Author's personal copy
Earth and Planetary Science Letters 277 (2009) 80–85
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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 e v i e r. c o m / l o c a t e / e p s l
A thin elastic core can control large-scale patterns of lithosphere shortening
F.O. Marques a,⁎, Y.Y. Podladchikov b
a
b
Univ. Lisboa, Fac. Ciências, Dept. Geologia and IDL, Lisboa, Portugal
University of Oslo, Physics of Geological Processes, Oslo, Norway
a r t i c l e
i n f o
Article history:
Received 30 April 2008
Received in revised form 30 September 2008
Accepted 1 October 2008
Available online 17 November 2008
Editor: T. Spohn
Keywords:
lithosphere rheology
lithosphere strength
thin elastic layer
stress levels
lithosphere folding
physical modelling
a b s t r a c t
Peak lithospheric strength should reside in the rocks that, under the applied stress, cannot either creep (due
to low temperature) or break (due to high confining pressure). The greatest resistance comes from dry
olivine/pyroxene-rich upper mantle/lowermost crust at Moho conditions (400–600 °C and N 1 GPa). We have
conducted laboratory experiments to investigate the importance of the unbreakable core of the lithosphere
in between its brittle and ductile parts and conclude that it can control the large-scale lithospheric
deformation pattern under shortening. Regardless of the thickness of the unbreakable core, it acts as a
restraining layer that is easily flexed but is unstretchable. This eliminates large scale brittle faulting or
homogeneous thickening as available shortening modes and results in irregular wrinkling of the unbreakable
layer. We discuss geodynamic implications of our laboratory experiments and advocate studies of large scale
buckling of the lithosphere as a relevant shortening mode.
© 2008 Elsevier B.V. All rights reserved.
1. Introduction
Deforming rocks exhibit complex rheological responses ranging from
strong quasi-rigid-like- to weak quasi-fluid-like-effective behaviour as a
function of temperature, stress, fluids and/or rock composition. Therefore,
there is a need for a simplified rheological model that can serve as
analogue of the complex heterogeneous lithosphere, while capturing its
effective large-scale rheological behaviour. A widely used conceptual
model of the lithosphere is the yield strength envelope (YSE — Fig. 1),
introduced by Goetze and Evans (1979) and based on laboratory rock
experimental data (see also Brace and Kohlstedt, 1980; Kirby, 1980). This
model incorporates both brittle rock strength, increasing with pressure
(depth), and viscous rock strength, which is a function of rock properties,
strain rate, and temperature, and generally decreases with depth (Fig. 1).
Rocks are assumed to fail by the weaker of the two criteria, resulting in a
branched strength envelope. For a standard parameter set, the YSE
includes effectively elastic cores between brittle and ductile layers.
Effectively elastic layers should exist if the stress level is smaller than
the brittle yield and viscosity is too high to allow creep and relaxation of
differential stresses (e.g. Kirby, 1983). The argument is that olivine and
pyroxene (or respective aggregates) are very strong, so that at Moho
temperature (400–600 °C depending on thermal gradient) they cannot
creep or break for differential stresses lower than 1 GPa, even when
the strain rate is corrected to geologically acceptable values, in the order of
⁎ Corresponding author. Tel.: +351 217500000; fax: +351 217500064.
E-mail address: fomarques@fc.ul.pt (F.O. Marques).
