doi: 10.1111/j.1365-3121.2004.00585.x
Structural softening of the lithosphere
Stefan M. Schmalholz, Yuri Y. Podladchikov and Bjørn Jamtveit
PGP, University of Oslo, PO Box 1048, Blindern, 0316 Oslo, Norway
ABSTRACT
Structural softening is a decrease in the amount of stress
needed to deform the lithosphere at a particular rate because
of its structural reorganization while all true rheological
properties remain constant. Structural softening is fundamentally different than material softening, where the decrease in
stress is generated by a change in rheological properties with
progressive deformation, such as grain size reduction resulting
from large shearing strain. We study structural softening
generated by folding of the crust-mantle boundary, which is a
structural instability that inevitably develops during compres-
Introduction
Rheological properties of the lithosphere vary significantly with depth
because of variations in mineral rock
composition and temperature dependence of rheological properties (Brace
and Kohlstedt, 1980; Carter and
Tsenn, 1987; Kohlstedt et al., 1995;
Fig. 1). Therefore, both oceanic and
continental lithospheres are considered mechanically layered. These
layers are never perfectly flat and
geometrical perturbations along lithospheric layers are expected because of
characteristics such as lateral variations of the depth of the crust-mantle
boundary. If a compression acts parallel to the mechanical layering of the
lithosphere, the deformation is sensitive to lateral perturbations and will
inevitably generate structural instabilities, such as folding (also often
termed buckling; Biot, 1961; Zuber,
1987; Martinod and Davy, 1992; Burov et al., 1993; Zuber and Parmentier, 1996; Burg and Podladchikov,
1999; Gerbault, 2000; Schmalholz
et al., 2002). However, the deformation of the lithosphere is frequently
modelled using thin-sheet models,
which assume that strain rates are
constant with depth and, hence, neglect structural instabilities such as
folding (England and McKenzie,
1982, 1983; Vilotte and Daigniere,
1982; Houseman and England, 1986;
Correspondence: Stefan M. Schmalholz,
PGP, University of Oslo, PO Box 1048,
Blindern, 0316 Oslo, Norway. Tel.: 00 47
22 85 6042; fax: 00 47 22 85 5101; e-mail:
s.m.schmalholz@fys.uio.no
66
sion of the mechanically layered lithosphere. For ductile
rheologies, the stress decrease represents a decrease of the
effective lithospheric viscosity, which is proportional to the
ratio of stress to lithospheric shortening strain rate. We present
analytical and numerical results quantifying the decrease in
stress and effective viscosity that occur during shortening at a
constant rate. The decrease in effective viscosity can be up to
10-fold.
Terra Nova, 17, 66–72, 2005
Flesch et al., 2001). The constant
strain rate assumption is also often
applied to construct yield strength
envelopes used to interpret the
mechanical strength of the lithosphere
(Brace and Kohlstedt, 1980; Kohlstedt
et al., 1995; Fig. 1). Yield strength
envelopes plot differential stress vs.
depth, and it is generally assumed that
differential stress can be directly related to mechanical strength.
In this study, we restrict the simplifying constant strain rate assumptions
to the far field only while allowing for
near field strain rate variations to
account for emerging lithospheric
structures. To keep the analysis simple, only ductile rheologies are considered to quantify the evolution of
the differential stress during lithospheric shortening. In this case, the
appropriate rheological property is
the effective lithospheric viscosity,
which is defined by the ratio of differential stress to twice the lithospheric
shortening strain rate. A decrease in
this effective viscosity with progressive
shortening strain is referred to here as
softening. A decrease in the effective
viscosity during progressive shortening while all rheological properties
(e.g. the Newtonian viscosity) remain
constant is termed here structural
softening. Structural softening is fundamentally different to material softening, where a decrease in stress
results from a change in rheological
properties during deformation (e.g.
Tackley, 1998; Braun et al., 1999;
Bercovici, 2003; Huismans and Beaumont, 2003).
