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Dislocation dynamics simulations of
slip systems interactions and forest
strengthening in ice single crystal
B. Devincre
a
a
LEM, CNRS–ONERA, 29 Avenue de la Division Leclerc, 92322
Châtillon, France
Version of record first published: 26 Jun 2012.
To cite this article: B. Devincre (2013): Dislocation dynamics simulations of slip systems
interactions and forest strengthening in ice single crystal, Philosophical Magazine, 93:1-3, 235-246
To link to this article: http://dx.doi.org/10.1080/14786435.2012.699689
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Philosophical Magazine, 2013
Vol. 93, Nos. 1–3, 235–246, http://dx.doi.org/10.1080/14786435.2012.699689
Dislocation dynamics simulations of slip systems interactions and
forest strengthening in ice single crystal
B. Devincre*
LEM, CNRS–ONERA, 29 Avenue de la Division Leclerc, 92322 Châtillon, France
Downloaded by [ONERA Documentation] at 08:19 12 April 2013
(Received 6 April 2012; final version received 4 May 2012)
The contribution of forest interactions to flow stress and strain hardening
in ice single crystals is evaluated from dislocation dynamics simulations.
The systematic mapping of dislocation–dislocation interactions and the
calculation of the interaction strength between slip systems suggest an
important contribution of collinear annihilation reactions. Comparison
with experiment shows that the forest strengthening induced by a small
density of collinear dislocation segments in cross-slip planes may have been
underestimated in current models for ice plasticity.
Keywords: ice; plastic flow; dislocation dynamics simulation; forest
strengthening
1. Introduction
The crystal structure of ordinary ice (Ih Ice) is reminiscent of HCP
pffiffiffiffiffiffiffiffi metals with a
temperature independent c/a ratio (1.628) close to the ideal value 8=3. As a result of
its non-metallic bonding, ice resembles in many respects Ge or Si, but it deforms
almost exclusively by basal slips [1]. Under stresses corresponding to those found in
glaciers and ice sheets, plastic deformation in ice single crystals is caused by the glide
Burgers vector. Creep
motion of dislocations in (0001) basal planes with h1120i
temperatures in ice are close to its melting temperature, and dislocation glide is
characterized by a low Peierls stress [2]. Generally, the observed dislocation microstructure is essentially made up of bundles of elongated dipolar loops, which look like
the microstructure observed in FCC crystals when deformed in single slip [3,4].
Recent X-ray diffraction experiments [5] provided evidence that in ice single
crystals with a low initial dislocation density, the plastic deformation is first
controlled by dislocation multiplication at a reduced number of sources. Such
heterogeneous dynamics is thought to rapidly promote long-range stresses in samples
deformed in torsion [5]. As strain increases, relaxation processes associated to
dislocation cross-slip (or climb) start to operate and few dislocation segments lying
out of the basal planes are observed. The latter density of dislocation segments is
expected to be forest obstacles restricting the mobility of basal dislocations. Again,
this feature exhibits some similitude with a recent model proposed for strain
hardening in FCC crystals when deformed in single slip conditions [6,7].
*Email: [email protected]
ß 2013 Taylor & Francis
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236
B. Devincre
Consequently, a detailed examination of the interaction strength between dislocations in different slip systems is of great interest to better understand the plastic
properties of ice.
slip are
In ice, the two expected alternative slip modes to basal (B) ð0001Þh1120i
prismatic (P) f1010gh1120i slip and first-order pyramidal (1) f1011gh1120i slip. As
the critical stress on prismatic and pyramidal planes is one or two orders of
magnitude larger than on basal planes, plastic deformation on those slip systems is
not possible in most experiments and the dislocation density observed in non-basal
slip systems may primarily result from local processes of plastic relaxation. The
possibility that basal dislocations preferentially cross-slip in prismatic rather than in
pyramidal planes is still a matter of investigation [8]. This is why both possibilities
are considered in the present work.
