Tectonophysics, 228 (1993) 199-210
Elsevier Science Publishers B.V., Amsterdam
Initiation of salt diapirs with frictional overburdens:
numerical experiments
A.N.B. Poliakov a.b, Yu. Podladchikov and C. Talbot
" HLRZ, KFA-Jiilich, Postfach 1913, D-5170 Jiilich, Germany
Hans Ramberg Tectonic Laboratory, Institute of Earth Sciences, Uppsala University, S-752 36 Uppsala, Sweden
(Received July 8, 1993; revised version accepted July 8, 1993)
ABSTRACT
We have studied the initiation of salt diapirs with frictional overburdens under lateral compression and extension. The
overburden is assumed to be elasto-plastic with Mohr-Coulomb yield criterion and non-associated flow rule. Plastic flow
localizes into narrow shear bands which may represent faults. The geometry and the evolution of these shear zones are
simulated for various rheological parameters. The dilational and frictional properties of frictional overburdens are found to
be important controls on the types of shear zones formed in lateral compression and extension. The friction and dilation
angles influence the angle of shear and the width, dip, spacing and development history of conjugate zones of shear-plus-dilation (fault zones) that define wedges and grabens.
Introduction
1
I
t
i
I1
This paper is devoted to a particularly active
topic in current tectonic research: the problem of
plastic rock failure, here in laterally shortened
and extended overburdens over inviscid salt. The
first systematic analog experiments on this subject
were published only recently (see Vendeville and
Jackson, 1992; Weijermars et al., 1993). These
studies showed interesting new relations between
faults and salt structures, while challenging the
idea that buoyancy forces are adequate for driving salt tectonics.
Numerical models are still two-dimensional
and not yet as realistic as profiles through threedimensional analogue experiments. Witlox (1988)
and Cundall(1990) studied the evolution of faults
in sedimentary rocks with different elastic properties and different strain-softening regimes, but
their models do not include a salt layer. We are
aware of only one earlier numerical study of salt
diapirism with a frictional overburden: the work
of Last (1988). The rarity of such studies reflects
the difficultly of including elasto-plastic (sediments) and viscous (salt) rheologies in the same
numerical model.
Here we model the initiation of salt diapirs in
a frictional (Mohr-Coulomb) overburden by using our new program PARAVOZ based on the
numerical technique used in FLAC (Fast Lagrangian Analysis of Continua) (Cundall and
Board, 1988; Cundall, 1989).
We study the evolution of fault zones in a
sedimentary overburden, compressed and extended a few percent, over a layer of inviscid salt.
We show that the friction and dilation angles of a
plastic material play an important role in its style
of faulting and consequently in the formation of
different diapiric structures under frictional overburdens.
Localization in frictional materials
One of the widely used rheological models for
sedimentary rocks is elasto-plastic material with
Mohr-Coulomb yield criterion and non-associated flow rule (Vermeer and De Borst, 1984).
This model has been confirmed experimentally
and has stimulated many theoretical and numerical studies. However, theoretical studies are still
not complete and numerical simulations are
0040-1951/93/$06.00 O 1993 - Elsevier Science Publishers B.V. All rights reserved
SSDI 0040-195 1(93)E0174-S
A.N.B. POLIAKOV ET AL.
rather simple due to the complexity of localization.
Here we explain some of the current aspects of
localization, before presenting the results of our
calculations.
Non-associated plasticity
Because of its novelty in geology, the term
non-associated plasticity deserves a brief explanation. It is well known from experiments that rocks
increase in volume (dilate), when distorted by
shear. Dilation occurs by tensile cracking and is
necessary to lift sliding blocks over asperities
(Vermeer and De Borst, 1984). A parameter suitable for characterizing a dilatant material is the
dilation angle $. For simple shear, 4 represents
the ratio of plastic volumetric strain rate divided
by plastic shear strain rate. For theoretical explanation of dilation angle see Hobbs et al. (1990).
Materials with $ = 0 are plastically incompressible and so conserve volume. Materials with higher
I)
dilate more with plastic strain. The type of
plasticity is thus characterized not only by the
yield criterion, but also by the plastic flow rule
which defines the vector of plastic strain rate.