The feasibility of structural softening is investigated in the case of
structural instabilities, such as folding,
emerging around the crust-mantle
boundary and lithospheric conditions
for which the upper crust is mechanically decoupled from the lithospheric
mantle. Single-layer folding is analysed as a representative process to
quantify the evolution of the effective
viscosity. Numerical simulations for
localized perturbations considering
gravitational stresses, depth-dependent effective viscosities and power-law
rheologies show that the analytical
solutions for single layer folding capture the essential mechanical behaviour of lithospheric structural
instabilities.
The main goals of this paper are to
present the basic mechanics of structural softening, to emphasize the
importance of structural instabilities
for the evolution of differential stress
during lithospheric shortening and to
show that differential stress is not
necessarily the suitable parameter that
quantifies the mechanical strength of
ductile rocks.
Structural softening: analytical
results
The fundamental structural softening
mechanism is sketched in Fig. 2.
Single-layer folding under pure shear
of a competent Newtonian layer
embedded in less competent Newtonian material is considered (Biot,
1961; Fletcher, 1974; Schmalholz
et al., 2002). The finite amplitude
solution (FAS) derived by Schmalholz
and Podladchikov (2000) describes the
evolution of the fold amplitude during
shortening. The main feature of the
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S. M. Schmalholz et al. • Structural softening of the lithosphere
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Fig. 1 Sketch of a typical yield strength
envelope for the continental lithosphere.
The differential stress is calculated from
experimentally derived flow laws assuming a constant strain rate. The vertical
profile of the differential stress corresponds to the vertical profile of the
rheological properties corresponding to
mechanical strength. Areas of high differential stress are considered to be
mechanically strong whereas areas of
low differential stress are considered
weak. Dashed line shows qualitatively
the effect of high-stress flow laws generating smaller variations in rheological
properties with depth.
FAS is that strain rates within the
folding layer can vary during shortening, although the overall background
shortening strain rate eB is constant.
The strain rate eL within the layer
(constant along and averaged across
the layer) is given by eL ¼ )(@L/@t)/L,
where L is the fold arc length approximated by the relation L/k ¼ 1 + b(A/
k)2, with b ¼ p2/[1 + 3(A/k)2], k
being the fold wavelength and A being
the fold amplitude (see Schmalholz
and Podladchikov, 2000). Transforming derivatives with respect to time
into derivatives with respect to strain
(see Schmalholz and Podladchikov,
2000) and approximating the resulting
expression for eL provides a ratio
between eL and the background shortening strain rate eB given by
eL
¼
eB
!
4 2
3 3 þ p2 Ak þ 6 þ p2 Ak þ1
4
2
3ð3 þ p2 Þ Ak þð2ap2 þ 6 þ p2 Þ Ak þ1
2
A
ð1Þ
12ap2
k
where a is the maximum amplification
rate (corresponding to the dominant
2005 Blackwell Publishing Ltd
Fig. 2 Schematic illustration of structural softening. A Newtonian layer, embedded in
material with smaller Newtonian viscosity, is shortened parallel to its layering. If
interfaces between layer and embedding material are perfectly straight, the layer
shortens and thickens with the same background shortening strain rate as the
embedding material. The deformation is homogeneous despite the viscosity contrast
between layer and embedding material and the stress (r) profile across the layer
represents the profile of the Newtonian viscosity. If small perturbations exist along
the layer (e.g. a sinusoidal perturbation), the shortened layer is mechanically unstable,
it folds and the strain rate within the folded layer is smaller than the background
shortening strain rate, because the fold arc length (L) shortens slower than the fold
wavelength (k). The deformation is heterogeneous and the stress profile across the
layer does not represent the profile of the Newtonian viscosity, because the strain rate
is not constant. Stress profiles for homogeneous and heterogeneous deformations are
different although in both cases the rheological properties are identical.
