GEOPHYSICAL RESEARCH LETTERS, VOL. 0, NO. 0, PAGES 0-0, M 0, 2001
Viscoelastic Folding: Maxwell versus Kelvin Rheology
S.M. Schmalholz
Geologisches Institut, ETH Zentrum, 8092 Zürich, Switzerland
Y.Y. Podladchikov
Geologisches Institut, ETH Zentrum, 8092 Zürich, Switzerland
Abstract. Folding of a viscoelastic layer embedded in a
viscous matrix is studied analytically using two viscoelastic rheological models: The Maxwell and the Kelvin model.
The layer deformation behaviour approximates the viscous
or elastic limits depending on the single parameter, R, which
is proportional to the viscosity contrast and the ratio of
layer-parallel stress to shear modulus. A layer with Maxwell
rheology approximates this limit that generates the fastest
amplification whereas a layer with Kelvin rheology approximates the slowest limit. For R < 1 the viscous limit is
fastest whereas for R > 1 the elastic limit is fastest. The
Kelvin rheology is suitable to describe the effective flexural
response of a lithospheric plate consisting of an elastic layer
overlying a viscous layer, since the Kelvin rheology yields
an identical bending moment. A critical elastic layer thickness, R2 H/3 (H=lithospheric thickness), is derived at which
the deformation behaviour of the lithospheric plate changes
from quasi-viscous to quasi-elastic.
(Maxwell rheology) or in parallel (Kelvin rheology), respectively (Fig. 1, e.g. [Findley, 1989; Ranalli, 1995]). The parameter R, defined in Schmalholz and Podladchikov [1999,
2000], suffices to determine whether a folding layer with
Maxwell rheology behaves similarly to a viscous layer (quasiviscous regime, R < 1) or an elastic layer (quasi-elastic
regime, R > 1, Figs. 2 and 3). R is given by
r
R=
3
µl
6µm
r
P
αdel
λdvi
=
=
G
αdvi
λdel
(1)
where µl , µm , P and G are the viscosity of the layer, the
viscosity of the matrix, the mean (averaged over layer thickness) layer-parallel stress and the shear modulus of the layer,
respectively, and λdvi , λdel , αdvi and αdel are the dominant
wavelength for a viscous layer, the dominant wavelength for
an elastic layer, the dominant (i.e., maximal) amplification
rate for a viscous layer and the dominant amplification rate
for an elastic layer (e.g., [Biot, 1961]). The aim of this
study is (i) to derive an analytical solution for folding of
a layer with Kelvin rheology, (ii) to point out the differences
in folding of layers with Maxwell and Kelvin rheology, (iii)
to present R as the suitable parameter for determining the
deformation behaviour of compressed viscoelastic layers and
(iv) to point out the geophysical significance of the Kelvin
rheology for lithospheric deformation.
Introduction
Deforming rocks generally behave viscoelastically, that
is, the rheology of rocks is a combination of time dependent
(e.g., viscous) and time independent (e.g., elastic) rheologies
(e.g., [Turcotte and Schubert, 1982; Ranalli, 1995]). However, under certain conditions the behaviour of viscoelastic
rocks can be closely approximated by either pure viscous
or pure elastic rocks. It is of great interest to know and
understand such conditions, because (i) they allow a reliable investigation of the deformation of viscoelastic materials using simpler rheologies, and (ii) they give a deeper
insight into the deformation process itself. The process investigated in this study is folding of a viscoelastic layer,
which is embedded in a viscous matrix (Fig. 1). Folding is
a common response of layered rocks to deformation and the
resulting structures, termed folds, occur frequently in nature
on all spatial scales (e.g., [Ramsay and Huber, 1987]). The
initial stages of folding have been intensively investigated
using analytical theories (e.g., [Biot, 1965; Ramberg, 1981;
Johnson, 1994]). The most important result is the identification of an amplification rate maximum for a certain
wavelength of the layer perturbation, which is designated
the dominant wavelength [Biot, 1961]. Most folds observed
in nature are thought to record dominant wavelengths. The
simplest viscoelastic rheologies can be built by connecting
a linear viscous and a linear elastic element either in series
Analytical thin-plate solution for
viscoelastic folding
A viscoelastic layer with an infinitesimal sinusoidal perturbation embedded in a low viscosity matrix subject to
layer-parallel compression is considered (Fig. 1). Assuming
the total layer-parallel stress, σ, to be constant along the
layer (i.e. neglecting shear stresses on the layer boundary)
the thin-plate theory provides a single equilibrium equation
for the folded layer (e.g., [Timoshenko, 1959; Reddy, 1999]):
∂2
∂x2
H/2
Z
−H/2
∂2W
yσdy +
∂x2
H/2
Z
σdy + q = 0
(2)
−H/2
where y and x are coordinates across and along the layer,
respectively, H and W are the thickness and the deflection
of the layer, respectively, and q is the vertical stress of the
viscous matrix. σ can be split into a mean layer-parallel
stress and a fiber stress:
Copyright 2001 by the American Geophysical Union.
