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

Contents 

Acknowledgements iii 

1 Introduction 1 

2 The rectangular models 5 

2.1 The ABAQUS model . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 

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 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

ehave 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 Lithgow- 

Bertelloni [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 

3

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 cratons 1 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 projection 1 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 × 10 10 m 2 each, giving 

the whole rectangle a total area of 1.28 × 10 8 km 2 . 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 

130˚W 

120˚W 

40˚N 

110˚W 

100˚W 

90˚W 

80˚W 

70˚W 

60˚W 

40˚N 

50˚W 

40˚W 

30˚N 

30˚W 

20˚W 

10˚W 

30˚N 

0˚ 

10˚E 

20˚E 

30˚N 

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 ABAQUS 2 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 elements 3 . 

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. 

30˚E 

20˚N 

40˚E 

10˚N 

0˚ 

10˚S

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 margin 5 (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-


Boundary force Forsyth & Uyeda Loohuis et al. Coblentz et al. 

FSU 

∼ 3 

2FRP 0.8 − 2.1 × 1012 FCC 

∼ 

- 

1 

2FRP 2.1 − 3.1 × 1012 2 × 1012 Table 2.1: Estimated values for the trench suction and continental collision forces FSU and FCC 

(in Nm −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 known 6 . 

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) Ridge push FRP [Nm −1 ] 

10 0.484×10 12 

20 0.968×10 12 

30 1.452×10 12 

40 1.937×10 12 

50 2.421×10 12 

60 2.905×10 12 

70 3.389×10 12 

80 3.873×10 12 

90 4.357×10 12 

100 4.841×10 12 

110 5.326×10 12 

120 5.810×10 12 

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 velocity 7 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 

h 

· (2 + 3 hL 

), (2.1) 

h 

where η ≈ 10 19 − 10 20 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.


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 × 10 22 P a · s × 0.94cm/yr = 3.17 × 10 12 Nm −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 intensity 8 (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 engi- 

neering): 

σν = 

where the σi are the principal stresses. 

1 

2 [(σ1 − σ2) 2 + (σ2 − σ3) 2 + (σ1 − σ3) 2 ], (2.3)

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


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


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


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


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


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.


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]) 

˙ɛij = ˙σij 

E 

σij 

+ , (2.4) 

2η 

where E is Young’s modulus and η is the viscosity 9 . Under constant strain the 

9 In an elastic (or Hooke) body stress is proportional to strain 

σij = Eɛij, (2.5) 

while in a linearly viscous (or Newton) body stress is proportional to the strain rate: 

σij = 2η ˙ɛij. (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 = 10 23 P a · s. 

solution of equation 2.4 becomes 

σij = σij,init · exp 

 

− E 

 

· t , (2.7) 

2η 

where σij,init is the initial stress. Equation 2.7 defines the so called Maxwell time 

τM = η 

, above which the viscous effects in a Maxwell body begin to dominate the 

E 

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


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


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


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


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


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 × 10 18 N, 2 × 10 18 N and 

3 × 10 18 N, for the African, Arabian and Indian collision, respectively, were applied. 

Atlantic ridge push on the other hand has a magnitude of 1.34 × 10 18 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 

EEP 

INT 

SC 

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 × 10 10 m 2 . 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 69×10 6 km 2 of which 51×10 6 km 2 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 collision 1 . 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).


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 × 10 5 km 2 (Himalayas and Pamir), 7.2 × 10 5 

km 2 (Zagros) and 1.2 × 10 5 km 2 (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]): 

˙ɛ = Aτ n exp(−Q/RT ), (3.1) 

where ˙ɛ 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): 

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 × 10 12 Nm −1 and 6 × 10 12 Nm −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]): 

˙ɛ ∝ exp(σ) or ˙ɛ ∝ 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: 

˙ɛ = ˙ɛY + A · (σ − σY ) n , (3.4) 

where ˙ɛY 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


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˙ɛ 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.


Figure 3.3: Orientation of compressional principal stress for the model eurasia27-01-04 6.


Figure 3.4: Orientation of extensional principal stress for the model eurasia27-01-04 6.


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


60˚ 

60˚ 

Method: 

focal mechanism 

breakouts 

80˚ 

80˚ 

100˚ 

drill. induced frac. 

