Download Dynamics of the Eurasian Plate

Dynamics of the Eurasian Plate
DIPLOMA THESIS
presented to the
Department of Physics
of the
Swiss Federal Institute of Technology Zurich
by
Mark T. Sargent
March 2004
Advisors:
Prof. Dr. Saskia Goes
Gabriele Morra
Institut für Geophysik
ETH Hönggerberg
Schaffmattstr. 30 (HPP)
CH-8093 Zürich
Switzerland
Acknowledgements
It is a pleasure to thank Saskia Goes for guiding me both skillfully and patiently
through this project and also Gabriele Morra for sharing his expertise in ABAQUS
with me. They and the many other friendly and helpful people I met at the Institute
of Geophysics provided me with a hospitable atmosphere in which I enjoyed working.
My thanks also to Professor Rice of the Institute of Theoretical Physics for agreeing
to be the co-referee for this thesis.
I am especially indebted to my mother and father whose support and encouragement throughout my studies I deeply appreciate. In particular I am grateful for the
subscription to Scientific American they gave me a couple of years ago, thanks to
which I happened across the article on plate tectonics that awakened my interest in
geophysics.
I am also thankful to my father for reading through this manuscript and reducing
its linguistic shortcomings.
Zurich, March 2004
Mark Sargent
iii
iv
Contents
Acknowledgements
iii
1 Introduction
1
2 The rectangular models
5
2.1
The ABAQUS model . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
Kinematic and dynamic boundary conditions . . . . . . . . . . . . . . 10
2.3
The effect of lateral strength variations . . . . . . . . . . . . . . . . . 13
2.4
The effect of different rheologies . . . . . . . . . . . . . . . . . . . . . 18
3 The model of Eurasia
5
25
3.1
The properties of the ABAQUS model . . . . . . . . . . . . . . . . . 26
3.2
Forces revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3
Rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4
A comparison with measurements . . . . . . . . . . . . . . . . . . . . 30
4 Conclusions
49
Appendix A
53
v
vi
CONTENTS
Appendix B
63
Bibliography
79
THE CAUSE IS HIDDEN.
THE EFFECT IS VISIBLE TO ALL.
Ovid
Chapter 1
Introduction
The past three decades have seen numerous attempts to numerically model stress
patterns in the lithosphere of the Earth on both global and regional scales. These
efforts have been indispensable in identifying the features we need to include in our
endeavour to develop better models of our planet’s lithosphere and they have also
raised our awareness for the many unresolved issues that need to be addressed in
the future.
One such issue is our generally still very modest understanding of the forces driving plate tectonics. In principle the forces that play a role are those due to density
contrasts within the lithosphere and the mantle. Tectonic plates are considered the
surface manifestation of mantle convection, but no mantle convection model to date
has been able to satisfactorily reproduce plate tectonics. Work conducted so far
indicates that nearly all aspects of plate generation require complex lithospheric
rheologies invoking elasticity, viscosity and plasticity (Bercovici [3]). However, current technical and computational limitations do not allow the implementation of
complicated rheological behaviour on the scale of the entire mantle. Furthermore,
the relevant parameters (e.g. viscosity or the importance of the water content in the
lithosphere and mantle) are still subject to considerable uncertainties.
Studies including complex rheologies become feasible if the effect of forces due to
density contrasts are parametrised on the scale of the lithosphere. The results of
such modeling have the advantage that they can be compared with actual observations and thus increase our understanding of surface deformation and its evolution.
They may also be useful in the development of self-consistent dynamic mantle models. In 1975 an article by Forsyth and Uyeda [11] listed the potential plate-driving
forces that should be reflected in such a parametrisation and (by assuming that their
values are the same worldwide) gave an estimate of their relative importance on a
global scale. These include:
1
2
CHAPTER 1. INTRODUCTION
1. Forces due to internal lithospheric density contrasts:
(a) the ridge push force FRP acting at divergent plate boundaries,
(b) the slab pull force FSP , which pulls oceanic plates towards the trench of a
subduction zone due to the tendency of the cold and heavier old oceanic
lithosphere to sink into the mantle, and
(c) the force FCM due to the difference in gravitational potential energy
across continental margins.
2. Forces due to (viscous) resistance
(a) the mantle drag force FDF which is due to the viscous coupling between
plates and the mantle beneath them,
(b) an additional drag force FCD beneath continental plates to account for
the depth-dependent rheological properties of the mantle immediately
underlying thin oceanic and thicker continental lithosphere,
(c) the collisional resistance FCR acting on plate boundaries of converging
continental lithosphere,
(d) the slab resistance FSR , due to the viscous resistance the subducting slab
encounters as it plunges into the mantle, and
(e) the transform fault resistance FT F which counteracts strike-slip displacement on faults joining offset mid-oceanic ridge segements.
3. Viscous driving forces:
(a) the trench suction force FSU which draws the overriding plate towards
subduction zones because of regional mantle flow patterns induced by
the subducting slab, and
(b) the forces FDF and FCD if one assumes that mantle flow drives plate
motions at the surface of the Earth rather than resisting them (c.f. 2.(a)
& 2.(b), above).
Only the ridge push force, which reflects lateral density variations caused by
spreading and cooling oceanic lithosphere, is understood well enough (see, for instance, Artyushkov [2] or Turcotte and Schubert [33]) to allow a quantitatively sound
application as a boundary constraint in models of individual plates. In addition to
these forces that generate what is sometimes called the first order stress field, more
recent approaches have also included the contributions of topography and lithospheric density variations to the stress field (e.g., Bird [4] or Lithgow-Bertelloni and
Guynn [22]).
The second important issue is that of the lithosphere’s rheological properties.
Often the lithosphere is modeled as a purely elastic (or even rigid, as in the case of
Forsyth and Uyeda [11]) thin shell, following the argument that plates as a whole
3
behave elastically away from the boundaries, and that viscous effects are negligible
on the short time scales needed to compute the current stress field (e.g., Gölke and
Coblentz [15], Grünthal and Stromeyer [16] or Lithgow-Bertelloni and Guynn [22]).
On the other hand, in their publications investigating plate motions from the Cenozoic to the present epoch, Lithgow-Bertelloni and Richards [23] and Conrad and
Lithgow-Bertelloni [8] resort to a purely viscous rheology for their lithosphere.
The concept of plasticity is used by Regenauer-Lieb and Petit [28] and Hochstein
and Regenauer-Lieb [19] in their models of the Alpine and Himalayan collisions and
Bird [4] also introduces a plastic yield limit. Furthermore Lithgow-Bertelloni and
Guynn [22] maintain that, by limiting the maximum harmonic degree of their fluid
velocity field, they too implicitly specify a yield strength.
Inevitably perhaps, given the complexity of the task, up until now all modelers
have decided to focus on those factors they expect to contribute most to their subject of investigation. Lithgow-Bertelloni and Richards [23], Conrad and LithgowBertelloni [8] and Lithgow-Bertelloni and Guynn [22] base their global models of
lithospheric stress patterns and plate motions primarily on models of density inhomogeneities in the mantle that generate mantle flow and thus traction on the base of
the lithosphere. Grünthal and Stromeyer [16] and Gölke and Coblentz [15], on the
other hand, neglect viscous coupling to the mantle altogether in their models of the
European part of Eurasia, because they expect it to be negligible due to Eurasia’s
slow motion.
In yet another approach Bird [4] imposes plate velocity patterns as the boundary
condition on the base of the lithosphere. This in turn raises the question of whether
it is more appropriate to apply kinematic or dynamic boundary conditions in these
kinds of models.
Concerning the forces acting on the surface and edges of the plates, Bird [4] and
Lithgow-Bertelloni and Guynn [22], for instance, incorporate the effects of topography and lateral density contrasts in the lithosphere and thus account for ridge push,
slab pull and topographically induces stresses, but in the case of the latter study
the collisional forces between the plates are omitted. The enigmatic trench suction
force is invoked by Loohuis et al. [24] and Conrad and Lithgow-Bertelloni [8], but
neither of these projects accounts for topography.
In the studies presented here we have tried to shed light on some of the open
questions pertaining to stress modeling. These range from issues as fundamental as
why it remains necessary to isolate the lithosphere from mantle flow in models of
plate dynamics, to specific questions concerning regional stress patterns. In the case
of the Eurasian plate one would like to know if the collision with India is responsible
for extensional tectonics in the region of Lake Baikal, how in China compression
can be aligned east-west when we would expect trench suction to lead to extension
along that axis and if the Eurasian plate is a single unit or rather made up of smaller
plates. In Western Europe itself it is not clear why stress directions trend NW-SE
when previous models (e.g., Goes et al. [14] and Loohuis et al. [24]) predict them
to be roughly east-west, and why the same area experiences not only compressional
4
CHAPTER 1. INTRODUCTION
tectonics as expected from its setting between ridge push and Alpine collision, but
also normal and strike-slip faulting.
We have chosen to concentrate on the importance of rheology and lateral strength
variations for lithospheric stress patterns and use our findings to build a model of the
Eurasian plate. In doing so we want to go beyond purely elastic models of the Central
European stress field and to develop a model for this area which is not artificially
cut at its eastern border as has been done by Gölke and Coblentz [15], Grünthal and
Stromeyer [16] or Regenauer-Lieb and Petit [28], who argue that the presence of the
East European platform permits such a simplification. By investigating the influence
of lateral strength contrasts such as cratons1 on stress trajectories we should be able
to determine if this technique is indeed justified or not.
We begin with a finite element analysis of rectangular pieces of elastic, plastic,
viscoelastic or elastoviscoplastic lithosphere in which we place regions of thinner or
thicker lithosphere in various locations and observe their response to either kinematic
or dynamic boundary constraints. By proceeding in this manner we hope to identify
the “fundamental” features of the stress field before we embark on a model of the
Eurasian plate in which geometrical effects may also contribute to the stress pattern.
1
A craton is a stable area of continental crust that has not undergone much plate tectonic or
orogenic activity for a long period. A craton includes a crystalline basement of rock (commonly
Precambrian) called a shield, and a platform in which flat-lying or nearly flat-lying sediments or
sedimentary rock surround the shield. (The Schlumberger Oilfield Glossary [31])
OUR MIDDLE CLASS CONSISTS OF EQUILATERAL OR EQUAL-SIDED TRIANGLES.
OUR PROFESSIONAL MEN AND GENTLEMEN ARE SQUARES [...] OR PENTAGONS.
Edwin A. Abbott, ’Concerning the Inhabitants of Flatland’ [1]
Chapter 2
The rectangular models
2.1
The ABAQUS model
Our first object of study provides us with a setting to explore the behaviour of the
stress field for different kinds of boundary conditions and rheological properties of
the lithosphere when we place structures of varying strength in it. By working with a
geometrically simple outline, in our case a rectangle, we do not have to worry about
the diverging stress concentrations that can occur in the corners of more complex
shapes in finite element modeling and which might mask the effects we are interested
in.
To enable rough comparisons of the stress patterns in our initial models with those
actually measured in the Eurasian plate, our rectangle approximately covers the
area of the latter. Figure 2.1 shows the rectangular finite element mesh used in
ABAQUS [18], superimposed on an oblique Mercator projection1 of Eurasia, the
boundaries of which are based on the NUVEL1 model of plate velocities (DeMets,
Gordon, Argus and Stein [10]).
The grid is composed of 1800 elements with an area of 7.1 × 1010 m2 each, giving
the whole rectangle a total area of 1.28 × 108 km2 . The distance between two neighbouring nodes is 300 km which corresponds to about 4.2◦ at 50.3◦ N , the latitude
of the oblique Mercator projection’s origin. The two-dimensional elements making
up the grid are so called plane stress elements that can be used when the thickness of a body is small relative to its lateral (in-plane) dimensions. The stresses
1
As we will be working with two-dimensional models, it is essential to find a projection that
changes the plate’s area as little as possible when going from the Earth’s spherical geometry to a
flat surface. The requirement of minimal distortion determined our choice of an oblique Mercator
projection with origin at 98.8◦ E / 50.3◦ N . (In an ordinary Mercator projection, for instance,
the importance of ridge push along Eurasia’s northern boundary in the Arctic ocean would be
exaggerated).
5
6
CHAPTER 2. THE RECTANGULAR MODELS
13
0˚
W
40
˚N
12
0˚W
110
˚W
100˚
W
90˚W
80˚W
70˚W
60˚W
˚N
40
W
5 0˚
˚W
40
˚W
˚N
30
30
0˚
10˚N
0˚
20
˚N
˚E
10
40
˚E
10˚S
30
˚N
30
20˚E
30
˚N
10˚E
˚W
˚W
20
Figure 2.1: The finite element mesh used for the rectangular models, superimposed on the
Eurasian plate (oblique Mercator projection with origin at 98.8◦ E / 50.3◦ N ).
are functions of planar coordinates alone, out-of-plane normal and shear stresses
are equal to zero and all loading and deformation are also restricted to this plane.
Hence, with our models, we will not be able to predict the height of mountains
forming in regions of continental collision and compare them with actual elevations.
In contrast to many earlier studies employing triangular elements (e.g., Bird [4],
Gölke and Coblentz [15], Loohuis et al. [24]) we work with four-sided elements. In
the Lagrangian formulation ABAQUS2 uses, deformation is always underestimated
(c.f. Getting Started with ABAQUS/Standard, p. 4-4) because shear locking causes
the elements to be too stiff. Shear locking is not a problem for reasonably regular
four-sided elements since their edges are able to curve, yet it does affect the results
if one wants to investigate deformation with triangular elements3 .
Eurasia borders on the North American, the Pacific, the Philippine, the Indian,
2
Appendix A contains a basic introduction into the theoretical concepts and the procedures of
the finite element analysis performed in ABAQUS.
3
It should be mentioned that the triangular elements used by Bird [4], Gölke and Coblentz [15]
and Loohuis et al. [24] can be expected to perform well if only used to model stresses.
2.1. THE ABAQUS MODEL
7
Figure 2.2: Map of the major tectonic plates of the world. (Courtesy of the U.S. Geological
Survey.)
the Australian, the Arabian and the African plates (see Figure 2.2)4 . Its boundaries
with them include divergent margins along the Mid-Atlantic Ridge and the Arctic
Mid-Ocean Ridge, collisional boundaries along most of Eurasia’s southern border
from the Mediterranean Sea to the Himalayas, subduction zones in Southeast Asia
and west of Japan and, finally, two segments with mostly strike-slip displacement
that link the Mid-Atlantic Ridge with the Mediterranean and the Arctic Mid-Ocean
Ridge with the Pacific subduction zones.