Plasticity with a dilation angle equal to the
angle of internal friction 4 is called "associated
plasticity". This means that, in stress space, the
vector of plastic strain rate remains perpendicuTar to the yield surface throughout an associated
plastic strain. Associated plasticity was originally
developed for metals, for which it is a good
model, because metals have a low friction angle
and also do not dilate. With associated plasticity,
many analytical solutions are available using limit
analysis theory.
"Non-associated plasticity" implies that Jr is
not equal to 4 and that the vector of plastic
strain rate is not normal to the yield surface
(instead, it is perpendicular to its own plastic
potential surface. In fact, the dilation angle is
always smaller than the friction angle for geomaterials.
+
Conditions on localization
Plastic flow in a frictional material is not uniform, even if the initial stress field is uniform. An
instability in plastic flow leads to strains concentrating in narrow shear bands which spontaneously appear in the initial continuum. These
bands, or zones of dilational damage with shear,
have finite widths, but, for brevity, we usually
refer to them as faults or fault zones. The bifurcation theory shows that the conditions promoting this destabilization of deformation are rough
(vertex-like) yield surfaces and a non-associated
plastic flow rule (Rudnicki and Rice, 1975; Hobbs
et al., 1990). This type of plasticity (MohrCoulomb yield and non-associated plastic flow) is
common in rock mechanics experiments on real
rocks (Vardoulakis, 1980; Vermeer and De Borst,
1984). Including this type of plasticity in numerical simulations of deforming rocks increases
model realism.
Orientation of shear bands
The orientation of shear bands in Coulomb
materials with non-associated flow rule is still not
well understood. Theories, experiments and numerical simulations give different values for 0,
the inclination angle subtended by a shear band
and the major compressive stress. There are two
end-member solutions for shear band inclinations:
and:
where 4 and
are the friction and dilation
angles, respectively. Eqn. (1) was derived by
Coulomb in 1776 and eqn. (2) was obtained by
Roscoe (1970).
Attempts to prove either of the results have
not been successful because there is experimental
evidence for both the Coulomb and the Roscoe
orientations. Vermeer (1990) showed that orientation of shear bands may depend on the grain
size of granular material and that Coulomb-type
shear bands occur in fine sands, whereas
Roscoe-type shear bands are observed in coarse
materials.
INITIATION OF SALT DIAPIRS WITH FRICTIONAL OVERBURDENS: NUMERICAL EXPERIMENTS
Arthur et al. (1977) and Vardoulakis (1980)
each reported experimental evidence for an intermediate orientation of shear bands:
A theoretical explanation of this observation
was given by Vermeer (1982). Later, Vermeer
(1990) analysed the post-bifuricational behaviour
of frictional material, allowing it to unload elastically outside of the shear band. He found that
that there is a wide range of admissible orientations of shear bands in between the two limits (1)
and (2).
Systematical numerical experiments on shear
band inclinations in a compressional uniaxial test
were carried out by Hobbs and Ord (1989) using
a FLAC program. They obtained a broad range
of shear band inclinations, not predicted by eqns.
(I), (2), or (3). However, they demonstrated that
the angle 0 decreases, as both 4 and increase.
The scattering of inclination angles in experiments may be due to friction at the end platens,
which can delay the inception of a shear band
(Vermeer and De Borst, 1984) or produce kinking (Dawson, 1993). In numerical experiments, a
similar effect can occur due to the finite size of a
model (i.e. influence of boundaries) which affects
the inclination angles.
For these reasons, we do not expect to find
unique shear band orientation. It is more instructive to study relative changes in shear orientation,
rather than absolute values.
+
Spacing of shear bands
Spacing of shear bands is difficult to obtain in
theoretical models and easier to observe in numerical or laboratory experiments. The most important control parameter is the dimensionless
bulk modulus (ratio of elastic bulk modulus K to
weight of the layer of rock (Cundall, 1990):
201
ing. Witlox (1988) demonstrated the same effect
numerically (using elastic shear modulus G instead of K ) and Vendeville et al. (1987) showed
it in sand box experiments. Witlox (1988) also
showed that fault spacing increases with an increase in the amount of strain softening.
In the numerical experiments of Wobbs and
Ord (1989) it is difficult to see the dependence of
fault spacing on friction and dilation angles. The
spacing changes during the calculations due to a
geometrical effect: buckling at the free surface
"locks" the shear bands and does not allow them
to adopt a natural spacing.