60˚ 60˚ 

borehole slotter 

overcoring 

hydro. fractures 

geol. indicators 

Regime: 

NF SS TF U 

Quality: 

A 

B 

C 

! (2003) World Stress Map 

100˚ 

120˚ 

120˚ 

140˚ 

40˚ 40˚ 

20˚ 20˚ 

0˚ 0˚ 

World Stress Map Rel. 2003 

Heidelberg Academy of Sciences and Humanities 

Geophysical Institute, University of Karlsruhe 

Projection: Mercator 

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 

140˚


45˚ 

45˚ 

50˚ 

40˚ 40˚ 

35˚ 35˚ 

30˚ 30˚ 

Method: 


breakouts 



overcoring 



Regime: 

NF SS TF U 

Quality: 

25˚ A 

25˚ 

B 

C 





50˚ 

55˚ 

55˚ 

Figure 3.6: World Stress Map data for Iran. 

60˚ 

60˚ 


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, 

65˚ 

65˚


Method: 


breakouts 



overcoring 



Regime: 

NF SS TF U 

Quality: 

A 

B 

C 


0˚ 

60˚ 60˚ 

40˚ 40˚ 

0˚ 




20˚ 

20˚ 

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 

40˚ 

40˚


Figure 3.8: Strain rates in the Himalayas and the Zagros mountains for the model eurasia27-01- 

04 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


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 = ˙ɛ 2 12 + ˙ɛ 2 23 + ˙ɛ 2 31 − (˙ɛ11 ˙ɛ22 + ˙ɛ11 ˙ɛ33 + ˙ɛ22 ˙ɛ33) (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


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 three 2 . 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 ˙ɛ = 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 .


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.


Figure 3.11: Orientation of compressional principal stress for the model eurasia01-03-04 1.


Figure 3.12: Orientation of extensional principal stress for the model eurasia01-03-04 1.


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 × 10 6 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 state 3 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.


Figure 3.13: Values of the second invariant of the strain rate tensor for the model eurasia01-03- 

04 1.


0.4 mm/yr 

0.08 mm/yr 

0.1 mm/yr 

0.3 mm/yr 

0.4 mm/yr 

0.5 mm/yr 

0.3 mm/yr 

0.5 mm/yr 

0.2 mm/yr 

0.5 mm/yr 

0.7 mm/yr 

1.2 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-


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.

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. 

51

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 

∂Xj 

· dXj = Fij · dXj 

(A.2) 

in the current configuration. The matrix Fij is called the deformation gradient 

matrix. 

The velocity of a material particle is vi = ∂xi , where the partial derivative with 

∂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 

∂xj 

· dxj = Lij · dxj 

(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 

˙ɛij = 1 

2 (Lij + Lji) = 1 

 

T 

∂vi ∂vi 

+ 

2 ∂xj ∂xj 

and an antisymmetric rotation rate matrix 

(A.4) 

Ωij = 1 

2 (Lij − Lji) = 1 

 

T 

∂vi ∂vi 

− . (A.5) 

2 ∂xj ∂xj 

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: 

 

S 

 

ti dS + 

The surface traction is related to the stress tensor σij by 

V 

fi dV = 0. (A.6) 

ti = σijnj, (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 

 

V 

∂σij 

∂xj 

 

+ fi dV = 0. (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: 

 

S 

 

ɛlkixkti dS + 

V 

ɛlkixkfi dV = 0. (A.9) 

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 

 

V 

∂σij 

∂xj 

 

+ fi · δvi dV = 0. (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 

so that 

 

V 

∂σij 

∂xj 

∂ 

∂xj 

· (σijvi) = ∂σij 

· δvi dV = 

= 

= 

 

 

 

V 

S 

S 

∂xj 

· δvi + σij · ∂δvi 

, (A.12) 

∂xj 

 

∂ 

· (σijvi) − σij · 

∂xj 

∂δvi 

 

dV 

∂xj 

 

njσijδvi dS − σij · 

V 

∂δvi 

dV 

∂xj 

 

tiδvi dS − σij · ∂δvi 

dV, 

∂xj 

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 

 

S 

 

tiδvi dS + 

V 

 

fiδvi dV = 

V 

V 

σij · ∂δvi 

∂xj 

dV. (A.13) 

The quantitiy ∂δvi is the virtual version δLij of the velocity gradient introduced 

∂xj 

in Equation A.3. It too may be expressed as the sum 

δLij = δDij + δΩij, 

of its symmetric and antisymmetric parts 

With these definitions 

and since σij is symmetric, 

δDij = 1 

2 (δLij + δLji) 

δΩij = 1 

2 (δLij − δLji). 