In order to make the preliminary models as realistic as possible in spite of their
simplified geometry, the different boundary conditions were applied to the sides of
the rectangle as follows:
1. ridge push FRP along the left edge (Figure 2.3) and along three quarters of the
upper edge, of a magnitude appropriate to the mean age of the oceanic lithosphere along Europe’s western continental margin5 (approximately 80 million
years) and in the Arctic ocean (50 million years),
4
Recently scientists (e.g., Bird [5]) have suggested the existence of another 38 small plates, of
which eight are within the area labeled as “Eurasian plate” in the map.
5
Note that we are not applying ridge push as a pressure distributed through the entire oceanic
lithosphere, as would be strictly correct, because the implementation of such a force in our two-
8
CHAPTER 2. THE RECTANGULAR MODELS
Boundary force
FSU
FCC
Forsyth & Uyeda
∼ 32 FRP
∼ 12 FRP
Loohuis et al.
0.8 − 2.1 × 1012
2.1 − 3.1 × 1012
Coblentz et al.
2 × 1012
Table 2.1: Estimated values for the trench suction and continental collision forces FSU and FCC
(in N m−1 ) as given by the three listed authors. Coblentz’ value for FCC is for the Himalayas, the
one of Loohuis for Eurasia as a whole. The range over which the values of FRP vary can be found
in the following table.
2. continental margin forces FCM opposing ridge push along the same segments,
3. a free border along what remains of the upper edge, representing a zone of
probably mostly strike-slip displacement (Gaina, Roest and Müller [12]) across
Siberia where Eurasia borders on the North American plate,
4. trench suction FSU , drawing Eurasia towards the subduction zones in the
Pacific and Indian Oceans, implemented along the rectangle’s right and a
segment of its lower edge,
5. continental collision where the Indian, Arabian and African plates converge
on Eurasia (applied to most of the lower edge).
To date, exact calculations of most of these boundary forces have not been carried
out because many of the physical processes they involve are not well known6 .
However, several authors (Forsyth and Uyeda [11], Loohuis et al. [24] and Coblentz,
Sandiford, Richardson, Zhou and Hillis [7]) have published estimates of their values
which they obtained from global or regional models of plate dynamics. Table 2.1
lists the values these authors obtained from their models and Table 2.2 provides
an overview of the dependence of the ridge push force - the only numerically well
constrained boundary force - on the age of the oceanic lithosphere.
Apart from the forces acting on the plate’s edges there is also the mantle drag
force, caused by the viscous coupling between lithosphere and the underlying mantle.
In all our models it resists plate motion, i.e. we assume that, even if mantle flow
patterns can play a role locally in trench suction, it is not driving the Eurasian
plate on a large scale. To implement basal drag with ABAQUS we assign so called
’dashpot elements’ to the nodes of the finite element mesh. Based on observations,
modelers of plate dynamics and kinematics often use the working hypothesis that
plate motions are not accelerated over long periods and ensure this by balancing
dimensional ABAQUS model is not convenient. Instead, we chose to take the edge of the slab
as the border of Eurasia’s continental lithosphere and apply to it (as a line force) the value of
ridge push as integrated over the whole oceanic lithosphere between the Mid-Oceanic ridge and
the continental margin.
6
Scholz and Campos [32] note that “The state of understanding of this topic [the nature of
forces in subduction zones] at present is probably best described as confused.”
2.1. THE ABAQUS MODEL
9
Age (in millions of years)
10
20
30
40
50
60
70
80
90
100
110
120
Ridge push FRP [N m−1 ]
0.484×1012
0.968×1012
1.452×1012
1.937×1012
2.421×1012
2.905×1012
3.389×1012
3.873×1012
4.357×1012
4.841×1012
5.326×1012
5.810×1012
Table 2.2: The dependence of the magnitude of FRP on the age of the oceanic lithosphere (after
Turcotte and Schubert [33])
the torques of the forces involved (e.g., Forsyth and Uyeda [11], Loohuis et al. [24]
or Lithgow-Bertelloni and Guynn [22]). The dashpots exert a resistive force on the
nodes that is proportional to the velocity7 of the lithosphere at that point and our
tests show that this leads to a steady state velocity field in our models as well, even
if we do not explicitly apply a torque balance.
A thorough understanding of the relevant boundary forces is mandatory for a realistic model of the Eurasian plate, but their precise values are of no great importance
for the rectangular models, the aim of which is to gain a qualitative understanding
of the relevance of rheology and lateral strength variations for the characteristics
of lithospheric stress. Actual values of the forces applied in the various rectangular
models of the coming sections are listed together with all other relevant model parameters for each model in Appendix B.
While the multifaceted issue of boundary forces will be discussed in more detail in
the next chapter, the subject of kinematic and dynamic boundary conditions will
be dealt with in the following section.
7
The input for the dashpot elements is the coefficient of friction that relates the force to the
velocity. It is a function of viscosity, which, in turn, determines the value of the shear stress τLA
between lithosphere and upper mantle (Turcotte and Schubert [33], p.230):
τLA =
−2ηvref
hL
· (2 + 3 ),
h
h
(2.1)
where η ≈ 1019 − 1020 P a · s and h ≈ 220 km are the viscosity and thickness of the low-viscosity
upper mantle and hL the thickness of the lithosphere. By multiplying this basal shear stress with
the area of an element one obtains the force acting on such an element and, after dividing the
result by vref (the average plate velocity), the value of the coefficient of friction that needs to be
assigned to the dashpots.
10
CHAPTER 2. THE RECTANGULAR MODELS
2.2
Kinematic and dynamic boundary conditions
While the magnitudes of boundary forces are subject to considerable uncertainties,
current plate velocities are quite well known. One could thus be tempted to use kinematic rather than dynamic boundary conditions, but considerable caution should
be exercised in doing so. The relative velocities at plate boundaries do not contain
any information on the prevailing mechanisms and the forces definitely do not scale
directly with the velocities.
The inconsistencies between the dynamic and the kinematic approach are clearly
visible when one compares two different segments where Eurasia possesses a convergent plate boundary: (1) the subduction zones in the Far East and (2) the Himalayas.
In the latter location the relative rate of convergence is 5 centimeters per year on
average, whereas in the former it is almost twice as high. Yet, if one assumes that
the Eurasian plate is being drawn towards the Pacific trenches by suctional effects
(Conrad and Lithgow-Bertelloni [8]), then the relevant quantity is the absolute and
outwardly directed velocity of Eurasia (approx. 1 centimeter per year). Due to the
high viscosity of the mantle, this modest speed implies a large force FSU acting on
the eastern margin of Eurasia and a simple calculation shows it to lie between one
and two times the magnitude of ridge push FRP (in approximate agreement with
Forsyth and Uyeda [11] and Loohuis et al. [24], see Table 2.1 above):
FSU ≈ η × (velocity of the overriding plate at trench)
= 4 × 1022 P a · s × 0.94cm/yr = 3.17 × 1012 N m−1 .
(2.2)
The force FCC acting in areas of continental collisions, however, is estimated
(Forsyth and Uyeda [11], Loohuis et al. [24]) to be only half the size of the ridge
push force FRP . One therefore faces the paradox that in the Himalayas one has lower
forces in spite of quite high velocities, but near the trenches exactly the opposite is
the case.
Depending on the kind of boundary conditions applied to the model slab, the
stress pattern in its interior could thus vary significantly, as is indeed the case
in Figures 2.3 and 2.4, which show the stress intensity8 (top) and stress direction
(bottom) in a purely elastic rectangle for the two cases described. When the relative
velocities of the neighbouring plates with respect to Eurasia are applied as boundary
8
The quantity plotted is the von Mises stress (a measure of stress intensity often used in engineering):
r
1
σν =
[(σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ1 − σ3 )2 ],
(2.3)
2
where the σi are the principal stresses.
2.2. KINEMATIC AND DYNAMIC BOUNDARY CONDITIONS
11
Figure 2.3: Stress pattern for the model kin23-02-04 1, in which kinematic boundary conditions
were applied. The values next to the colour scale in the upper half are in units of Pa. Yellow lines
in the lower half of the picture give the orientation of extensional principal stress, red lines that of
compressional principal stress.
constraints, one receives an almost purely compressional stress pattern, whereas in
the case of dynamic boundary conditions one finds that extensional stresses dominate
on the right side of the plate, due to the application of trench suction. A comparison
of the stress fields of both models shows not only a different stress pattern, but
also substantially higher stresses in the kinematic case. Velocities are very strong
constraints forcing a region to move as prescribed, even if this induces unrealistically
high stress values. Furthermore, being instantaneous quantities and probably only
valid in the crust of the Earth, velocities are less suited as long term boundary
conditions (we run the majority of our models for 1.5 million years) than are forces
which are less susceptible to rapid changes (leaving aside events like the onset of
continental collision). The additional benefit of dynamic boundary constraints is
that - at least in the case of ridge push and slab pull - they are averages over the
entire thickness of the lithosphere, or are held to be of the same order of magnitude
as FRP and FSP . Hence, forces differ among each other at most by a factor of two
or three, which is a smaller range than with plate velocities, that can vary over two
12
CHAPTER 2. THE RECTANGULAR MODELS
Figure 2.4: Stress pattern for the model dsm elastic, in which dynamic boundary conditions were
applied.
orders of magnitude; from 1 millimeter per year to 10 centimetres a year.
In spite of this, the application of kinematic boundary conditions should still be
permissible if dynamic consistency is ensured. A way to achieve this is to verify
that they do not put more energy into a system than can be extracted from it
(see Han and Gurnis [17] for a discussion of this issue in the context of subduction).
Nevertheless, we believe that modeling stresses with the help of dynamic constraints
is more consistent than doing so by applying kinematic boundary conditions and
mixing kinematics and dynamics in the process. This choice is reflected throughout
the rest of this thesis, which is dominated by models to which boundary forces,
rather than velocities, have been applied.
The finite element program ABAQUS offers two different ways to implement
boundary forces; one can apply them as concentrated loads to the nodes on the edge
of the plate or as distributed loads to the edge of an element. For the concentrated
loads a direction may be specified while the distributed loads are basically a pressure
that acts perpendicularly to the boundary surface. By applying the concentrated
loads at right angles to the boundary (at closely spaced nodes) and comparing the
2.3. THE EFFECT OF LATERAL STRENGTH VARIATIONS
13
Figure 2.5: Map of the Eurasian plate (adapted from Villaseñor et al. [34]), showing the location
of the two major cratons in the region studied; the East European platform (EEP) and the Siberian
craton (SC). The other labels are: AP - Arabian peninsula; AR - Andaman ridge; C - Caucasus;
H - Himalayas; HD - Hangay dome; HK - Hindu-Kush; IS - Indian shield; TB - Tarim basin; TIP
- Turkish-Iranian Plateau; TP - Tibetan Plateau; TS - Tien-Shan; U - Urals; Z - Zagros.
result with a model applying distributed loading, we found that the two alternatives
are equivalent.
2.3
The effect of lateral strength variations on the
stress field
In the past several studies of the European stress field have truncated their modeled
area at approximately 40◦ East longitude (e.g., Gölke and Coblentz [15], Grünthal
14
CHAPTER 2. THE RECTANGULAR MODELS
Figure 2.6: Stress pattern for the model dsm elasticrat. The two cratons are visible as dark blue
patches of lower stress intensity.
and Stromeyer [16] or Regenauer-Lieb and Petit [28]) because they believe that a
region of old and stable continental crust known as the East European platform a so called craton - effectively shields Europe from the processes in the remaining
parts of the Eurasian plate (see Figure 2.5). By placing regions of thicker or thinner
lithosphere in our rectangular slab and observing the reaction of the stress field to
these inhomogeneities we can determine if: (1) stress levels change (get reduced in
the case of thicker and thus mechanically stronger cratonic lithosphere) in the ’lee’
of such structures, (2) stress directions get bent and (3) this happens in any setting
or if the forces prevailing in a given region play a role too.
We can also investigate if raising the dashpots’ coefficients of friction beneath the
cratons, thus simulating the higher drag that the deeper lithospheric roots of a
craton experience due to viscosity growing with depth, anchors the plate at that
point, thereby justifing the method of Gölke and Coblentz [15] who pin Europe’s
Eastern margin.
If one alters the model shown in Figure 2.4 by assigning a thickness of 150 km
(compared with 60 km for the surrounding continental lithosphere) to the rectangle’s
2.3. THE EFFECT OF LATERAL STRENGTH VARIATIONS
15
Figure 2.7: Stress pattern for the model dsm elastohypocrat1, which contains an ’artificial’
craton next to the boundaries experiencing trench suction instead of the East European platform
and the Siberian craton.
elements that coincide with the East European platform and the Siberian craton on
the map in Figure 2.1, then the stress field depicted in Figure 2.6 results. The
stress intensity map in the upper half of the pictures reveals more structure than in
Figure 2.4. Both cratons are visible as dark blue patches of lower stress intensity in
the upper half of the rectangle, a bit to the left and right of the centre. The East
European platform is also clearly visible as a region of compressive stress oriented
at roughly 45◦ from the upper left to the lower right in the lower half of the figure.
In Figure 2.4 this area was dominated by a stress field at right angles to the lower
border. On the other hand, the Siberian craton, while visible in the stress intensity
plot, cannot be distinguished in the perpendicular stress pattern generated by Arctic
ridge push and continental collision of India with Eurasia.
The same observation can be made in Figure 2.7; in this model there is only one
craton, located in the area that is experiencing extensional stresses due to trench
suction. It too shows up in the stress intensity map but much less so in the lower part
of the figure. Why does the influence of cratons vary in what, at first glance, might
seem to be an unpredictable fashion? The results suggest that thicker and thus
16
CHAPTER 2. THE RECTANGULAR MODELS
Figure 2.8: Stress pattern for the model dsm elastithincrat, which contains thinner rather than
thicker lithosphere in the areas of the East European platform and the Siberian craton.
stronger lithosphere always reduces stress values in the area of the craton. However
it affects stress orientation only if the structure is located in an area under the
influence of more than one source of stress (in Figure 2.6 ridge push and continental
collision) that lead to two competing fields of comparable magnitude. In this case,
stress directions do tend to change at the edge of the craton and thereby induce
so called ’stress bending’. Such bending of stress orientations has been invoked by
Grünthal and Stromeyer [16] and Müller et al. [26] to explain European variations
in stress style.