0012-821X/$ – see front matter © 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.epsl.2008.10.009
10− 15 s− 1 (e.g. Griggs et al., 1960; Avé Lallemant, 1978; Goetze, 1978; Evans
and Goetze, 1979; Kirby and Kronenberg, 1984; Karato, 1997; Dorner and
Stöckhert, 2004; Li et al., 2004; Boettcher et al., 2007; Renshaw and
Schulson, 2007; Korenaga and Karato, 2008). The validity of great
extrapolation from laboratory length and time scales have been frequently
questioned, and rock strength at geological time scales has been suggested
to be significantly weaker (Rutter and Brodie, 1991). However, all
quantified weakening mechanisms, such as grain size reduction, are
inefficient at low temperatures and small strains, the conditions prevalent
in stable lithosphere (e.g. Chopra and Paterson, 1981, 1984; Karato, 1984;
Tsenn and Carter, 1987; Karato and Wu, 1993). Moreover, the direct
evidence for long-term integrated strength of the stable lithosphere,
which does not significantly relax differential stress on geological time
scale, has been inferred by persistence of density anomalies causing largescale gravity anomalies in old terrains (e.g. Artemjev and Artyushkov,
1971; Burov et al., 1998), or by rheology independent topographic force
balance at active mountain belts (e.g. Jeffreys, 1959), or by absence of
expected viscoelastic topography adjustments in various tectonic settings,
such as foreland basins or depressions of the oceanic lithosphere loaded by
volcanic islands (e.g. Watts and Talwani, 1974; Watts and Cochran, 1974;
Watts and Burov, 2003;Watts and Zhong, 2000).
Since elastic strain is assumed to be negligible compared to typical
mountain building strains, then the presence of an effectively elastic core
seems to be inconsistent with large-scale lithospheric deformation. This
can only be initiated after the elastic layer has vanished and yield
conditions over the whole of the strength versus depth profile to be
attained, a concept named whole lithosphere failure (WLF) by Kusznir
and Park (1982). They considered a number of geodynamic scenarios
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F.O. Marques, Y.Y. Podladchikov / Earth and Planetary Science Letters 277 (2009) 80–85
81
Fig. 1. Sketch with the YSE for compression of oceanic lithosphere prior to (solid line) and after WLF (dashed line).
possibly responsible for the vanishing of the quasi-elastic core of the
lithosphere and identified major controlling parameters like duration of
loading, stress level and heat flow. The evolution of the intra-lithospheric
elastic core can be visualized as wedging out to a vanishing point, the WLF
(Fig.1). It was not the aim of this study to investigate mechanisms that can
lead to WLF, or lithospheric deformation under WLF.
There is another possibility for the onset of large-scale deformation
while the elastic core is still present. The unbreakable layer can be
wrinkled or folded away to allow for shortening of the horizontally
compressed plates, a concept named structural softening of the
lithosphere by Schmalholz et al. (2005). Strong layers within a
shortening section of the rheologically stratified lithosphere can either
Fig. 2. A — sketch of typical geometry and layer distribution in the experimental models. B to D — Top views of two stages of deformation of Model 1 (LDP thin layer). Images show
upper surface of models: B — intermediate stage; C — final stage; D — oblique view of PDMS top surface after sand removal. Comparison with c shows that fold frequency in the elastic
layer is much greater than in sand, especially in the antiformal, acute hinges.
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F.O. Marques, Y.Y. Podladchikov / Earth and Planetary Science Letters 277 (2009) 80–85
fold in a creeping mode or buckle quasi-elastically (Schmalholz et al.,
2002). Our main objective is to test if the onset of the large-scale
deformation is possible while a thin effectively elastic layer is still
present at the crust-mantle boundary and how it affects the lithospheric deformation patterns.
Our approach was to build a scaled experimental model in which
we put together, side-by-side, model lithosphere with and without
the thin unbreakable layer in between a brittle and a ductile layer. We
first present and discuss model materials, boundary conditions and
scaling, then we present the experimental results, and finally we
discuss experimental results and large scale implications for lithosphere shortening to conclude that a thin layer can control the large
scale deformation pattern.
The experiments were performed in a Perspex rectangular box, in
which three lateral walls were fixed and one was mobile. This acted as a
piston and was pushed by a computer controlled stepping motor at a
steady velocity of about 1.3×10− 5 m s− 1 so compressing the model. The
models comprised, from bottom to top, a horizontal layer of silicone putty
ca. 25 mm thick on average, a thin layer strongly adherent to PDMS and a
sand layer ca. 15 mm thick on average with an initial flat top surface (no
intentional initial perturbations) (Fig. 2). Model materials were in very low
friction contact with the walls by using liquid soap. The thin layer occupied
only a central strip of the silicone surface for comparison purposes in
individual experiments: deformation with and without the thin layer.