Thickness of shear bands and mesh size
The thickness of shear bands observed in experiments with sand is about 13-18 grain diameters (Muhlhaus and Vardoulakis, 1987), depending on the coarseness of the sand. The theory of
classical plasticity does not include any internal
length scale and therefore can not predict the
thickness of shear bands (it is assumed to be
zero). To avoid this ambiguity, Miihlhaus and
Vardoulakis (1987) used Cosserat theory for a
granular medium to introduce an internal grain
size and predict the thickness and orientation of
shear bands consistent with observations. However, the thickness of a shear band is very small
compared with the size of a laboratory specimen
or the characteristic length scale of a tectonic
problem. Thus it is always desirable to keep modelled shear bands as thin as possible.
There are two basic problems in modelling
localization in frictional materials:
- First, are the results dependent on mesh size,
for materials with strain softening rheological behaviour ?
- Second, if we assume that material has no
strain softening, can it represent true rock behaviour and what is the mesh size which can
correctly resolve shear band arrays?
Mesh dependence for strain-softening materials
where pSedi, is the density of sediments, g is
acceleration of gravity and H is the thickness of
the sedimentary layer. Cundall(1990) showed that
increasing this ratio gives closer shear band spac-
The introduction of strain softening (softening
of cohesion or/and friction angle) in a model of
plasticity makes the study of localization a very
difficult task. Results from standard numerical
A.N.B. POLIAKOV ET AL.
models become mesh dependent, because estimation of the strain at every time step depends on
the size of the computational element. In order
to avoid this problem, several numerical techniques have been introduced:
(1) a displacement discontinuity (or interface)
in the finite element (Pietruszak and Mr6z, 1981;
Rots, 1988; Klisinksi et al., 1990);
(2) an internal length with non-local or gradient continuum theories (Bazant et al., 1984; De
Borst and Muhlhaus, 1991);
(3) an internal length using a micro-polar
(Cosserat) continuum (De Borst and Muhlhaus,
1991);
(4) rate effects in the constitutive model
(Needleman, 1988; Sluys and De Borst, 1991), a
technique not applicable for quasi-static problems;
(5) "displacement softening" rnultiplyng the
plastic strain by characteristic size of an element
(Dawson, 1993).
Unfortunately, each of these techniques is applicable only to a restricted class of problems (De
Borst and Miihlhaus, 1991). There is no universal
model which can handle the diverse problems of
localization. In this paper we use a model without
strain softening to avoid the problem of mesh
dependence.
Mesh sensitivity for materials without strain softening
Cundall (1990) demonstrated that his FLAC
algorithm (and here also our code, based on his
algorithm) gives shear bands which are 3-4 elements wide, for materials without dilation and
without strain softening. H e assumed that the gap
between shear bands is of the same order and
thus results of FLAC calculations are reliable as
long as the spacing between bands is resolved
with 6-10 elements approximately. Hobbs and
Ord (1989) and Cundall(1990) checked the FLAC
algorithm for different element sizes under the
same physical conditions. Their results differed
on the exact positions of shear bands, but the
general features such as inclination angle and
spacing remained the same. After comparing the
geometry of faulting with that of sand box experi-
ments (Cundall, 1990) and checking results with
analytical solutions for the yield criterion (Hobbs
and Ord, 1989) we infer that the FLAC algorithm
is reliable and can be used for simulating sedimentary rock behaviour.
Thickness of shear bands as a finction of friction
and dilation angles
Hobbs and Ord (1989) found that increasing
the dilation angle $ increases the thickness of the
shear band at fixed friction angle 4 . This fact can
be understood in the sense that material with
high dilation expand in volume and this expansion takes place only in shear bands, making
them thicker.
Another observation of Hobbs and Ord (1989)
is that deformation is strongly localized for small
friction angles, but becomes more diffuse as friction increases. In our experience, this is not a1ways true. Instead, the difference between frictional and dilation angles controls the localization. Thus increasing the difference ( 4 - $1 gives
stronger localization. This is consistent with the
theory of Rudnicki and Rice (1975), showing that
one of the conditions for localization is the nonassociated flow rule. Thus increasing the difference ( 4 - $) increases non-associativity, making
plastic flow more unstable and giving narrower
shear bands. This dependence will be demonstrated below.
