It has been assumed that the orientation of the maximum horizontal
compressive stress (SHmax) in the upper crust is governed on a
regional scale by the same forces that drive plate motion. However, several
regions are identified where stress orientation deviates from the expected
orientation due to plate boundary forces (first-order stress sources), or the
plate wide pattern. In some of these regions, a gradual rotation of the
SHmax orientation has been observed.
Several second- and third-order stress sources have been identified in the
past, which may explain stress rotation in the upper crust. For example, lateral heterogeneities in the crust, such as density and petrophysical
properties, and discontinuities, such as faults, are identified as potential
candidates to cause lateral stress rotations. To investigate several of these
candidates, generic geomechanical numerical models are set up with up to five
different units, oriented by an angle of 60∘ to the direction of
shortening. These units have variable (elastic) material properties, such as
Young's modulus, Poisson's ratio and density. In addition, the units can be
separated by contact surfaces that allow them to slide along these vertical
faults, depending on a chosen coefficient of friction.
The model results indicate that a density contrast or the variation of Poisson's ratio alone hardly rotates the horizontal stress
(≦17∘). Conversely, a contrast of Young's modulus allows
significant stress rotations of up to 78∘, even beyond the vicinity of
the material transition (>10km). Stress rotation clearly
decreases for the same stiffness contrast, when the units are separated by low-friction discontinuities (only 19∘ in contrast to 78∘). Low-friction discontinuities in homogeneous models do not change the stress
pattern at all away from the fault (>10km); the stress pattern is
nearly identical to a model without any active faults. This indicates that
material contrasts are capable of producing significant stress rotation for
larger areas in the crust. Active faults that separate such material
contrasts have the opposite effect – they tend to compensate for stress
rotations.
Introduction
Knowledge of the stress tensor state in the Earth's upper crust is important
for a better understanding of the endogenous dynamics, seismic hazard or
exploitation of the underground. Therefore, several methods have been
developed to estimate the stress tensor orientation and the stress magnitudes.
Stress orientation data are compiled globally in the World Stress Map database
.
Based on such data compilations, it was assumed that patterns of stress
orientation on a regional scale are more or less uniform within tectonic
plates .
The plate-wide pattern is overprinted on a regional scale by the contemporary
collisional systems. Recent examples in Europe are the Alps
, the Apennines or the
Carpathian Mountains . Closely related to that
is the variability of crustal thickness, density and topography
. It was
suggested that remnant stresses due to old plate tectonic events are able to
overprint stress orientation on a regional scale e.g.. Such old basement structures also present
geomechanical inhomogeneities and discontinuities, which have the potential to
perturb the stress pattern. However, pre-Cenozoic orogens (or “old” suture
zones), often covered and hidden by (thick) sediments, were rarely indicated
as causes of significant stress rotation. In many cases it is the opposite: old
orogens have apparently no impact on the present-day crustal stress pattern,
e.g. the Appalachian Mountains or Fennoscandia
. Deviations from the assumed uniform plate-wide stress
pattern (here called stress rotations) are observed recently in several
regions, such as in Australia, Germany and North America . However, these effects can only be
partly explained by the topography or lithospheric structures.
The complex stress pattern in central–western Europe was a subject of several
numerical investigations in recent decades . Apart from a recent 3-D model , the
previous models were limited to 2-D. These 2-D models were able to reproduce
some of the observed stress patterns by considering variable lateral elastic
material properties or discontinuities.
However, 2-D models have some limitations: they have to integrate topography,
crustal thickness and stiffness to one property, and they potentially
overestimate the horizontal stress magnitude .
Furthermore, none of these previous studies investigated the impact of the
influencing factors separately.
In this work, a series of large-scale 3-D generic geomechanical models is used
to determine which properties can cause significant stress rotations at a
distance (>10km) from material transitions or discontinuities.
The model geometry is inspired by the crustal structure and the stress pattern
in the German Central Uplands, where the SHmax orientation is 120
to 160∘. This is in contrast to a N–S orientation (∼0∘)
of SHmax to the north and to the south of the uplands
Fig. ,. The basement structures there
are striking 45 to 60∘, which is almost perpendicularly to the observed
SHmax orientation. The influence of the structures on the stress
field will be tested with a generic variation of Young's modulus, Poisson's ratio, the density and vertical low-friction discontinuities, which
separate the crustal blocks. Each property is tested separately first, to
avoid interdependencies; possible interactions are tested afterwards.
Stress rotation in the upper crustConcept of stress rotation
This study focuses on stress rotations that occur horizontally, i.e. in the
map view. A vertically uniform stress field is assumed, which is consistent
with previous studies . Stress
rotations with depth are occasionally observed within deep wells
, due to evaporites
e.g. or man-made activities in
the underground e.g..
On a map view, several potential sources of stress can superpose on another
and the resulting stress at a certain point comprises the sum of all stress
sources from those plate-wide to very local stress sources. Differences between the
resulting stress orientation and the regional stress source can be described
by the angular deviation , which can be substantial and can
lead to a change of the stress regime . The stress regime is defined
by the relative stress magnitudes, which are a normal faulting regime
(SV>SHmax>Shmin), strike slip regime
(SHmax>SV>Shmin) and thrust faulting regime
(SHmax>Shmin>SV), where SV is the
vertical stress and Shmin and SHmax are the minimum- and
the maximum horizontal stress, respectively. The difference between the
largest and smallest principal stress is the differential stress (σD=σ1-σ3), while the deviatoric stress is the difference between
the stress state and the mean stress δσij=σij-σm;
Comparison of selected previous observations or models on the subject of stress rotation in the context of faults, elastic material properties, density or topography variation. The characters “X” and “V” indicate whether the property is included or varied; “(X)” means that the subject is included indirectly. The characters “ < ” and “ > ” indicate that significant rotation occurs near (<10km) or at greater distance (>10km) from the fault or material transition.
PublicationModel (M) orDensity/Max. observedYoung'sPoisson'sFaultsSignificant rotationobservation (O)thicknessrotation [∘]modulusratio> or <10kmM–90XX–>M–∼25VV–>O–90––X<MV90––––M–90VXX>M–90VXX>M–90VX–>M–58VV–>MX∼45XX–>MX50XXX<M(X)90VXV>MX∼35–––>O–90––X<M(X)90VXV>O–50––X>
Stress rotation within this study means an angular deviation of the
SHmax orientation from the large-scale stress pattern. In the
following subsections, previous observations and models on the respective
causes are reviewed and also summarized in Table .
Density contrast and topography
Variability of density within the crust or lithosphere has a significant
impact on the stress state . showed that
local stress increases due to topography and crustal inhomogeneities are in the
order of tens of MPa, which is on the order of stresses resulting from the
plate boundary forces.