wavelength and derived by linear
stability analysis) normalized by eB
(Biot, 1961; Fletcher, 1974). The ratio
eL/eB is identical to the stress ratio
P/P0, where P is the deviatoric layerparallel stress within the folding layer
(P ¼ 2 lLeL, with lL being the Newtonian viscosity of the layer) and P0
is the stress for pure shear deformation of the layer (P0 ¼ 2 lLeB). We
can alternatively express the ratio
eL/eB as
eL 2lL eL
1 P
l
P
¼
¼
¼ eff ¼ :
eB 2lL eB lL 2eB
lL
P0
ð2Þ
The effective viscosity leff is defined as
the ratio of stress within the layer to
twice background shortening strain
rate (P/2eB). In the analytical solution,
the stress and strain rate within the
embedding material remain constant
and, therefore, a change in leff during
shortening represents a change in the
effective viscosity of the total system
including layer and embedding material.
In eqn (1), the ratio A/k increases
during shortening (e ¼ 100(k0 ) k)/k0,
with k0 being the initial wavelength,
Fig. 2) representing fold amplification
(Fig. 3). If the ratio A/k is zero (no
perturbation) then eB ¼ eL and the
deformation is homogeneous. In addition, if a ¼ 0 then geometrical perturbations do not grow (no instability)
and again eB ¼ eL. The equation
shows that both a competence contrast (to provide a > 0) and a geometrical perturbation (to provide
A/k > 0) are necessary to generate
a stress decrease (Fig. 3B).
For a comparison with elastic rheologies, we consider the classical solution for post-buckling of a compressed
elastic bar, known as the Elastica (e.g.
Euler, 1744; Bazant and Cedolin,
1991). An approximate relationship
between amplitude and shortening for
the Elastica is derived in Bazant and
67
Structural softening of the lithosphere • S. M. Schmalholz et al.
Terra Nova, Vol 17, No. 1, 66–72
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Fig. 3 Amplification and structural softening during folding of a single layer. The
legend for both graphs A and B is displayed in figure B. The numbers displayed in the
legend represent the viscosity contrast (vc) between layer and embedding material. The
shortcut FAS represents the analytical finite amplitude solution (derived in Schmalholz
and Podladchikov, 2000) and the shortcut FEM represents a numerical finite element
solution. (A) The evolution of the fold limb dip (h ¼ arctan(2pA/k)180/p, see Fig. 2) is
plotted vs. shortening (e). The Elastica is the solution for post-buckling of an elastic
column and represents an upper bound for viscous folding solutions. (B) The averaged
layer-parallel stress P (normalized by the value for pure shear deformation P0) is plotted
vs. e times vc2/3 using eqn (1). Multiplying by the viscosity contrast to the power 2/3
collapses the results onto a narrow zone in the space P/P0-evc2/3. The ratio P/P0 also
represents the ratio of the effective layer viscosity to the true Newtonian layer viscosity
(see eqn 2). Note that the decrease in stress and effective viscosity can be more than one
order of magnitude.
Cedolin (1991, their equation 1.9.13)
and is also plotted in Fig. 3A. Interestingly, a comparison between the
FAS and the Elastica shows that the
Elastica provides an upper bound for
viscous folding and corresponds to
folding with infinitely high viscosity
contrast.
Effects of gravity on folding are
quantified using analytical results of
Schmalholz et al. (2002). Gravity effects on lithospheric shortening are
best evaluated using dimensionless
numbers such as the Argand number.
For folding processes, the Argand
number represents the ratio of gravitational stress acting against folding
to compressive stress driving folding.
The Argand number is given by Ar ¼
DqgH/(2 lLeB), where Dq, g and H are
the density difference between the
material below and above the folding
layer, the gravitational acceleration
and the layer thickness, respectively.