Method
Computer simulation of salt diapirism with a
frictional overburden is difficult, because to treat
materials with elasto-plastic and viscous rheologies in the same numerical model leads to severe
computational problems. FLAC (Fast Lagrangian
Analysis of a Continua) provides a powerful tool
for studies of this kind (Cundall and Board, 1988;
Cundall, 1989). FLAC is based on an explicit,
time-marching scheme for the solution of the full
dynamic equations of motion. The main advantage of this method is that it uses an explicit form
of the constitutive (rheological) relation between
stress and strain. This feature allows several complex rheologies to be incorporated in the same
INITIATION OF SALT DIAPIRS WITH FRICTIONAL OVERBURDENS: NUMERICAL EXPERIMENTS
modeI. Thus materials that creep, and/or are
visco-elastic can be combined with materials that
exhibit various types of plasticity and either strain
soften or strain harden.
Our program PARAVOZ is based on FLAC.
We first used PARAVOZ to model RayleighTaylor instability in Maxwell visco-elastic continua (Poliakov et al., 1993). Here we use the
same program to model the initiation of lowviscosity diapirs (e.g., salt) under frictional overburdens approximated as Mohr-Coulomb plastic
material with a non-associated flow rule.
Figure 1 shows the geometry of the problem
solved by PARAVOZ. For computational simplicity and efficiency, we consider only a single
elasto-plastic layer with density pSedi,= 2500
kg/m3, that rests on an inviscid fluid, representing low-viscosity salt with density p,,,, = 2200
kg/m3. In this paper, we use a dimensionless
elastic bulk modulus of the elasto-plastic layer
K ' = 166.7 or 1667.0 (eqn. 4) and its Poisson's
ratio v = 0.25. K' = 166.7 represents typical parameters for a sedimentary layer: K = 0.1667 .
10" Pa, p,,,,, = 2500 kg/m3 and H = 4000 m. In
some numerical experiments, we attempted to
reproduce the natural heterogeneity of rocks, by
assigning random plastic properties to each element in the numerical grid (as in Cundall, 1991).
In these experiments, the mean friction angle
4 = 40" with standard deviation 2", but these
heterogeneties appeared to play no significant
role in the style of faulting. We varied the dilation angle 51, from 0" (no volume change) up to
the 15" which is typical for dense sand (Hettler
and Vardoulakis, 1984). Cohesion of the material
was taken to be zero for simplicity.
The boundary conditions of the problem are as
+
Free surface
Compression
203
follows: the upper boundary remains stress-free
while constant velocities are applied to the lateral
boundaries. Hydrostatic pressures calculated for
the buoyant underlying salt are applied to the
bottom boundary. An initial sinusoidal perturbation is applied to the bottom of the frictional
sediments. The horizontal length of perturbation
is equal to half of the length of the whole layer
and the amplitude is 10% of the thickness of the
whole elasto-plastic layer.
Because the viscosity of the salt layer is assumed to be zero, the elasto-plastic problem has
no intrinsic time scale. Thus the maximum strain
rate and the time printed beneath the illustrated
results have relative, not absolute significance.
The grey scale in the figures indicates the magnitude of the square root of the second invariant of
the strain rate, which is calculated for each element as follows:
where €ij is the ij-component of the strain-rate
tensor. The second invariant of the strain rate
approximately equals the maximum shear strain
rate in each element (the deviatoric part of the
strain rate tensor is much larger than the rate of
area change). Thus, darker regions experience
larger shear rates (i-e. they are in plastic state)
and lighter regions experience smaller shear rates
and move as nearby rigid elastic blocks. For better understanding of shear band dynamics, we
show also the velocity field with black arrows.
Our main aim here is to study the geometry of
fault zones during lateral compression and extension. To study the development of a population
of spontaneous fault zones out of the initial continuum, we used a numerical grid with 160 x 40
square elements. Because we do not use remeshing when the grid becomes too deformed, our
results are limited to initial stages of diapiric
growth.
Results for lateral compression
Fig. 1. Geometry and boundary conditions for problem-initiating salt diapirs. Brittle sediments are approximated by a
Lateral compression with large elastic moduli
Mohr-Coulomb material with non-associated flow rule and
the salt is assumed to be an inviscid fluid of low density that
exerts buoyancy forces from below.