Gravitational forces are also derived by surface topography . Within mountains, SHmax is oriented parallel to the
ridge and perpendicular to the ridge at the base of the mountain chain. Along passive
continental margins, effects similar to those due to topography can be observed
.
investigated the interaction of different regional
deviatoric stress regimes (δσij) with stresses
arising from buoyancy forces (σG) and observed a rotation
of SHmax of up to 90∘. According to that,
SHmax rotates toward the normal trend of the density anomaly. If
regional stresses are large, compared to stresses driven by a density anomaly
(δσij/σG≫1), the influence
of a density anomaly is small and vice versa: if the regional stress is small
compared to the stress driven by the density anomaly (δσij/σG≪1), the impact of a density
anomaly on the resulting stress field is large. In the case that both stress
sources are on a similar level (δσij/σG≈1), small changes of one of the stress sources
are able to change the stress regime, and thus potentially the stress
orientation.
Stiffness contrast
Mechanical stiffness describes the material behaviour under the influence of
stress and strain. The focus here is on linear elastic material properties,
characterized by Young's modulus and Poisson's ratio. Stress
refraction between two elastic media can be calculated, but only at the
interface of the two media, based on the known stress state on one side of the
interface and Young's modulus on both sides . Stress
rotation due to stiffness contrast is for example reported for the Peace River Arch
in Alberta, Canada . Potential stress
rotation is supported by several numerical studies e.g..
Discontinuities
Discontinuities are planar structures within or between rock units, where the
shear strength is (significantly) lower than that of the surrounding rock.
Genetically, discontinuities can be classified into bedding, schistosity,
joints and fault planes. In the context of this study the term discontinuity
refers to fault planes or fault zones. Similar to the Earth surface, (nearly)
frictionless faults without cohesion act like a free surface in terms of
continuum mechanics . One of the three
principal stresses must be oriented perpendicular to the frictionless fault;
the two remaining ones are parallel to the discontinuity. For this reason,
the stress tensor rotates near a frictionless fault, depending on its
orientation. Significant stress rotation in the context of faults is reported
. However,
assumes that stress rotation occurs only within several
kilometres from the fault. Large differential stress leads to a more stable
stress pattern , whereas low differential
stresses allow a switch of the stress regime caused by faults. The impact of
faults on stress rotation has been investigated analytically
and by numerical models e.g..
Regional settingStress orientation in central Europe
Crustal stress data from Europe have been collected since the 1960s
e.g., later as part of the World
Stress Map database from and more recently by
.
SHmax orientation in western Europe is 145∘±26∘ and rotates clockwise by about 17∘ to
the direction of absolute plate motion from . This is in
agreement with , who obtained a better fit for relative plate
motion between Africa and Europe than for absolute plate motion. As the
major causes of the observed stress pattern in western and central Europe, the
ridge push of the Mid-Atlantic ridge and the collisional forces along the
southern plate margins are identified .
A fan-like stress pattern has been observed in the western Alps and Jura
mountains, where SHmax in front of the mountain chain is
perpendicular to the strike of the orogen (Fig. ).
assume that these structures only locally overprint the
general stress pattern. However, in light of the recently available data,
it is assumed that the SHmax orientation is rather controlled by
gravitational potential energy of the alpine topography than by plate boundary
forces .
The stress pattern in western and central Europe has been the subject of
several modelling attempts in the last three decades . In particular, these previous studies investigated the
impact of a lateral stiffness contrast in the crust , the
elastic thickness of the lithosphere , the stiffness
contrast of the mantle , a lateral density contrast or
topographic effects , the post-glacial
rebound in Scandinavia , and activity on faults
.
Stiffness variation in the lithosphere, e.g. in the Teisseyre–Tornquist Zone
(TTZ) or the Bohemian Massif (BM), has been identified as a potential cause
for the observed stress rotation in Central Europe . One example is the fan-shaped stress pattern in the North German
Basin (NGB), with a rotation of SHmax from north-west in the
western part to north-east in the eastern part of the basin as a product of
the TTZ, which is the boundary between the Phanerozoic Europe (Avalonia) and
the much stiffer Precambrian Eastern European Craton (Baltica).
came to the conclusion that active tectonic zones and
topography have major effects, whereas the stiffness contrast leads only to
minor effects. Lateral variation of density does not have a significant
impact on the stress pattern ; it causes only local effects.
Finally, low differential stress allows significant stress rotation
.
Basement structures in Germany
In large part, Germany consists of Variscan basement units, either exposed or
covered by Post-Paleozoic basin sediments. The Variscan orogen is a product of
the late-Paleozoic collision of the plates Gondwana and Avalonia (Laurussia)
in late Devonian to early Carboniferous time, which lead to closure of the
Rheic Ocean , and finally the formation of the
super-continent Pangaea. Despite the fact that the European Variscides are
well investigated in the last century and decades e.g., it is for example still a matter of
debate whether several microplates have been amalgamated in between or not.
Stress orientation in the German Central Uplands with the basement structural elements (separated by black lines), political boundaries (red) and major rivers (blue). Bars represent orientation of maximum horizontal compressional stress (SHmax); line length is proportional to quality. Colours indicate stress regimes, with red for normal faulting (NF), green for strike–slip faulting (SS), blue for thrust faulting (TF) and black for unknown regime (U). The Variscan basement structures introduced by are visualized; the regional segmentation is as follows:
BM = Bohemian Massif,
MGCH = Mid-German Crystalline High,
MZ = Moldanubian Zone,
NPZ = Northern Pyllite Zone,
RHZ = Rheno-Hercynian Zone,
STZ = Saxo-Thuringian Zone,
and VDF = Variscan Deformation Front.
Other structures are as follows:
MB = Molasses Basin;
NGB = North German Basin,
TS = Thor Suture,
TTZ = Teisseyre–Tornquist Zone
redrawn after.
published the structural zonation of the European
Variscides, which is still widely used (Fig. ). The parts
to the north-west of the Rheic Suture Zone are the Rheno-Hercynian Zone (RHZ)
with the sub-unit of the Northern Phylite Zone (NPZ), both of Laurussian
origin. South-east of the suture zone are the Mid-German Crystalline High
(MGCH), the Saxo-Thuringian Zone (STZ) and the Moldanubian Zone (MZ); all except the MGCH were exclusively part of Gondwana.