The present paper aims at investigating dislocation reactions between noncoplanar slip systems in ice and at quantifying the resulting forest strengthening
[9,10]. The Dislocation Dynamics (DD) simulation code (microMegas1) used for this
work is described in [11] and references therein. All calculations are made within the
framework of isotropic elasticity. First, the orientation dependence of the interaction
between two initially straight segments gliding in different slip systems is simulated
(Section 2). As already shown in earlier work dedicated to other materials [12–16],
this type of simulation allows the construction of angular interaction mappings that
provide systematic information on the nature and symmetry of the reactions
occurring during forest interactions.
Secondly, model DD simulations of latent hardening have been made to calculate
the interaction coefficients entering the definition of constitutive laws for the plastic
flow stress and the dislocation storage rate at the level of slip systems (Section 3).
These simulation results are finally compared with experimental data using a simple
model for strain hardening in Section 4.
2. Dislocation–dislocation interactions
With the view of investigating forest strengthening in ice at high homogeneous
temperatures, we first determine in a systematic manner the distinct types of mutual
interactions occurring between two dislocation segments in non-coplanar slip
systems. This first part of the study has a double interest. It allows checking that all
the crystallographic properties of slip systems are correctly taken into account in the
DD simulation code. Also, it enables establishing a first ranking of the slip system
interactions by listing the different types of relaxed configurations that can be
observed as a function of dislocation character.
The ice slip systems considered in the simulations are listed in Table 1. For
reasons of symmetry, it is sufficient to differentiate only seven pair combinations of
slip systems to explore all possible forest reactions that may occur in an ice crystal.
An illustration of the interaction mapping procedure we used is presented in
Figure 1. Following the methodology defined in [13], we consider initially two
intersecting straight segments of length 1 mm in different glide planes and with
variable orientations. The line orientations are defined by their angles 1 and 2
with respect to the reaction direction at the intersection of the two tested slip systems.
237
Philosophical Magazine
Table 1. Nature of the slip systems considered in the present study.
Type
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Basal (B)
Prismatic (P)
Pyramidal (1)
Sys. Sym.
No. of Slip Sys.
ð0001Þh1120i
f1010gh11
20i
f1011gh11
20i
3
3
6
Figure 1. Schematic view of a junction made from two attractive non-coplanar dislocation
segments with the same finite length. The junction segment (in bold) is formed by a zipping
process taking place at the intersection of the two dislocation glide planes. The angles 1 and
2 considered in the interactions maps are defined.
After elastic relaxation, three types of configurations are obtained: (i) when the
interactions are strong and attractive, junction formation or collinear annihilation
[17] occurs; (ii) when the interaction is weakly attractive, a non-local pinning
configuration called crossed-states [12,18] occurs; and (iii) the outcome of repulsive
interactions is the formation of repulsive states, i.e. non-contact dislocation
configurations. Each final configuration is in equilibrium under zero applied stress
and does not depend on kinetics. In other words, the equilibrium states are only
function of the material parameters controlling the elastic energy minimization, i.e.
the shear modulus ¼ 4 GPa, Poisson’s ratio ¼ 0.33 and the Burgers vector
amplitude b ¼ 0.45 nm.
Figures 2 and 3 show, respectively, the mappings obtained for the four types of
slip system interactions including the formation of junctions and the three types of
slip system interactions including collinear annihilation. The boundaries between the
configuration domains associated to junction or annihilation, crossed and repulsive
states are approximated using two simplified elastic models, similar to those used in
previous studies [12,13]. These calculations are not detailed here for the sake of
brevity. One should only notice that simulation results are in good agreement with
the two simple elastic model predictions.