2. Experimental method
We chose a length ratio (model over nature) lr = lm / ln = 5.33 × 10− 7,
so that a lithosphere 100 km thick scaled down to a model layer ca.
53 mm thick. The models were 330 mm long, 375 mm wide and
35 mm deep (Fig. 2a), thus representing 620 km × 700 km × 65 km in
nature. The time had to be taken into account in the scaling of a ductile
layer, because its stress depends upon the applied strain rate. We chose
a strain rate ratio (vm /lm) / (vn /ln) ≈ 1010, so that a velocity of about 8 cm
yr− 1 (ca. 2.5× 10− 9 m s− 1) scaled to ca. 1.3 × 10− 5 m s− 1 in the models. The
reciprocal of strain rate ratio defines the time ratio of ca. 10− 10 (1 h
representing about 1 Ma). Because inertial forces are negligible, the ratio
of viscous (viscosity × strain rate) to gravitational (density× acceleration
due to gravity × length) forces was the only force ratio to respect for
model to nature scaling. For the experiments this means that the
viscosity ratio equals the product of density, length and time ratios,
becoming (0.37) × (5.33 × 10− 7) × (1 × 10− 10) ≈ 2 × 10− 17. Therefore, to
represent a lithospheric layer having a viscosity of about 1021 Pa s, we
used a model material with a viscosity of about 4 × 104 Pa s (Table 1).
The ratio of elastic (Young's modulus E) to gravitational forces per
unit area (ρgz) was approximately respected for plasticine in
the model to nature scaling, but not for LDP. When the scaling is
respected, the E ratio is equal to the product of density and length
ratios, becoming (0.39) × (5.33 × 10− 7) ≈ 2.1 × 10− 7 for plasticine, and
(0.27)×(5.33×10− 7)≈1.4×10− 7 for LDP. Accordingly, plasticine represents
a lithospheric layer having a high but still realistic E≈2×1011 Pa (for olivine
E≈1.2×1011 Pa, e.g. Maxisch and Ceder, 2006), and the LDP represents a
non-realistically rigid layer having E=5.1×1013 Pa when scaled up to
nature. Using the length ratio 5.0×10− 7, the thin elastic model layer scales
up to nature to a layer 20 m thick, which is also unrealistic but in an
opposite way. The natural elastic layer should be kilometres in thickness,
depending on age and thermal profile of the oceanic lithosphere
(temperatures between 400 and 600 °C). Anyway, above a certain rigidity
level, achieved by the two model thin layers, the emerging pattern does
not depend qualitatively on thickness or rigidity of the layer as shown
below. We therefore conclude that the ratio of elastic rigidity to gravity is
not important for our analogue modelling strategy.
Taking as basis the YSE constructed by Goetze and Evans (1979) for
a given geotherm and strain rate, we used a 3-layer model of the
lithosphere: one brittle layer overlying a viscous layer, with a thin
unbreakable layer in between. We did not model a complex
continental lithosphere because that was not the aim. We wanted to
make the model simple to clearly show the influence of the thin elastic
layer, unmasked by the interplay of many variables and parameters
inherent to a more complex model. Consequently, the present models
apply more directly to an oceanic lithosphere. However, they should
also apply to continental lithospheres where a brittle layer overlies a
viscous layer, with a thin unbreakable layer in between.
2.1. Model materials and boundary conditions
To model the brittle upper crust, we used a Coulomb material,
natural quartz sand from Fontainebleau, with grain size of ca. 300 μm,
density of ca. 1.3 × 103 kg m− 3, very low cohesion and an angle of
internal friction between 30° and 40°, as in brittle rock (e.g. Hubbert,
1951; Krantz, 1991; Mourgues and Cobbold, 2003). For the viscous
lithosphere we used polydimethylsiloxane (PDMS — DC SGM36),
which has a density of ca. 0.965 × 103 kg m− 3, is Newtonian and has a
viscosity in the order of 104 Pa s. In order to keep a correct density
profile, the density was increased by addition of wolframite powder.