The RHZ is exposed in the Rhenish Massif, in the Harz
mountains and in the Flechtingen Hills. Dominant are Devonian to lower
Carboniferous clastic shelf sediments . These
low metamorphic slates, sandstones, greywacke and quartzite are supplemented
with continental and oceanic volcanic rocks, reef limestones and a few older
gneisses. Further to the north of the RHZ are the sub-Variscan foreland
deposits, consisting of clastic sediments and coal seams.
The NPZ is uncovered at the southern edge of the
low mountain ranges Hunsrück, Taunus and eastern Harz. Petrologically it
is probably the greenschist facies equivalent of the
Rheno-Hercynian shelf sequence , consisting of
meta-sediments and within-plate metavolcanic rocks .
The MGCH is open in the Palatinate Forest,
Odenwald, Spessart, Kyffhäuser, Ruhla Crystalline (Thuringian Forest)
and Flechtingen Hills. It has been interpreted previously as a magmatic arc of
the Saxo-Thuringian Zone. But assumes that the MGCH is
composed of both Saxo-Thuringian and Rheno-Hercynian rocks. Composition and
metamorphic grade vary considerably along the strike of the MGCH
. It consists of late-Paleozoic sediments, meta-sediments,
volcanic rocks, granitoides, gabbros, amphibolite and gneisses.
The Saxo-Thuringian Zone (STZ) is exposed in the
Thuringian-Vogtlandian Slate Mountains, Fichtel Mountains, Ore
Mountains, Saxonian Granulite Massif, Elbe Valley Slate Mountains and the
Lausitz. It consists of Campro-Ordovician mafic and felsic magmatic rocks and
late Ordovician to early Carboniferous marine and terrestrial sediments
. These rocks underwent metamorphic
overprint up to the early Carboniferous with different metamorphism stages up
to eclogite- or granulite facies. These units are interspersed by late- or
post-orogenic granites.
The MZ is exposed in the Bohemian Massif, the Bavarian
Forest, the Münchberg Gneiss Massif, the Black Forest and the Vosges.
They consist of mostly high-grade metamorphic crystalline rocks (gneisses,
granulite, migmatite) and Variscan granites .
Model set-upModel geometry
The chosen model geometry is inspired by the geometrical situation in the
German Central Uplands (Fig. ), but the overall intention
is a generic model. To make it easy to understand, compass directions are
used for the model description. The model geometry has a north–south extent of
400 and 300 km in the east–west direction, with a thickness of
30 km (Fig. ). In the centre of the model, three
diagonal units each with a width of 50 km are oriented 60∘
from the north. The unit boundaries are vertically incident. A model variant is
generated in which the unit boundaries allow free sliding, depending on a
chosen friction coefficient. For each of the three central units, different
material properties can be applied. The northernmost and southernmost block
has always the same (reference) material properties, except for the realistic
rock property scenario.
Reference model with the applied boundary conditions, used for all models, in map view and from the south.
The model has a lateral extent of 300km×400km and a thickness of 30 km.
It consists of five interconnected units, which have the same material properties.
Blue visualizes the reference material (Table ).
The boundary conditions ban motion in the x direction on the western side, in the y direction on the northern side and in the z direction at the model base.
A push of 400 m from the south and a pull of 60 m to the east is applied.
The resulting SHmax orientation (north–south) at a depth of 1000 m is illustrated by the black bars.
The red point (and line) indicates the location of the virtual well (Figs. and ).
The four diagonal boundaries can be used as vertical faults with a chosen friction coefficient.
Selection of common elastic rock properties (Young's modulus and Poisson's ratio) and density . Coloured vertical bars indicate applied material properties; see Table .
Solution of the equilibrium of forces
The stress orientations in the models are investigated using the finite-element method (FEM). The usage of 3-D FEM models to investigate the stress
state in the crust is a well-established technique e.g.. The major reason that
complex 2-D or 3-D models can be computed is the opportunity to use
unstructured meshes.
The method in general computes the equilibrium of stresses arising from
boundary forces (via displacement boundary conditions) and body forces
(gravity) acting on the rock whose mechanical behaviour is characterized by a
constitutive law and associated material parameters. The equilibrium of
forces is represented by partial differential equations, which are solved
numerically.
δσijδxj+ρxi=0,
where δσij is the variation of total stress,
δxj is the spatial change and ρxj represents the weight of the
rock section (ρ=density). Linear elastic material behaviour expressed by
Hooke's law is assumed. Two material properties, Young's modulus (E)
and Poisson's ratio (ν) are essential. The stress state in this study
will be calculated based on defined displacement boundary conditions
(Fig. ).
The lateral resolution of the model is about 3 km, consisting primarily
of hexahedrons and some wedge elements (degenerated hexahedrons). Resolution
into depth ranges from 0.44 km near the surface to about
3.4 km at the base of the model. In total, about 166 000 elements were
used. The model version with contact surfaces uses 1725 contact elements
along each contact surface. Model discretization was performed with
HyperMesh® v.2019. The equilibrium of forces (body forces
and boundary condition) is computed numerically using the
Abaqus®/Standard v.6.14-1 finite-element software.
Mechanical properties
The main subject of this study is to investigate the impact of the variation
of elastic rock properties, density and friction along faults on stress
orientation in the upper crust in the given geometrical setting outlined in
the previous sections (Fig. ). To do this, each parameter is
tested individually. Figure visualizes the range of density
(ρ), Young's modulus (E) and Poisson's ratio (ν) of representative
rocks, taken from a textbook .
The reference material for this investigation has a density of
ρ=2.7gcm-3, a Poisson's ratio of ν=0.25 and a Young's
modulus of E=50GPa. Such a material could represent for example
granite or limestone. Based on this reference material, a lower and
higher material value is always defined (Table ), which is within
the range of common rock properties (Fig. ). The material
with a low density (ρ=2.2gcm-3) may represent sediments
(sandstone, limestone, shale etc.), whereas the high-density material
(ρ=3.2gcm-3) could represent a rock from the lower crust or
the upper mantle. A low Poisson's ratio (ν=0.15) may represent sediments
(sandstone or shale), and a high Poisson's ratio (ν=0.35) could represent
ultramafic rocks. Soft material with a low Young's modulus
(E=10GPa) may represent sediments, pre-damaged rock or weathered rock.
Again ultramafic rock is an example of a stiff rock, having a large Young's
modulus (E=100GPa).
Young's modulus, Poisson's ratio and densities used in the models. Bold numbers indicate the properties used, which differ from those of the reference material.
Young'sPoisson'sNamemodulusratioDensity[GPa][–][gcm-3]Reference material (B)500.252.7Low density (g)500.252.2High density (G)500.253.2Low Poisson (p)500.152.7High Poisson (P)500.352.7Low stiffness (e)100.252.7High stiffness (E)1000.252.7Upper mantle1300.253.25
The stress ratio k (Eq. ) is plotted vs. depth. Stress in the reference model is marked with the bold green line. Additionally, several data and defined stress ratios from the literature are visualized for comparison .