σijδLij = σijδDij + σijδΩij,

A.1. THEORY 57 

σijδΩij = 1 

2 (σijδLij − σijδLji) = 1 

2 (σijδLij − σjiδLji) = 0. 

Thus, in its classical form, the virtual work statement that will be used for the 

finite element analysis is 

 

V 

 

σijδDij dV = 

S 

 

tiδvi dS + 

Recall that ti, fi, and σij are an equilibrium set: 

ti = σijnj, 

∂σij 

∂xj 

and that δDij and δvi are compatible: 

+ fi = 0, σij = σji; 

δDij = 1 

 

∂δvi 

+ 

2 ∂xj 

∂δvj 

 

. 

∂xi 

V 

fiδvi dV. (A.14) 

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 

 

σijδDij 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: 

 

V 

 

σijδDij dV = 

V 0 

J · σijδDij dV 0 

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 

 

V 0 

τijδ ˙ɛij dV 0 = 

 

S 

 

tiδvi dS + 

V 

fiδvi dV, (A.16) 

where τij and ɛij are any conjugate pairs of material stress and strain measures. 

In a first discretisation step a finite element interpolator is introduced, the i th component 

of which is: 

ui = 

N nodes 

N N i u N , 

where the u N are nodal variables (e.g. displacement or temperature), the N N i 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 u N 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 N N i , which can be regarded as directional 

weighting functions 2 . 

The virtual field, δvi, must be compatible with all kinematic constraints. Introducing 

the above interpolation implies that it must have an identical spatial form: 

δvi = 

N nodes 

N N i δv N . 

Now δ ˙ɛij 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 gives 3 

δ ˙ɛij = 

N nodes 

β N ij δv N , 

) is a matrix that depends, in general, on the current 

where βN ij = βN ij (xk, N N l 

position xi and the interpolation functions N N i . The equilibrium equation A.16 is 

approximated as 

δv N 

 

V 0 

β N ij τij dV 0 = δv N 

 

tiN 

S 

N i dS + 

 

fiN 

V 

N i dV 

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: 

 

V 0 

β N ij τij dV 0 = 

 

tiN 

S 

N i dS + 

 

 

. 

fiN 

V 

N i dV. (A.17) 

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 N N i : 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 1 

8 for each node. 

3Only 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 

 

V 0 

(dτijδ ˙ɛij + τijdδ ˙ɛij) dV 0 

− 

S 

 

dtiδvi dS − 

 

 

− dfiδvi dV − fiδvidJ 

V 

V 

1 

J 

S 

tiδvidAr 

1 

Ar 

dS 

dV = 0, (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 u N in Equation A.18, and after a 

lengthy calculation one obtains 

K MN 

= 

Here the two quantities 

 

− 

V 0 

β M ij Hjkβ N ki dV 0 + 

S 

N M i Q N 

S,i dS − 

Q N S,i = ∂ti 

+ ti 

∂uN Q N V,i = ∂fi 

+ fi 

∂uN 

V 0 

∂β 

τij 

M ij 0 

dV 

∂uN N 

V 

M i Q N V,i dV. (A.19) 

1 

Ar 

∂Ar 

∂u N 

1 ∂J 

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 = Hikdɛkj + 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 (u M ) = 0, (A.20) 

where F N is the force component conjugate to the N th variable in the problem 

and u M is the value of the M th nodal variable. 

The basic idea of Newton’s method is the following. Let u M n be an approximation 

to the solution of Equation A.20 reached after the n th iteration and let c M n+1 denote 

the difference between this solution and the exact solution, meaning that 

F N (u M n + c M n+1) = 0. 

The left hand side of this equation can be expanded in a Taylor series about 

the incremental solution u M n . If u M n is a good approximation of the solution, the 

magnitude of each c M 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 (u M n ) + 

P nodes 

∂F N 

∂u P (uM n ) · c P n+1 = 0. 