The models indicate that both orientation and magnitude of the stress fields on
the two sides of the structure are not necessarily independent. While cratons do lead
to reduced stress intensities within the craton, they do not seem to create a stress
shadow, in which the magnitude of the stress field is significantly lower on the edge
farther from the source. As can be seen in Figure 2.6, the cratons are symmetrically
surrounded by regions of higher stress values, and Figure 2.7 reveals that the stress
on the left of the hypothetical craton reaches the same values as to the right of it.
In view of these findings we are led to the conclusion that the valitity of Gölke and
2.3. THE EFFECT OF LATERAL STRENGTH VARIATIONS
17
Figure 2.9: Full displacement field of the model dsm creepcrat after 32000 years; no discontinuities are observed at the edges of the cratons.
Coblentz ’ [15], Grünthal and Stromeyer ’s [16] and Regenauer-Lieb and Petit’s [28]
method mentioned above, in which they cut the Eurasian lithosphere at the East
European platform, is questionable.
Having looked at the result of thickening a region of the lithosphere, we now
consider thinner pieces of lithosphere. In Figure 2.8 the areas formerly making up
the East European platform and the Siberian craton now have a thickness of 50
km, while the rest of the plate is 100 km thick. (The 40 km increase in lithosphere
thickness results in the stress levels in Figure 2.8 being generally lower than in the
preceding models). The regions that displayed lower stress intensities in Figure 2.6
now have values higher than the surroundings. As before, the thickness variation
in the area of the Siberian craton does not appear in the diagram showing stress
directions, but the area of the East European platform tends to deflect stresses.
Still, the stress pattern resembles the model without lateral structure in Figure 2.4
more than the one with the cratons in Figure 2.6. The main effect of the thinner
lithosphere in Figure 2.9 is a less abrupt change from horizontal to vertical stress
orientations in the area above and below its left edge. (’Horizontal’ and ’vertical’
refer to alignment along the long and short edge of the rectangle, respectively.)
In a further model we tested the effect of both increasing the thickness of a piece of
lithosphere and raising the asthenospheric drag forces beneath it. Keeping in mind
that Gölke and Coblentz [15] fix the eastern edge of their model of Eurasia, we are
especially interested if raising the drag forces below the cratons might anchor them
or induce jumps in the displacement field at their edges. The boundary constraints
of the model shown in Figure 2.9 are identical with those of dsm elasticrat in Figure
2.6 and the absence of discontinuities at the craton’s edges after a period of 32000
years suggests that it would be more realistic not to pin the eastern margin of the
finite element mesh when modeling the stress field in Europe.
18
CHAPTER 2. THE RECTANGULAR MODELS
If one wishes to make a more detailed model of the Eurasian plate to the west of
the Urals, one should begin by determining the stresses along the foreseen cut using
a model of the whole Eurasian plate and then apply these stresses as boundary
constraints on the eastern border of the refined model.
As a final comment it should be emphasized that, although the conclusions of this
section were reached assuming a purely elastic lithosphere, the observations remain
correct in the viscoelastic, elastoplastic and elastoviscoplastic rheologies discussed
in the upcoming section.
2.4
The effect of different rheologies on the stress
field
To date the majority of the models of the Earth’s lithosphere published in scientific literature have employed either a purely elastic (e.g., Gölke and Coblentz [15],
Grünthal and Stromeyer [16], Lithgow-Bertelloni and Guynn [22]) or viscous (e.g.,
Bird [4], Lithgow-Bertelloni and Richards [23], Conrad and Lithgow-Bertelloni [8])
material behaviour. In the previous section I presented a model for a rectangular piece of purely elastic lithosphere that was run for approximately 32000 years
and contains two regions of thicker continental crust, representing the East European platform and the Siberian craton (Figure 2.6). This section deals with three
additional rheologies, namely viscoelasticity, plasticity and elastoviscoplasticity to
see how these material properties modify the stress pattern obtained in the purely
elastic case by applying ridge push and continental margin forces, as well as trench
suction and continental collision along the boundaries.
First we will consider a lithosphere under uniaxial stress that behaves like a
viscoelastic (Maxwell) body with the rheological equation (c.f. Turcotte and Schubert [33])
σ̇ij σij
+
,
(2.4)
˙ij =
E
2η
where E is Young’s modulus and η is the viscosity9 . Under constant strain the
9
In an elastic (or Hooke) body stress is proportional to strain
σij = Eij ,
(2.5)
while in a linearly viscous (or Newton) body stress is proportional to the strain rate:
σij = 2η ˙ij .
(2.6)
By taking the derivative of Equation 2.5 with respect to time and combining it with 2.6 one gets
Equation 2.4
2.4. THE EFFECT OF DIFFERENT RHEOLOGIES
19
Figure 2.10: Stress pattern for the model dsm viscoelasticrat, in which the lithosphere behaves
like a Maxwell body with a viscosity of ηLIT H = 1023 P a · s.
solution of equation 2.4 becomes
E
σij = σij,init · exp − · t ,
2η
(2.7)
where σij,init is the initial stress. Equation 2.7 defines the so called Maxwell time
τM = Eη , above which the viscous effects in a Maxwell body begin to dominate the
elastic ones. The Maxwell time is important for our calculations, since we have to
know for how long our models should be run to investigate the effects of viscosity on
the stress field. Using the common values for Young’s modulus (E = 1011 Pa) and
viscosity (η = 1023 P a · s) in the lithosphere to estimate the Maxwell time one finds
it to be of the order of 1012 seconds or around 32000 years (i.e. the computation
time of the purely elastic models in the preceding section) and I have raised the
computation time to 1.5 × 1013 seconds (approximately 1.5 million years) for the
models in this section.
The stress intensity distribution and the stress directions for a viscoelastic rheology are given in Figure 2.10 (apart from rheology the same input parameters were
used as in Figure 2.6). A comparison of these two figures clearly shows that both
20
CHAPTER 2. THE RECTANGULAR MODELS
Figure 2.11: Stress pattern for the model dsm elastoplasticrat, which has a plastic rheology with
a yield stress of 100 MPa.
stress levels and orientations are very similar. According to Equation 2.7 Maxwell
bodies exhibit exponential stress relaxation, with the relaxation time τM . Once the
viscoelastic plate has reached a steady state - as we expect it to do after running the
model for 15 Maxwell times - stresses relax at the same rate as new ones build up
due to the flowing material, so stress values should no longer change. Furthermore,
because viscoelastic bodies behave elastically on Maxwell-time scales the residual
stress levels should be those of a purely elastic model that is already at equilibrium at the Maxwell time. The features of the stress field in Figure 2.10 are thus
consistent with viscoelastic lithosphere in a steady state.
In keeping with these considerations we do not anticipate any changes in the
stress field of the purely elastic plate after 32000 years, even if the forces were
applied for 1.5 million years. A closer look at Figure 2.11 appears to reveal a flaw
in this reasoning. It depicts the stress pattern for a rectangle of material with a
yield threshold of 100 MPa. Since yielding is limited to small areas in three of the
four corners, the largest part of the plate should have behaved purely elastically and
thus look exactly like in Figure 2.6. Yet the stress intensity over much of the plate
2.4. THE EFFECT OF DIFFERENT RHEOLOGIES
21
Figure 2.12: Stress pattern for the model dsm elastoviscoplasticrat, which has a viscoelastic
rheology with a yield stress of 100 MPa.
has reached ∼50 MPa, 30 MPa higher than in Figure 2.6. This is most likely caused
by the dashpots. Since they add a viscous component to the a priori purely elastic
piece of lithosphere, they delay the transmission of stresses into the plate’s interior
and cause stress values to continue growing. There is further evidence to support
this hypothesis. First of all, in a snapshot of the elastoplastic model in Figure 2.11
after 32000 years stress values are still the same as in Figure 2.6. Furthermore,
increasing the frictional coefficients of the dashpots produces even more extensive
areas of relaxed lithosphere in the plate’s interior, compatible with a still slower
transmission of the stresses towards the inside.
While the characteristics of the dashpots probably influence stress levels, they do
not alter stress directions significantly. One of the main issues of my thesis was to
find how one might alter the stress field in Europe, which in purely elastic models is
dominated by Mid-Atlantic ridge push and continental collision in the Himalayas,
by prescribing different rheological properties to the lithosphere. We supected that
plasticity might the key to the question. Yielding should localize all deformation
and, because only the yield stress can be propagated beyond the area of highest
deformation, reduce stress levels in the remaining plate. In this manner the rest of
22
CHAPTER 2. THE RECTANGULAR MODELS
Figure 2.13: Stress pattern for the model ksm elastoplasticrat, which has the same plastic rheology as the rectangle in Figure 2.11, but which uses kinematic rather than dynamic boundary
conditions.
Eurasia could be decoupled to a certain extent from the stresses generated by the
indentation of the Indian plate. This would allow the weaker collisional forces from
the convergence of Africa on Europe to gain importance in influencing the orientation European stress field.
So far there is no grounds to believe that more complex rheologies resolve the problem. From Figures 2.6, 2.10 and 2.11, the stress in the left quarter of the rectangle,
a region we can identify with Europe in our simple model, is seen to be horizontal.
There is thus no indication whatsoever of the African collision acting on the lower
edge in that portion of the lithosphere. As illustrated in Figure 2.12 the situation is
no different in an elastoviscoplastic rheology; indeed, one can barely distinguish any
differences in the stress patterns of the elastic, viscoelastic and elastoviscoplastic
rectangles.
For comparison, consider a kinematically constrained rectangle again. The model
of Figure 2.13 has the same plastic rheology as the one in Figure 2.11, but the
boundary forces for trench suction and continental collision have been replaced by
2.4. THE EFFECT OF DIFFERENT RHEOLOGIES
23
Figure 2.14: Stress pattern for the model ydsm elastoplasticrat0. The collisional forces have
been chosen such that they lead to roughly the same extent of yielding as in Figure 2.13.
kinematic constraints. Now the formerly horizontal stress orientations are replaced
by a field at roughly 45◦ , consistent with the large scale stress orientations in Europe
which are aligned NW-SE, apparently reflecting the influence of both ridge push and
African convergence. Could this then imply that we have been applying the wrong
forces in the dynamic models?
In the section on kinematic and dynamic boundary conditions we found that velocities lead to (too) high boundary loads which, in the purely elastic models, created
unrealistically high stress intensities throughout the plate. By prescribing a yield
stress in the model of Figure 2.13, the deformation is localized and stress levels
outside the area of yielding (in grey, in Figures 2.13 and 2.14) are as low as in
the models with dynamic constraints, with the important difference that the stress
patterns seem to be closer to reality than in the dynamic models.
In view of this discovery we now are led to the conclusion that plasticity is the
key to creating a realistic stress map of the Eurasian plate, because it allows the
application of sufficiently large forces without raising the stress levels in the interior
of the plate to unrealistically high values. Figure 2.14 shows a dynamical model
24
CHAPTER 2. THE RECTANGULAR MODELS
in which the boundary forces have been chosen such that the characteristics of the
stress field are quite similar to those in the preceding kinematically constrained
model. To achieve this, forces with a magnitude of 1.5 × 1018 N, 2 × 1018 N and
3 × 1018 N, for the African, Arabian and Indian collision, respectively, were applied.
Atlantic ridge push on the other hand has a magnitude of 1.34 × 1018 N. On page 8
it was mentioned that previous studies predict collisional forces to be about half as
large as ridge push, while in our case they are equal or larger. How can these two
observations be reconciled? We believe that the forces listed in Table 2.1 are the
stresses that are transmitted to the interior of the plate at the border of the area
of yield. Since perfectly plastic materials do not offer any resistance to deformation
once their yield limit has been attained, this yield stress acts on the border and the
rest of the plate and by doing so contributes to its torque balance. We assume that
this yield force, rather than the actual collisional forces, was used for the calculation
of the relative force magnitudes in the results listed in Table 2.1.
With the insights on rheology, strength variations and forces gained in the models
presented in this section we will move on to consider the ’real’ Eurasian plate now.
In the next chapter we investigate whether the promising results live up to our
expectations and are able to produce a model of Eurasia that satisfactorily matches
actual data.
MOTIONS UP IN THE HEAVENS ARE ORDERLY, PRECISE, REGULAR AND MATHEMATICAL, THOSE
DOWN ON EARTH MESSY AND IRREGULAR AND CAN BE DESCRIBED ONLY QUALITATIVELY...
The Aristotelian view of the universe [9]
Chapter 3
The model of Eurasia
Before discussing more realistic models of the Eurasian plate than those treated in
Chapter 2, we should sumarize which of the features observed in our best rectangular model (Figure 2.14) already agree with actual measurements.
Introducing plasticity causes stress directions to be determined by the nearest sources
of stress; in the portion of the rectangle that we identify with Europe this leads to a
field matching the NW-SE maximal horizontal stress in Western Europe as given by
The 2003 release of the World Stress Map [30]. The area east of the Indian continental collision displays extensional stresses which are consistent with the extrusion
of large blocks of lithosphere along the major Chinese strike-slip fault systems, such
as the Red River fault and the Altyn Tagh fault that is located on the border between the Tibetan Plateau and the Tarim Basin. In fact, even in our rectangle we
predict strike-slip faulting along the band where both compressional (in red) and
extensional (in yellow) stresses prevail. It lies in a line linking the right edge of the
Indian indenter to the beginning of the subduction zone “further north” and thus
agrees quite well with the orientation of a region of high seismic activity (c.f. The
Global Seismic Hazard Map [13]) running from Afghanistan north to Lake Baikal.
Finally, the models predict reasonable stress values (between 20 and 60 MPa) in
much of the plate’s interior, and the displacement velocities (between 1.9 cm and 8
mm a year) also coincide well with the range of velocities measured in Eurasia.
The results in a model using the Eurasian plate’s real outline will have to stand
up to more trying tests than the rectangle, which allows qualitative comparisons at
best. In the new setting we will be able to compare our findings with actual data
for stress, strain rate and deformation velocities. The prerequisites for achieving the
higher level of accuracy made necessary by the increased complexity of the problem
are: (1) an improved understanding of the boundary conditions in the areas of
continental collision and subduction, and probably also (2) a rheological model that
is closer to reality than perfect plasticity.