Laboratory rock experiments in the past delivered friction coefficients of
about μ=0.6 to 0.85 . However, recent investigations
using realistic slip rates for earthquakes decreased estimated friction
coefficients by 1 order of magnitude down to μ<0.1.
Faults are represented by cohesionless contact surfaces in the models. The
used friction coefficients are 0.1, 0.2, 0.4, 0.6, 0.8 and 1.0, which covers
both slow and fast slip rates.
Initial stress state
The present-day stress state in the crust is a complex product of several
stress sources from the past to the present. In order to model the stress
state an initial stress state is defined, which is in equilibrium with the
body forces (gravity) and which subsequently undergoes lateral straining to
account for tectonic stress. provided a simple
semi-empirical function (Eq. ) for the stress ratio k
(Eq. ), where E is Young's modulus and z is the depth in kilometres.
2k=0.25+7E0.001+1z3k=SHmeanSV=SHmax+Shmin2SV
Sheorey's equation (Eq. ) is a reliable stress ratio
vs. depth estimation, when compared to real-world data
(Fig. ). The model is pre-stressed with zero horizontal
strain boundary conditions. The pre-stressing method used here has so far been
used several times . The model
is allowed to compact several times under application of the body forces
(gravity) using a Poisson's ratio of ν=0.396 during that procedure only.
During the pre-stressing procedure, models with contact surfaces have a very
large friction coefficient (μ=10) to prevent slip. At a virtual well in
the centre of the model (x=150km; y=200km) stress was
extracted from the model and compared to the stress magnitude data, which are
visualized in Figs. and , showing a good
fit to stress–depth distribution assumptions and measured magnitude ratios also .
Boundary conditions
The overall SHmax orientation on a virtual profile along
longitude 11∘ (Fig. ) displays a north–south
orientation in the North German Basin (NGB) and in the Molasse Basin (MB)
north of the Alps, except the Variscan basement units in between.
Correspondingly, a north–south orientation of SHmax is intended
for the reference model. In order to generate a meaningful stress state in
the model, appropriate boundary conditions are required, which are technically
applied by a defined lateral displacement. Results from a virtual well in the
model centre are compared with data from deep wells. An extension of
60 m (ϵx=2×10-4) in the east–west direction and a
shortening of 400 m (ϵy=-1×10-3) in the north–south
direction (Fig. ) provide a good fit of the reference model to
stress magnitudes from selected deep wells Fig. ,
. By fitting the data, the focus was
more on the observed Shmin magnitudes and to a lesser extent on the
SHmax magnitudes. The latter are less reliable, as they are
usually not measured; they are calculated on the basis of several assumptions.
The determined boundary conditions are used for all models.
The stress magnitudes are plotted as a function of depth.
The stress components from the virtual well in the model are illustrated by the coloured lines.
The location of the virtual well and boundary conditions used are shown in Fig. .
Due to the applied initial stress conditions, the stress regime changes from thrust faulting at a depth of 400 m to strike slip faulting, and finally to a normal faulting regime at a depth greater than 5500 m.
Published stress magnitude data are shown for comparison .
Influence of density on the stress orientation. Black bars represent the orientation of SHmax at a depth of 1000 m. Colours indicate the material properties used. The medium blue area uses the reference material properties (ρ=2.7gcm-3), the light blue material uses a lower density (g: ρ=2.2gcm-3), the dark blue a larger density (G: ρ=3.2gcm-3).
Generic model scenario's
The model geometry consists of five units (Fig. ). The
northern- and southernmost blocks are always assigned the reference material
properties (Table ). In between there are three diagonal
units in which material properties are varied. Along the vertical borders
within the model, friction properties can be used. The lower (L) or higher
(H) values of the material properties with respect to the reference material
(B) will be varied in the following way: LLL, HHH, LBL, BLB, etc. When the
model geometry mimics discontinuities using contact surfaces, all contacts
have the same friction coefficient. In the figures showing the results the label “|”
indicates contact. For example, HLH with four contacts is |H|L|H|.
The SHmax orientation is visualized at a depth of 1000 m
below the surface using a pre-defined grid, where the lateral distance to the
material transition or discontinuity is >12.5km, as far-field
effects are the main interest of this study. The variation of density,
Poisson's ratio, Young's modulus and friction coefficient will be tested
first. In addition, the variation of Young's modulus is tested in interaction
with low-friction contacts.
Material properties used for the scenario using realistic rock properties for Variscan basement units; properties are estimated based on . MGCH = Mid-German Crystalline High,
MZ = Moldanubian Zone,
NPZ = Northern Pyllite Zone,
RHZ = Rheno-Hercynian Zone,
STZ = Saxo-Thuringian Zone.
Young'sPoisson'sVariscanDensitymodulusratiounitsρEν[gcm-3][MPa][ ]RHZ2.10200.15NPZ2.20300.15MGCH2.75700.30STZ2.60500.25MZ2.75700.30Realistic rock property scenario
A reality-based rock property scenario, inspired by the structural zonation of
the European Variscides according to , is tested. The
RHZ and the NPZ are dominated
by clastic shelf sediments with a low- or mid-metamorphic overprint, which is made of slate (RHZ) and phyllite (NPZ). This zone, the RHZ and the NPZ together, is
the most flexible one and will have the lowest Young's modulus
(Table ). The MGCH consists of
granitoids or gabbros and their metamorphic equivalents (gneiss, amphibolite),
meta-sediments, and some volcanites. Therefore, this zone is a stiff unit. The
Saxo-Thuringian Zone (STZ) is dominated by meta-sediments, mafic and felsic
magmatites and their metamorphosed equivalents, and some high-grade
metamorphic rocks (granulite, eklogite). Taking all the different rock types
into account, the STZ is stiffer than the RHZ and softer than the MGCH.
Mechanically, the MZ can be represented by high-grade
metamorphic rocks (gneiss, granulite, migmatite) and granitoids and will be a
stiff unit, similar to the MGCH. Therefore, the unit stiffnesses are
different: they are from slightly deformable to rigid in the following order:
RHZ≈NPZ<STZ<MGCH≈MZ. Material properties used are
estimated based on typical rock values (Table ). The same
initial stress procedure, boundary condition and visualization procedure are
applied as previously described.
Influence of Poisson's ratio on the stress orientation. Black bars represent the orientation of SHmax at a depth of 1000 m. Colours indicate the material properties used. The blue area uses the reference material properties (ν=0.25), the light purple area is characterized by a low Poisson's ratio (p: ν=0.15) and the dark purple one by a large Poisson's ratio (P: ν=0.35).