Figure 2 shows the development of spontaneous fault zones for a model with large elastic
204
A.N.B. POLIAKOV ET AL.
modulus (K' = 1667). The main feature of this
model is that faults have no steady position. Instead, they migrate, disappear from one place
and appear in another. The shape and inclina-
tions of these faults also changes with time. However, despite this somewhat chaotic behaviour of
the shear bands, a wedge is formed in the central
part of the model.
Strain rate 2nd Invariant
$1=40k 0°, $= 0°, c= 0
Fig. 2. Development of shear zones in a material with large elastic modulus ( K ' = 1667). Results are for horizontal compression of
a frictional overburden over inviscid salt. Frictional angle 4 = 40"; dilation angle II/ = 0" (meaning no increase in volume during
plastic flow). Shades of grey color indicate magnitude of shear strain rate (second invariant of strain rate). Black arrows represent
velocity field. T is a nondimensional time.
INITIATION OF SALT DIAPIRS WITH FRICTIONAL OVERBURDENS: NUMERICAL EXPERIMENTS
Another feature of Figure 2 is the formation of
a second pair of conjugate fault zones outside the
first pair. These new shear zones led to widening
both of the wedge on the surface and of the
initiated salt intrusion. We believe that the second pair of reverse faults form because the weight
of the central wedge is balanced by buoyancy
forces and it cannot continue to rise. It is easier
for the system to create a new set of faults along
which sliding can occur.
A possible problem with this model is its limited length. The flexural rigidity of the elastoplastic layer is large due to the large elastic
205
modulus. Therefore, relatively large bending
stresses can arise, due to boundary effects and
these may influence the orientation of shear
bands.
Lateral compression with small elastic moduli
Figure 3 shows the results for compression of a
frictional material with small elastic modulus ( K '
= 166.7), for different friction and dilation angles, after 4% shortening.
Small elastic moduli increase the fault spacing
(see section about shear bands spacing) and
Fig. 3. Results for compression (4% of shortening) of frictional material with small elastic modulus ( K ' = 166.7) for different
dilation and friction angles. With increasing dilation angle, shear bands become more diffuse and dip less steeply. Size of wedge
increases in (B) compare to (A). A decrease in friction angles does not significantly change the inclination angle but make the
localization zone more wide.
206
A.N.B. POLIAKOV ET AL.
therefore only one wedge forms. The flexural
rigidity is ten times smaller than in the previous
simulation. Faults do not migrate and are almost
steady state during compression. Therefore
wedges are better developed for the same amount
of compression.
Figure 3A and B shows that an increase in
dilation angle $ produces wider shear bands and
Strain rate 2nd Invariant
q5=40* 0°, += 0°, c = 0
Fig. 4. Development of extensional shear zones in the frictional material with large elastic modulus ( K t = 1667). Note closer
spacing of shear bands compared with compressional examples and not monotonic slip on the bands ("flip-flop" effect, Cundall,
1990).
INITIATION OF SALT DIAPIRS WITH FRICTIONAL OVERBURDENS: NUMERICAL EXPERIMENTS
decreases the dip. These observations are consistent with theory and earlier numerical studies
(see sections about orientation and thickness of
shear bands). Thus, localization with high dilation
angle decreases the inclination angle towards major compressive stress (which is horizontal) and
makes horizontal size of wedge longer. Additionally the volume of the layer increases progressively with compression. This feature is not realistic because dilation in real rocks is not constant,
but decreases with progressive deformation
(Vermeer and De Borst, 1984).
Figure 3A and C shows that decreasing the
friction angles makes wider shear bands. This
observation is opposite to the conclusion of Hobbs
and Ord (1989). Our explanation is given in section about shear bands thickness. Surprisingly,
there is almost no difference in the inclination
angle in these two examples with different friction angles. This means that our shear zones are
closer to the Roscoe type than the Coulomb type,
An interesting feature of these models is the
initial shape of the salt diapir. It has either a very
broad and almost flat shape (Fig. 3A) or a smaller
and rounder shape (Fig. 3B and C).
Results for lateral extension
For lateral extension, shear bands dip more
steeply because the major compressive stress is
vertical. Unfortunately, calculations were terminated at = 1.0-1.5% of extension, much earlier
than for compression because of element distortion and overlap.
207
can become active again. Such a "flip-flop7' behaviour was reported by Cundall (1990). We believe that it is typical of frictional materials with
very strong localization 6.e. large difference between friction and dilation angles). For materials
with diffuse shear zones, this effect is much
weaker.
