SESolid EarthSESolid Earth1869-9529Copernicus GmbHGöttingen, Germany10.5194/se-6-497-2015Fracturing of ductile anisotropic multilayers: influence of material
strengthGomez-RivasE.e.gomez-rivas@abdn.ac.ukhttps://orcid.org/0000-0002-1317-6289GrieraA.https://orcid.org/0000-0003-4598-8385LlorensM.-G.Department of Geology and Petroleum Geology, University of
Aberdeen, Aberdeen, Scotland, UKDepartament de Geologia, Universitat Autònoma de
Barcelona, Barcelona, SpainDepartment of Geosciences, Eberhard Karls University of
Tübingen, Tübingen, GermanyE. Gomez-Rivas (e.gomez-rivas@abdn.ac.uk)19May20156249751431December201429January201510April201520April2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://se.copernicus.org/articles/6/497/2015/se-6-497-2015.htmlThe full text article is available as a PDF file from https://se.copernicus.org/articles/6/497/2015/se-6-497-2015.pdf
Fractures in rocks deformed under dominant ductile conditions typically form
simultaneously with viscous flow. Material strength plays a fundamental role
during fracture development in such systems, since fracture propagation can
be strongly reduced if the material accommodates most of the deformation by
viscous flow. Additionally, the degree and nature of anisotropy can
influence the orientation and type of resulting fractures. In this study,
four plasticine multilayer models have been deformed under coaxial boundary
conditions to investigate the influence of strength and anisotropy on the
formation of fracture networks. The experiments were made of different
mixtures and had two types of anisotropy: composite and composite-intrinsic.
The transition from non-localised deformation to systems where fracture
networks control deformation accommodation is determined by the ability of
the material to dissipate the external work and relax the elastic strain
during loading either by viscous flow or by coeval flow and failure.
Tension cracks grow in experiments with composite anisotropy, giving rise to
a network of shear fractures when they collapse and coalesce with
progressive deformation. The presence of an additional intrinsic anisotropy
enhances the direct nucleation of shear fractures, the propagation and
final length of which depend on the rigidity of the medium. Material strength
increases the fracture maximum displacement (dmax) to fracture length
(L) ratio, and the resulting values are significantly higher than those from
fractures in elastic–brittle rocks. This can be related to the low
propagation rates of fractures in rocks undergoing ductile deformation.
Examples of brittle deformation localisation in a ductile dominant
system. (a) small-scale shear fractures (F1 to F4) with flat/ramp
segments and roll-over geometries in deformed banded quartzites of the
Rabassers outcrop, at Cap de Creus (E Pyrenees, Spain) (Gomez-Rivas et al.,
2007); (b) the interpreted structures. Fracture planes are smooth.
Antithetic, synthetic and double-sense drag folds can be observed. Note that
same layers at both sides of fractures do not present the same drag fold
pattern. Displacements along fault surfaces are not constant and do not show
an elliptical distribution, as expected for an isolated fracture. Maximum
displacement–length relationships (dmax/L) range between 0.10 and
0.15. These fractures are interpreted as formed by segment linkage and
growth during coetaneous brittle and ductile deformation. Fracture offsets
of reference layers are 0 cm (L1), 3.5–3.8 cm (L2), 2.5–3.0 cm (L3) and
3.3–3.7 cm (L4). This view is perpendicular to the foliation and fracture
planes. Anisotropy of this rock is a consequence of grain size differences
between dark and white layers and preferred orientation of phyllosilicates.
The diameter of the Eur 0.05 coin is 21 mm.
Introduction
The deformation behaviour of Earth's crust rocks is often seen as a
transition from frictional and elastic–brittle behaviour at shallow depths
to ductile crystal–plastic flow at deeper levels. The change from brittle
and discontinuous deformation (i.e. fracture-dominated) to ductile and
continuous deformation (i.e. flow-dominated) is known as the
brittle-to-ductile transition and is typically characterised by systems in
which displacement is accommodated by networks of shear zones where brittle
and ductile deformation coexist and compete (e.g. Paterson, 1978; Passchier,
1984; Hobbs et al., 1986; Mancktelow, 2008). It is assumed that an increase
in depth progressively reduces pressure-dependent plasticity and increases
viscous flow, which is mainly controlled by strain rate and temperature.
However, field observations and experiments suggest that coeval fracturing
and rock flow are not restricted to a certain zone in the middle crust but
can affect rocks at a wide range of depths and deformation conditions from
the Earth's surface to the upper mantle (e.g. Simpson, 1985; Pennacchioni
and Cesare, 1997; Guermani and Pennacchioni, 1998; Passchier, 2001; Exner et
al., 2004; Kocher and Mancktelow, 2005, 2006; Mancktelow, 2006; 2009).
Moreover, brittle fractures can be precursors of ductile shear zones in
certain cases (e.g. Segall and Simpson, 1986; Pennacchioni, 2005; Mancktelow
and Pennacchioni, 2005; Pennacchioni and Mancktelow, 2007; Fusseis et al.,
2006; Misra et al., 2009). It is therefore of crucial importance to
recognise the main parameters controlling systems in which brittle and
ductile deformation coexist. We understand brittle behaviour here as
deformation showing loss of cohesion along discrete surfaces, and we
therefore assume that it results in strongly localised systems. In a
different manner, we consider that ductile strain can be localised or
distributed. Ductile localisation is characterised by zones of localised
deformation with continuous variations of strain across their width (i.e.
without discontinuity) and without loss of cohesion (e.g. Twiss and Moores,
1992). In this contribution, the term ductile is not associated with any
particular constitutive relationship (e.g. viscous or plastic behaviour). We
refer here to viscous deformation when stress is dependent on the strain
rate. Figure 1 shows an illustrative example of the interaction of brittle and
ductile deformation in a deformed quartzite (Cap de Creus, eastern Pyrenees,
northern
Spain) in a greenschist environment (Gomez-Rivas et al., 2007). Layers,
defined by grain-size variations and preferential orientation of
phyllosilicates, can be used as markers to track the displacement field.