ResultsDensity influence
To identify the influence of a density variation, the reference density
(ρ=2.7gcm-3) in blue is varied using a small density
(g: ρ=2.2gcm-3), which is coloured in light blue, and a
large density (G: ρ=3.2gcm-3), which is dark blue (Fig. ).
The low-density anomaly (ggg) results in a slight counter-clockwise
(-6∘) rotation of the SHmax orientation in the reference
material near the anomaly (Fig. ). Within the low-density
units near the reference material, nearly no rotation is observed
(-1∘), but SHmax orientation turns counter-clockwise
(-8∘) in the centre of the material anomaly. The angular variation
of SHmax crossing the units is of the order of 7∘. The
high-density anomaly (GGG) results in a slightly clockwise rotation
(+7∘) in the reference material near the anomaly. In the high-density unit near the reference material, SHmax is minimally
influenced (+1∘) but rotates further clockwise (+12∘) in
the centre of the anomaly. Based on that, the variation across the units is
about 11∘. The models with mixed densities in the three units show
a clockwise rotation (+10∘) of SHmax within the lighter
material next to the denser units. The high-density units show a
counter-clockwise rotation (-7∘) next to the low-density unit;
therefore, the total variation of SHmax is 17∘.
In general, SHmax tends to be oriented parallel to the anomaly in
low-density units and perpendicular to the anomaly in large density units. In
the centre of the low-density units (ggg), the stress orientation becomes
perpendicular to the overall structure. In the centre of the high-density
units (GGG) the opposite is true, and SHmax becomes parallel to the
structure.
Influence of Poisson's ratio
The influence of Poisson's ratio on the stress rotation is tested by
variation of the reference Poisson's ratio (ν=0.25) using a lower one
(p: ν=0.15) in light purple and a larger one (P: ν=0.35) in dark
purple (Fig. ). The models with only a lower (ppp:
-1.5∘) and only a higher Poisson's ratio (PPP: +2.2∘) show
only little SHmax rotation (Fig. ). Mixed models
with largest Poisson's ratio variation (pPp and PpP) have some
counter-clockwise rotation in the low Poisson's ratio units (-3.0∘)
and a clockwise rotation in the high Poisson's ratio units (+4.2∘).
Therefore, the total variance of SHmax is about 7.5∘.
Impact of Young's modulus
Influence of Young's modulus variation on the stress orientation. Black bars represent the orientation of SHmax at a depth of 1000 m. Colours indicate the used Young's modulus; the blue area uses the reference material properties (B: E=50GPa), the green material uses a low Young's modulus (e: E=10GPa) and the red material has a large Young's modulus (E: E=100GPa).
The impact of Young's modulus is investigated using the reference material
(B: E=50GPa) in contrast to a softer material
(e: E=10GPa) in green and a stiffer material
(E: E=100GPa) in red (Fig. ). The models with
the soft units (eee, eBe and BeB) exhibit a strong clockwise SHmax
rotation (+56∘) in the units with the reference material and a
counter-clockwise rotation in the softer units (-22∘) near the
material transitions. For the models with three soft units (eee) the
SHmax orientation decreases to -5∘ in the centre of the
units. This means that the SHmax variation within the soft units
is considerable (17∘). The resulting total variation is 78∘.
The models with the stiff units (EEE, EBE and BEB) exhibit a gentle
counter-clockwise rotation in the units with the reference material (-5.5 to
-7∘) next to the stiff units. Within the stiff units, a significant
clockwise rotation (+20 to +25∘) is apparent next to the reference
units. In the model with three stiff units (EEE), the SHmax
orientation decreases to (+5∘) in the centre. This is a
considerable SHmax variation of 15∘ within the stiff
units. The total variation is 31∘.
For the models with alternating soft and stiff material units (EeE and eEe),
the soft units exhibit a counter-clockwise SHmax rotation (-19
to -22∘), whereas the stiff units display a clockwise rotation
(+53 to +56∘). Consequently, the total variation between the soft
and stiff units is 72 to 78∘. The general observation is that next
to the material transition, SHmax rotates perpendicular to the
anomaly for the compliant units and parallel for the stiff units.
Influence of faults
Several models with the reference material properties separated by three
discontinuities (|B|B|B|) with a friction coefficient (μ) from 0.1 to
1 are tested. The low-friction coefficient (μ=0.1) leads to a
counter-clockwise SHmax rotation of only -3∘
(Fig. ). The maximum observed fault offset is about
16 m. By increasing the friction coefficient to μ=0.2, the
SHmax rotation is -2∘; for μ=0.4, SHmax
rotation is only -1∘. For larger friction coefficients, the
SHmax rotation is below -1∘. As the SHmax
rotation is too small for a visual differentiation, only the μ=0.1 model
is shown in Fig. .
Influence of low-friction faults on the far-field stress orientation. Black bars represent the orientation of SHmax at a depth of 1000 m. All areas have the properties of the reference material (Table ). White lines indicate cohesionless discontinuities (vertical faults). The model using a friction coefficient of μ=0.1 along the three discontinuities is shown. The other models with a larger friction coefficient (up to 1 and larger) have similar results; they are waived out because of the visual similarity.
Influence of Young's modulus in interaction with low-friction faults on the far-field stress orientation. Black bars represent the orientation of SHmax at a depth of 1000 m. Colours indicate the material properties used. The blue area uses the reference material properties, the green material uses a low Young's modulus and the red material has a larger Young's modulus; see Table . White lines indicate cohesionless vertical discontinuities (faults) with a friction coefficient of μ=0.1.
Stiffness variation combined with low-friction faults
The interaction between a significant Young's modulus contrast and a
cohesionless contact with a low-friction coefficient (μ=0.1) is tested
along all four discontinuities. The model with three stiff units (|E|E|E|)
provides only little counter-clockwise rotation (-4∘) in the
reference material near the material transition
(Fig. ). Similar clockwise rotation occurs in the
stiff units (+4∘) near the material transition and decreases to the
centre of the units (+1∘). The total SHmax variation is
about 8∘.
The model with the soft units and the low-friction discontinuities (|e|e|e|)
shows larger rotations than the model with stiffer units. Clockwise rotation
of +19∘ occurs in the reference material and counter-clockwise
rotation of -13∘ in the soft units. This decreases towards the
centre of the soft units (-9∘). Overall rotation is about
32∘.