Because wolframite has a density of ca. 7.5, a relatively small amount
was added to PDMS to correct the density without changing the
viscosity appreciably. As analogues of the thin unbreakable layer, we
used two types of materials, which readily fold under layer parallel
compression: plasticine (e.g. Weijermars, 1986; Zulauf and Zulauf,
2004 for rheological properties) ca. 1 mm thick, with ρ ≈ 1.32 × 103 kg
m− 3 and E ≈ 4.3 × 104 Pa, and a low density polyethylene film (LDP) ca.
10 μm thick, with ρ ≈ 0.915 × 103 kg m− 3 and E ≈ 7.2 × 106 Pa. Under the
applied experimental conditions, plasticine behaves as viscoelastoplastic and LDP as pure elastic.
2.2. Scaling
Table 1
Summary of dimensional analysis
Independent
Dependent
Variable
Scale
Dimensionless ratio
Model
(m) (SI units)
Scaling factor
Natural
(n) prototype (SI units)
Depth
Layer thickness
Velocity
Pressure
Density
Time (t)
Strain rate (ɛ)
Viscosity (μ)
Elasticity (E) plasticine
Elasticity (E) LDP
Layer thickness (l) plasticine
Layer thickness (l) LDP
H
Z
V
ρgh
ρ
h/v
v/h
ρgh2/v
ρgh
ρgh
H
H
z/h
5.33 × 10− 2
hr = hm / hn = 5.33 × 10− 7
1 × 105
NA — not applicable.
tv/h
ρgz/ρgh
tv/h
ɛh/v
μv/ρgh2
E/ρgh
E/ρgh
l/h
l/h
−5
1.3 × 10
NA
1.28 × 10+ 3
3.6 × 103
4 × 10− 5
4 × 104
4.3 × 104
7.2 × 106
1 × 10− 3
1 × 10− 5
3
vr = vm / vn = 5.3 × 10
NA
ρr = ρm / ρn = 0.37
tr = (h / v)m / (h / v)n = 1 × 10− 10
ɛr = (vm / lm) / (vn / ln) = 1010
ρr × hr × tr = 1 × 10− 17
Er = ρr × hr = 1.9 × 10− 7
Er = ρr × hr = 1.9 × 10− 7
hr = hm / hn = 5.33 × 10− 7
hr = hm / hn = 5.33 × 10− 7
Given
−9
2.5 × 10
NA
3.4 × 103
3.6 × 1013
4 × 10− 15
1 × 1021
2.3 × 1011
3.8 × 1013
1.86 × 103
19
Computed
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3. Experimental results
4. Discussion and conclusions
3.1. Model 1 — low-density polyethylene layer
In the models, the rheology of the thin layer varied from elastic to
plastic, and its thickness and Young's modulus varied by 2 orders of
magnitude; however, the final outcome was very similar. Therefore,
we conclude that none of these parameters are responsible for the
pattern selection. The thin layer acts as an unbreakable restraining
sheet that is easily flexed but is unstretchable. The fixed arc length of
an embedded layer kinematically eliminates faulting or homogeneous
thickening as admissible modes of deformation. The only available
shortening mode is hence folding of the arc length-preserving layer
(e.g. Schmalholz and Podladchikov, 2000; Schmalholz et al., 2002;
Kaus and Schmalholz, 2006).