They show that these rocks coevally fractured and flowed, as evidenced by
the presence of isoclinal folds outside fracture zones and drag folds
associated with small-scale faults. The resulting fractures present
relatively high values of maximum displacement (dmax) to length (L)
ratios, with strong gradients along them. It is a common observation that
dmax/L ratios are significantly higher in systems dominated by ductile
deformation (on the order of ∼ 10-1, e.g. Gomez-Rivas and Griera,
2011; Grasemann et al., 2011) than in brittle media (which range from ∼ 10-2 to ∼ 10-4) (e.g. Walsh and Watterson, 1987; Kim and
Sanderson, 2005). This can be explained by the low propagation rate and
rotation of fractures in ductile media compared to those in elastic–brittle
rocks (e.g. Exner et al., 2004; Kocher and Mancktelow, 2005; Pennacchioni
and Mancktelow, 2007).
The formation of brittle fractures in elastoplastic materials is a
relatively well-known process (e.g. Mandl, 2000, and references therein).
However, there are still many open questions about how fractures and shear
bands form and evolve in rocks deformed by dominant viscous flow.
Conceptually, brittle fractures will normally develop in a ductile medium
when
viscous flow is not able to relax the loading stress, therefore reaching the
strength limit of the material. Under these conditions, fracture propagation
has to be studied as a time-independent process, because plastic strain work
at crack tips is significantly increased during propagation and can result
in a reduction of fracture propagation rates (e.g. Perez, 2004). A number of
factors determine the characteristics of the resulting fracture network in
such complex coupled systems (e.g. amount of fractures, formation
mechanisms, orientations, type, connectivity, displacement). One of these
factors is material stiffness, which defines how rigid a material is and
can therefore determine the relative ratio of loading rate to the rate of
stress relaxation by viscous flow, which at the end would control material
strength in a ductile system (i.e. how the rock resists deformation).
Indeed, viscosity defines the rock's resistance to deformation by shear and
tensile stresses and can therefore control the brittle-to-ductile
transition. Another relevant factor is the degree of anisotropy, which can
induce a directional dependence of the resistance to deform. Transverse
anisotropy is a very common type of rock heterogeneity and can arise from
the stacking of layers with different properties (i.e. composite anisotropy;
Treagus, 1997) and/or from the presence of preferentially oriented planar
minerals (i.e. intrinsic anisotropy; e.g. Griera et al., 2013).
Sketch of a multilayer experiment. The arrows indicate
the direction of the principal stresses applied by the deformation
apparatus. The initial layer thickness was ∼ 4–5 mm. After
Gomez-Rivas and Griera (2011).
This contribution presents an experimental study of the influence of
material strength and the degree and type of anisotropy on the formation of
brittle fractures in ductile multilayers under low effective confinement and
plane strain conditions. Plasticine multilayers with different mechanical
properties and anisotropies have been coaxially deformed at a constant strain
rate to visualise the transition from non-localising systems to models
where deformation is strongly localised along a few large fractures. Layers
in these experiments are oriented parallel to the extension direction and
perpendicular to the maximum compression. We aim to address cases where
effective confining pressure is relatively low, like ductile rocks at
shallow depths (e.g. clays, salt bodies, etc.) or middle- to lower-crust
rocks with high fluid pressures or subjected to local effective tensional
stress (e.g. Fagereng, 2013). We aim to (1) analyse the influence of
material strength on the transition from non-localising to strongly
localised systems using the same deformation conditions and very similar
analogue materials, (2) address the role of different types of transverse
anisotropy (composite and composite-intrinsic) on the degree of localisation
and developed structures, (3) understand how coeval ductile–brittle
deformation is visualised in terms of stress–strain relations and (4) capture
the key factors controlling the style and characteristics of the
resulting structures (tension and shear fractures, pinch-and-swell) and how
they evolve towards well-developed fracture networks with different
properties (orientations, displacement–length ratios, etc.). In order to
provide proper dynamic scaling and define the mechanical reference
framework, the rheology of the analogue materials was characterised prior to
experiments with uniaxial compression and relaxation tests.
Materials and methodsDeformation apparatus
A strain rate and temperature-controlled apparatus (BCN-stage; Carreras et
al., 2000) was used to deform the plasticine models. The prototype is based
at the Universitat Autònoma de Barcelona (Spain) and can apply
deformations from pure to simple shear (0 < Wk < 1) at
variable temperatures. This apparatus has been used for several analogue
modelling studies (Druguet and Carreras, 2006; Bons et al., 2008; Druguet
and Castaño, 2010; Gomez-Rivas, 2008; Gomez-Rivas and Griera, 2009,
2011, 2012).
Experimental setup and deformation conditions
Plasticine is an ideal analogue of rocks undergoing coeval ductile and
brittle deformation, because it can flow and also fracture at the same time
depending on its composition and deformation conditions (temperature, strain
rate, boundary conditions). It therefore presents elastoviscoplastic
behaviour. Two kinds of commercial plasticine were utilised in this study.
They were sold under the trademarks OCLU-PLAST and JOVI, both manufactured
in Barcelona (Spain). Using them as a base, four different mixtures were
created in order to build four models: type A (white and purple OCLU-PLAST
pure plasticine), type B (white and purple OCLU-PLAST plasticine mixed with
10 % paper flakes), type C (white and green JOVI pure plasticine) and type
D (blue and red JOVI plasticine mixed with 10 % paper flakes). Please note
that the type A plasticine was the same material used for the experiments of
Gomez-Rivas and Griera (2011, 2012). Flakes were made of differently coloured
paper and had a maximum size of ∼ 2 mm and a density of
80 g m-2. The density of these plasticines is ∼ 1100 kg m-3.
The models were created by stacking layers (∼ 4.25 to ∼ 4.50 mm thick)
of alternating colours, oriented perpendicular to the
Z direction (Fig. 2). This thickness could vary up to ±0.5 mm for
models with paper flakes. Two consecutive layers of the same (intermediate)
colour were inserted twice in each model (at Z= 10 cm and Z= 20 cm) in order
to have reference markers for the quick identification of structures and
other layers. Materials were mixed by hand at room temperature and then
flattened with an adjustable hand rolling mill for engraving art (∼ 50 cm long
and with a diameter of ∼ 15 cm). In this way, paper flakes were
preferentially oriented parallel to layers and therefore perpendicular to
the compression direction Z. This procedure also avoided the presence of air
bubbles within the models.