25
26
CHAPTER 3. THE MODEL OF EURASIA
SC
INT
EEP
Figure 3.1: The finite element mesh used for the models of the Eurasian plate (same projection
as in Figure 2.1). Areas of thicker continental lithosphere are marked in blue (150 km) and green
(200 km): EEP - East European Platform; INT - stable platform in central Eurasia; SC - Siberian
craton.
3.1
The properties of the ABAQUS model
The whole mesh as shown in Figure 3.1 has 4013 elements with an average area of
1.719 × 1010 m2 . The nodes are separated by a distance of approx. 130 km which,
at the latitude of the projection’s origin, corresponds to 2.1◦ . Building the mesh for
a complex outline like the Eurasian plate is trickier than in a simple and symmetric
shape like a rectangle, because all elements should be as regular as possible for a
stable and fast finite element analysis. To construct a mesh fulfilling this requirement
we used the mesh building program PATRAN [25], which generates a file that can
be read by ABAQUS. As in the rectangular models we employ four sided continuum
elements with plane stress characteristics.
The Eurasian plate has an area of 69×106 km2 of which 51×106 km2 are continental
lithosphere. Due to the limitations of the two-dimensional elements (c.f. footnote
on page 7) that keep us from applying ridge push as a distributed load we will
3.2. FORCES REVISITED
27
restrict our analysis to continental Eurasia. This leaves us with 3723 elements with
a thickness of 200 km in the East European platform and the Siberian craton, and
thicknesses of 150 km in the quiescent area between these two continental shields
and 100 km in the rest of the plate.
3.2
Forces revisited
In the rectangular models we implemented four forces, namely: basal drag, ridge
push, trench suction and continental collision1 . The first two of these we will apply
in the same conceptual way as before, albeit in a slightly refined manner. Thus we
refer to the Digital age map of the ocean floor [27] for detailed data on the age of
the ocean floor along Eurasia’s Atlantic and Arctic continental margins to calculate
a realistic ridge push field. From it we then subtract the continental margin force
(which we assume to be of the same magnitude along all relevant border segments)
to get the boundary loads along the perimeter all the way from the Algarve to the
Laptev Sea. We no longer apply basal drag as a uniform force but, accounting for
the depth dependence of viscosity, calculate it individually for the cratons, continental lithosphere and the region between the cratons. The values of shear stress acting
on the base of the lithosphere then range from 2.42 MPa underneath the continent
to 34.04 MPa beneath the cratons.
As far as trench suction and continental collision are concerned, our lack of understanding of the underlying physical processes has so far prevented quantitative
predictions of the magnitude of the forces involved. In the following paragraphs I
will discuss concepts that could help estimate the values of the boundary constraints
in these environments.
As continental collision is due to interaction between the plates involved, it should
act on both plates with equal strength. This does not mean that the same amount
of deformation need occur on each of them; if one of the plates participating in the
event is stronger - as for instance India probably is, because of the presence of a
craton - it will likely suffer less deformation.
Forsyth and Uyeda [11] argue that during collision a higher strain rate does not
increase the stress, but merely reduces the length of time required to reach the
level of stress at which brittle failure or plastic yielding occurs. Thus, they take
the average stress over a period of time to be independent of the relative motion
at the plate boundary. I believe this point of view can be reconciled with our
assumption that, for the reasons set forth on page 24, the stress acting on the
plates’ interior is the yield stress. However, the varying sizes of Eurasia’s orogens is
a strong indication that the relative rate of convergence nevertheless is an important
1
We continue to neglect the resistance along strike-slip boundary segments (which also includes
transform fault resistance on the strike-slip faults joining different segments of the mid-oceanic
ridges).
28
CHAPTER 3. THE MODEL OF EURASIA
factor. India, Arabia and Africa are, on average, moving at speeds of 4.6 cm, 2.9
cm and 0.6 cm with respect to Eurasia today and the mountainous regions along
their borders cover a surface of 19.2 × 105 km2 (Himalayas and Pamir), 7.2 × 105
km2 (Zagros) and 1.2 × 105 km2 (Alps). Consequently it seems appropriate to scale
the collisional forces that induce yielding with velocities.
Laboratory results suggest that rocks under lithospheric conditions are most likely
to obey a power law dislocation creep relationship unless they behave in a brittle
manner (Brace and Kohlstedt [6]):
˙ = Aτ n exp(−Q/RT ),
(3.1)
where ˙ is the strain rate, τ is the deviatoric stress, T is the absolute temperature,
R the universal gas constant and A, Q and n are material-dependent constants.
The activation energy, Q, ranges from ca. 500 kJ mol−1 for dry olivine to ca. 160
kJ mol−1 for quartz and the power-law exponent, n, is in the range 3-5 (Kirby
and Kronenburg [20]). When relating stresses to the velocities of the neighbouring
plates we thus assume that the following relation holds (strain rate being the spatial
gradient of velocity):
r
σ = σref
3
v
vref
.
(3.2)
The value for the reference stress σref can be gained by considering the European
stress field. Judging by its orientation it is governed by ridge push and a force of
about the same magnitude acting along its southern perimeter. Hence, we take σref
to be equal to the value of ridge push transmitted by the piece of oldest oceanic
crust along Europe’s western continental margin. The reference velocity vref going
with σref corresponds to the convergence rate of the African plate on Europe in the
Central Mediterranean Sea. Having fixed these values we can deduce the stresses
along all collisional borders by scaling them with the local velocities according to
Equation 3.2. The actual forces that get applied parallel to the velocity at each
boundary node then are computed by multiplying the stresses by the thickness of
the lithosphere and the distance between the individual nodes.
The source of trench suction is a secondary hydrodynamic flow induced in the
upper mantle above a sinking slab, and this flow exerts shear tractions at the base
of nearby plates. Since the subduction of the slab gives rise to the secondary flow it
appears reasonable to assume that the forces acting on the overriding plate scale with
its rate of descent. The basal drag forces mentioned at the beginning of this section
are proportional to the velocity at which the plate drifts over the upper mantle.
Whether the mantle moves with respect to the lithosphere (as is the case in trench
suction) or vice versa (as happens in basal drag) is merely a question of reference
frame and thus we expect trench suction to vary linearly with the velocity of the
3.3. RHEOLOGY
29
subducting plate at the trench. Such a linear dependence is also found by Scholz
and Campos [32], although it is not a dependence on the velocity of the subducting
plate alone, but a combination of overriding and subducting plate velocities and
trench roll back speed instead.
Just as in the case of continental collision we still need a reference force for the
scaling relationship. The estimates listed in Table 2.1 on page 8 allow for values
of FSU between 0.8 × 1012 N m−1 and 6 × 1012 N m−1 . Our own estimate given in
equation 2.2 lies about half way between these two values and logically offers itself as
a compromise. We chose it to correspond to the mean subduction velocity along all
Eurasian trenches in the Far East and apply the forces so calculated perpendicular
to the plate boundary in the corresponding regions.
3.3
Rheology
The perfect plasticity used in the rectangular models is an idealisation. In reality,
two mechanisms, diffusion creep and dislocation creep, are active during the purely
elastic phase below the yield stress of 100 MPa. Diffusion creep only occurs if the
rocks are made up of very small grains or at stresses significantly lower than those in
our models. We can therefore ignore this kind of creep and restrict ourselves to the
implementation of dislocation creep. Laboratory results suggest that rocks under
lithospheric conditions are most likely to obey a power law creep as in Equation 3.1
(commonly with n = 3), provided they do not behave in a brittle manner.
Dislocation creep breaks down when stress and strain rates are increased sufficiently
so that dislocations start to appear in considerable number inside the individual
grains of the material, thereby ending the domain of low power viscous creep. At
stresses higher than the yield limit σY the activation of different dominant glide
planes leads to an exponential dependence of strain rate on stress known as Peierl’s
creep or low temperature plasticity (see Regenauer-Lieb and Yuen [29]):
˙ ∝ exp(σ)
or
˙ ∝ sinh(σ).
(3.3)
In purely fluid dynamic models the laws describing this regime are frequently
approximated by a power law dislocation glide of the form:
˙ = ˙Y + A · (σ − σY )n ,
(3.4)
where ˙Y is the value of the strain rate at σY , attained in the rheological regime
valid in the stress range below the yield limit. The larger the exponent n is, the
more the material behaves like a perfectly plastic body. Thus in our elastoplastic
models we calculate the response of a solid with a very high value of n at stresses
30
CHAPTER 3. THE MODEL OF EURASIA
larger than σY and for which viscous creep below this limit is neglected. Strictly
taken this is probably only appropriate if the model is run for less than the Maxwell
time introduced in Equation 2.7 of the preceding chapter, and not for 1.5 million
years as we do.
Power law creep and plasticity are at opposite ends of the rheological spectrum
in that plasticity can be described by a viscosity that jumps from infinity below σY
to zero after yielding whereas the nonlinear (or effective) viscosity η = σ/2˙ varies
continuously between these two values.
In the next section we present two models of Eurasia, one using a viscoelastic power
law rheology and the other using plasticity.
3.4
A comparison with measurements
To begin with we investigate how well a model with a perfectly elastoplastic rheology
matches actual data for stress directions, displacement velocities and strain rates
within Eurasia.
The stress field of continental Eurasia, as depicted in Figures 3.2 to 3.4, results
from the following boundary conditions:
1. ridge push counteracted by continental margin forces along the Atlantic and
Arctic continental margin of Eurasia; the ridge push is largest off the coast
of Portugal, where the oceanic lithosphere is about 100 million years old and
smallest in the Laptev Sea where it is only half as old,
2. forces due to continental collision with the African, Arabian and Indian plates
along the southern border of Eurasia and the North American plate in Siberia
(scaled with relative velocities according to Equation 3.2),
3. trench suction scaled linearly with the convergence rate of the neighbouring
slab being subducted in all Far Eastern subduction zones and in the Aegean
Sea where the African plate is diving beneath Eurasia,
4. strike-slip segments are left free; they include
(a) the segment running from the coast of Burma to the Kingdom of Bhutan,
(b) the segment joining the coast of Pakistan to Nepal,
(c) the segment between the Strait of Gibraltar and the Atlantic continental
margin,
(d) a short segment in northern Siberia
3.4. A COMPARISON WITH MEASUREMENTS
31
Figure 3.2: Stress intensity map for the elastoplastic model eurasia27-01-04 6 (von Mises stress,
c.f. Equation 2.3 on page 10). In the grey areas the yield limit of 100 MPa has been exceeded.
32
CHAPTER 3. THE MODEL OF EURASIA
Figure 3.3: Orientation of compressional principal stress for the model eurasia27-01-04 6.
3.4. A COMPARISON WITH MEASUREMENTS
Figure 3.4: Orientation of extensional principal stress for the model eurasia27-01-04 6.
33
34
CHAPTER 3. THE MODEL OF EURASIA
5. an unconstrained segment approximately 300 km in length along the coast of
Pakistan where Arabian oceanic lithosphere is being subducted underneath
Eurasia (as mantle flow is probably not important in that area due to the
surrounding regions undergoing continental collision, we do not apply trench
suction to it).
Looking at Figure 3.2 one can easily recognize the two cratons as bluish regions of
low stress intensity. The two areas in grey mark those regions where the yield limit
has been exceeded and orogeny should occur. In the case of the Himalayas the extent
of this area is consistent with the presence of actual mountains and, as a fascinating
detail, there is an area within it where the lithosphere has not yet yielded and that
one could tentatively identify with the Tibetan Plateau. This feature could be due
to the shape of the Eurasian border which does not present the indenting Indian
plate with a straight border, but one that has a convex bend to it. To the north
of the boundary with Arabia the area of yield is far too large compared with real
mountain ranges. Forces along this zone are applied more obliquely to the border
of the plate than in the Indian collision and we suspect this induces an unrealistic
reaction of the boundary elements that seem to deform too easily when sheared.
Since the boundary loads for trench suction applied to Southeast Asia are weaker
than the collisional forces by a factor of up to three, that area of the Eurasian plate
also has quite low stress levels.
Let us now turn to the orientation of the compressional and extensional stresses
in the elastoplastic model, given in Figure 3.3 and 3.4. They can be compared with
the maps in Figures 3.5 to 3.7 that show data gathered in the framework of the
World Stress Map Project [30].
Beginning in the east, Figure 3.5 displays compressional stresses parallel to the motion of India in the central Himalayas, and stresses that rotate to WNW-ESE in
the Hindu-Kush. Our model in Figure 3.3 matches these observations well. To the
east of the Himalayas and all the way to the Japan and Ryukyu Trenches compressional stresses are aligned roughly E-W. However, as can be seen in Figure 3.4, we
predict extensional stresses to have that orientation and compressional stresses run
N-S. Similarly, our compression is perpendicular to the observed directions along
the Philippine Trench. Along the Java Trench we do predict extensional stresses at
90◦ to the observed compression, but this is probably irrelevant since our models
show vanishingly small compression in that area. In fact, although our model predicts that the whole of Southeastern Asia will be dominated by extension, there is
no indication of such a regime in the World Stress Map (henceforth referred to as
WSM).
A comparison with Iranian WSM data in Figure 3.6 reveals that the orientation
of compressional stresses in Figure 3.3 are generally correct in this region. The
prominent extensional stresses along the plate boundary from Nepal to Iraq are
probably an artifact of the too easy deformability of the elements under shearing
loads, so we need not be concerned that we predict strike slip faulting rather than
3.4. A COMPARISON WITH MEASUREMENTS
60˚
80˚
100˚
35
120˚
140˚
Method:
focal mechanism
breakouts
drill. induced frac.
borehole slotter
overcoring
hydro. fractures
geol. indicators
60˚
60˚
Regime:
NF
SS
TF
U
Quality:
A
B
C
! (2003) World Stress Map
40˚
40˚
20˚
20˚
0˚
0˚
60˚
80˚
100˚
World Stress Map Rel. 2003
120˚
140˚
Projection: Mercator
Heidelberg Academy of Sciences and Humanities
Geophysical Institute, University of Karlsruhe
Figure 3.5: World Stress Map data for Asia. The lines indicate the direction of maximal compression. Stress indicators in red, green and blue are based on focal mechanism of earthquakes: NF
- normal faulting; SS - strike-slip faulting; TF - thrust faulting.
the predominantly observed thrust faulting.