The parameters in the Argand number
depend on the assumed deformation
behaviour. Originally, the Argand
number was derived for thin-sheet
models using different parameters (referred to here as ArTS; England and
McKenzie, 1982). Thin-sheet models
neglect structural instabilities and
consider homogeneous thickening
only, and ArTS represents the ratio
68
of gravitational stress acting against
thickening to compressive stress driving thickening. The characteristic
length scale in Ar is the layer thickness, whereas the characteristic length
scale in ArTS is the thickness of the
entire lithosphere. Therefore, Ar and
ArTS can be considerably different
although the same geological setting
is considered. Values of a larger than
10 indicate that vertical, lithospheric
velocities as a result of structural
instabilities are one order of magnitude larger than vertical, lithospheric
velocities for homogeneous thickening
(Fig. 4). Figure 4A shows that for
values of Ar smaller than 0.25 and
viscosity contrasts larger than around
40 values of a are always larger than
10. Assuming a density difference
between lithospheric mantle and lower
crust of 450 kg m)3 and folding of the
top 10 km of the mantle lithosphere,
the gravitational stresses are around
45 MPa. To reach values of Ar smaller than 0.25 the compressive stress
must be larger than 180 MPa, which is
a stress value expected in compressional settings (e.g. Gerbault, 2000;
Fig. 4B).
Structural softening: numerical
results
Structural instabilities are studied in
the framework of continuum mechanics (Mase, 1970; Sedov, 1994). The
equilibrium equations and the continuity equation for incompressible flow
are solved numerically using the finite
element method (e.g. Thomasset,
1981; Cuvelier et al., 1986; Girault
and Raviart, 1986). Pure shear shortening in the horizontal x-direction is
applied. Lateral and bottom model
boundaries are kept straight and shear
stresses at the bottom and lateral sides
are zero. The top boundary is a free
surface. The rheology is power-law
using an effective power-law viscosity
given by geff ¼ BE(1/n)1) (England and
McKenzie, 1982). B is a material
Fig. 4 Analytical solutions showing the effects of gravity on folding. (A) The maximal
amplification rate (a, normalized by the background shortening strain rate, eB) is
plotted vs. the effective viscosity contrast (vc) for different values of the Argand
number (Ar, figures printed next to the lines). A value for a of 10 means that vertical
lithospheric velocities because of folding are 10 times larger than vertical lithospheric
velocities because of homogeneous thickening. (B) Contours of Ar are plotted in the
space gravitational stress (DqgH) – compressive stress (2 leB).
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S. M. Schmalholz et al. • Structural softening of the lithosphere
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Fig. 5 Numerical solutions for single-layer folding (power law, n ¼ 3) showing effects of gravity. (A) The limb dip (h) is plotted vs.
shortening (e). Fold amplification with gravity (Ar ¼ 0.3) is slower than without gravity. (B) The averaged differential stress within
the layer Drav (normalized by the value for pure shear deformation Drav0) is plotted vs. e. To generate the same amount of stress
decrease, folding with gravity has to accommodate more shortening than folding without gravity. (C) The ratio Drav/Drav0 is
plotted vs. h. The stress decrease for folding with and without gravity is more similar if h and not e is used as a measure for
deformation.
constant, n is the power-law exponent
and E is the second invariant of the
strain rate tensor (E equals one for
pure shear). For n ¼ 1, B corresponds
to the Newtonian viscosity using
sxx ¼ 2geffexx, with sxx being the deviatoric stress tensor component and exx
being the deviatoric strain rate component, both in the horizontal,
x-direction.
Single-layer folding with and without gravity is simulated for n ¼ 3. The
layer exhibits initially a sinusoidal
perturbation corresponding to the
dominant
wavelength
(Fletcher,
1974; Schmalholz et al., 2002) and
the initial ratio of amplitude to layer
thickness is 0.01. The initial contrast
between the effective power-law viscosities is 100. For folding with gravity,
Ar equals 0.3, which corresponds to a
situation where gravitational stresses
and compressive stresses have an
equal impact on folding (i.e. S ¼ 1,
where S ¼ 61/29/4lmn/(geffAr3/2) with
lm being the viscosity of the matrix;
see Schmalholz et al., 2002). Despite
the strong effect of gravity, the compressive stresses decrease significantly
during shortening (Fig. 5).