One thus is left with a linear system of equations 

where KNP n = ∂F N 

solution is then 

c P n+1 = 

P nodes 

 

− K NP −1 N 

n Fn (u M 

n ) , (A.21) 

∂u P is the Jacobian matrix. The next approximation to the 

u M n+1 = u M n + c M n+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 × 10 10 m 2 

4. Computation time: 1 × 10 12 s (approx. 32000 years) 

5. Minimal time step: 1 × 10 5 s 

6. Maximal time step: 1 × 10 11 s 

7. Rheology: purely elastic with E = 1 × 10 11 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): ηCONT = 4 × 10 19 P a · s 

10. Coefficient of friction in dashpots: kCONT = 9.86 × 10 25 kg/s 

11. Dynamic boundary constraints: 

(a) FRP along atlantic continental margin: 1.34 × 10 18 N (line force for 100 

million year old oceanic crust: 4.841 × 10 12 N/m) 

(b) FRP along arctic continental margin: 6.31 × 10 17 N (line force for 50 


63

64 Appendix B 

(c) FCM opposing ridge push: 10 12 N/m 

12. Kinematic boundary constraints: relative velocities of neighbouring plates 

from Siberia to Gibraltar 

dsm elastic (Fig. 2.4, p. 12): 







7. Rheology: purely elastic with E = 1 × 10 11 Pa & ν = 0.3 

8. Lithospheric thickness (uniform): hL = 60 km 










(d) FCC due to India: 9.08 × 10 17 N (line force: 2.5 × 10 12 N/m) 

(e) FCC due to Arabia: 3.55 × 10 17 N (line force: 1.67 × 10 12 N/m) 

(f) FCC due to Africa: 1.74 × 10 17 N (line force: 0.56 × 10 12 N/m) 

(g) FSU line force: 3 × 10 12 N/m

dsm elasticrat (Fig. 2.6, p. 14): 







7. East European platform and Siberian craton 

8. Rheology (identical for cratons and continental lithosphere): purely elastic 

with E = 1 × 10 11 Pa & ν = 0.3 

9. Thickness of continental lithosphere: hL = 60 km 

10. Thickness of lithosphere within cratons: hCRAT = 150 km 













(g) FSU line force: 3 × 10 12 N/m 

dsm elastohypocrat1 (Fig. 2.7, p. 15): 


65

66 Appendix B 






7. Craton in the region dominated by the subduction zones’ stress field 


with E = 1 × 10 11 Pa & ν = 0.3 
















dsm elastithincrat (Fig. 2.8, p. 16): 



3. Area of ABAQUS-element: 7.1 × 10 10 m 2




7. East European platform and Siberian craton with thinner lithosphere 


with E = 1 × 10 11 Pa & ν = 0.3 


10. Thickness of thinned lithosphere: hCRAT = 50 km 














dsm creepcrat (Fig. 2.9, p. 17): 




4. Computation time: 5 × 10 13 s (approx. 1.5 million years) 


67

68 Appendix B 



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 × 10 11 Pa & ν = 0.3 



11. Viscosity of the upper mantle under continental lithosphere: 

ηCONT = 4 × 10 19 P a · s 

12. Viscosity of the upper mantle beneath cratons: ηCRAT = 5 × 10 20 P a · s 

13. Coefficient of friction in continental dashpots: kCONT = 7.98 × 10 25 kg/s 

14. Coefficient of friction in dashpots under cratons: kCRAT = 1.31 × 10 27 kg/s 











dsm viscoelasticrat (Fig. 2.10, p. 19): 




4. Computation time: 5 × 10 13 s (approx. 1.5 million years)




8. Rheology (identical for cratons and continental lithosphere): viscoelastic with 

E = 1 × 10 11 Pa, ν = 0.3 & ηLIT H = 10 23 P a · s 
















dsm elastoplasticrat (Fig. 2.11, p. 20): 







69

70 Appendix B 


8. Rheology (identical for cratons and continental lithosphere): elastoplastic with 

E = 1 × 10 11 Pa & ν = 0.3 and the yield stress σY = 100 MPa 
















dsm elastoviscoplasticrat (Fig. 2.12, p. 21): 







7. East European platform and Siberian craton


a plastic yield stress of σY = 100 MPa (E = 1 × 10 11 Pa, ν = 0.3 & ηLIT H = 

10 23 P a · s 
















ksm elastoplasticrat (Fig. 2.13, p. 22): 










71

72 Appendix B 












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): 