In Europe, the region to which we wanted to pay special attention, data is abundant (Figure 3.7). In its western part and in the British Isles compressional stresses
trend NW-SE. There is evidence that their orientation rotates around the arc defined
by the Alps to NNE-SSW, although this tendency is masked to a certain extent by
36
CHAPTER 3. THE MODEL OF EURASIA
45˚
50˚
55˚
60˚
65˚
40˚
40˚
35˚
35˚
30˚
30˚
Method:
focal mechanism
breakouts
drill. induced frac.
borehole slotter
overcoring
hydro. fractures
geol. indicators
Regime:
NF
SS
TF
U
Quality:
25˚
25˚
A
B
C
! (2003) World Stress Map
45˚
50˚
55˚
60˚
World Stress Map Rel. 2003
65˚
Projection: Mercator
Heidelberg Academy of Sciences and Humanities
Geophysical Institute, University of Karlsruhe
Figure 3.6: World Stress Map data for Iran.
regional patterns which are probably due to topographical effects. In southeastern
Europe stress directions and focal mechanisms reflect the motion of Turkey towards
Greece and in Scandinavia compression appears to be parallel to ridge push.
Our model contains evidence of a rotation of stresses into NNE-SSW to the east
of the Alps, albeit only in the proximity of the southern border, and we successfully predict compressional stress directions in the interior of western Europe as far
north as Great Britain. However along the continental margin and in Scandinavia
we fail to match the observations. Here the predicted compressional stresses are
parallel to the continental margin. Although the East European platform is in the
vicinity, we do not believe it to be the source of the phenomenon since we have no
evidence of such abrupt stress bending from the rectangular models presented in the
last chapter. As seen in to Figure 3.4, extensional stresses are negligible in Europe,
3.4. A COMPARISON WITH MEASUREMENTS
0˚
20˚
37
40˚
Method:
focal mechanism
breakouts
drill. induced frac.
borehole slotter
overcoring
hydro. fractures
geol. indicators
Regime:
NF
SS
TF
U
Quality:
A
B
C
! (2003) World Stress Map
60˚
60˚
40˚
40˚
0˚
20˚
World Stress Map Rel. 2003
40˚
Projection: Mercator
Heidelberg Academy of Sciences and Humanities
Geophysical Institute, University of Karlsruhe
Figure 3.7: World Stress Map data for Europe.
suggesting that the strike-slip faulting observed in central Europe may result from
local effects superimposed on the large scale stress distribution. Such a change in
38
CHAPTER 3. THE MODEL OF EURASIA
Figure 3.8: Strain rates in the Himalayas and the Zagros mountains for the model eurasia27-0104 6. The values of the second invariant of the strain rate tensor are given in s−1 .
regime would probably only be possible if stress levels are generally low in this part
of Europe and indeed the calculations presented in Figure 3.2 show them to be in
the range of 30 to 40 MPa.
For much of northern and central Eurasia stress measurements are sparse, making
it difficult to verify the validity of our model in these settings.
In summary we can say that, with respect to stresses, our elastoplastic model does
well in predicting the stress orientations in much of Europe, as well as from Iran to
Tibet. On the other hand the inability to predict the correct stress patterns along
much of the eastern margin is a serious failure which raises the question if we are
applying the correct forces. This suspicion is supported by the fact that Figure 3.2
implies high stress levels and even smallish areas of yield long the eastern boundary,
whereas in reality one does not observe any mountain building; the mountains in
these parts are of volcanic origin.
One might argue that the issue could be resolved by integrating the high Himalayan
3.4. A COMPARISON WITH MEASUREMENTS
39
Figure 3.9: Global strain rates published by Kreemer et al. [21], based on a compilation of
geodetically measured plate velocities.
topography which leads to gravitational spreading and thus induces compression
at the foot of the mountain range. While the negligence of topographical effects
admittedly is a shortcoming of our simple model (indeed, Lithgow-Bertelloni and
Guynn [22] have shown that topography may even be dominant in certain places),
it is unlikely that this would influence the stress field all the way to the eastern
plate boundary. Since there is no obvious way to get E-W compression in the Far
East while trench suction is pulling on the eastern margin of Eurasia, one is led to
the conclusion that trench suction is not the appropriate boundary condition. In
the second model presented after the next paragraph we will investigate the effect
of leaving those boundaries unconstrained.
Having treated the stress field we now discuss the strain rate. Figure 3.8 is a
contour plot of values of the second invariant of the strain rate tensor
I2 = ˙212 + ˙223 + ˙231 − (˙11 ˙22 + ˙11 ˙33 + ˙22 ˙33 )
(3.5)
in the Himalayas and the Zagros mountains. A comparison with the global strain
rate distribution calculated by Kreemer et al. [21] (Figure 3.9) reveals that the values
in the model are of the correct magnitude, but their distribution is not consistent
with actual data. In the Himalayas the areas with the range of strain rates expected
in regions undergoing continental collision are far smaller than in Figure 3.9 and
also smaller than those of the model in Iran. The latter fact once again points to
40
CHAPTER 3. THE MODEL OF EURASIA
unrealistic deformation caused by excessive, local shearing of the elements of the
finite element mesh and seems to indicate that, without this artificial phenomenon,
strain rates would be lower than are observed.
Figures 3.10 to 3.12 show the stress levels and orientations in a viscoelastic
Eurasian plate in which dislocation creep is described by a power law rheology
of order three2 . To improve the faulty features in the stress and strain maps of the
elastoplastic model we change the following boundary constraints:
1. Sections formerly experiencing trench suction (i.e. the trenches in the Far East
and in the Aegean Sea) are now left free. Doing so could be justified if the
collisional resistance - a quantity we neglected so far - across the interface of
the subducting slab and the overriding plate is comparable in magnitude to
trench suction, so that the two forces effectively cancel out.
2. In Siberia, the convergence of North America on Eurasia happens at quite an
oblique angle. We thus leave this segment unconstrained, interpreting it as a
strike-slip boundary rather than a collisional zone.
3. In an attempt to match the lateral extent of the areas with realistic strain
rates in the Himalayas we apply collisional forces corresponding to an Indian
indendation velocity of 15 cm per year. Albeit completely arbitrary on the
grounds of current observations this value is the estimated velocity of the
Indian plate at the onset of continental collision about 40 million years ago.
There are two major differences between the von Mises stress maps of the elastoplastic (Figure 3.2) and the power-law model (Figure 3.10). First of all, both areas
of yield have shrunk considerably, with the result that Iran now has a mountain
range consistent with the lateral dimensions of the Zagros. Since boundary conditions in this part of the plate were not adapted it is obviously a consequence of
the change of rheology. The lithosphere, which is perfectly elastic below σY in the
elastoplastic model, propagates stresses into the interior more easily than the lithosphere described by a viscous power law relationship. For this latter case, some of
the work done by the boundary loads is dissipated by frictional effects. Secondly,
stress values east of 95◦ E have decreased now that we are not pulling on the eastern
margin and a glance at Figure 3.11 reveals that leaving it unconstrained results in
E-W directed compression in China, in agreement with WSM data plotted in Figure
3.5. The model even predicts the orientation of compressional stresses around 20◦
2
For a power law as in Equation 3.1 the constant A has been determined experimentally. If
we take the law to be of the form ˙ = Aσ n instead, then A contains the dependence of strain
rate on temperature and pressure that was formerly described by the exponential factor. Since A
is no longer constant in this case and our models do not incorporate the effects of pressure and
temperature we were forced to calculate it with the observation that strain rates in areas of yield
is on the order of 10−15 s−1 . Inserting these two values into the power law gives us A ≈ 10−39
P a−3 · s−1 .
3.4. A COMPARISON WITH MEASUREMENTS
41
Figure 3.10: Stress intensity map for the model eurasia01-03-04 1 which obeys a power law
rheology with a stress exponent of three.
42
CHAPTER 3. THE MODEL OF EURASIA
Figure 3.11: Orientation of compressional principal stress for the model eurasia01-03-04 1.
3.4. A COMPARISON WITH MEASUREMENTS
Figure 3.12: Orientation of extensional principal stress for the model eurasia01-03-04 1.
43
44
CHAPTER 3. THE MODEL OF EURASIA
N/110◦ E, where they rotate south into the southeast Asian protrusion. Southeast
Asia itself has low stress levels in the new rheology, making it difficult to predict
compressional stress directions based on Figure 3.11 or 3.12. Nevertheless extensional stresses do run parallel to the Java trench as the WSM tells us it should.
Along much of the northern perimeter of Eurasia, the modification of the boundary conditions results in compressional stresses lying parallel to the plate boundary
although one would expect them to be perpendicular to the edge because of ridge
push. We have no explanation for this observation, however the fact that in Figure
3.11 it also happens in areas where no cratons are found nearby supports our assumption that they are not responsible for it. The effect appears to occur in regions
of low stress intensities where a change of boundary conditions (e.g. the higher
collisional forces in the Himalayas in the power-law model) might be able to rotate
stress directions.
In Figure 3.13 we plot the spatial variation of the values of the second invariant of
the strain rate tensor. Here power law rheology is responsible for a much shallower
gradient of strain rate values, with the consequence that, for instance, the area
bounded by the 5.8 × 106 s−16 contour (18.3 × 10−9 in the more common units of
yr−1 ) agrees well with the extent of similar values in Figure 3.9. It is interesting
to note that the southern edges of both cratons within Eurasia define the trend of
the 8.3 × 10−17 s−1 contour, supporting the notion that they are not as easy to
deform as normal continental lithosphere. The isolines of second invariant strain
rate are much closer together in Figure 3.8 than in Figure 3.13 because power laws
with higher exponents (of which perfect plasticity is just the most extreme version)
result in deformation which is more localized around the stress sources.
The introduction of new boundary conditions has improved the stress orientations
to such an extent that it is appears justified to claim that the power-law model agrees
well with the large scale features of the Eurasian stress field. It has also lead to a
realistic distribution of strain rates. The third comparison we can make is to test if
the deformation velocities can be matched. Unfortunately I am not able to present
a figure of the deformation fields of either the elastoplastic or the power-law model
due to technical problems encountered as this manuscript was nearing completion.
Instead Figure 3.14 depicts the velocity field during the last final 160000 years of
another model of the Eurasian plate (remember that the model is run for a total
of approximately 1.5 million years). The boundary conditions are identical to those
applied in the power-law model, but the rheology is such that it behaves elastically
up to the yield limit and, if it is exceeded, obeys a power law creep of order three.
In calculating the velocity field two important issues arise: (1) Is the velocity field
in a steady state3 and (2) in which reference frame should the field be plotted?
An inversion for a component of rigid rotation was performed for eight consecutive
periods in the second half of the computation time of the model. The pole of
3
If we are to interpret the velocity as a physically meaningful quantity arising from the applied
boundary constraints, the motion of the plate should not be accelerating. If this requirement is
satisfied the torques of the forces acting on our plate are balanced.
3.4. A COMPARISON WITH MEASUREMENTS
45
Figure 3.13: Values of the second invariant of the strain rate tensor for the model eurasia01-0304 1.
46
CHAPTER 3. THE MODEL OF EURASIA
1.2 mm/yr
0.5 mm/yr
0.5 mm/yr
0.4 mm/yr
0.1 mm/yr
0.7 mm/yr
0.2 mm/yr
0.08 mm/yr
0.5 mm/yr
0.4 mm/yr
0.3 mm/yr
0.3 mm/yr
Figure 3.14: Deformation velocity during the last 160000 years of the model eurasia02-02-04 1.
rotation varied over an area corresponding to 1/100000 of the area of the Eurasian
plate and the angular frequency strayed from its mean by at most four percent.
Having confirmed that our plate is in a steady state we filtered out the component
of rigid rotation for the time span of interest and chose a reference frame such that
a node between the East European platform and the Siberian craton is at rest.
The result obtained does not agree with the data from GPS and other measurements.
For example the magnitudes of the velocities are all too small by a factor of ten. Such
an error might be caused by too high coefficients of friction in the dashpots used
to simulate basal drag forces and could be overlooked if the pattern of the velocity
field matched measurements. Yet this is not the case. Referring to Bird ’s [5] plate
boundary model we see that only the directions of velocities east of the Himalayas
are in rough agreement with observations. In Europe we predict motion to the west
whereas according to Bird it is towards the northeast, while in the Himalayas our
deformation velocities are lower by a factor of thirty than suggested by Kreemer et
al. [21].
Since we work with a simple two dimensional model, we cannot expect to repro-
3.4. A COMPARISON WITH MEASUREMENTS
47
duce the detailed properties of the stress, strain and velocity fields. The velocity
field depends strongly on the location of plate boundaries. Bird [5] proposes the
existence of several smaller plates within the area of our model, in view of which
it is not surprising that we fail to predict a small scale feature such as the motion
of the Aegean plate along the North Anatolian fault in either the velocity or the
stress field. Similarly accounting for topographical forces other than the continental
margin force would lead to local corrections of our generally satisfactory stress field.
Finally it should be mentioned that, as far as we are aware, none of the studies to
date has even attempted to match the whole set of stress, strain rate and velocity
data with a single model. Our efforts, even if carried out at a simple two-dimensional
level, have rewarded us with a model (presented in the Figures 3.10 to 3.13) that
makes satisfactory to good predictions with respect to large scale properties of stress
and strain rate within the Eurasian plate. It will be the subject of further research
to determine if the velocity field can also be made to agree with observations.
48
CHAPTER 3. THE MODEL OF EURASIA
THE CASE HAS, IN SOME RESPECTS, BEEN NOT ENTIRELY DEVOID OF INTEREST.
Sherlock Holmes, ’A Case of Identity’
Chapter 4
Conclusions
The main aim of the work presented in the preceding chapters was to determine how
different rheological models and lateral strength variations influence the characteristics of the lithospheric stress field. We limited our research to two dimensional finite
element models carried out in the program ABAQUS, with which the concepts of
viscoelasticity, plasticity and more complex material behaviour like power-law rheologies can be implemented. The Eurasian plate is the setting underlying our tests
and we ultimately propose a model for this area based on the experience gathered
in geometrically simple rectangles.
The rectangular models of Chapter 2 allowed us to first investigate these issues
on a qualitative basis, without obliging us to match the details of the stress field in
Eurasia. We found that thicker regions of lithosphere reduce stress levels within the
structure and that, at the edges of such regions, stress orientations can change. Such
stress bending, however, occurs only if the structure is situated in an area where
different sources of stress (e.g. boundary conditions along different edge segments
of the plate) contribute roughly equally to the local stress field. Regions of thinner
lithosphere than their surroundings display higher stress intensities and they also
tend to bend stress directions, but less so than thicker lithosphere. Furthermore
there is no evidence that thicker regions such as cratons act as stress barriers in
that they shield areas on opposite sides of the structure from the influence of one
another.