A layer of light material rests on a
layer of denser material (Fig. 6A). The
initial model height is normalized to
one and the initial width is 10 times
the initial height. The initial height of
the bottom layer is 0.7. Both layers
exhibit n ¼ 3 with different values of
B decreasing exponentially with depth
(Fig. 6B). B decreases with depth
because of its temperature dependence
2005 Blackwell Publishing Ltd
(Fletcher and Hallet, 1983) using B ¼
B0exp()z/f), where B0 is the effective
viscosity at the top of the layer, z is the
vertical distance from the top of the
layer and f is the e-fold length (Fletcher and Hallet, 1983). f for the top
layer is 1/30th of the total model
thickness and f for the bottom layer
Fig. 6 Numerical results for initial localized perturbations. (A) Model set-up. (B)
Profile of the rheological property B (see text) vs. z (vertical coordinate). (C) Values of
the vertically averaged differential stress (Drav, see text) vs. shortening (e). Drav is
normalized by Drav0, the value corresponding to homogeneous pure shear deformation. Structural softening becomes significant after a shortening of about 30 for the
given perturbation. The strongest stress decrease occurs for the vertical section across
the initial perturbation, where the fold develops (see Fig. 7). The evolution of the
decrease in differential stress for structural instabilities arising from localized
perturbations is similar to the stress decrease for single-layer folding (Figs 3 and 5).
Note that the decrease in differential stress occurs along the entire model domain,
although the perturbation is localized, and that the state of stress in areas far from the
initial perturbation is affected by the structural instability.
69
Structural softening of the lithosphere • S. M. Schmalholz et al.
Terra Nova, Vol 17, No. 1, 66–72
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Fig. 7 Numerical results for the model shown in Fig. 6. Contours of the basis 10
logarithm of the second invariant of the strain rate tensor, log10(E), are displayed for
different amounts of shortening (e). log10(E) ¼ 0 corresponds to pure shear. Results
show that E varies more than 1.5 orders of magnitude. The black line represents the
boundary between top and bottom layer. With increasing shortening, strain rate
variations become significant and occur within a domain significantly larger than the
domain of the initial perturbation.
is 1/24th of the total thickness (if the
total thickness represents a 120 km
thick lithosphere then f for the bottom
layer is 5 km). The viscosity profile
does not exhibit contrasts in effective
power-law viscosities larger than 100,
which is achieved by limiting the value
of the minimum viscosity, which is
normalized to one. Ar ¼ 0.1 using the
effective power-law viscosity at the top
of the bottom layer to calculate the
compressive stress, f of the bottom
layer as characteristic length scale and
the densities of the top and bottom
layer to calculate the density difference (Schmalholz et al., 2002). In the
middle of the model domain a small
geometrical perturbation is introduced
(Fig. 6A). Differential stresses are calculated using Dr ¼ 2(0.25(sxx)szz)2 þ
s2xz )1/2 (e.g. Turcotte and Schubert,
1982). The averaged differential stress
(Drav) is calculated at a specific horizontal x-position by integrating Dr
vertically along the z-direction and
dividing by the thickness of the integrated domain. During shortening,
values of Drav decrease at every horizontal x-position representing structural softening of the entire model
lithosphere (Fig. 6C). Similar to eqn
70
(2), the ratio of Drav to Drav0 (the
constant value for pure shear deformation) corresponds to the ratio of
the effective lithospheric viscosity to
the power-law viscosity of the layer.
The second invariant of the strain rate
tensor, E, measures the deviation from
homogeneous, pure shear deformation for which E ¼ 1, and is displayed
in Fig. 7. E ¼ (e2xx + e2xz )1/2, using
exx ¼ )ezz for incompressible materials.