All mappings are periodic with an angular period of 2 as this rotation leaves the
initial configuration unchanged. Except for the basal-prismatic interactions, all
calculated interaction mappings present strong similitudes with other mappings
already published for other crystal symmetries like in FCC materials [13]. The
domains of junctions in Figure 2 take the form of extended lobes encircling the origin
238
B. Devincre
(b) 180
120
120
60
Φ2(degrees)
Φ2(degrees)
(a) 180
0
-60
60
0
-60
-120
-120
-180
-180
-180 -120 -60
0
-180 -120 -60
60 120 180
120
120
Φ2(degrees)
(d) 180
60
0
-60
-120
0
60 120 180
Φ1(degrees)
(c) 180
Φ2(degrees)
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Φ1(degrees)
60
0
-60
-120
-180
-180
-180 -120 -60
0
60 120 180
Φ1(degrees)
-180 -120 -60
0
60 120 180
Φ1(degrees)
Figure 2. Interaction maps with two non-coplanar dislocation segments for slip system
reactions forming junctions. Interaction map between (a) prismatic–basal, (b) prismatic–
prismatic, (c) pyramidal–basal and (d) prismatic–pyramidal slip systems. Filled rectangles:
junctions; crosses: crossed-states; empty circles: repulsive states. The thin lines represent the
calculated boundaries between attractive and repulsive states and the thick lines the calculated
neutral conditions for junction formation.
(1 ¼ 2 ¼ 0). Therefore, junction formation is observed with many dislocation
combinations and is potentially an efficient strengthening mechanism in ice crystals.
Also, collinear annihilation in Figure 3 forms square domains larger than the
junction domains, which suggests that this dislocation–dislocation reaction should be
the strongest of all. From the dimensions of the reaction domains it is not possible to
quantitatively evaluate the strength of slip system interactions; nevertheless, their
comparison suggests a first classification. Hence, interactions between prismatic–
prismatic and prismatic–pyramidal slip systems are expected to give stronger
(junction-controlled) forest interactions than basal–prismatic and basal–pyramidal
slip system interactions.
Also, the prismatic–prismatic interaction mapping reproduced in Figure 2b can
be compared with two other calculations published in the case of zirconium and
magnesium crystals [15,16]. As the junction zipping and unzipping process is
controlled by elastic energy relaxation and is rather insensitive to the dislocation core
Philosophical Magazine
120
120
60
60
Φ2(degrees)
(b) 180
Φ2(degrees)
(a) 180
0
-60
0
-60
-120
-120
-180
-180
-180 -120 -60
0
60
Φ1(degrees)
-180 -120 -60
0
60
Φ1(degrees)
120 180
120 180
(c) 180
120
Φ2(degrees)
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239
60
0
-60
-120
-180
-180 -120 -60
0
60
120 180
Φ1(degrees)
Figure 3. Interaction maps for two non-coplanar dislocation segments in slip systems leading
to collinear annihilation. Interaction mapping between (a) prismatic–basal, (b) pyramidal–
basal and (c) prismatic–pyramidal slip systems. Same drawing convention as in Figure 2.
energy [19], the interaction mapping of ice is mostly identical to that of other HCP
materials.
Another interesting result is found in the basal–prismatic interaction mapping
when including the formation of junctions (Figure 2a). Indeed, in this case, a very
particular feature is observed. In all the other reported calculations, a periodic
symmetry of the interaction results is observed with respect to the two diagonals of
the mapping. This symmetry is important as it illustrates that to any combination of
1–2 interaction, a 2–1 analog exists and gives the same interaction result.
Surprisingly, the basal–prismatic calculations in Figure 2a do not reproduce such a
symmetry, but rather a periodicity along the vertical and horizontal axes. For this
reason, one can immediately conclude that it is statistically not equivalent to expand
basal dislocation loops in a prismatic forest rather than the opposite. Asymmetry in
the interaction strength between basal and prismatic slip systems can then be
anticipated.
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B. Devincre
3. Slip system interactions
In conformity with the forest model analysis in materials with low lattice friction,
Franciosi et al. [20] proposed to evaluate the strength of slip system interactions
through coefficients of an interaction matrix. This interaction matrix is defined with
a generalized Taylor equation measuring the critical shear stress, i, needed to
activate a slip system i in a forest microstructure made of slip system density j:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
ð1Þ
i ¼ b
aij j :
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j
Whereas measuring the interaction coefficients aij is a difficult task from
experiments, several recent works have proved that DD simulations have the
capacity to do it [17,21,22]. For this purpose, model simulations of the latent
hardening test can be devised to individually calculate each coefficient. This is made
by measuring the critical stress needed to force a mobile density of dislocations of a
given slip system to glide through a 3D forest density made of a single selected slip
system. Details of the computation procedure used for this type of simulations can be
found in [21] and are illustrated in Figure 4.