While cratons do not decouple different parts of the lithosphere, introducing plasticity by imposing a yield strength to the lithosphere does. Stress patterns for purely
elastic and viscoelastic rheologies are generally very similar in that the stress pattern is dominated by the largest boundary load (e.g., the collision with India in
Eurasia) in both cases. Only in an elastoplastic rheology can weaker stress sources
(an example of which is the collision with Africa) make a noticeable contribution to
the stress orientations in the regions where they act.
With respect to boundary conditions we maintain that dynamic boundary conditions
49
50
CHAPTER 4. CONCLUSIONS
are the better approach for models of the lithosphere than kinematic constraints.
The latter are based on measurements of relative plate motions which are instantaneous quantities and probably also only are valid at the surface of the Earth.
Furthermore, the relative motion of the plates contains no information of the processes involved at the interfaces of tectonic plates. Concerning dynamic boundary
conditions we believe that collisional forces are significantly higher than proposed
by previous studies which calculate the relative magnitudes of boundary forces by
assuming that their torques are balanced. In keeping with this we suggest that the
forces contributing to the torque balance are probably the yield forces acting on the
interior of a plate at the boundary of an area of yield. While this distinction may
not be crucial to global models seeking to explain large scale dynamics, it is likely
to be important when it comes to setting up detailed models of individual plates or
specific regions within them.
Models employing the real shape of the Eurasian plate entail a higher level of
sophistication. The prediction of a stress field that agrees well with observations necessitates a quantitative understanding of the boundary forces. Based on rheological
and dynamic arguments we developed scaling relations between relative plate velocities and the corresponding forces in collisional and subduction zones. Throughout
most of the Eurasian plate the application of the boundary constraints according to
our assumptions leads to satisfactory results. Contrary to our expectations, stress
orientations in the Far East are at odds with observations if we apply trench suction.
Leaving the eastern margin of Eurasia unconstrained results in a good match with
World Stress Map data for the region and probably indicates that collisional resistance along the interface of the overriding and the subducting plate approximately
cancels trench suction.
Models of the Eurasian plate were run using either (1) a purely elastic plate in which
the material deforms without resistance once the plastic yield limit is exceeded, or
(2) a third order viscous power law rheology. While stress orientations are generally insensitive to the different material behaviour if identical boundary constraints
are applied, stress levels, and consequently the extent of lithospheric yielding, do
change. Due to frictional dissipation, which drains the system of some of the energy furnished by the boundary loads, stress levels remain lower in the interior of
the model described by the power law rheology. At the same time, prescribing a
power law behaviour seems to remove a tendency of purely elastoplastic lithosphere
to deform too easily under shear (e.g., oblique boundary loads), resulting in a more
realistic prediction of the size of orogens.
The analysis of our rectangular models provided us with strong evidence that the
implementation of plasticity is the prerequisite for a good stress model of Eurasia.
Interestingly, a power law rheology with appropriately chosen parameters seems
to have very similar effects on the outcome of the stress pattern and it thus also
appears to be suited for the task. To determine whether this is generally valid or
simply a result of the geometry of the Eurasian plate we will need to run additional
rectangular models employing the same viscoelastic power-law behaviour.
51
We believe that realistic models of continental lithosphere require the application of
some kind of nonlinear rheological behaviour. For models of the more rigid oceanic
lithosphere on the other hand, an elastic rheology might suffice.
Our best model can account for the observed directions of maximal horizontal
compression and for stress levels in the Eurasian plate. It also predicts a reasonable
distribution of strain rates, but is inadequate when it comes to forecasting the
deformation velocities. It is possible that a calculation of the velocity field for our
favoured model presented on pages 41 et sqq. will allow us to modify our judgement
in this respect. If this is not the case, the mismatch might be due to either a
conceptual or technical flaw in our extraction of the actual deformation from the
whole displacement. It might also be a hint that by rejecting an active role of
the mantle there may be regions where we have not applied the correct boundary
conditions at the base of the lithosphere.
52
CHAPTER 4. CONCLUSIONS
Appendix A
The contents of this appendix is based on the first two chapters of the ABAQUS
Theory Manual.
A.1 Theory
In structural analysis one is interested in the deformation of an initial configuration
throughout the history of loading. A material particle initially located at some position Xi (i=1, 2, 3) in space will move to a new position xi (“current configuration”):
if one assumes material cannot appear or disappear, there will be a one-to-one correspondence between Xi and xi , so it is always possible to write the history of a
particle’s location as
xi = xi (Xn , t)
(A.1)
and this relationship can be inverted. Two neighbouring particles, located at Xi
and at Xi + dXi in the initial configuration, must satisfy
dxi =
∂xi
· dXj = Fij · dXj
∂Xj
(A.2)
in the current configuration. The matrix Fij is called the deformation gradient
matrix.
i
, where the partial derivative with
The velocity of a material particle is vi = ∂x
∂t
respect to time t means the rate of change of the spatial position, xi , of a particular
particle. ABAQUS thus takes a Lagrangian viewpoint: it follows a material particle
through the motion, rather than looking at a fixed point in space and watching
the material flow through this point. The Lagrangian perspective makes it easy to
record and update the state of a material point since the mesh is embedded in the
53
54
APPENDIX A
material.
The velocity difference between two neighbouring particles in the current configuration is
dvi =
∂vi
· dxj = Lij · dxj
∂xj
(A.3)
where Lij is the velocity gradient. It is composed of a rate of deformation plus a
rate of rotation and can be split into a symmetric strain rate matrix
1
1
˙ij = (Lij + Lji ) =
2
2
T !
∂vi
∂vi
+
∂xj
∂xj
(A.4)
T !
∂vi
∂vi
−
.
∂xj
∂xj
(A.5)
and an antisymmetric rotation rate matrix
1
1
Ωij = (Lij − Lji ) =
2
2
Many of the problems to which ABAQUS is applied involve finding an approximate (finite element) solution for the displacements, deformations, stresses and
forces in a solid body subjected to some history of loading. The exact solution of
such a problem requires that both force and moment equilibria be maintained at all
times over any arbitrary volume of the body.
Let V denote the volume occupied by a part of the body in the current configuration and S be the surface bounding this volume. Force equilibrium for the volume
states that the integral of the surface tractions ti a over S and the integral of the
body forces fi over the volume be equal but opposite in magnitude. For the i-th
component this yields:
Z
Z
ti dS +
S
fi dV = 0.
(A.6)
V
The surface traction is related to the stress tensor σij by
ti = σij nj ,
(A.7)
where nj is the unit outward normal to S. Using this definition and Gauss’ theorem
to rewrite the surface integral as a volume integral we get
A.1. THEORY
55
Z V
∂σij
+ fi dV = 0.
∂xj
(A.8)
Since the volume is arbitrary, the integrand must vanish everywhere, thus providing three differential equations of translational equilibrium.
Moment equilibrium is most simply written by taking moments about the origin.
The vector product of Equation A.6 with xk is:
Z
Z
lki xk ti dS +
S
lki xk fi dV = 0.
(A.9)
V
With the definition of the stress tensor σij in A.7 and the help of Gauss’ theorem
one can prove that the stress tensor is symmetric:
σij = σji .
(A.10)
By chosing the stress matrix to be symmetric, moment equilibrium is satisfied automatically and one only needs to consider translational equilibrium when explicitly
writing the equilibrium equations.
ABAQUS approximates the equilibrium requirement by replacing it with a weaker
requirement, that equilibrium must be maintained in an average sense over a finite
number of divisions of the volume’s body. To develop such an approximation the
three equations represented by Equation A.8 are replaced by an equivalent “weak
form” - a single scalar equation over the entire body. It is obtained by multiplying
the pointwise differential equations by an arbitrary, vector-valued test function,
defined over the entire volume, and integrating. The test function can be imagined
to be a “virtual” velocity field, δvi , which is completely arbitrary except that it
must obey any prescribed kinematic constraints and have sufficient continuity: the
scalar product of this test function with the equilibrium force field then represents
the “virtual”1 work rate.
Taking the scalar product of the equation describing translational equilibrium with
δvi and integrating over the entire body gives
Z V
∂σij
+ fi · δvi dV = 0.
∂xj
(A.11)
From this expression we need to derive a basic equilibrium statement for the finite
element formulation that will be introduced in the next section (“Procedures”). The
1
Virtual quantities are infinitesimally small variations of physical measurements.
56
APPENDIX A
chain rule allows us to write
∂σij
∂δvi
∂
· (σij vi ) =
· δvi + σij ·
,
∂xj
∂xj
∂xj
(A.12)
so that
Z
V
∂σij
· δvi dV
∂xj
Z ∂
∂δvi
=
· (σij vi ) − σij ·
dV
∂xj
∂xj
ZV
Z
∂δvi
=
nj σij δvi dS −
σij ·
dV
∂xj
S
V
Z
Z
∂δvi
ti δvi dS −
σij ·
=
dV,
∂xj
S
V
where Gauss’ theorem and the definition of the stress tensor were applied in the
first and second equalities respectively. Thus, the virtual work statement, Equation
A.11, can be written
Z
Z
ti δvi dS +
S
Z
σij ·
fi δvi dV =
V
V
∂δvi
dV.
∂xj
(A.13)
i
is the virtual version δLij of the velocity gradient introduced
The quantitiy ∂δv
∂xj
in Equation A.3. It too may be expressed as the sum
δLij = δDij + δΩij ,
of its symmetric and antisymmetric parts
1
(δLij + δLji )
2
1
(δLij − δLji ).
=
2
δDij =
δΩij
With these definitions
σij δLij = σij δDij + σij δΩij ,
and since σij is symmetric,
A.1. THEORY
57
1
1
(σij δLij − σij δLji ) = (σij δLij − σji δLji ) = 0.
2
2
σij δΩij =
Thus, in its classical form, the virtual work statement that will be used for the
finite element analysis is
Z
Z
σij δDij dV =
V
Z
ti δvi dS +
S
fi δvi dV.
(A.14)
V
Recall that ti , fi , and σij are an equilibrium set:
ti = σij nj ,
∂σij
+ fi = 0,
∂xj
σij = σji ;
and that δDij and δvi are compatible:
1
δDij =
2
∂δvi ∂δvj
+
∂xj
∂xi
.
The virtual work statement has a simple physical interpretation: the rate of
work done by the external forces ti and fi subjected to any virtual velocity field
δvi is equal to the rate of work done by the equilibrium stresses σij , acting at the
rate of deformation δDij of the same virtual velocity field. The advantage of this
formulation is that it is cast in the form of an integral over the volume of the body:
it is possible to introduce approximations by choosing test functions for the virtual
velocity field that are not entirely arbitrary, but the variation of which is restricted
to a finite number of nodal values. This approach provides a stronger mathematical
basis for studying the approximation than the alternative of direct discretisation
of the derivative in the differential equation of equilibrium at a point, which is the
typical starting point for a finite difference approach to the same problem.
58
APPENDIX A
A.2 Procedures
In Equation A.14 the internal virtual work rate
Z
σij δDij dV
V
was expressed directly in terms of the current volume V. The elasticity of a
material is derivable from a thermodynamic potential written about a reference
state to which it returns upon unloading. Therefore, for isothermal deformations,
there will be a potential function for the elastic strain energy per unit of the natural
reference volume. The internal virtual work rate may be rewritten as an integral
over the natural reference volume:
Z
Z
J · σij δDij dV 0
σij δDij dV =
V0
V
where the Jacobian J = dV /dV 0 is the ratio of the volume of the material in the
current and the natural configurations. It is then convenient to define the stress
measure
τij = Jσij
(A.15)
as the work conjugate to the strain measure, the rate of which is the rate of deformation, Dij . Employing the natural reference volume V 0 , Equation A.14 becomes
Z
0
Z
τij δ ˙ij dV =
V0
Z
ti δvi dS +
S
fi δvi dV,
(A.16)
V
where τij and ij are any conjugate pairs of material stress and strain measures.
In a first discretisation step a finite element interpolator is introduced, the ith component of which is:
ui =
X
NiN uN ,
N nodes
where the uN are nodal variables (e.g. displacement or temperature), the NiN are
interpolation functions that depend on some material coordinate system and the
A.2. PROCEDURES
59
summation is carried out over all nodes in the finite element mesh. For example,
the vector ~u might represent the displacement field of the solid, in which case the
individual values of uN would be the magnitudes of the displacement at the different
nodes. The contribution of each node (in direction and amplitude) to the total
displacement field is determined by the NiN , which can be regarded as directional
weighting functions2 .
The virtual field, δvi , must be compatible with all kinematic constraints. Introducing
the above interpolation implies that it must have an identical spatial form:
X
δvi =
NiN δv N .
N nodes
Now δ ˙ij is the virtual rate of material strain associated with δvi and, because it
is a rate form, it must be linear in δvi . Hence, the interpolation assumption gives3
X
δ ˙ij =
βijN δv N ,
N nodes
where βijN = βijN (xk , NlN ) is a matrix that depends, in general, on the current
position xi and the interpolation functions NiN . The equilibrium equation A.16 is
approximated as
δv
N
Z
V0
βijN τij
0
dV = δv
N
Z
ti NiN
Z
dS +
S
fi NiN
dV .
V
Since the δv N are independent variables, each one can be chosen to be nonzero
and all others zero in turn, to arrive at a system of nonlinear equilibrium equations:
Z
V0
βijN τij
0
Z
dV =
S
ti NiN
Z
dS +
fi NiN dV.
(A.17)
V
This system of equations forms the basis for the finite element analysis procedure.
However, for the Newton algorithm used in ABAQUS/Standard, one needs to know
2
Consider the following example which shows that the contributions of the different nodes must
be weighted by the NiN : let a circle be approximated by an octagon and each node be displaced by
the same magnitude and in the same direction. The octagon as a whole moves by the same distance
as the individual nodes, so one cannot simply sum all nodal displacements but must divide the
sum by eight to obtain the correct displacement. By doing so one effectively introduces a weighting
factor of 18 for each node.
3
Only the sums over the nodal variables are written out explicitly. For the subscripts Einstein’s
summing convention is applied.