9. Thickness of continental lithosphere: hL = 60 km











(d) FCC due to India: 3 × 10 18 N 

(e) FCC due to Arabia: 2 × 10 18 N 

(f) FCC due to Africa: 1.5 × 10 18 N 


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 










10. Thickness of lithosphere between cratons: hINT = 150 km 


ηCONT = 4 × 10 19 P a · s 

73

74 Appendix B 


13. Viscosity of the upper mantle at 150 km depth: ηINT = 2.5 × 10 20 P a · s 



16. Coefficient of friction in dashpots between cratons: kINT = 1.580 × 10 26 kg/s 


(a) ridge push field (DLOAD) along the atlantic & arctic continental margins 

(line forces for 100 × 10 6 , 75 × 10 6 , 55 × 10 6 & 50 × 10 6 years old oceanic 

lithosphere, averaged over the corresponding lithospheric thicknesses and 

acting perpendicular to the margin) 

(b) continental margin line forces opposing ridge push: 10 12 N/m (also averaged 

over the thickness of the oceanic lithosphere) 

(c) collisional forces scaled with velocity ranging from 2 × 10 18 N (for a velocity 

of 5.25 cm/year) in the Eastern Himalayas to 5.48 × 10 16 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 × 10 12 N/m) and 

averaged over the thickness of the continental lithosphere (DLOADs) 

(f) trench suction in the Aegean Sea (DLOAD) 

(g) free segments: 

i. the stretch between Gibraltar and the continental margin 

ii. the line linking Burma and Bhutan 

iii. the coast of Pakistan 

iv. part of the line linking Pakistan’s shore with Nepal 

eurasia01-03-04 1 (Figs. 3.10 to 3.13, p. 41 et sqq.): 



3. Computation time: 5 × 10 13 s (approx. 1.5 million years)





E = 1 × 10 11 Pa & ν = 0.25 and a power law (n = 3 & A = 10 −39 P a −3 s −1 ) 

describing viscosity 





ηCONT = 4 × 10 19 P a · s 










acting perpendicular to the margin) 



(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: 




75

76 Appendix B 


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): 








E = 1 × 10 11 Pa & ν = 0.25 and the yield stress σY = 100 MPa, followed by 

a power law (n = 3) for dislocation creep above σY 





ηCONT = 4 × 10 19 P a · s 










acting perpendicular to the margin)



(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: 





v. the boundary with North America running across Siberia 

vi. the trenches in the Far East and the Aegean Sea 

77

78 Appendix B

Bibliography 

[1] E.A. Abbot. Flatland: a romance of many dimensions by A. Square. Number 

ISBN 0 14 04.3531 X. Penguin Books Ltd., 1884. 

[2] E.V. Artyushkov. Stresses in the lithosphere caused by crustal thickness inhomogeneities. 

J. Geophys. Res., 78:7675–7708, 1973. 

[3] D. Bercovici. The generation of plate tectonics from mantel convection. Earth 

Planet. Sci. Lett., 205:107–121, 2003. 

[4] P. Bird. Testing hypotheses on plate-driving mechanisms with global lithosphere 

models including topography, thermal structure and faults. J. Geophys. Res., 

103:10115–10129, 1998. 

[5] P. Bird. An updated digital model of plate boundaries. Geochem. Geophys. 

Geosyst., 4(3), 2003. 

[6] W.F. Brace and D.L. Kohlstedt. Limits on lithospheric stress imposed by laboratory 

experiments. J. Geophys. Res., 85:6248–6252, 1980. 

[7] D.D. Coblentz, M. Sandiford, R.M. Richardson, S. Zhou, and R. Hillis. The 

origins of the intraplate stress field in continental Australia. Earth Planet. Sci. 

Lett., 133:299–309, 1995. 

[8] C.P. Conrad and C. Lithgow-Bertelloni. How mantle slabs drive plate tectonics. 

Science, 298:207–209, 2002. 

[9] R.P. Crease. The prism and the pendulum: the ten most beautiful experiments 

in science. Number ISBN 1-4000-6131-8. Random House, 2003. 

[10] C. DeMets, R.G. Gordon, D.F. Argus, and S. Stein. Current plate motions. 

Geophys. J. Int., 101:425–478, 1990. 

[11] D. Forsyth and S. Uyeda. On the relative importance of the driving forces of 

plate motion. Geophys. J. R. astr. Soc., 43:163–200, 1975. 

[12] C. Gaina, W.R. Roest, and R.D. Müller. Late Cretaceous-Cenozoic deformation 

of northeast Asia. Earth Planet. Sci. Lett., 197:273–286, 2002. 

79

80 BIBLIOGRAPHY 

[13] The Global Seismic Hazard Assessment Program. Global Seismic Hazard Map. 