Recent research has paid special attention to lithospheric scale folding
and argues that it is a common response of the lithosphere to shortening
(e.g. Weissel et al., 1980; McAdoo and Sandwell, 1985; Cloetingh et al.,
1999; Burg and Podladchikov, 1999; Gerbault, 2000; Schmalholz et al.,
2002; Schmalholz et al., 2005). However, large-scale lithosphere buckling
is still enigmatic, because it has been shown that available tectonic forces
are insufficient for the buckling of the entire elastic lithosphere, i.e. if the
thickness of the elastic core is of the same order of magnitude of the total
thickness of the lithosphere (e.g. Turcotte and Schubert, 2002). Actually,
while running physical experiments to study buckling, we have identified
a possible solution to the enigma of lithospheric buckling by considering
the thickness of the buckling layer much smaller than the thickness of the
entire model. Once the layer is thin, the magnitude of the acting force
should not be of major concern because large interlayer stresses do not
contribute significantly to the total force (depth integrated stress) needed
to deform the entire lithosphere. Regardless of the unknown origin and
magnitude, the availability of the forces is well documented by large-scale
lithospheric deformation at considerable distances from plate boundaries.
Moreover, non-periodic and localized elastic buckling modes may be
nucleated at small strains, and at integrated stress levels much lower than
those needed to allow the buckling of the entire model lithosphere in
strictly periodic and cylindrical mode (Hunt et al.,1996,1997; Whiting and
Hunt,1997; Fletcher,1991; Kaus and Schmalholz, 2006; Schmalholz, 2008;
Jeng and Huang, 2008). We indeed observe the non-periodic, noncylindrical, highly irregular and localized morphologies of the experimental wrinkles, which nucleated and were observable at very small
strains. Similarly, seismic studies reveal irregular wrinkled patterns of
Alpine and Tibetan Moho topography (Waldhauser et al., 2002; Lombardi
et al., 2008; Shin et al., 2007). Although spectral analysis of gravity
anomalies over the Tibetan plateau was interpreted as documenting
folding frequencies in a statistical sense (Jin et al., 1994), it is actually hard
to recognise an individual periodic and cylindrical fold train anywhere in
the area (see, however, Shin et al., 2007). Lack of clear periodic cylindrical
waveforms in addition to the high stress argument is frequently used as an
argument against large scale folding. We, therefore, suggest that irregular
wrinkling of a thin internal layer at stress levels lower than those implied
by a standard periodic linear stability analysis (e.g. Turcotte and Schubert,
2002) is the solution for both theoretical and observational enigmas.
Nevertheless, in order to buckle even a thin layer, the stress level within
this layer must be high enough to ensure sufficient growth rate of the
buckling instability in order to achieve significant amplitudes within
geologically realistic time and strains. If the growth rate is small the layer
stays nearly flat even after significant shortening. However, exactly in this
situation, the sufficient stress level within elastic layer is quickly (i.e. in a
percent of total shortening strain) and unavoidably built by elasticity, with
negligible negative feedback on plate convergence and overall stress
balance if the elastic layer is thin. Particular quantitative features of the
buckling pattern such as wavelength, amplitude and degree of localization
are controlled by the choice of the material for the thin layer and its
thickness, which is being investigated by our ongoing studies.
Seemingly, the worst case scenario for buckling to develop is thus
when the integrated stress must be kept exactly at a constant level for
bulk force balance reasons, and when this level is insufficient to
The experimental results of this model (Fig. 2) are characterized by:
(i) folding concentrated exclusively in the thin film in the initial stages of
deformation; (ii) folding propagating into lateral areas without LDP at
intermediate stages of deformation; (iii) higher frequency of folds in the
LDP layer than in overlying sand; (iv) fore limb of fold closer to piston
gradually transforming into a scarp of a well-developed fore thrust; (v)
folds in sand overlying the LDP layer developing into pop-ups limited by
fore and back thrusts; (vi) overall structure of a fold-and-thrust belt.
Higher frequency of folds in the LDP layer than in overlying sand can
be the result of: (i) the sand layer being much thicker than the LDP layer;
(ii) the sand layer undergoing homogeneous shortening before and
during buckling. On the one hand, homogeneous shortening is the only
mode of strain accommodation in the thick sand layer in the absence of
the elastic layer. On the other hand, buckling is the only mode of strain
accommodation in the LDP thin layer. In the folded areas, sand also
fractures along fold hinges, and thrust faults develop from the inflection
points in fold limbs (e.g. Davy and Cobbold, 1991; Marques, 2008).