Paper number 1999GL000000.
0094-8276/01/1999GL000000$05.00
σ = −P + τ y cos(ωx)
1
(3)
2
SCHMALHOLZ AND PODLADCHIKOV: VISCOELASTIC FOLDING
(e.g., [Biot, 1961])
q = −4µm ω Ẇ
(5)
where µm is the viscosity of the matrix and dots above symbols denote differentiation with respect to time. The Kelvin
rheology for total stresses is given by (e.g., [Findley, 1989;
Ranalli, 1995]):
σ = 4Gε + 4µl ε̇
(6)
where ε is the total layer-parallel strain. The factor 4 appears in eqn. (6), because the total layer-parallel stress is
two times the deviatoric stress by setting the total vertical
stress to zero and assuming incompressible materials (e.g.,
[Turcotte and Schubert, 1982]). Neglecting shear stresses
within the layer ε is given by (e.g., [Timoshenko, 1959])
ε = εB − y
Figure 1. Sketch of viscoelastic folding and presentation of the
two simplest viscoelastic rheologies: the Maxwell and the Kelvin
rheology. In a Maxwell model strains are additive whereas in a
Kelvin model stresses are additive. The equations are given for
deviatoric stresses.
where τ is the fiber stress coefficient and ω = 2π/λ with λ
being the wavelength of the layer perturbation. The deflection of the layer is assumed as
W = A(t) cos(ωx)
(4)
where A(t) is the amplitude of the layer perturbation dependent on time, t. The vertical stress exerted by the infinitely
thick, viscous matrix onto the layer boundaries is given by
A)
∂2W
∂x2
(7)
where εB is the mean (averaged over the layer thickness)
layer-parallel strain and the second term on the right hand
side represents the fiber strain. Substituting eqns. (3), (4),
(5) and (7) into eqn. (2), collecting the coefficients in front
of cos(ωx) and expressing τ through A(t) using eqns. (3)
and (6) yields (in non-dimensional form)
1
3R2
−
3
2λV
2λV
∗
∗
A (t ) +
1
+1
2λ3V
∂A∗ (t∗ )
=0
∂t∗
(8)
where λV =p
λ/λdvi is the non-dimensional wavelength with
λdvi = 2πH 3 µl /(6µm ) (the dominant wavelength for a viscous layer), A∗ = A/A0 is the non-dimensional amplitude
with A0 being the initial amplitude and t∗ = tG/µl is the
non-dimensional time. The solution of eqn. (8) is of the
form A∗ (t∗ ) = exp(αke t∗ ), where αke is a non-dimensional
B)
α dke
λdke
Figure 2. The non-dimensional amplification rates, α, as a function of the ratio wavelength to thickness, λ/H, for different
rheologies. A) For R < 1 and De = µl ε̇B /G = 0.02 the amplification rate spectra for a layer with Maxwell rheology is close to the
viscous spectrum, whereas the Kelvin spectrum is close to the elastic spectrum. B) For R > 1 and De = 0.02 the Maxwell spectrum
is close to the elastic spectrum, whereas the Kelvin spectrum is close to the viscous spectrum. Note, that R controls the deformation
behaviour of a viscoelastic layer and the Deborah number De, which is the same in both cases, does not.
3
SCHMALHOLZ AND PODLADCHIKOV: VISCOELASTIC FOLDING
amplification rate given by
2
3 (λV R) − 1
2λ3V + 1
αke =
(9)
where the subscript “ke” denotes that the amplification rate
is valid for a layer exhibiting a Kelvin rheology. Amplification rates for layers with viscous and elastic rheologies
are derived for example in [Biot, 1961] and for layers with
Maxwell rheology in [Schmalholz and Podladchikov, 1999].
The amplification rate αke yields a maximum for a certain
λV , designated as the dominant wavelength (Fig. 2), which
is given by
λdke =
λdvi
√
3
2
≈
q
3
1+
p
1−
4
27R6
q
+
3
1−
p
1−
material properties due to thermal and pressure gradients
raises the question on an “effective” rheology of a lithospheric layer. The Maxwell rheology, which causes elastic
stresses to relax with time, has been examined as an effective rheology of the lithosphere (e.g., [De Rito et al., 1986]).
However, there are studies of lithospheric flexure suggesting
that the upper part of the lithosphere may behave elastically
over long time scales and exhibits long-term strength (e.g.,
[De Rito et al., 1986; Watts and Zhong, 2000] and references
therein). The Kelvin rheology does sustain long-term elastic
stresses and may be a more suitable rheological model than
the Maxwell model for specific tectonic settings.
A frequently used simple lithospheric model consists of
an elastic layer overlying a viscous layer resting on a low
4
27R6
λdel 1 + R3 /2 for R < 1
λdvi 1 + 1/ 3R2
for R > 1
(10)
The dominant amplification rate, αdke , can then be obtained
by substituting eqn. (10) into (9) (Fig. 2A).
