrowski and Victoria Yarushina, all at PGP, for their constructive
comments and criticisms. The authors thank Bjørn Jamtveit (PGP)
for his suggestions which helped improve previous versions of this
paper. The authors are thankful for the critical reviews of Leonard
Germanovich and an anonymous reviewer. This study was funded by
a Centre of Excellence grant from the Norwegian Research Council
to PGP.
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