60
APPENDIX A
the Jacobian of the finite element equilibrium equations. It can be developed by
taking the variation of Equation A.16, giving
Z
Z
Z
0
dti δvi dS −
(dτij δ ˙ij + τij dδ ˙ij ) dV −
V0
S
Z
S
Z
−
dfi δvi dV −
V
ti δvi dAr
fi δvi dJ
V
1
dV = 0,
J
1
dS
Ar
(A.18)
where d( ) represents the linear variation of the quantity ( ) with respect to the
basic variables (the degrees of freedom of the finite element model). In the above
expression J = |dV /dV 0 | is the volume change between the reference and the current
volume occupied by a piece of the solid and, likewise, Ar = |dS/dS 0 | is the surface
ratio between the reference and the current configuration. The Jacobian is obtained
by allowing only variations of the nodal variables uN in Equation A.18, and after a
lengthy calculation one obtains
K
MN
Z
=
V0
Z
−
N
βijM Hjk βki
NiM QN
S,i
0
Z
dV +
τij
V0
Z
dS −
∂βijM
dV 0
N
∂u
NiM QN
V,i dV.
(A.19)
V
S
Here the two quantities
QN
S,i =
∂ti
1 ∂Ar
+ ti
N
∂u
Ar ∂uN
QN
V,i =
1 ∂J
∂fi
+ fi
N
∂u
J ∂uN
stand for the variation of the load vectors with nodal variables, and based on
mechanical constitutive theory it is assumed that dτij can be expressed as
dτij = Hik dkj + gij ,
where the matrices Hij and gij depend on the properties of the material being
loaded.
ABAQUS/Standard uses the Newton incremental method for solving the nonlinear equilibrium equations A.17 and A.19. They can symbolically be written as
A.2. PROCEDURES
61
F N (uM ) = 0,
(A.20)
where F N is the force component conjugate to the N th variable in the problem
and uM is the value of the M th nodal variable.
The basic idea of Newton’s method is the following. Let uM
n be an approximation
th
to the solution of Equation A.20 reached after the n iteration and let cM
n+1 denote
the difference between this solution and the exact solution, meaning that
M
F N (uM
n + cn+1 ) = 0.
The left hand side of this equation can be expanded in a Taylor series about
M
the incremental solution uM
n . If un is a good approximation of the solution, the
magnitude of each cM
n+1 will be small, and it is thus possible to neglect terms of
second and higher orders in the series expansion, such that:
F N (uM
n )+
X ∂F N
P
(uM
n ) · cn+1 = 0.
P
∂u
P nodes
One thus is left with a linear system of equations
cPn+1
=
X n
N P −1 N M o
− Kn
Fn (un ) ,
(A.21)
P nodes
where KnN P =
solution is then
∂F N
∂uP
is the Jacobian matrix. The next approximation to the
M
M
uM
n+1 = un + cn+1
and the iteration continues until the solution to Equation A.20 converges.
62
APPENDIX A
Appendix B
Appendix B gives an overview of all relevant input parameters for each of the models
mentioned in this thesis. The models are listed in the order in which they appear
in the text. (E stands for Young’s modulus, ν for the Poisson ratio and σY for the
plastic yield limit).
kin23-02-04 1 (Fig. 2.3, p. 11):
1. Rectangle length: 16000 km (60 elements)
2. Rectangle width: 8000 km (30 elements)
3. Area of ABAQUS-element: 7.1 × 1010 m2
4. Computation time: 1 × 1012 s (approx. 32000 years)
5. Minimal time step: 1 × 105 s
6. Maximal time step: 1 × 1011 s
7. Rheology: purely elastic with E = 1 × 1011 Pa & ν = 0.3
8. Lithospheric thickness (uniform): hL = 60 km
9. Viscosity of the upper mantle (at 100 km depth, the average thickness of
oceanic, continental and cratonic lithosphere): ηCON T = 4 × 1019 P a · s
10. Coefficient of friction in dashpots: kCON T = 9.86 × 1025 kg/s
11. Dynamic boundary constraints:
(a) FRP along atlantic continental margin: 1.34 × 1018 N (line force for 100
million year old oceanic crust: 4.841 × 1012 N/m)
(b) FRP along arctic continental margin: 6.31 × 1017 N (line force for 50
million year old oceanic crust: 2.421 × 1012 N/m)
63
64
Appendix B
(c) FCM opposing ridge push: 1012 N/m
12. Kinematic boundary constraints: relative velocities of neighbouring plates
from Siberia to Gibraltar
dsm elastic (Fig. 2.4, p. 12):
1. Rectangle length: 16000 km (60 elements)
2. Rectangle width: 8000 km (30 elements)
3. Area of ABAQUS-element: 7.1 × 1010 m2
4. Computation time: 1 × 1012 s (approx. 32000 years)
5. Minimal time step: 1 × 105 s
6. Maximal time step: 1 × 1011 s
7. Rheology: purely elastic with E = 1 × 1011 Pa & ν = 0.3
8. Lithospheric thickness (uniform): hL = 60 km
9. Viscosity of the upper mantle (at 100 km depth, the average thickness of
oceanic, continental and cratonic lithosphere): ηCON T = 4 × 1019 P a · s
10. Coefficient of friction in dashpots: kCON T = 9.86 × 1025 kg/s
11. Dynamic boundary constraints:
(a) FRP along atlantic continental margin: 1.34 × 1018 N (line force for 100
million year old oceanic crust: 4.841 × 1012 N/m)
(b) FRP along arctic continental margin: 6.31 × 1017 N (line force for 50
million year old oceanic crust: 2.421 × 1012 N/m)
(c) FCM opposing ridge push: 1012 N/m
(d) FCC due to India: 9.08 × 1017 N (line force: 2.5 × 1012 N/m)
(e) FCC due to Arabia: 3.55 × 1017 N (line force: 1.67 × 1012 N/m)
(f) FCC due to Africa: 1.74 × 1017 N (line force: 0.56 × 1012 N/m)
(g) FSU line force: 3 × 1012 N/m
65
dsm elasticrat (Fig. 2.6, p. 14):
1. Rectangle length: 16000 km (60 elements)
2. Rectangle width: 8000 km (30 elements)
3. Area of ABAQUS-element: 7.1 × 1010 m2
4. Computation time: 1 × 1012 s (approx. 32000 years)
5. Minimal time step: 1 × 105 s
6. Maximal time step: 1 × 1011 s
7. East European platform and Siberian craton
8. Rheology (identical for cratons and continental lithosphere): purely elastic
with E = 1 × 1011 Pa & ν = 0.3
9. Thickness of continental lithosphere: hL = 60 km
10. Thickness of lithosphere within cratons: hCRAT = 150 km
11. Viscosity of the upper mantle (at 100 km depth, the average thickness of
oceanic, continental and cratonic lithosphere): ηCON T = 4 × 1019 P a · s
12. Coefficient of friction in dashpots: kCON T = 9.86 × 1025 kg/s
13. Dynamic boundary constraints:
(a) FRP along atlantic continental margin: 1.34 × 1018 N (line force for 100
million year old oceanic crust: 4.841 × 1012 N/m)
(b) FRP along arctic continental margin: 6.31 × 1017 N (line force for 50
million year old oceanic crust: 2.421 × 1012 N/m)
(c) FCM opposing ridge push: 1012 N/m
(d) FCC due to India: 9.08 × 1017 N (line force: 2.5 × 1012 N/m)
(e) FCC due to Arabia: 3.55 × 1017 N (line force: 1.67 × 1012 N/m)
(f) FCC due to Africa: 1.74 × 1017 N (line force: 0.56 × 1012 N/m)
(g) FSU line force: 3 × 1012 N/m
dsm elastohypocrat1 (Fig. 2.7, p. 15):
1. Rectangle length: 16000 km (60 elements)
66
Appendix B
2. Rectangle width: 8000 km (30 elements)
3. Area of ABAQUS-element: 7.1 × 1010 m2
4. Computation time: 1 × 1012 s (approx. 32000 years)
5. Minimal time step: 1 × 105 s
6. Maximal time step: 1 × 1011 s
7. Craton in the region dominated by the subduction zones’ stress field
8. Rheology (identical for cratons and continental lithosphere): purely elastic
with E = 1 × 1011 Pa & ν = 0.3
9. Thickness of continental lithosphere: hL = 60 km
10. Thickness of lithosphere within cratons: hCRAT = 150 km
11. Viscosity of the upper mantle (at 100 km depth, the average thickness of
oceanic, continental and cratonic lithosphere): ηCON T = 4 × 1019 P a · s
12. Coefficient of friction in dashpots: kCON T = 9.86 × 1025 kg/s
13. Dynamic boundary constraints:
(a) FRP along atlantic continental margin: 1.34 × 1018 N (line force for 100
million year old oceanic crust: 4.841 × 1012 N/m)
(b) FRP along arctic continental margin: 6.31 × 1017 N (line force for 50
million year old oceanic crust: 2.421 × 1012 N/m)
(c) FCM opposing ridge push: 1012 N/m
(d) FCC due to India: 9.08 × 1017 N (line force: 2.5 × 1012 N/m)
(e) FCC due to Arabia: 3.55 × 1017 N (line force: 1.67 × 1012 N/m)
(f) FCC due to Africa: 1.74 × 1017 N (line force: 0.56 × 1012 N/m)
(g) FSU line force: 3 × 1012 N/m
dsm elastithincrat (Fig. 2.8, p. 16):
1. Rectangle length: 16000 km (60 elements)
2. Rectangle width: 8000 km (30 elements)
3. Area of ABAQUS-element: 7.1 × 1010 m2
67
4. Computation time: 1 × 1012 s (approx. 32000 years)
5. Minimal time step: 1 × 105 s
6. Maximal time step: 1 × 1011 s
7. East European platform and Siberian craton with thinner lithosphere
8. Rheology (identical for cratons and continental lithosphere): purely elastic
with E = 1 × 1011 Pa & ν = 0.3
9. Thickness of continental lithosphere: hL = 100 km
10. Thickness of thinned lithosphere: hCRAT = 50 km
11. Viscosity of the upper mantle (at 100 km depth, the average thickness of
oceanic, continental and cratonic lithosphere): ηCON T = 4 × 1019 P a · s
12. Coefficient of friction in dashpots: kCON T = 9.86 × 1025 kg/s
13. Dynamic boundary constraints:
(a) FRP along atlantic continental margin: 1.34 × 1018 N (line force for 100
million year old oceanic crust: 4.841 × 1012 N/m)
(b) FRP along arctic continental margin: 6.31 × 1017 N (line force for 50
million year old oceanic crust: 2.421 × 1012 N/m)
(c) FCM opposing ridge push: 1012 N/m
(d) FCC due to India: 9.08 × 1017 N (line force: 2.5 × 1012 N/m)
(e) FCC due to Arabia: 3.55 × 1017 N (line force: 1.67 × 1012 N/m)
(f) FCC due to Africa: 1.74 × 1017 N (line force: 0.56 × 1012 N/m)
(g) FSU line force: 3 × 1012 N/m
dsm creepcrat (Fig. 2.9, p. 17):
1. Rectangle length: 16000 km (60 elements)
2. Rectangle width: 8000 km (30 elements)
3. Area of ABAQUS-element: 7.1 × 1010 m2
4. Computation time: 5 × 1013 s (approx. 1.5 million years)
5. Minimal time step: 1 × 105 s
68
Appendix B
6. Maximal time step: 1 × 1011 s
7. East European platform and Siberian craton
8. Rheology (identical for cratons and continental lithosphere): diffusion creep
until the yield stress of 50 MPa is reached, then power-law creep (dislocation
glide), E = 1 × 1011 Pa & ν = 0.3
9. Thickness of continental lithosphere: hL = 80 km
10. Thickness of lithosphere within cratons: hCRAT = 150 km
11. Viscosity of the upper mantle under continental lithosphere:
ηCON T = 4 × 1019 P a · s
12. Viscosity of the upper mantle beneath cratons: ηCRAT = 5 × 1020 P a · s
13. Coefficient of friction in continental dashpots: kCON T = 7.98 × 1025 kg/s
14. Coefficient of friction in dashpots under cratons: kCRAT = 1.31 × 1027 kg/s
15. Dynamic boundary constraints:
(a) FRP along atlantic continental margin: 1.34 × 1018 N (line force for 100
million year old oceanic crust: 4.841 × 1012 N/m)
(b) FRP along arctic continental margin: 6.31 × 1017 N (line force for 50
million year old oceanic crust: 2.421 × 1012 N/m)
(c) FCM opposing ridge push: 1012 N/m
(d) FCC due to India: 9.08 × 1017 N (line force: 2.5 × 1012 N/m)
(e) FCC due to Arabia: 3.55 × 1017 N (line force: 1.67 × 1012 N/m)
(f) FCC due to Africa: 1.74 × 1017 N (line force: 0.56 × 1012 N/m)
(g) FSU line force: 3 × 1012 N/m
dsm viscoelasticrat (Fig. 2.10, p. 19):
1. Rectangle length: 16000 km (60 elements)
2. Rectangle width: 8000 km (30 elements)
3. Area of ABAQUS-element: 7.1 × 1010 m2
4. Computation time: 5 × 1013 s (approx. 1.5 million years)
69
5. Minimal time step: 1 × 105 s
6. Maximal time step: 1 × 1011 s
7. East European platform and Siberian craton
8. Rheology (identical for cratons and continental lithosphere): viscoelastic with
E = 1 × 1011 Pa, ν = 0.3 & ηLIT H = 1023 P a · s
9. Thickness of continental lithosphere: hL = 60 km
10. Thickness of lithosphere within cratons: hCRAT = 150 km
11. Viscosity of the upper mantle (at 100 km depth, the average thickness of
oceanic, continental and cratonic lithosphere): ηCON T = 4 × 1019 P a · s
12. Coefficient of friction in dashpots: kCON T = 9.86 × 1025 kg/s
13. Dynamic boundary constraints:
(a) FRP along atlantic continental margin: 1.34 × 1018 N (line force for 100
million year old oceanic crust: 4.841 × 1012 N/m)
(b) FRP along arctic continental margin: 6.31 × 1017 N (line force for 50
million year old oceanic crust: 2.421 × 1012 N/m)
(c) FCM opposing ridge push: 1012 N/m
(d) FCC due to India: 9.08 × 1017 N (line force: 2.5 × 1012 N/m)
(e) FCC due to Arabia: 3.55 × 1017 N (line force: 1.67 × 1012 N/m)
(f) FCC due to Africa: 1.74 × 1017 N (line force: 0.56 × 1012 N/m)
(g) FSU line force: 3 × 1012 N/m
dsm elastoplasticrat (Fig. 2.11, p. 20):
1. Rectangle length: 16000 km (60 elements)
2. Rectangle width: 8000 km (30 elements)
3. Area of ABAQUS-element: 7.1 × 1010 m2
4. Computation time: 5 × 1013 s (approx. 1.5 million years)
5. Minimal time step: 1 × 105 s