Transverse anisotropy in all models was defined by the stacking of beds,
which created a composite layering. Additionally, experiments containing
paper flakes (B and D) also presented an intrinsic anisotropy defined by
their preferred orientation. It is important to note that these flakes were
inserted in order to increase the degree of anisotropy and do not try to
simulate the role of individual minerals in rocks. Each model had an initial
size of 30 × 15 × 10 cm and was compressed in the Z direction and extended
in the X direction, while the Y direction remained constant using a reinforced
transparent glass. Strain rate and temperature were kept constant at 2 × 10-5 s-1 and 26 ∘C respectively. The samples
were deformed until a bulk finite strain ratio of RX/Z∼ 4 (i.e. ∼ 50 % shortening). Stress was recorded using gauges parallel to X and Z.
Digital pictures of the upper surface (X–Z plane) were taken every 1 %
shortening. Frictional effects were minimised by lubricating the press
boundaries with Vaseline and the top surface with glycerine. Each
model was biaxially compressed during 24 h at ∼ 10 kPa
prior to deformation in order to bond layers and avoid further interlayer
slipping.
Properties calculated from compression tests for the different
mixtures used as rock analogues. The values of axial stress, dynamic
effective viscosity, stress exponent and material constant were calculated
for 10 % shortening.
Summary of material properties calculated with relaxation tests for
the different mixtures used as rock analogues. Only tests deformed at
similar strain rates than multilayer experiments are displayed. The values
of elastic shear modulus (G) and Deborah number (De) were calculated assuming a
Maxwell body.
Stress vs. axial strain (expressed in % shortening) curves for
the materials used in this study at different strain rates. Dashed lines
indicate the strain reference value used for comparison of material
properties (10 % shortening).
Values of strain rate and effective viscosity of mid-crustal rocks
and the less and more viscous analogue materials used in this study (Types A
and D mixtures respectively). The strain rate value for mid-crustal rocks
was taken from Pfiffner and Ramsay (1982) and Weijermars (1997). The
viscosity of schists was taken from Talbot (1999) and Davidson et al. (1994).
Log strain rate (ε˙) vs. log effective dynamic
viscosity (η∗) comparing the mixtures used in this study with
other kinds of commercially available plasticine used by other authors.
Effective dynamic viscosity values were taken at 10 % shortening. Modified
from Gomez-Rivas and Griera (2011).
A 100 cm2 area at the centre of each model was used for acquiring
fracture data and observations, thus avoiding boundary effects such as
friction with walls. The following parameters were systematically measured
on all fractures at 10 % shortening intervals: fracture length (L), angle
between fracture and the Z axis (δ) and cumulative fracture
displacement (dmax). In order to minimise personal bias effects while
collecting data, the two first authors independently acquired measurements
and the root mean square deviation normalised to mean values was
systematically calculated for each parameter.
The degree of localisation was macroscopically estimated with a strain
localisation factor (Iloc). We defined this parameter as the ratio of
the maximum to the minimum shortening measured using reference layers for
each model (see Sect. 3), which were used as normal shortening markers.
Homogeneous deformation would result in Iloc= 1.
Mechanical properties of the experimental materials
Prior to carrying out the experiments, the mechanical properties of each
analogue material were characterised with uniaxial compression and
relaxation tests at variable strain rates and temperatures. A total of 30
tests were performed by deforming 10 cm cubes up to a minimum 20 %
shortening at variable strain rates, using the same deformation apparatus
and procedure as for the final experiments. The cubes were shaped using a
wooden plate to form the six faces and then cut with a saw. The methods and
equations for these tests are described in detail in Gomez-Rivas and Griera (2011)
and are based on the studies of McClay (1976), Weijermars and
Schmeling (1986), Mancktelow (1988), Ranalli (1995), Schöpfer and Zulauf (2002) and Zulauf and Zulauf (2004).
The conditions at which the tests were run, and the parameters resulting
from them are summarised in Tables 1 and 2 and Figs. 3 and 4. Parameters,
including volumetric strain, were calculated at 10 % shortening. It is
important to note that the samples did not fracture during uniaxial
compression. The results indicate that these mixtures behave as non-linear
elastoviscous materials with stress exponents ranging from
n∼ 3–4 for pure plasticine (types A and C) to
n∼ 4.5–5 for mixtures containing plasticine and paper flakes
(types B and D). The correlation coefficients (R2) calculated from the
log diagrams of strain rate vs. stress range between 0.73 and 0.98. The reason
why stress exponents are higher in mixtures containing paper flakes than in
those composed of pure plasticine is because flakes produce an increase of
strain partitioning and heterogeneities of the flow field. A marked strain
hardening can be identified form strain–stress curves for most of the tests
(Fig. 3). However, the amount of shortening reached during these tests was
not high enough to produce macroscopic failure. Clear yield stresses can be
detected in most of the tests made of OCLU-PLAST and in some made of JOVI
plasticine. However, not all tests present yield stress, especially those
with a higher viscoelastic response (Fig. 3f). The viscosities of the
mixtures made of JOVI plasticine (C, D) are considerably higher than those
of OCLU-PLAST (A, B). Adding paper flakes to plasticine makes the material
stiffer and increases the non-linearity behaviour of the mixture. The
addition of dye (purple or green) to plasticine makes it slightly softer and
more non-linear than the white one, although its rheology does not
significantly change. Strain vs. effective viscosity curves reveal a marked
strain-rate softening (Fig. 4). These two types of plasticine have an
effective viscosity between ∼ 0.6 × 109 and
∼ 3 × 109 Pa s at low strain rates and thus behave in a
similar way than other kinds of plasticine used by other authors (see Fig. 4, references included in the figure). Volumetric strain values (Table 1)
indicate that these mixtures loose between ∼ 1 and
∼ 6 % of their volume at 10 % shortening. Mixtures
composed of OCLU-PLAST plasticine undergo more volumetric strain when paper
flakes are inserted. On the contrary, mixtures made of JOVI plasticine
present a similar volumetric strain regardless of whether they contain paper
flakes or not. A dependence of volumetric strain on strain rate is not
observed.
