Environmental Engineering Reference
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when low elastic modulus values are assumed for
the Flcl unit.
The mean values assumed in the Mohr-coulomb
models well represent the rheological behaviour of
the system. in fact, when the hoek-Brown crite-
rion, which enables to compute the equivalent
Mohr-coulomb c and φ as a function of the mini-
mum stress σ
3
is applied, the strain field doesn't
show any significant difference in relation to deep-
seated instabilities.
These evidences support the choice of imple-
menting the analysis using the asymmetrical
model with stepped hP/Flcl contact and Mohr-
coulomb constitutive law for Ve and Flcl.
The complexity of the system was then increased
dyke complex (
Fig. 5F
). The presence of the dyke
complex has a similar buttressing effect of mak-
ing the boundary closer to the volcano centre,
but the strain field in the area of interest is too
conditioned.
once these results enabled to choose the model
which best fits the observed etna flank dynam-
ics, the third phase of analysis was the applica-
tion of magma pressure along the feeding system.
an interface was introduced in the model, in
order to apply magmatic pressures. such inter-
face is characterized in terms of physical and
mechanical properties (normal and shear stiff-
ness k
n
= k
s
= 3.21e8 Pa/m; c and φ equal to those
of the host rock). Magma pressure is considered
in terms of: i) magmastatic component (P
m
);
ii) overpressure component (P
0
). The magmastatic
component has a triangular distribution and is
obtained as P
m
= γ
m
z, where z is the depth from
the summit craters and γ
m
is the magma bulk den-
sity, which ranges in values along the conduit in
relation to gas content, exolution and vesiculation
processes, from 0.02 g/cm
3
to 2.57 g/cm
3
. The over-
pressure component is applied as a constant of 2
MPa all along the interface, according to iverson
(1995) and apuani & corazzato (2009), and refer-
ences therein.
Figure 5. sketch of the conceptual models considered
in the sensitivity analysis. a. symmetric topography,
equal boundary distance. B. Real topography, different
boundary distance, one unit (Flcl). c. Model with
3 units and flat hP/Flcl contact. D. Model with 3 units
and stepped hP/Flcl contact. e. Model with 4 units
and stepped hP/Flcl contact. F. Model including the
dyke complex as a rigid elastic unit.
iv) presence of a unit representing the dyke com-
plex as a rigid elastic body.
The results of this sensitivity analysis were
expressed in terms of stress-strain field, displace-
ment pattern, plasticity states and shear strain
increments.
symmetrical stress and strain fields, the choice of
in asymmetrical fields. This assumption conditions
all the following modelling results, but it seems to
be an adequate tool to take into account the differ-
ent tectonic regimes (buttressed by the apennine-
Maghrebian chain to the nW, and transtensional
to the se). The attention was then focused to se
flank of the volcano, neglecting the strain field to
the nW, which is affected by this boundary. The
strain pattern shows a predisposition to se-ward
displacement, both when flat or stepped hP/Flcl
second case the tendency to develop subsidence
and eastward movement is enhanced, especially
3.1.2
FDM main results
The main results of elasto-plastic model assuming
Mohr-coulomb properties are here presented.
The results of the analysis at elasto-plastic con-
dition in the case of a 3 units-model, presented
in Figure 6, are obtained using the worst values
(in bold) of Table 1. The stress field, expressed in
terms of horizontal and vertical stress contours, is
conditioned by the rheological contrasts and by the
stepped geometry of the Flcl/hP contact. The
se-ward directed displacement vectors also in the
western portion of the model and maximum val-
ues of 8 m. The maximum shear strain increment