6. Maximal time step: 1 × 1011 s
70
Appendix B
7. East European platform and Siberian craton
8. Rheology (identical for cratons and continental lithosphere): elastoplastic with
E = 1 × 1011 Pa & ν = 0.3 and the yield stress σY = 100 MPa
9. Thickness of continental lithosphere: hL = 60 km
10. Thickness of lithosphere within cratons: hCRAT = 150 km
11. Viscosity of the upper mantle (at 100 km depth, the average thickness of
oceanic, continental and cratonic lithosphere): ηCON T = 4 × 1019 P a · s
12. Coefficient of friction in dashpots: kCON T = 9.86 × 1025 kg/s
13. Dynamic boundary constraints:
(a) FRP along atlantic continental margin: 1.34 × 1018 N (line force for 100
million year old oceanic crust: 4.841 × 1012 N/m)
(b) FRP along arctic continental margin: 6.31 × 1017 N (line force for 50
million year old oceanic crust: 2.421 × 1012 N/m)
(c) FCM opposing ridge push: 1012 N/m
(d) FCC due to India: 9.08 × 1017 N (line force: 2.5 × 1012 N/m)
(e) FCC due to Arabia: 3.55 × 1017 N (line force: 1.67 × 1012 N/m)
(f) FCC due to Africa: 1.74 × 1017 N (line force: 0.56 × 1012 N/m)
(g) FSU line force: 3 × 1012 N/m
dsm elastoviscoplasticrat (Fig. 2.12, p. 21):
1. Rectangle length: 16000 km (60 elements)
2. Rectangle width: 8000 km (30 elements)
3. Area of ABAQUS-element: 7.1 × 1010 m2
4. Computation time: 5 × 1013 s (approx. 1.5 million years)
5. Minimal time step: 1 × 105 s
6. Maximal time step: 1 × 1011 s
7. East European platform and Siberian craton
71
8. Rheology (identical for cratons and continental lithosphere): viscoelastic with
a plastic yield stress of σY = 100 MPa (E = 1 × 1011 Pa, ν = 0.3 & ηLIT H =
1023 P a · s
9. Thickness of continental lithosphere: hL = 60 km
10. Thickness of lithosphere within cratons: hCRAT = 150 km
11. Viscosity of the upper mantle (at 100 km depth, the average thickness of
oceanic, continental and cratonic lithosphere): ηCON T = 4 × 1019 P a · s
12. Coefficient of friction in dashpots: kCON T = 9.86 × 1025 kg/s
13. Dynamic boundary constraints:
(a) FRP along atlantic continental margin: 1.34 × 1018 N (line force for 100
million year old oceanic crust: 4.841 × 1012 N/m)
(b) FRP along arctic continental margin: 6.31 × 1017 N (line force for 50
million year old oceanic crust: 2.421 × 1012 N/m)
(c) FCM opposing ridge push: 1012 N/m
(d) FCC due to India: 9.08 × 1017 N (line force: 2.5 × 1012 N/m)
(e) FCC due to Arabia: 3.55 × 1017 N (line force: 1.67 × 1012 N/m)
(f) FCC due to Africa: 1.74 × 1017 N (line force: 0.56 × 1012 N/m)
(g) FSU line force: 3 × 1012 N/m
ksm elastoplasticrat (Fig. 2.13, p. 22):
1. Rectangle length: 16000 km (60 elements)
2. Rectangle width: 8000 km (30 elements)
3. Area of ABAQUS-element: 7.1 × 1010 m2
4. Computation time: 5 × 1013 s (approx. 1.5 million years)
5. Minimal time step: 1 × 105 s
6. Maximal time step: 1 × 1011 s
7. East European platform and Siberian craton
8. Rheology (identical for cratons and continental lithosphere): elastoplastic with
E = 1 × 1011 Pa & ν = 0.3 and the yield stress σY = 100 MPa
72
Appendix B
9. Thickness of continental lithosphere: hL = 60 km
10. Thickness of lithosphere within cratons: hCRAT = 150 km
11. Viscosity of the upper mantle (at 100 km depth, the average thickness of
oceanic, continental and cratonic lithosphere): ηCON T = 4 × 1019 P a · s
12. Coefficient of friction in dashpots: kCON T = 9.86 × 1025 kg/s
13. Dynamic boundary constraints:
(a) FRP along atlantic continental margin: 1.34 × 1018 N (line force for 100
million year old oceanic crust: 4.841 × 1012 N/m)
(b) FRP along arctic continental margin: 6.31 × 1017 N (line force for 50
million year old oceanic crust: 2.421 × 1012 N/m)
(c) FCM opposing ridge push: 1012 N/m
14. Kinematic boundary constraints:
(a) average collisional velocity of the Indian plate: 1.43 × 10−9 m/s
(b) average collisional velocity of the Arabian plate: 9.51 × 10−10 m/s
(c) average collisional velocity of the African plate: 3.17 × 10−10 m/s
(d) velocity due to trench suction: 2.91 × 10−10 m/s
ydsm elastoplasticrat0 (Fig. 2.14, p. 23):
1. Rectangle length: 16000 km (60 elements)
2. Rectangle width: 8000 km (30 elements)
3. Area of ABAQUS-element: 7.1 × 1010 m2
4. Computation time: 5 × 1013 s (approx. 1.5 million years)
5. Minimal time step: 1 × 105 s
6. Maximal time step: 1 × 1011 s
7. East European platform and Siberian craton
8. Rheology (identical for cratons and continental lithosphere): elastoplastic with
E = 1 × 1011 Pa & ν = 0.3 and the yield stress σY = 100 MPa
9. Thickness of continental lithosphere: hL = 60 km
73
10. Thickness of lithosphere within cratons: hCRAT = 150 km
11. Viscosity of the upper mantle (at 100 km depth, the average thickness of
oceanic, continental and cratonic lithosphere): ηCON T = 4 × 1019 P a · s
12. Coefficient of friction in dashpots: kCON T = 9.86 × 1025 kg/s
13. Dynamic boundary constraints:
(a) FRP along atlantic continental margin: 1.34 × 1018 N (line force for 100
million year old oceanic crust: 4.841 × 1012 N/m)
(b) FRP along arctic continental margin: 6.31 × 1017 N (line force for 50
million year old oceanic crust: 2.421 × 1012 N/m)
(c) FCM opposing ridge push: 1012 N/m
(d) FCC due to India: 3 × 1018 N
(e) FCC due to Arabia: 2 × 1018 N
(f) FCC due to Africa: 1.5 × 1018 N
(g) FSU line force: 3 × 1012 N/m
eurasia27-01-04 6 (Figs. 3.2 to 3.5 & Fig. 3.8, p. 31 et sqq.):
1. No. of elemements in continental lithosphere: 3723
2. Area of ABAQUS-element: 1.719 × 1010 m2
3. Computation time: 5 × 1013 s (approx. 1.5 million years)
4. Minimal time step: 1 × 105 s
5. Maximal time step: 1 × 1011 s
6. East European platform and Siberian craton
7. Rheology (identical for cratons and continental lithosphere): elastoplastic with
E = 1 × 1011 Pa & ν = 0.25 and the yield stress σY = 100 MPa
8. Thickness of continental lithosphere: hL = 100 km
9. Thickness of lithosphere within cratons: hCRAT = 200 km
10. Thickness of lithosphere between cratons: hIN T = 150 km
11. Viscosity of the upper mantle under continental lithosphere:
ηCON T = 4 × 1019 P a · s
74
Appendix B
12. Viscosity of the upper mantle beneath cratons: ηCRAT = 5 × 1020 P a · s
13. Viscosity of the upper mantle at 150 km depth: ηIN T = 2.5 × 1020 P a · s
14. Coefficient of friction in continental dashpots: kCON T = 2.627 × 1025 kg/s
15. Coefficient of friction in dashpots under cratons: kCRAT = 3.695 × 1026 kg/s
16. Coefficient of friction in dashpots between cratons: kIN T = 1.580 × 1026 kg/s
17. Dynamic boundary constraints:
(a) ridge push field (DLOAD) along the atlantic & arctic continental margins
(line forces for 100 × 106 , 75 × 106 , 55 × 106 & 50 × 106 years old oceanic
lithosphere, averaged over the corresponding lithospheric thicknesses and
acting perpendicular to the margin)
(b) continental margin line forces opposing ridge push: 1012 N/m (also averaged over the thickness of the oceanic lithosphere)
(c) collisional forces scaled with velocity ranging from 2 × 1018 N (for a velocity of 5.25 cm/year) in the Eastern Himalayas to 5.48 × 1016 N (for a
velocity of 0.39 cm/year) at Gibraltar (CLOADs); the scaling follows the
rule (strainrate) ∝ (stress)3 , i.e. the stresses scale as the velocities to
the 3rd root
(d) collisional forces due to the North American plate in Siberia
(e) trench suction scaled with subduction velocity (taking the average velocity of 8.29 cm/year to correspond to a line force of 3 × 1012 N/m) and
averaged over the thickness of the continental lithosphere (DLOADs)
(f) trench suction in the Aegean Sea (DLOAD)
(g) free segments:
i.
ii.
iii.
iv.
the stretch between Gibraltar and the continental margin
the line linking Burma and Bhutan
the coast of Pakistan
part of the line linking Pakistan’s shore with Nepal
eurasia01-03-04 1 (Figs. 3.10 to 3.13, p. 41 et sqq.):
1. No. of elemements in continental lithosphere: 3723
2. Area of ABAQUS-element: 1.719 × 1010 m2
3. Computation time: 5 × 1013 s (approx. 1.5 million years)
75
4. Minimal time step: 1 × 105 s
5. Maximal time step: 1 × 1011 s
6. East European platform and Siberian craton
7. Rheology (identical for cratons and continental lithosphere): viscoelastic with
E = 1 × 1011 Pa & ν = 0.25 and a power law (n = 3 & A = 10−39 P a−3 s−1 )
describing viscosity
8. Thickness of continental lithosphere: hL = 100 km
9. Thickness of lithosphere within cratons: hCRAT = 200 km
10. Thickness of lithosphere between cratons: hIN T = 150 km
11. Viscosity of the upper mantle under continental lithosphere:
ηCON T = 4 × 1019 P a · s
12. Viscosity of the upper mantle beneath cratons: ηCRAT = 5 × 1020 P a · s
13. Viscosity of the upper mantle at 150 km depth: ηIN T = 2.5 × 1020 P a · s
14. Coefficient of friction in continental dashpots: kCON T = 2.627 × 1025 kg/s
15. Coefficient of friction in dashpots under cratons: kCRAT = 3.695 × 1026 kg/s
16. Coefficient of friction in dashpots between cratons: kIN T = 1.580 × 1026 kg/s
17. Dynamic boundary constraints:
(a) ridge push field (DLOAD) along the atlantic & arctic continental margins
(line forces for 100 × 106 , 75 × 106 , 55 × 106 & 50 × 106 years old oceanic
lithosphere, averaged over the corresponding lithospheric thicknesses and
acting perpendicular to the margin)
(b) continental margin line forces opposing ridge push: 1012 N/m (also averaged over the thickness of the oceanic lithosphere)
(c) collisional forces (CLOADs) scaled with current velocities along the borders with Africa and Arabia; the scaling follows the rule (strainrate) ∝
(stress)3 , i.e. the stresses scale as the velocities to the 3rd root
(d) collisional forces (CLOADs) scaled as above but corresponding to a relative velocity of 15 cm/yr in India
(e) free segments:
i. the stretch between Gibraltar and the continental margin
ii. the line linking Burma and Bhutan
iii. the coast of Pakistan
76
Appendix B
iv. part of the line linking Pakistan’s shore with Nepal
v. the boundary with North America running across Siberia
vi. the trenches in the Far East and the Aegean Sea
eurasia02-02-04 1 (Fig. 3.14, p. 46):
1. No. of elemements in continental lithosphere: 3723
2. Area of ABAQUS-element: 1.719 × 1010 m2
3. Computation time: 5 × 1013 s (approx. 1.5 million years)
4. Minimal time step: 1 × 105 s
5. Maximal time step: 1 × 1011 s
6. East European platform and Siberian craton
7. Rheology (identical for cratons and continental lithosphere): elastoplastic with
E = 1 × 1011 Pa & ν = 0.25 and the yield stress σY = 100 MPa, followed by
a power law (n = 3) for dislocation creep above σY
8. Thickness of continental lithosphere: hL = 100 km
9. Thickness of lithosphere within cratons: hCRAT = 200 km
10. Thickness of lithosphere between cratons: hIN T = 150 km
11. Viscosity of the upper mantle under continental lithosphere:
ηCON T = 4 × 1019 P a · s
12. Viscosity of the upper mantle beneath cratons: ηCRAT = 5 × 1020 P a · s
13. Viscosity of the upper mantle at 150 km depth: ηIN T = 2.5 × 1020 P a · s
14. Coefficient of friction in continental dashpots: kCON T = 2.627 × 1025 kg/s
15. Coefficient of friction in dashpots under cratons: kCRAT = 3.695 × 1026 kg/s
16. Coefficient of friction in dashpots between cratons: kIN T = 1.580 × 1026 kg/s
17. Dynamic boundary constraints:
(a) ridge push field (DLOAD) along the atlantic & arctic continental margins
(line forces for 100 × 106 , 75 × 106 , 55 × 106 & 50 × 106 years old oceanic
lithosphere, averaged over the corresponding lithospheric thicknesses and
acting perpendicular to the margin)
77
(b) continental margin line forces opposing ridge push: 1012 N/m (also averaged over the thickness of the oceanic lithosphere)
(c) collisional forces (CLOADs) scaled with current velocities along the borders with Africa and Arabia; the scaling follows the rule (strainrate) ∝
(stress)3 , i.e. the stresses scale as the velocities to the 3rd root
(d) collisional forces (CLOADs) scaled as above but corresponding to a relative velocity of 15 cm/yr in India
(e) free segments:
i.
ii.
iii.
iv.
v.
vi.
the stretch between Gibraltar and the continental margin
the line linking Burma and Bhutan
the coast of Pakistan
part of the line linking Pakistan’s shore with Nepal
the boundary with North America running across Siberia
the trenches in the Far East and the Aegean Sea
78
Appendix B
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