Relaxation tests (i.e. stress evolution under constant strain) revealed that
the estimated elastic shear modulus (G) ranges between 2.4 × 106
and 4.3 × 106 Pa for mixtures made of the softer plasticine (types A and B)
and between 8.1 × 106 and 1.2 × 107 Pa for mixtures made of the
harder plasticine (types C and D) (Table 2). The Deborah number (De) (Reiner,
1964) is a non-dimensional factor that defines how fluid a material is, and
it is equivalent to the ratio of the time of relaxation (Maxwell time,
τm) and the time of observation (strain rate, ε˙). Estimated De values (between 3.8 × 10-3 and
5.8 × 10-3; Table 2) indicate that our experimental materials have the typical relaxation
behaviour of a viscoelastic solid (e.g. Poliakov et al, 1993).
Models A and C have a relative low degree of anisotropy because of the low
viscosity contrast between alternating layers. On the contrary, models B and
D are significantly more anisotropic, since they contain preferentially
oriented paper flakes. Gomez-Rivas and Griera (2009) estimated a degree of
anisotropy of ∼ 6 for experiments that used a very similar
composition to type B.
Model scaling
The experiments presented in this study are scaled based on the geometrical
and dynamic similarity of the observed deformation, following the methods
described by Ramberg (1981). The normal stress ratio between model and
nature is σ∗=ρ∗g∗l∗,
where ρ∗, g∗ and l∗ are the model / nature
ratios of density, gravity and length respectively. Since we address
fracture formation at the mesoscale, we have considered a ratio of 1 : 1
between the experimental and natural scales in space. When the same gravity is
assumed for both the experiments and nature, then the normal stress scaling
ratio is only determined by the density ratio. As mentioned in Sect. 2.2 the
models have a density of ∼ 1100 kg m-3. Therefore, the
value of σ∗ is 0.41 when we consider that schists have an
average density of 2700 kg m-3 (e.g. Smithson, 1970). The model / nature
ratio of strain rate ε˙∗ is 2 × 109, assuming
a natural strain rate of 10-14 s-1 (e.g. Pfiffner and Ramsay,
1982). In this way, 1 experimental second (at ε˙=2 × 10-5 s-1) is approximately equivalent to ∼ 60 natural years.
Therefore, the value of natural viscosity that would correspond to these
experiments is expressed by
ηnature=ηmodel⋅ε˙∗σ∗,
and therefore scaled natural viscosities of 3 × 1018 Pa s and 1.7 × 1019 Pa s would correspond to the softer (model A) and harder (model D)
experiments respectively. Viscosity and strain rate values of experimental
and natural materials are presented in Table 3 for their dynamic scaling.
These values of scaled viscosities are slightly lower in magnitude than the
estimated ones for schists in the middle crust, although within the
published ranges (∼ 1019–1020 Pa s: e.g. Talbot, 1999;
Davidson et al., 1994) (Table 3).
Compilation of shear-fracture data at shortening intervals of
20, 30, 40 and 50 %. δ is the average shear-fracture
orientation with respect to the Z axis, measured in degrees. Fracture length
(L) and maximum displacement (dmax) of shear fractures are measured in
cm. Standard deviations are in parentheses. N and Ntension indicate the
total number of shear fractures and tension cracks respectively. Length
(L) values do not include data from tension cracks and refer to individual
fracture segments. Fault zones composed of several segments are therefore
longer than individual ones. The number of tension cracks for models B and D
is not displayed, since cracks were too small to trace them accurately. Note
that data corresponding to 50 % shortening in model D were actually
measured at 44 % shortening, when this experiment finished.
Photographs of the initial (0 % shortening) and final (∼ 50 % shortening) stages of the four multilayer models. Maps of analysed
fracture networks are displayed on the right side. Red dashed lines indicate
reference layers. Grey shadowed areas in type D model show the location of
large fault zones. Only structures located in the central area of each
model, indicated with a rectangle, are systematically studied. Iloc is
the localisation factor (see Sect. 2.2). The degree of localisation
progressively increases from model A to model D and depends on the
strength, effective viscosity and degree of anisotropy of the models. Marked
differences between the resulting fracture networks can be clearly
identified.
Stress vs. strain curves for the four multilayer experiments.
Yielding followed by slight strain softening can be clearly identified for
models C and D, while experiments A and B record a progressive stress
increase that tends towards steady state with increasing strain.
Rose diagrams showing the orientation and number of fractures (at
orientation intervals of 10∘) of experiments: (a) type A,
(b) type B, (c) type C and (d) type D. N is the number of data measurements
in each diagram. Only fractures measured within the sampling area are
included. Note that horizontal scales are logarithmic.
Detailed photographs showing the evolution of structures in the
four different experiments. (a–c) Type A experiment: (1) a shear-fracture
forms from a slightly pinched layer; (2) coeval development of a tension
crack and a shear fracture; (3) tension cracks keep forming until ∼ 40 % shortening. (d–f) Type B experiment: (4) mixed-mode fracture with
associated drag folds. The propagation rate was very low due to plastic
deformation at fracture tips; (5) tension crack that collapsed and gave rise
to a shear fracture; (6) in-plane shear fracture with enhanced propagation
at one of the tips; (7, 8) linkage of fracture segments produced larger
fractures with heterogeneous displacements. (g–i) Type C experiment: (9) nucleation and propagation of a tension crack that evolved to a mixed-shear
mode fracture; (10) progressive enlargement of individual fractures as a
result of progressive tip-line propagation and linkage with the nearest
fracture segments; (11) nucleation of conjugate shear fractures from tension
cracks, resulting in sharp fracture segments; (12) relatively long shear
fracture that formed by linkage of a tension crack and a shear fracture;
(13) small tension crack that evolved to form a heterogeneous asymmetric
boudinage-like structure. (j–l) Type D experiment: (14) formation of two
parallel dextral fractures that propagate and join, and their lower tip ends at
a larger conjugate fracture; (15) development of a tension crack at the
intersection between large shear fractures; (16) a very large sinistral
fracture with zigzag geometry forms by segment linkage from side to side of
the model. Solid lines represent fractures, while dashed lines indicate
layering.
Log maximum fracture length (dmax) vs. log maximum fracture
displacement (L) graphs for models (a) type A, (b) type B, (c) type C and (d) type D. Data correspond to shear fractures measured at 20, 30,
40 and 50 % of shortening within the sampling area. Grey lines
indicate linear relationships in the log–log graph for different c values in
Eq. (2).