For R < 1 a layer with Maxwell rheology behaves quasiviscous and a layer with Kelvin rheology behaves quasielastic, and vice versa for R > 1 (Fig. 2). All dominant
amplification rates for layers with viscous, elastic, Maxwell
and Kelvin rheology increase with increasing R (Fig. 3A). A
layer with Maxwell rheology always amplifies faster than viscous and elastic layers, whereas a layer with Kelvin rheology
always amplifies slower than viscous and elastic layers (Fig.
3). The transition of viscoelastic layers from quasi-viscous
to quasi-elastic behaviour is sharp and occurs at R = 1.
For values of R smaller around 0.4 and larger around 2.2
the viscous or elastic layers provide dominant amplification
rates that are less than 5% different to the rates provided
by viscoelastic layers. However, at R = 1 the difference
of the dominant amplification rates for viscoelastic folding
and elastic or viscous folding is around 30% (Fig 3B) and,
therefore, for R around one the viscoelastic folding theories
should be used.
d
Results
α dvi / α del
Discussion and Conclusion
α del / α dke
α dvi / α dke
For folding, R but not the conventionally employed Deborah number, De = µl ε̇B /G, is the suitable parameter to
determine the deformation behaviour. De involves the time
scale of observation (here 1/ε̇B ) and of stress relaxation
(µl /G) [Reiner, 1964]. However, viscoelastic folding is additionally characterized by a third time scale, which is the
time scale of fold amplification. All three characteristic time
scales are included in R but not in De.
The effective property of rheological elements arranged
in parallel is dominated by the strongest (most resistant)
element. In contrast, the effective response of rheological
elements arranged in series is dominated by the weakest element. This explains why the folding of the Maxwell layer is
faster (all physical and geometrical parameters being identical) then the Kelvin one (Fig. 3).
The Maxwell rheology is frequently used for rocks deformation since it allows accommodation of large shear strains.
Therefore, small-scale folding is best described using the
Maxwell rheology. At larger scales, the strong variation of
α dvi / α dma
α del / α dma
Figure 3. The dominant (i.e. maximal) amplification rates for
viscous , αdvi , elastic, αdel , Maxwell, αdma , and Kelvin, αdke ,
rheologies versus R. A) αdma approaches the maximum out of
αdvi and αdel . αdke is approximated best by the minimum out
of αdvi and αdel . B) For R < 1 (i.e. αdvi > αdel ) αdvi /αdma
approaches asymptotically the value 1 that is αdvi ∼ αdma . For
R > 1 (i.e. αdvi < αdel ) αdel /αdma approaches asymptotically
the value 1 that is αdel ∼ αdma . For R < 1 αdel ∼ αdke whereas
for R > 1 αdvi ∼ αdke .
4
SCHMALHOLZ AND PODLADCHIKOV: VISCOELASTIC FOLDING
viscosity half space. Possible crust-mantle decoupling leads
to “jelly sandwich” or “leaf spring” models (e.g., [McNutt
et al., 1988; Burov and Diament, 1995; van Balen et al.,
1998]). The bending moment or flexural response of a sandwich plate consisting of elastic and viscous layers can be
identically represented by one layer with Kelvin rheology,
i.e. from the flexural perspective the layers are added in
parallel. The bending moment (i.e. the integral in the left
term in eqn. (2)) for a two-layered lithosphere is identical
to the bending moment for an effective single layer lithosphere, having same thickness H and Kelvin rheology, if the
effective shear modulus and viscosity used for the Kelvin
rheology are given by
Gef f = Gδe 3 − 6δe + 4δe2
µef f = µ 1 − 3δe + 6δe2 − 4δe3
(11)
(12)
where G is the shear modulus of the elastic layer, µ is the
viscosity of the viscous layer and δe is the ratio of the elastic
layer thickness to the total thickness, H. The effective R
parameter, Ref f , for a folded two-layered lithosphere (elastic
and viscous layer) that is simulated by one layer with Kelvin
rheology is given by
√
3
1
1 − 3δe + 6δe2 − 4δe3
Ref f = R p
∼ R√
(for δe << 1)
3δe
δe (3 − 6δe + 4δe2 )
(13)
Negligibly small δe implies Ref f >> 1 and viscous effective
response (Fig. 3). Progressive increase of the relative elastic
layer thickness, δe , results in monotonous decrease of Ref f
and causes a transition from the viscous to the elastic deformation mode at Ref f ∼ 1 yielding the critical thickness
of the elastic layer as R2 H/3 (cf., eqn. (13)).
Acknowledgments.
We thank two anonymous reviewers for helpful and constructive reviews. S. M. Schmalholz was
supported by ETH project Nr. 0-20-499-98.
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S. M. Schmalholz, Geologisches Institut, ETH Zentrum, 8092
Zürich, Switzerland, (e-mail: stefan@erdw.ethz.ch)
Y. Y. Podladchikov, Geologisches Institut, ETH Zentrum, 8092
Zürich, Switzerland, (e-mail: yura@erdw.ethz.ch)
(Received August 2, 2000; revised October 26, 2000;
accepted October 31, 2000.)