Here, all simulations were made by assuming that the temperature and the
dislocation density are high enough to assume that lattice friction is almost
completely overcome by thermal activation. Then, the dislocation mobility can be
represented by a simple viscous drag velocity law (v ¼ b/B) accounting for phonondamping [11]. Such a law is consistent with the viscous power law observed for the
plastic strain rate in ice single crystals during creep experiements [1,2]. In the
following, as little is known about dislocation mobility in ice, the parameter B was
empirically set to 5 105 Pa s, a value similar to the one experimentally observed in
Figure 4. (colour online). Illustration of the initial dislocation microstructure used in the
simulations of the latent hardening test. Periodic boundary conditions are applied to a volume
element of dimensions 9.3 9.9 11.5 mm3. Forest segments appear in green (f ¼ 1012 m2)
and primary segments in red (p ¼ f/ 5).
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Philosophical Magazine
FCC materials. Tests were performed to verify that all the DD simulation results
reported in this article are numerically unchanged for B values lower than 102 Pa s.
We first report on the results of the simulations dedicated to the calculation of
ajonc, the interaction coefficients between slip systems involving junction formations.
These interaction coefficients are obtained from Equation (1) and the steady-state
flow stress, i, calculated in the simulation of the latent hardening test. All calculated
values are reported in Table 2. In agreement with the previous section results, the
interaction between basal–prismatic and basal–pyramidal slip systems are weaker
than the prismatic–prismatic and the prismatic–pyramidal ones. More generally, the
ajonc coefficients calculated in the last two cases are similar to the strength of forest
interactions found in FCC crystals [21]. In contrast, the strength of the interactions
between basal slip systems and the prismatic or pyramidal slip systems is much
lower. For this reason, a weak influence of junction zipping–unzipping on
dislocation dynamics is expected in ice plasticity whatever the loading conditions.
In Figure 5, a justification is given for the ajonc asymmetry observed with the
basal–prismatic slip system interactions. The reason why ajonc(B ! P) ajonc(P ! B) is not a trivial result and cannot be explained from a static analysis
of the dislocation microstructure. The origin of this property is essentially dynamic.
In the two simulations involving basal and prismatic dislocations, the propagation of
Table 2. Interaction coefficient values associated to the formation of junctions between noncoplanar slip systems. The considered slip system interaction is defined in the first line. Arrows
point from the primary slip system to the forest slip system tested. A double arrow means
aij ¼ aji. The average numerical uncertainty is approximately 0.005.
Interaction
B!P
P!B
B ! 1
1 ! B
P$P
P $ 1
ajonc
0.034
0.082
0.05
0.06
0.1
0.08
Figure 5. (colour online). Thin foils of thickness 1 mm extracted from the two simulations
dedicated to the calculation of ajonc between basal and prismatic slip systems. Simulation
conditions are identical as in Figure 4. Panel (a) corresponds to a mobile dislocation density in
a basal slip system interacting with a forest microstructure made of segments randomly
distributed in a selected prismatic slip system. Image (b) is the opposite construction. Forest
segments appear in light blue, primary mobile dislocations in black and junctions as straight,
thick red lines. Note that the two configurations are dynamically not equivalent since the
character of dislocation segments zipping and unzipping junctions in the two calculations is
different.
242
B. Devincre
Table 3. Interaction coefficient values associated to collinear
annihilation between non-coplanar slip systems. The considered
slip system interaction is defined in the first line. The double arrow
means aij ¼ aji. The average numerical uncertainty is approximately
0.005.