The models with alternating stiffnesses and low-friction discontinuities
(|E|e|E| and |e|E|e|) generate a counter-clockwise rotation of about -10 to
-12∘ in the soft units. Within the stiff units, the
SHmax orientation is in the range of +2 to +7∘. The
total variation is up to 19∘. The maximum observed fault offset is
about 10 to 15 m.
Stress rotation for realistic material properties
The resulting SHmax orientation (Fig. a) of the
model using realistic material properties (Table ) indicates
counter-clockwise rotation in the RHZ and NPZ and clockwise rotation within the MGCH and MZ units. The overall pattern of the simple
model (Fig. a) shows only limited similarity with the
observed and the mean SHmax orientation on a regular grid using a
search radius of 150 km and a quality and distance weight
(Fig. b and c). However, some similarities can be observed.
For example, the simple model (Fig. a) shows a clockwise
rotation from the NPZ to the MGCH and counter-clockwise from the MGCH to the
STZ. In Figure b these areas show similar, but less
pronounced, rotation of SHmax. The north-north-east
SHmax orientation within the central part of the MGCH is similar
between the model, the data and the mean SHmax orientation
(Fig. a–c).
Comparison of orientations of the maximum horizontal stress (SHmax). The equivalent regions are the RHZ (Rheno-Hercynian Zone), NPZ (Northern Phyllite Zone), the MGCH (Mid-German Crystalline High), the STZ (Saxo-Thuringian Zone) and the MZ (Moldanubian Zone). (a) Model results, application of estimated material properties of the Variscan units (Table ).
Black bars represent the SHmax orientation at a depth of
1000 m. (b) Bars indicate the SHmax orientation
data , and quality is indicated by shades of grey; see
legend. (c) Mean SHmax orientation on a 150 km
search radius with a distance and quality weight (n>3) using the tool
stress2grid . Panels (b) and (c) have the same extent as Fig. .
DiscussionModel simplification
This study investigates the influence of elastic material properties, density
and friction coefficient at vertical faults on the orientation of
SHmax. The focus is not on stress rotation close to the material
transition or discontinuity (<10km), the priority is on the far-field effects (>10km). Although the model is inspired by a
particular region, the goal is to gain a better understanding on how the
variable material properties affect the stress orientation. For this reason,
the model geometry is very simple and some of the material properties used may
have no proper natural equivalent.
Chosen properties are constant over a depth of 30 km, which is
unlikely. Even for a given lithology, the properties can change with depth,
as a result of the acting gravity and compaction, especially for sediments.
Each lithological unit is at least partially affected by these changes.
Linearly increasing rock properties with depth would account for this and be a
more realistic representation. But this would not affect the resulting stress
pattern, especially since a vertically uniform stress field is assumed
, with a few exceptions.
The simple generic models neglect various rheological processes in the crust
by applying linear-elastic material law. However, the overall geometry seems
reasonable, as the brittle domain or elastic thickness of the lithosphere
(Te), which is a measure of the integrated stiffness of the lithosphere, is
of the order of 30 km and more in central Europe
. The Moho depth in Germany or central Europe is also about
30 km. for
example used a range of Te=30–100 km for their model of central
Europe. Furthermore, results are represented and discussed mainly for a depth
of 1000 m where elastic behaviour is certainly predominant.
The scenario models were tested with an additional very stiff mantle
(Table ) with a thickness of 30 km. This had no
influence on the observed stress pattern at a depth of 1000 m.
However, the models with the same geometry but a total thickness of only
10 km resulted in much lower stress rotation. Therefore, the elastic
thickness of the lithosphere and the aspect ratio of thickness and width of
the units are important constraints for the possible stress rotation. The
depth at which the stress orientation is plotted is also important, as the
stress rotation decreases with depth (Fig. ), so that it
disappears at about 10 km depth for the used configuration. As
homogeneous material properties are used, smaller scaling of results seems to
be reasonable, considering the aspect ratio.
North–south depth profiles displaying the SHmax orientation colour-coded for models with a variable Young's modulus. In the model without the discontinuities (eEe), SHmax is oriented around 40∘ in the stiffer units next to the softer units near the Earth surface. A similar orientation can be observed in the soft units in the deepest parts. In contrast to that, in the model with the same material properties but low-friction faults (|e|E|e|), the SHmax orientation is nearly north–south for all units and depths. (Small coloured dots are artefacts.) The discontinuities with a low-friction coefficient counterbalance stress rotation due to the stiffness contrasts.
All models were loaded with the same displacement boundary conditions
(Fig. ). This results in slightly different stress magnitudes
due to the variable material properties. Since these models have different
mechanical properties depending on the unit, the question would arise, in
which of the units identical stress magnitudes should be achieved? Even if
each model were calibrated individually, this would not significantly change
the results, as both the stress regime and stress orientation would remain
nearly constant for slightly different boundary conditions. Therefore,
constant boundary conditions are reasonable and applied to all scenarios.
Stress rotation by density contrast
The lateral variation of the density is responsible for SHmax
rotation in the range of 7 to 17∘ (Fig. ).
In general, the SHmax rotates in the low-density units slightly
toward parallel to the high-density unit (+10∘), whereas
SHmax rotates in the high-density units a little bit in the
direction to the low-density units (-7∘).
Taking a broad range of sediments into account (evaporites, shale, sandstone or
limestone), they could have even a lower density than the lowest value used
(ρ=2.2gcm-3). Most probably, models with a lower stiffness
would result in larger stress rotation. However, sediments with a low
stiffness could reach a thickness of several thousand metres, but not of the order of the model depth of 30 km or with such a low density due to
increasing compaction with depth. Therefore, the impact of density variation
on the stress orientation in nature will be much smaller, or on a very local
scale. This agrees with the results of , where a lateral
density variation did not have a significant impact on the stress pattern;
only local effects are observed.
This assumption seems to be a contradiction to the fact that the
gravitational load is one of the main sources of stress in the Earth's crust.
However, a density anomaly is a much smaller influencing variable on the
stress state than density. According to , the resulting
stress rotations depend on the relative influence of regional stress sources
as opposed to the density anomaly. Depending on the model scenarios used, the
influence of the boundary conditions (regional stress sources) appears to be
greater than that of the density anomaly. Therefore, the model results are
probably not representative for regions with small horizontal differential
stresses.
Stress rotation due to a variation of Poisson's ratio and Young's modulus
Model results suggest that the variation of Poisson's ratio can be
responsible for a SHmax rotation of up to 7.5∘
(Fig. ). This is below the uncertainties of stress
orientation estimations of about ±15∘ and more
. Therefore, the variation of Poisson's ratio can be
neglected as a potential source of significant stress rotation.