Discussion
Folding is a type example for structural instabilities. This study shows
that differential stresses during folding
of the crust-mantle boundary can be
significantly (10-fold) smaller than
stress estimates based on constant
strain rate assumptions, such as applied for constructing yield strength
envelopes or thin-sheet models. In a
recent paper, Jackson (2002) explored
compositional (wet vs. dry) dependences of rock properties to evaluate
the Ômechanical strengthÕ of the Indian
mantle lithosphere. Our results show
that a competent lithospheric layer
around the crust-mantle boundary in
a compressive tectonic setting may
exhibit a significant decrease of its
stress level without any loss of the
Ômechanical strengthÕ, which is because of structural rather than material softening. Neglecting horizontal
deformation, Jackson (2002) concluded that stresses are small and, therefore, the mantle lithosphere is
Ômechanically weakÕ. The analysis of
Jackson has been criticized by Lamb
(2002) for not considering any compressive stresses, which are expected
to be present during a continental
collision. The magnitude of these
compressive stresses can be accurately
estimated by rheology independent
force balance calculations to be
around 100 MPa at average in order
to maintain the topographic elevation
of the Himalaya (Jeffreys, 1976). This
high level of average compressive
stress is sufficient to drive large scale,
localized lithospheric folding of the
Indian lithosphere (similar to scenarios presented by Burg and Podladchikov, 1999). Moreover, it is
essential for discussions about the
strength of rocks to identify the property which actually quantifies the
strength. For ductile rheologies, the
parameter quantifying the relative
strength or weakness is viscosity,
which is proportional to the ratio of
stress to strain rate. Therefore, both
stress and strain rate values are needed to estimate the effective viscosity
(i.e. the mechanical strength) of the
lithosphere, which can vary significantly during lithospheric deformation as demonstrated in this study
(see also Watts and Burov, 2004).
Folding is often neglected in lithospheric shortening models, because
lithospheric viscosities vary exponentially with depth and not discontinuously as considered for single-layer
folding. However, several studies
showed that shortening layers with
exponentially decreasing viscosities
generates folding (Biot, 1961; Zuber
and Parmentier, 1996; Schmalholz
et al., 2002). In addition, flow laws
for high stresses indicate a smaller
decrease of rheological properties
with depth compared with an exponential decrease (e.g. Regenauer-Lieb
and Yuen, 2003; Fig. 1) and generate, together with power-law flow
laws, a distribution of rheological
properties closer to the single-layer
folding setup. Finally, developing
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S. M. Schmalholz et al. • Structural softening of the lithosphere
.............................................................................................................................................................
structural instabilities is preferable to
homogeneous thickening, because
lithospheric deformation involving
structural instabilities requires less
compressive stress than lithospheric
deformation involving thickening
only.
Only ductile rheologies have been
considered in this study. Leroy et al.
(2002) showed that for an elastic layer
over an inviscid and buoyant fluid the
lateral compressive force decreases
during shortening, indicating structural softening also occurs with pure
elastic rheologies.
Conclusions
Structural instabilities are likely to
emerge around the crust-mantle
boundary during lithospheric shortening and decrease the effective lithospheric viscosity while all rheological
properties remain constant (Fig. 6).
This structural softening is also felt far
from the lithospheric domain where
the structural instability actually
develops (Figs 6 and 7). Effects of
gravity, depth-dependent rheological
properties and limited strength of
rocks do not prohibit structural instabilities and structural softening. Neglecting structural instabilities a priori
in models for lithospheric shortening
is not justified. Dependences of effective rheological properties on shortening strain, similar to those presented
in Fig. 3B, must be introduced to
account for the softening caused by
structural instabilities.
Acknowledgements
We thank M. Gerbault and E. Burov for
their helpful and constructive reviews,
Boris Kaus for discussions and Nate Onderdonk for improving the English of the
manuscript. This study was supported by a
centre of excellence grant to PGP from the
Norwegian Research Council.
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Received 26 August 2004; revised version
accepted 27 October 2004
2005 Blackwell Publishing Ltd