Experimental results
The results of the four multilayer experiments indicate that the mechanical
behaviour and the resulting deformation pattern are notably different
depending on the material used (Fig. 5). There is a marked transition from a
model in which deformation is almost homogeneously distributed (type A) to a
system controlled by a few large shear fractures (type D). At the end of the
experiments (∼ 50 % bulk shortening), model A accommodated
deformation mainly by homogeneous flattening. Increasing the material
strength resulted in a larger number of macroscopic fractures (models B, C
and D), although the characteristics of the resulting fracture networks
strongly varied between these three experiments. The stress–strain curves
(Fig. 6) reveal that recorded stresses increased systematically from model A
to model D. Deformation was mainly accommodated by homogeneous flow during
the first deformation stages in all models. Such flow was associated with a
stress increase. At about 10 % shortening the yield stress was reached for
the stiffer models (C and D) and progressively decreased in these
experiments up to the end without reaching a clear steady state. The first
macroscopic fractures were not visible until ∼ 15–18 %
shortening. In the softer models (A and B) a sharp yield stress was not
identified, and stress progressively rose with strain until a steady state
was reached. This steady state behaved slightly different in each
experiment, as stress kept slowly growing in model B while it slightly
decreased in model A.
The type of fractures and their orientations with respect to the deformation
axes are also significantly different depending on the material (Figs. 7, 8, 9). Strain localisation and material embrittlement are enhanced when
stiffness (or viscosity) is increased, and therefore the density and type of
developed fractures strongly depend on how stiff the analogue material is.
After 50 % bulk shortening, deformation in experiment A was mainly
accommodated by homogeneous flattening associated with viscous flow (Figs. 5, 8a–c). The estimated strain localisation factor was Iloc∼ 1.04
and normal shortening measured using the reference layers of Fig. 5 ranged
between 50 and 52 %. Traction structures along layer interfaces associated
with potential inter-layer slipping were not observed. Layers were thinned
with increasing deformation, and only a very small number of tension cracks
and shear fractures could develop in this experiment. Structures of both
types only started to be macroscopically visible after 30 % shortening.
However, tension cracks were formed until ∼ 40 % shortening and after
that only shear fractures developed. The collapse of cracks gave rise to the
formation of hybrid fractures (or mixed mode I–II fractures). They evolved
to become shear fractures organised in two conjugate sets (Fig. 8a–c). With
very few exceptions, shear fractures formed at angles of ∼ 40 to 50∘ with respect to Z and tended to
rotate towards X at a rate significantly slower than a passive line (Fig. 7a,
Table 4). At the end of the experiment, the length of fractures within the
sampling area varied between 0.33 and 1.47 cm, following an exponential
distribution. The cumulative fracture slip was always less than 20 % of
the fracture length. Relatively variable maximum displacement (dmax)–length (L) ratios could be found in this case (Fig. 9a).
The behaviour of model B, which was made of a mixture of soft plasticine and
paper flakes, was significantly different. The presence of heterogeneities
associated with flakes enhanced the nucleation of a large population of
small-scale shear fractures (Fig. 5). Small tension cracks were also
recognisable within the sample (Fig. 8d–f), although flakes prevented their
propagation in a way that large cracks could not form. In this case, a large
number of millimetre-scale cracks formed at the interfaces between layers or
between flakes and plasticine. Pinch-and-swell and boudinage-like structures
also started to develop in the first deformation stages. Pinch-and-swell
formed from heterogeneities in areas where layers were slightly thinner or
thicker, or where microscopic cracks already existed. Boudinage-like
structures developed when several tension cracks nucleated in the same
layer. Two symmetrical sets of conjugate shear fractures formed in three
different ways with increasing strain: (1) they directly nucleated (i.e.
without precursors) enhanced by the heterogeneity of the two-phase
(plasticine-paper flakes) system; (2) they formed by progressive necking of
pinch-and-swell and boudinage-like structures; and (3) they formed by coalescence and
collapse of tension cracks (Fig. 8d–f). Shear fractures formed at an angle
higher than 45∘ with regard to the Z axis (Fig. 7b). The
percentage of fractures oriented at more than 45∘ with Z
ranged between 88 % at 20 % shortening to more than 94 % for
30–50 % shortening, thus indicating that fractures slightly rotated
towards X. Average orientations increased between ∼ 49
and 56∘ at 20 and 50 % shortening respectively
(Table 4). At the end of the experiment, shear-fracture lengths ranged
between 0.5 and 1.9 cm, and the cumulative fracture slip was approximately
25 % of the total length. The ratio between maximum displacement
(dmax) and length (L) was considerably higher than that of model A (Fig. 9), even though fracture propagation was not very high in model B since new
fractures nucleated all the time until the end of the experiment (see N –
total number of fractures per set in Table 4). At the model scale,
deformation was approximately homogenously distributed, as evidenced by a
strain localisation factor of Iloc∼ 1.16 and shortening normal to
the reference layers ranged between 47 and 54 %.
The evolution of model C resembles that of model A but with a considerably
higher amount of fractures. In this case, strain localisation was related to
the nucleation and growth of a very large population of relatively long
tension cracks, which evolved to form two conjugate sets of shear fractures
with increasing strain (Fig. 7c, Table 4). Tension cracks formed during the
first experiment stages and up to ∼ 30 % shortening. When deformation
increased, their nucleation and propagation was aborted and they started to
quickly collapse and rotate towards the extension direction, thus enhancing
the formation of two conjugate sets of shear fractures. This process took
place by crack collapse and coalescence and by fracture segment linkage
(Fig. 8g, i). Such mechanisms enhanced fracture connectivity, thus amplifying
the direct nucleation of secondary shear fractures. Boudinage-like
structures did not develop in this experiment because tension cracks were
not constrained to one layer but cut several consecutive layers. Fracture
statistics illustrate the clear transition from a tension- to a shear-fracture-dominated system with progressive deformation. Almost no shear
fractures were observed at 20 % shortening, while many tension cracks
developed. At 30 % shortening there were still more tension than shear
fractures, which were oriented at an average of ∼ 35∘
with respect to Z (with a standard deviation of ∼ 7∘;
Table 4). A marked change in the properties of the fracture network took
place between 30 and 40 % shortening. At 40 % shortening only a few
tension cracks remained active, while a dense network of shear fractures was
observed. Such fractures had at this stage widely variable orientations with
respect to Z (from ∼ 25 to ∼ 55∘), with 43 % of them oriented at angles higher than 45∘.