In contrast to that, the lateral variation of Young's modulus can lead to
significant SHmax rotation (Fig. ). For the
geometry and material parameters used, the relative rotations are up to
78∘, which is not far from the maximal possible rotation of
90∘. The largest rotation occurs in the units with a lower Young's
modulus, for example the eee model has a total rotation of 78∘,
whereas the EEE model causes only a 31∘ rotation. This is not
surprising as Young's modulus is simply a measure of the stiffness.
Therefore, the largest stress rotation due to stiffness contrast will happen in
the soft units, not in the rigid ones. From this, it can be deduced that for
units with smaller Young's modulus, the stress rotation is even greater.
SHmax will be oriented parallel to the structure for stiff units
and perpendicular for soft units, which agrees with the literature
. The largest stress rotation occurs nearest to
the material transition and decreases with distance to the material
transition, similar to other models . Similar impacts of
stiffness contrast have been described in previous studies
. In
contrast to that, found that a stiffness contrast has
only minor effects. But they did not test the stiffness contrast separately;
they applied it only in combination with active faults in between the units.
However, this agrees with the results of this study, as active faults balance
stress rotation by stiffness contrast.
Within the units with a small Young's modulus, significant deformation is
possible. For example, within the eEe model (Fig. ), the soft
units in green will be sinistrally deformed. The stiff unit in red cannot be
deformed in the same way. But as the units are connected, the stiff unit is
affected by the tangentially acting stress source. This leads to a
SHmax orientation parallel to the structure, within the stiff
units. As the soft one allows such deformation, SHmax will be
oriented normally to the stiff unit.
At the interface between stiff and soft units differential stresses are
greatest, as both units are differently deformable. This fits with the
observation of concentrated intra-plate earthquakes around cratons
. On a smaller scale this has been observed for stiff
sedimentary layers or rigid dykes, which attracts the occurrence of seismicity
.
The observed radial stress pattern to the south of the Bohemian Massif
agrees well with this study, where SHmax in
the soft sediments of the Upper and Lower Austrian basin is perpendicular to
the stiff crystalline Bohemian Massif. This is more ambiguously the case
for the fan-shaped pattern in the western and northern part of the Alpine
molasse basin . The reason for this could be a lateral stiffness contrast of the rock, next to the
topographic features of the mountain chain and the overall crustal structure.
When comparing the stress rotation, it is important to consider the respective
depth (see Fig. ). For example, data in the north-western
Alps originate from focal mechanisms, and in the foreland of the central Alps,
the majority of data are from wells, which are more shallow
.
Substantial stress rotations are not observed along major pre-Mesozoic
boundaries and sutures in the eastern United States, like the Greenville
front, a suture from Missouri to New York, or in the Appalachian Mountains
. reports the same for
Fennoscandia. In the case that these tectonic boundaries did not provide a
significant stiffness transition, it is not a contradiction to this study.
The mechanical contrast is important, not the relative ages.
Comparison of stress rotation due to elastic material properties
The rotation of SHmax perpendicular (counter-clockwise) to the
structure can be observed most clearly in material with a lower Young's modulus next to a
material transition, up to -22∘. Rotation in the same
direction, but with a lower amount, is observed in rocks with a larger density
or a smaller Poisson's ratio. Within the units with a greater Young's
modulus, SHmax rotates significantly parallel (clockwise) to the
material transition, up to 56∘. Similar rotation with a smaller
magnitude can be observed in the low-density units or in the units with a
larger Poisson's ratio. As rocks with a larger Young's modulus will usually
have a larger density and vice versa (Fig. ), real rocks will
have less SHmax rotation as suggested by these generic models.
But the aim of this study is to test and combine the possible range of
variation, in order to identify the most important causes
(Fig. ).
Comparison of resulting maximum stress rotation, based on the geometry used (Fig. ) and the varied properties (Table ).
Failure criteria
As only elastic material properties are used, failure is not possible. To
study the influence of this simplification, two models (EEE and eee) have been
calculated using two different Coulomb failure criteria. The models are run
first with a cohesion (C) of 30 MPa and a friction angle (FA) of 40∘
and in addition with C=10 MPa and a FA of 30∘.
For the EEE model with C=30 MPa and FA = 40∘, no failure will be
reached (yield criteria <1). For the eee model using that criteria and for
both models (EEE and eee) using C=10 MPa and FA = 30∘ failure occurs.
Conditions of failure or close to failure (Yield criteria ∼1 or >1)
occur only near the surface (a few kilometres) and close to the material transition
(∼20–30 km). Around the material transition, (near) failure
can be observed within the stiff units only. For the EEE model with C=10 MPa
and FA = 30∘, failure is more spaciously distributed near the surface.
In the case of failure, the SHmax orientation will be balanced,
which means that SHmax rotates back in the north–south
orientation, similar to the applied boundary conditions. In general, failure
compensates for stress rotation in the same way as low-friction contact
surfaces (faults). However, the stress orientation in the models that
account for failure shows a similar stress pattern in the pre-failure phase to
the models without failure. As a conclusion from this observation, the model
results showing significant stress rotation are still valid for solid rocks.
Effect of faults on stress orientation
According to the model results, the influence of low-friction faults can be
neglected concerning the orientation of the far-field stress pattern for
homogeneous units. The low-friction faults (C=0, μ=0.1; FA 5∘)
lead to only 3∘SHmax rotation at a distance of about
12.5 km next to the fault zone. Observed stress rotation is lower
than 1∘ for a friction coefficient μ>0.4. This is not in
contrast to the strong stress perturbation, observed in the vicinity of faults; as one of the three principal stresses must be oriented perpendicular to a
fault, the two remaining ones are parallel to the discontinuity . Observations from meso-scale
outcrops indicate stress perturbation within 2 km or
less than 1 km to a fault ; larger stress
perturbation can be observed at the termination of the fault
(2–3 km). If SHmax is parallel next to the fault, it
will rotate by 90∘ at the termination of the fault .
suggests significant stress rotation as a product of active
faults within a distance of several hundred metres for large differential
stress provinces and several kilometres for regions with small differential
stresses. This is supported by observed stress rotations near a fault within
a range of a few hundred metres to a few kilometres . However, not all observed stress rotation agrees with the
presented models, like observations offshore of eastern Canada , where stress rotation occurs at a distance of about
10–15 km to a fault. Whether this is due to an inaccurate
localization of the fault, a low Young's modulus or other causes cannot be
clarified here.
Numerical models investigating stress rotations near a fault provide stress
rotation between 20 and 60∘, next to the fault, depending on the fault
strike, the boundary conditions, and the friction or weakness of the fault.