Lateral extension with small elastic moduli
The development of shear bands in a material
with small elastic modulus ( K t = 166.7) (Fig. 5) is
similar that for large elastic modulus. The main
difference is that the number of shear bands
decreases. An interesting feature is the point of
intersection between conjugate shear bands moves
down at the beginning of the simulation and then
stops (bottom figure). At this intersection, the
mesh is most distorted. It causes termination of
the program at relatively early stages of extension. We believe that the migration of this intersection point is caused by a slight change in the
geometry of the bottom boundary. Distortion of
the bottom boundary causes the shear bands to
move closer, thus moving the intersection point.
The localization geometry for various friction
and dilation angles is presented in Figure 6.
The results for lateral extension are qualitatively the same as for compression. Thus widening of shear bands increases with an increase in
the dilation angle and decrease in the friction
angle. An increase in dilation angle makes the
shear bands steeper.
Conclusions
Lateral extension with large elastic moduli
Figure 4 shows the development of shear bands
in a material with a large elastic modulus ( K t =
1667).
The spacing of shear bands is much smaller
than for compression. This, combined with the
fact that the extensional shear bands dip more
steeply than the compressional bands, causes the
grabens to have smaller linear dimensions than
the wedges (compare Figs. 2 and 4).
Sliping on faults is time dependent. Anyone
fault may stop slipping for some time and then
The numerical experiments reported here are
among the first to simulate the generation of
spontaneous fault zones by shortening or extending elasto-plastic slabs underlain by inviscid fluid.
Most previous numerical models of faulting studied the reactivation or propagation of pre-existing
faults.
We have shown the difference between faulting in lateral compression or extension. In extension, the spacing of faults is much smaller. The
grabens formed in extension are wider than than
the wedges formed in compression. Extensional
208
A.N.B. POLIAKOV ET AL.
faults are steeper than compressional ones, because of the change in orientation of the maximum compressive stress, $ la Anderson's theory
of faulting.
The inclination of shear bands is controlled by
the dilation angle, rather than by the friction
angle (similar to the Roscoe type of shear band).
In materials without strain softening shear
bands become wider with increasing dilation angle. The width of the localization zone decreases
if the difference between friction and dilation
angles (Qj - $) increases.
Strain rate 2nd Invariant
Fig. 5 , Development of extensional shear zones in a frictional material with small elastic modulus ( K ' = 166.7). The point of
intersection of the shear bands moves down with time and the final geometry is slightly nonsymrnetric.
UTIATION OF SALT DIAPIRS WITH FRICTIONAL OVERBURDENS: NUMERICAL EXIJEKI?\.IkNTS
T h e generation of spontaneous fault zones in
equences of rocks with other realistic rheological
,roperties will be the subject of future work.
3xamples might be strain-softening overburdens
vith various angles of internal friction and vari)us elastic moduli, deforming over viscous
iecollements (of e.g., salt, anhydrite or clay) to
;imulate non-planar curved faults.
From the present results on initiation of diapirs, it is very difficult to draw any conclusions
about the shapes of well-developed diapirs. Modelling of further diapiric development is a complex problem, requiring remeshing and reequlibrating of the stress state. This is currently
under investigation.
3 Fl
Peter Cundall is gratchlly thanked for his
original algorithm for solving the problem of plusticity and his patient explanation of FLAC. We
thank Peter Cobbold for his patience as thc Ciuest
Editor and very tidious correction of our
manuscript. Ethan Damon kindly explained thc
problem of mesh dependence and made many
suggestions. Alison Ord and an anonymous rcviewer provided constructive criticism. Mans Herm a n is thanked for supporting of A. Poliakov
during completion of this paper and discussions
about localization. All calculations were done on
Sparc stations in the Hochstleistungsrechen-
Fig. 6. Results for extension (= 1.0-1.5%) in a frictional material with small elastic modulus ( K ' = 166.7) for various dilation and
friction angles. With increasing dilation angle, shear bands become more wide and slightly steeper. Decreasing of the friction
angles widens the shear bands and does not change the inclination angle.
A.N.B. POLIAKOV ET AL.
zentrum (HLRZ). Anne-Claire Bourdessol at Sun
Microsystems France, S.A. provided an important
technical support during calculations. Michael
Leibig kindly improved the English of this paper.
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