Available online at seismo.ethz.ch/GSHAP/, 1999. 

[14] S. Goes, J.J.P. Loohuis, M.J.R. Wortel, and R. Govers. The effect of plate 

stresses and shallow mantle temperatures on tectonics of northwestern Europe. 

Global Planet. Change, 27:23–38, 2000. 

[15] M. Gölke and D. Coblentz. Origins of the European regional stress field. 

Tectonophysics, 266:11–24, 1996. 

[16] G. Grünthal and D. Stromeyer. The recent crustal stress field in central Europe: 

trajectories and finite element modellling. J. Geophys. Res., 97:11805–11820, 

1992. 

[17] L. Han and M. Gurnis. How valid are dynamic models of subduction and 

convection when plate motions are prescribed? Phys. Earth Planet. Inter., 

110:235–246, 1999. 

[18] Hibbit, Karlsson, and Sorenson Inc. ABAQUS/Standard, version 6.3, 2002. 

[19] M.P. Hochstein and K. Regenauer-Lieb. Heat generation associated with the 

collision of two plates: The Himalya Geothermal Belt. J. Volc. Geotherm. Res., 

83:75–92, 1998. 

[20] S.H. Kirby and A.K. Kronenburg. Rheology of the lithosphere: selected topics. 

Rev. Geophys., 25:1219–1244, 1987. 

[21] C. Kreemer, W.E. Holt, and A.J. Haines. An integrated global model of presentday 

plate motions and plate boundary deformation. Geophys. J. Int., 154:8–34, 

2003. 

[22] C. Lithgow-Bertelloni and J.H. Guynn. Origin of the lithospheric stress field. 

J. Geophys. Res., 109:B014008, doi:10.1029/2003JB002467, 2004. 

[23] C. Lithgow-Bertelloni and M.A. Richards. The dynamics of Cenozoic and Mesozoic 

plate motions. Rev. Geophys., 36:27–78, 1998. 

[24] J.J.P. Loohuis, M.J.R. Wortel, and P.Th. Meijer. A first order stress model for 

the Eurasian plate with an assessment of the role of the Alpine collision zone. 

1998. 

[25] The MacNeal-Schwendler Corporation. MSC/Patran, version 7, 1997. 

[26] B. Müller, M.L. Zoback, K. Fuchs, L. Mastin, S. Gregersen, N. Pavoni, 

O. Stephansson, and C. Ljunggren. Regional patterns of tectonic stress in 

Europe. J. Geophys. Res., 97:11783–11803, 1992. 

[27] R.D. Müller, W.R. Roest, J.-Y. Royer, L.M. Gahagan, and J.G. Sclater. Digital 

isochrons of the world’s ocean floor. J. Geophys. Res., 102:3211–3214, 1997.

BIBLIOGRAPHY 81 

[28] K. Regenauer-Lieb and J.-P. Petit. Cutting of the European continental lithosphere: 

plasticity theory applied to the present Alpine collision. J. Geophys. 

Res., 102:7713–7748, 1997. 

[29] K. Regenauer-Lieb and D.A. Yuen. Modeling shear zones in geological and 

planetary sciences: solid- and fluid-thermal-mechanical approaches. Earth Sci. 

Rev., 63:295–349, 2003. 

[30] J. Reinecker, O. Heidbach, and B. Mueller. The 2003 release of the World Stress 

Map. Available online at www.world-stress-map.org, 2003. 

[31] Schlumberger. The Oilfield Glossary: Where the Oil Field Meets the Dictionary. 

Available online at www.glossary.oilfield.slb.com, 2004. 

[32] C.H. Scholz and J. Campos. On the mechanism of seismic decoupling and back 

arc spreading at subduction zones. J. Geophys. Res., 100:22103 – 22115, 1995. 

[33] D.L. Turcotte and G. Schubert. Geodynamics, applications of continuum 

physics to geological problems. Number ISBN 0-521-66186-2. Cambridge University 

Press, 2002. 

[34] A. Villaseñor, M.H. Ritzwoller, A.L. Levshin, M.P. Barmin, E.R. Engdahl, 

W. Spakman, and J. Trampert. Shear velocity structure of central Eurasia from 

inversion of surface wave velocities. Phys. Earth Planet. Sci. Int., 123:169–184, 

2001.

Dynamics of the Eurasian Plate

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