Near the termination of the fault, stress rotation increases to
50–90∘. However,
rotation is observed by these models only within 2–3 elements, away from the
discontinuity, which are anyway needed to distribute the deformation by such
numerical models. Therefore, observed distances of rotation within these
models are not considered here. To avoid this influence of an overly coarse mesh,
the orientation of SHmax in this study is displayed at least four
elements away from the contact surface.
Effect of faults combined with stiffness contrasts on stress orientation
The models with low-friction faults and a variable stiffness
(Fig. ) illustrate much lower stress rotation than the
models without the faults (Fig. ). The reason is that the
soft units cannot transfer tangential shear stresses to the stiffer units.
Therefore, each unit can be deformed independently from each other.
It seems to be that discontinuities play an important role in reducing stress
rotations, produced by lateral Young's modulus variation (or other reasons).
Regarding the used model geometry and materials, the SHmax
variation is reduced for the soft models from 78 to 32∘, for a
comparison of eee and |e|e|e| in Figs.
and , using a friction coefficient of μ=0.1.
Also for the mixed models, a reduction from 78 to 19∘ is significant.
Much lower is the rotation for the stiff model, with a reduction from 31 to
8∘ (EEE in contrast to |E|E|E|).
The influence of low-friction faults in combination with variable mechanical
properties was also investigated for the models with a density contrast. The
observed effects are limited; therefore, presentation of the results is
omitted.
Depth variation
The interaction of units with a variable Young's modulus and presence or
non-presence of low-friction faults is visible in Fig. .
While the model without active faults displays significant stress rotation,
the same setting with low-friction faults shows only little rotation. The
observed stress rotation within the model without faults strongly depends on
the depth. In the soft units, SHmax rotates counter-clockwise
near the surface (0 to -8km). In contrast to that a clockwise
rotation can be observed at greater depth (18–30 km). The stiffer
units show a clockwise rotation in the upper part, about 0 to
-8km, which changes slightly to a counter-clockwise orientation in
the deeper part.
This could be an indication that stress rotations due to stiffness contrasts
in or near sedimentary basins can be significantly greater than in deeper
material transitions, such as in a buried crystalline basement. This is all
the more likely as sediments tend to be less stiff than crystalline rock.
Model using the Variscan zone rock properties
Figure presents comparatively the results of
SHmax orientation of the model using the chosen material
properties, inspired by the Variscan units (a), the SHmax
orientation data (b) and the averaged orientation of SHmax on a
regular grid (c). A limitation of the mean stress orientation on a regular
grid (Fig. c) is the calculation based on distance, and not
depending on the specific unit. The similarity of the SHmax
orientation is not very convincing, because the overall pattern cannot be
reproduced. Some of the deviations from the trend are similar, some are not.
There are probably several reasons why the simple model is not able to
reproduce the observed stress pattern in the German Variscides
(Fig. ). First of all, only one single elastic material
composition represents each of the Variscan units. The model did not
reproduce the complex and uncertain vertical variability of the deeper
structures , as no deep wells
are present there. Only refraction seismic profiles from the 1980s (DEKORP)
and their interpretations are available . Consequently,
besides the uncertain structures, the material properties and the dip of the
unit boundaries are also uncertain. More complex geometries and variable dip
angle may result in different stress patterns to the ones obtained for
vertical discontinuities. However, studying such variability is beyond the
scope of this study.
Of course, it could also be possible that the units are decoupled. Thus, a
scenario was calculated in which the units are separated by low-friction faults. But this did not provide a better fit of the stress orientation in
contrast to the observation. Furthermore, there are no seismic or geodetic
indications for such a decoupling of the units in that region.
Structures outside the Variscan zonation may also play a role. To the south,
the stress pattern in the Molasses Basin is probably more governed by the
structure of the Alpine chain than older structures.
The fan-shaped stress pattern in the eastern part of the North German Basin
has been explained as an effect of the close boundary to the stiff Eastern
European Craton along the north-west to south-east striking
TTZ . This
interpretation agrees well with the results of the models, where
SHmax becomes perpendicular in a soft unit (NGB) directed to a
stiff region, like the East European Craton (e.g. model eee in
Fig. ).
Conclusions
The effect of varying elastic material parameters (Young's modulus and
Poisson's ratio), density and low-friction discontinuities on the stress
pattern in map view is investigated. Each property is tested separately to
avoid interdependencies. This is performed with generic 3-D models using the
finite-element method. Three units of variable material properties are
included within the models, with boundary conditions determining the overall
SHmax orientation. The variation of density and Poisson's ratio lead to small rotation (17 and 7.5∘) of
SHmax. In contrast, stiffness variation is able to produce
significant stress rotation of 31 to 78∘. Therefore, variation of Young's modulus in the upper crust is a potent explanation for observed stress
rotation. Faults are represented in the models by cohesionless contact
surfaces. The observed stress rotation in the far field due to low-friction faults (μ=0.1) is less than 3∘. Implementation of low-friction discontinuities in models with a Young's modulus anomaly results in much
smaller SHmax rotation, of the order of 8 to 32∘. It
follows that faults do not produce far-field stress rotation, but rather
compensate for stress rotation that is an effect of the Young's modulus
anomaly or other causes. Comparison of the model results with the observed
stress orientation in the region that inspired the models provides only
limited consistency. Nevertheless, the studies clearly show that fault systems
are hardly the source of stress rotations on length scales of 100 km or
larger. Furthermore, the study indicates that strength contrasts are promising
candidates that have the potential to explain the slight stress pattern
rotations in intraplate settings where topography and low-friction fault
systems are missing.
Code availability
Model discretization was performed with the
commercial software HyperMesh® v.2019; the equilibrium of forces is
computed numerically using the commercial software Abaqus®/Standard
v.6.14-1. Model input files are available online at 10.48328/tudatalib-560. Maps and model illustrations (except Fig. ) were generated using Generic Mapping Tools (GMT) software .
Data availability
Stress orientation data are available online at
10.5880/WSM.2016.001.
Competing interests
The author declares that there is no conflict of interest.
Acknowledgements
I want to thank Oliver Heidbach, Tobias Hergert,
Birgit Müller and Moritz Ziegler for fruitful discussions about
mechanics in the upper crust and the observed stress rotation pattern as
well as comments on the paper. Also I would like to thank two anonymous
reviewers for their constructive annotation, which improved the paper.
Financial support
This research has been supported by the Federal
Ministry for Economic Affairs and Energy (BMWi) (grant no. 02E11637A).
This study is part of the SpannEnD Project
(http://www.SpannEnD-Projekt.de, last access: 8 June 2021).
Review statement
This paper was edited by David Healy and reviewed by two anonymous referees.
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