These variable orientations remained at 50 % shortening, when all tension
cracks have disappeared. At this stage 67 % of shear fractures were
oriented at more than 45∘ with Z. Despite the differences in
material behaviour and type of fractures, the ratio between fracture length
and accumulated displacement was similar to the one observed for the type B
model (Fig. 9). At the model scale, strain localisation by the fracture
network was resolved at a length scale smaller than the sample length, since
fractures did not cut across the entire experiment. The calculated normal
shortening ranged between 45 and 55 %, and the strain localisation factor
was therefore relatively low (Iloc∼ 1.2).
Finally, the stiffer model (type D) experienced a very different deformation
history than the previous three experiments. Despite this, it presents some
similarities with model B, mainly associated with the presence of a second
phase (i.e. paper flakes). Large tension cracks were not observed in model D
(Fig. 5). Instead, a small number of very large shear fractures developed,
with lengths ranging between 2.5 and 9 cm. It is important to note that
these measurements refer to individual fracture segments, but fracture zones
composed of several segments were of course significantly longer than that.
Some of them propagated up to the limits of the model and were able to
accommodate considerably larger displacements than the ones registered in
the other three experiments (Fig. 9). This observation is clearly supported
by the fact that the maximum fracture displacement was approximately 40 %
of the total fracture length. Another special feature of this model is that
the two conjugate shear-fracture sets were not symmetric, since the
sinistral set nucleated earlier than the dextral one, which subsequently
cross-cut and displaced the early sinistral fractures (Fig. 8j–l). Shear
fractures in this model were, on average, oriented at 43 to 50∘
with Z. However, these angles were very variable and some large fractures
formed a lower angle with the maximum compression axis. Fractures in this
model tended to accommodate deformation by slip, instead of rotating towards
X (Fig. 7d, Table 4), in a way contrary to the other three experiments in
which
the two sets were always symmetrical with respect to the X and Z axes. At the
model scale, deformation was heterogeneously distributed and strong necking
was observable at the central part of the experiment, where relative large
shear fault zones cross-cut. A strong strain partitioning was detected
between high and low strain domains, where layer-normal shortening was about
63 and 30–36 % respectively. The strain localisation factor
(Iloc) was higher than 2.0.
Discussion
The experimental results obtained in this study indicate that the mechanical
properties of an elastoviscoplastic material have a strong influence on the
degree of brittle deformation and how deformation is accommodated by a
fracture network (Figs. 5, 8). The style of developed structures and their
properties strongly depend on the material mechanical behaviour (Figs. 7, 9, Table 4). A marked transition from distributed to strongly localised systems
can be observed when variants of the same materials are deformed under the
same conditions. Our experiments are made on two commercial types of
plasticine (OCLU-PLAST and JOVI), which have a similar stress exponent when
they are not mixed with other components (Table 1). The effective viscosity
of pure JOVI plasticine is about 3 times higher than that of pure
OCLU-PLAST, while elastic shear modulus (G) values of mixtures made of JOVI
plasticine are between 2 and 6 times higher than those made of OCLU-PLAST
(Table 2). These variations are already high enough to result in two very
different deformation systems, since model A (made of pure OCLU-PLAST
plasticine) mostly accommodated deformation by homogeneous flattening
associated with viscous flow while a dense network of tension and shear
fractures coeval with ductile flow developed in model C (made of pure JOVI
plasticine). The stronger elastoviscous behaviour of type C plasticine does
not allow an efficient stress relaxation by viscous flow, even for high
deformation values (> 25 % shortening), thus enhancing fracture
formation. The addition of paper flakes as a second phase, statistically
oriented parallel to layering in the initial model, produced again a
remarkably different mechanical behaviour. A dense network of small shear
fractures formed in model B (made of OCLU-PLAST plasticine and flakes),
while a few large fractures controlling the system developed in the most
rigid experiment (model D, made of JOVI plasticine and paper flakes).
As explained in Sect. 2.3, no shear bands or fractures formed in the
uniaxial compression tests. This is evidenced by the lack of material
discontinuities and the absence of pronounced and sharp yield points in the
stress–strain curves (Fig. 3). This applies not only to tests made of
pure plasticine but also to tests composed of mixtures B and D, which
include randomly oriented paper flakes. These observations suggest that the
presence of heterogeneities within the material is required to produce
fracture onset at relatively low deformation stages (less than 20 %
shortening). In our experiments, heterogeneities are associated with two
types of transverse anisotropy: (1) composite anisotropy (Treagus, 1997)
defined by stacking of layers with slight contrasting properties and (2) intrinsic anisotropy produced by the preferred orientation of elements of a
second phase (i.e. paper flakes) statistically oriented parallel to
layering. All experiments are composites, but models B and D include an
additional intrinsic anisotropy. The type and degree of anisotropy
(composite vs. intrinsic) can play a fundamental role on the resulting
structures and the bulk material behaviour (Griera et al., 2011, 2013).
The transition from ductile to coeval ductile–brittle behaviour is
determined by the ability of the material to dissipate the imposed external
work and relax the elastic strain energy stored as a consequence of loading
(e.g. Anderson, 2005). This relaxation can take place either by viscous or
coeval viscous–brittle deformation. New fractures can only grow when the
strain energy released during fracture growth exceeds the sum of the surface
energy of the new crack segment and the plastic deformation energy at the
crack tip (e.g. Perez, 2004). These processes strongly depend on the
material strength. Stress–strain curves are used to establish a qualitative
relationship between the strain localisation pattern and the work necessary
to deform the sample. Such curves reveal a higher degree of localisation in
the harder model (type D), which registered a marked strain softening
behaviour following the stress peak (∼ 12 % shortening) (Fig. 6). The
localisation of fracture networks is related to a reduction of the active
volume that is being deformed and an increase on the efficiency of the
accommodation of the imposed shortening by fracture slip. The growth of a
network of a few large fractures in the most viscous model (type D), or the
development of a well arranged but segmented fracture network in experiment
C, results in strain softening after yielding (Figs. 5, 6). Fracture
networks in these two models were able to accommodate the displacement
imposed by the boundary conditions, although experiment C also deformed
coevally by dominant viscous flow. Model A basically deformed by viscous
flow, and the resulting stress–strain curve displays first a slight
increase, and then a steady-state flow with gentle strain softening after
∼ 25 % shortening. Contrarily, model B evolved by coeval
small-scale fracturing and viscous flow. Deformation was distributed in a
large population of small shear fractures with low propagation rates.