3.2. Model 2 — plasticine layer
The end result of this experiment (Fig. 3) shows a very simple
deformation pattern: the plasticine layer irregularly folded, with
wavelength gradually increasing (and amplitude decreasing) away
from piston. Silicone putty to either side of folded plasticine shows no
folding for the strain at which the experiments were halted.
Fig. 3. Final stage of deformation of Model 1 (plasticine layer 1 mm thick). Photographs are
top views showing upper surface of models. A — sand surface; B — top surface of PDMS
after sand removal. Note the absence of folding in silicone putty adjacent to plasticine layer.
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overcome lithospheric strength and to produce large scale folding. There
is, however, an intrinsic mechanism of stress amplification in a thin
elastic core even at constant total load. Kusznir and Bott (1977) and
Kusznir and Park (1984) analyzed the intraplate state of stress of the
lithosphere when a constant horizontal load is applied at the ends of a
lithospheric plate. Such loads typically originate at plate boundaries and
comprise the horizontal component of gravity. They can be tensile (e.g.
slab pull) or compressive (e.g. ridge-push), and their magnitudes can be
reasonably well estimated and used in the models. Kusznir and Bott
(1977) showed that, in a lithosphere comprising an elastic layer on top of
underlying viscoelastic medium, viscoelastic stress relaxation amplify
the stresses in the elastic layer by a factor equal to the ratio between
lithosphere and elastic layer thickness. Therefore, if the thickness of the
elastic layer is small enough it can reach a value for which the applied
stress is high enough to buckle this layer, while keeping the total load
exactly constant. Kusznir and Park (1984) model lithosphere comprises
brittle and ductile layers with elastic cores in between (as in the present
experimental models). While brittle and ductile layers yield under
constant load, stress redistribution induces significant stress amplification in the elastic layer, which has to support the constant integrated
over depth horizontal stress.
In search for a simplified rheological model that can serve as analogue
of the complex heterogeneous lithosphere we, as commonly done in
sandbox laboratory experiments, fix rheological types to the analogue
material layers (e.g. Weijermars and Schmeling, 1986; Weijermars et al.,
1993). Deforming rocks at any level of the lithosphere may exhibit
rheological responses ranging from strong quasi-rigid-like- to weak quasifluid-like-effective behaviour. These effects can readily be examined by
means of numerical modelling (e.g. Burg and Schmalholz, 2008) in order
to verify the robustness of our fixed multi-layer analogue model to capture
the effective large-scale response of the lithosphere to shortening.
In conclusion, lithosphere deformation history has been previously
assumed to start only when the thickness of the quasi-elastic core of
the lithosphere has vanished. However, there is certainly a moment
when the thickness of the quasi-elastic core is non-zero, but greatly
reduced compared to the initial thickness. Lithosphere may start to
deform at this stage, before complete vanishing of its quasi-elastic
core. Regardless of details of buckling dynamics, the strength of the
thin layer will not stop the whole lithosphere failure because it is too
thin to resist deformation and significantly contribute to the overall
force balance. However, it is able to rule the large-scale pattern and
the way the whole lithosphere fails or deforms, as it does in the
present experiments. While the weak model viscous layer flows, the
thin unbreakable layer folds and controls failure localization in the
brittle layer above. The experimental patterns controlled by the thin
layer look familiar on Earth, whereas thickening without the thin layer
produces a featureless and flat landscape. The Earth-like pattern
controlled by thin layer is characterized by strongly non-periodic,
irregular and non-cylindrical folding. It appears like mobile fold belts
separated by relatively undeformed plateaux, the ‘plate tectonics’
pattern. Moreover, thin layers do not bring any stress problems while
buckling. Therefore, it seems that we have experimentally obtained a
familiar pattern with no mechanical paradox.
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