Viscous relaxation was able to soften the increase of stress and inhibit the
propagation of large faults in this case, and the stress–strain curve thus
registered a very slight hardening associated with steady-state flow.
When loading started, ductile deformation was dominant in all models and
macroscopic fractures only began to nucleate after 15–20 % shortening. The
type of early structures strongly depends on whether the anisotropy is only
composite (modes A and C) or composite plus intrinsic (models B and D). In
composite anisotropic experiments, the first developed structures were
pinch-and-swell and relatively large tension cracks that evolved to
mixed-mode and shear fractures when fault planes rotated towards the
extensional direction or when cracks collapsed and coalesced (Figs. 7,
8a–c). Shear fractures in these cases are oriented at ∼ 45∘ with regard to the maximum compression axis Z.
Schmalholz and Maeder (2012) suggested that pinch-and-swell structures can
develop in materials with power-law rheologies and low viscosity ratios and
showed with numerical simulations how conjugate shear zones can nucleate
from such precursors. The presence of hard flakes prevented the propagation
of large tension cracks in composite-intrinsic experiments but enhanced the
nucleation of numerous small ones. Their fast collapse favours the formation
of a conjugate shear-fracture network oriented at relative high angles from
the maximum compressive stress. Very small cracks were visible in model B
but they were not the only precursors of shear fractures, since the latter
could also nucleate directly from heterogeneities associated with the
presence of flakes. Not many tension cracks were observed in model D,
suggesting that this material accumulated stresses until brittle yielding
was reached and then relatively large shear fractures directly nucleated.
Tension cracks are related to low effective confining pressure and low
differential stress conditions, which enhance the presence of tensional
stresses. Such cracks stop nucleating in these models when viscous flow
and/or slip along the shear-fracture network are more efficient in
dissipating the applied stress. Gomez-Rivas and Griera (2011) presented
experiments with the model A configuration, but performed at different
strain rates, and found that tension cracks can form until the end of the
experiment if the strain rate is high enough (10-4 s-1). This is
an indication that the strain rate at which we performed our experiments
allows a transition from extensional- to shear-fracture-dominated systems
with progressive deformation.
The collapse and coalescence of tension cracks formed as a result of local
layer-perpendicular extension or by cavitation processes (Bons et al., 2004,
2008, 2010; Arslan et al., 2008; Rybacki et al., 2008; Fusseis et al., 2009)
can result in the development of shear fractures in middle- and lower-crust
rocks. A key question here is how the development of cracks and open voids
in our models scales to nature, since these experiments have been run under
low confining pressure conditions and in the absence of fluids. Rybacki et
al. (2008) suggested that cavitation processes in natural shear zones might
not only be related to high pore fluid pressures but also depend
on local stress concentrations and strain compatibility problems at grain
boundaries, triple junctions and matrix/particle interfaces. In the case of
our experiments, heterogeneities associated with the presence of paper
flakes and layer interfaces are probably causing local problems in
maintaining strain compatibility and therefore enhance crack opening. Figure 8
illustrates how shear fractures form as a consequence of single crack
collapse (structures 1 and 5), rotation of a single tension crack
(structures 2 and 9), coalescence and linkage of several cracks (structures
11 and 12), direct nucleation from small-scale heterogeneities (structures
6, 14 and 15) or linkage of pre-existing mixed-mode or shear fractures
(structures 7, 8, 10, 12, 14 and 15). The variety of fracture formation
mechanisms indicates that caution has to be taken when using failure
criteria to predict fracture formation in ductile and anisotropic rocks.
However, in spite of the complex localisation mechanisms, most shear
fractures form at orientations close to 45∘ with regard to Z, as
predicted by the Tresca criterion (e.g. Twiss and Moores, 1992) in models A,
C and D. This implies that shear fractures develop in the same orientation
as the maximum shear stress, which is evidence for a very low frictional
behaviour of plasticine. However, when intrinsic anisotropy is present and
the material is relatively soft (i.e. model B), shear fractures form at an
angle higher than 45∘ with Z. This phenomenon has been observed in
a variety of field and experimental studies, in which the principal compressive
stress σ1 is parallel to the obtuse bisector between conjugate
shear band or shear-fracture sets (e.g. Cobbold et al., 1971; Platt and
Vissers, 1980; Berhmann, 1987; Harris and Cobbold, 1984; Hanmer et al.,
1996; Kidan and Cosgrove, 1996; Mancktelow and Pennacchioni, 2005;
Gomez-Rivas et al., 2007; Pennacchioni and Mancktelow, 2007; Gomez-Rivas and
Griera, 2012). Such large angles can be related to a variety of factors,
including fracture rotation towards the extension direction, re-activation
of pre-existing structures (especially when they are frictionally weak
surfaces), cataclastic grain size reduction or pressure-driven processes.
Since some of these processes do not operate in our models, we consider that
the high dihedral angles in our experiment B are associated with the almost
absent frictional behaviour of plasticine (e.g. McClay, 1976; Zulauf and
Zulauf, 2004) combined with low fracture propagation due to dominant viscous
flow and presence of planar heterogeneities (as in the experiments presented
in Gomez-Rivas and Griera, 2012).
As demonstrated by average fracture lengths (L) and number of fractures per
set (N) in Table 4, fracture propagation and connectivity are low in models
A, B and C, while they are relatively higher in model D. Both conjugate sets
of shear fractures have similar lengths, displacements and a symmetric
arrangement with respect to the compression axis in experiments A, B and C
(Figs. 5, 7, 9). This is coherent with the imposed coaxial deformation
conditions and a symmetrical orientation of layers with regard to the
principal stress axes Z and X. In these experiments large fractures
cross-cutting the entire model did not develop. Although stresses were high
enough to activate the onset of brittle tensile and shear fractures, the
high material toughness reduced the ability of fractures to propagate,
making it easier for them to grow by linkage and dissipate imposed stresses
by nucleation of new fractures rather than propagating existing ones.
Contrarily, the stiffer experiment (model D) did not display a symmetric
distribution of both sets of fractures. In this case the dextral shear sense
array is predominant over the sinistral one, despite the coaxial boundary
conditions and the initial orientation of layers parallel to X. A strong
strain partitioning between low and high strain bands can be observed in
this model. Large fracture zones cross-cutting the model dominate deformation
in this experiment. They accommodate the imposed deformation and prevent the
growth of a small-scale conjugate shear-fracture network. The formation of
large fractures controlling the system is related to a release of the stored
elastic energy during crack growth (e.g. Perez, 2004) and an associated
increase of their growth rate. This can be explained by the higher stiffness
and viscoelastic behaviour of the material that require longer times to
relax elastic stresses by viscous flow. The large stored elastic energy
during the shear fractures onset is probably the driving force that can
explain the relative higher fracture propagation rates in this experiment
and therefore the associated development of a few long fractures.
Experimental observations show that fracture growth is a consequence of a
combination of fault tip propagation with slip increase and segmentation
linkage (Walsh et al., 2002). The strength of the deforming material
partially controls the ratio between the maximum displacement of each fault
(dmax) and its length (L). The relationship between these two parameters
depends on the fault displacement, expressed in the parameter c, and an
exponent m (Kim and Sanderson, 2005):
dmax=cLm.
This relationship in our models is approximately linear in a log–log graph,
although the limited data range prevents extrapolating these data to larger
scales. The dmax/L ratios observed in this series of experiments range
between 0.12 and 0.23 (Fig. 9) and are therefore higher than the ones
inferred from natural faults, which vary between ∼ 10-2
and ∼ 10-4; e.g. Kim and Sanderson, 2005). There is a
progressive increase in the dmax/L ratio when the material strength is
raised, systematically from model A (softer) to model D (tougher).
Gomez-Rivas and Griera (2011) also reported that strain rate increases the
displacement/length relationship. Such relationships in our experiments are
considerably higher than in other cases previously reported in the
literature (e.g. Kim and Sanderson, 2005 and references therein), mostly
from brittle rocks. However, Grasemann et al. (2011) found similar values in
small-scale fractures associated with flanking folds within a low-grade
ductile shear zone. This suggests that fault displacement–length
relationships are probably higher in rocks that undergo dominant ductile
deformation. Since shear fractures in our experiments are easy slip zones
(i.e. weak faults) and do not normally heal, a significant amount of
deformation can be accommodated by displacement along fractures, especially
in the harder models. However, the high material toughness prevents
fractures from propagating, thus resulting in relatively high
dmax/L values. Fracture propagation requires the release of a certain
amount of energy at crack tips, which are areas of strain hardening. The
dominant viscous deformation of models A, B and C does not allow enough
energy accumulation at crack tips for propagation. On the contrary, such
energy is high enough in model D, where fractures could grow.
The results of the experiments presented in this study contribute to the
understanding of the main controls on fracture localisation in ductile
materials as well as the accommodation of deformation by different fracture
networks depending on the rock mechanical properties. These models
illustrate how strain localisation processes operate in a dominant ductile
regime and allow visualising the transition from brittle to ductile
behaviour using materials with a similar rheology. The progressive onset,
interaction and evolution of different types of structures (tension cracks,
pinch-and-swell, hybrid fractures, shear fractures) define a progressive
change in the behaviour of the system. The presence of composite or combined
composite-intrinsic transverse anisotropy plays a fundamental role, since it
enhances brittle behaviour, promoting fracture formation and helping to
dissipate the applied stress.
Conclusions
This contribution presents an experimental study on the influence of
material strength on the formation of fracture networks in materials that
are deformed by dominant viscous flow. Four plasticine multilayers, made of
different mixtures, were deformed under coaxial boundary conditions at a
constant strain rate and temperature. The following main conclusions arise
from these experiments:
The increase of material strength causes a progressive transition from a
non-localising end member, where deformation is mostly accommodated by
homogeneous flattening, to a strongly localised system where a few fractures
accommodate displacement. This ductile-to-brittle transition is controlled
by the ability of the material to dissipate the external work and relax the
elastic strain during loading, either by viscous flow or coeval flow and
failure. Shear fractures, which are oriented at ∼ 45∘ from
σ1 in most experiments, form through the collapse and coalescence of
tension cracks, the evolution of pinch-and-swell structures or by
direct nucleation associated with heterogeneities.
Stress–strain curves record the progressive transition from
ductile-dominated to fracture-dominated systems. Models deformed by dominant
viscous flow are characterised by the absence of yield points and a slight
stress increase followed by steady state behaviour. Contrarily,
localising systems record higher stress magnitudes and clear yield points
followed by subsequent strain softening associated with deformation
accommodated by fractures.
Additional intrinsic anisotropy, resulting from the presence of paper flakes
statistically oriented parallel to layers, produces a change in the
deformation behaviour inhibiting the nucleation of tension cracks. Enhanced
transverse anisotropy in the soft model reduces fracture propagation and
favours the formation a dense network of small-scale shear fractures
oriented at high angles (> 45∘) with σ1.
Contrarily, flakes significantly increase the rigidity of the material
when added to the harder plasticine and promote the formation of an
asymmetric arrangement of a reduced number of large fractures controlling
the system.
Material strength increases the fracture maximum displacement
(dmax) to length (L) ratios. Such values are relatively high compared to
those resulting from fractures formed in elastic–brittle media. This is
associated with the low propagation rates of fractures in rocks undergoing
ductile deformation and also with the presence of anisotropy.
Acknowledgements
This work was financed through the research project CGL2004-03657, funded by
the Spanish Ministry of Education and Science. We thank J. Carreras, E. Druguet and L. M. Castaño for discussions on some aspects related to this
work. We gratefully acknowledge G. Zulauf and T. Duretz, whose constructive
reviews greatly improved the manuscript, and the editorial
guidance of N. Mancktelow.
Special Issue: “Deformation mechanisms and ductile strain localization in the
lithosphere”
Edited by: L. Menegon, G. Pennacchioni, M. Stipp, N. Mancktelow, and R. Law
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