Interaction
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acol
B$P
B $ 1
P $ 1
0.65
0.5
0.35
mobile dislocations through a forest dislocation density induces an accumulation of
many junctions restricting the plastic deformation. Intrinsically, the stability of
junctions in both cases is identical, but as an effect of dislocation line tension
anisotropy the bowing of dislocation segments is found to be easier in the basal
planes. Indeed, the screw character of the dislocations is then parallel to the junction
direction and therefore the stress needed to bow-out a segment connected with two
junctions is a minimum.
The dislocation structure observed in the simulations involving collinear
annihilation is specific and strongly differs from the dislocation microstructures
formed when the slip system interactions are controlled by junctions (like in
Figure 5). In the former case, the active slip system density is made of short segments,
which are strongly bowed out between collinear forest segments. Hence, the critical
stresses must be large to force plastic slip. The results of the 2 3 simulations
involving collinear annihilation used to calculate the acol coefficients are listed in
Table 3. As expected, these coefficients are much stronger than the ajonc equivalents.
Hence, a large forest strengthening is expected when two collinear slip systems
coexist in a dislocation microstructure. In addition, it can be noted that the
classification of the interaction strengths between slip systems is reversed. Now, the
strongest acol coefficients are calculated in the simulations testing dislocations glide
in basal slip systems.
4. Basal slip hardening in ice single crystals
The general shape of stress/strain curves observed in deformation test of ice single
crystals consists of a yield point followed by a strong softening. It is agreed that the
upper yield point corresponds to the onset of extensive mobile dislocation
multiplication on one or two basal slip systems, which results in a significant
decrease of their velocities at constant strain rate. At larger strains, a steady state is
reached, corresponding to an almost constant flow stress. Creep data for basal glide
of single crystals at 10 C, give axial flow stresses in the order of 0.2 and 2 MPa
depending on the stain rate [23]. A few reports mention a non-zero work hardening
rate at large strains, but when observed, the corresponding hardening coefficient is
always very low [24].
Such observations explain why most existing models for the plasticity of ice
assume a weak contribution of forest strengthening and why the self-interaction
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Philosophical Magazine
243
coefficient employed in the Taylor relation for isotropic forest hardening with
pffiffiffi
basal slip, ¼ b , is generally set to low values ( 0.1) [5,25,26].
The multiscale methodology used in Section 3 is now applied and compared with
experimental data to evaluate the precise contribution of forest strengthening and
hardening in basal slip. This evaluation is made assuming that the dislocation
processes controlling plastic flow in ice at large strain have some similitude with
those observed in FCC crystals when deformed in single slip conditions.
The main characteristic feature of single slip deformation is the occurrence of a
microstructure containing dipolar or multipolar dislocation bundles made up of long
primary edge segments (mostly part of prismatic loops) and short collinear jogs in
the cross-slip planes of the active slip system [4]. As discussed in [6,27], DD
simulations have shown that such a microstructure is formed through complex
dynamics. First, a few collinear long segments are produced in cross-slip planes by
screw dislocation annihilation with the help of the cross-slip mechanism, which is
mostly stress-driven at low stress. Further, the collinear edge segments formed in the
cross-slip planes behave as forest obstacles to mobile dislocations and are therefore
repeatedly chopped into smaller segments by the collinear annihilation mechanism
[17]. This process leads rapidly to an accumulation of many small collinear superjogs
along the mobile dislocations and therefore to forest strengthening via collinear
superjog pinning and dragging. In FCC materials, the experimentally observed
density of superjog segments is a constant fraction, typically k ¼ 10%, of the
dislocation density on the active slip system [28].
As the interaction between collinear slip systems is strong in ice, the possible
influence of collinear superjogs on the dislocations dynamics deserves some
attention. Following a suggestion initially made for plastic deformation in FCC
single slip [29], the strengthening effect expected from the presence of collinear
superjogs on dislocation lines can be cast into the form:
pffiffiffiffiffiffiffi
ð2Þ
¼ b a00 where a00 kacol is an effective self-interaction coefficient accounting for both the
constant ratio of collinear superjog density and the strength of the interaction
between mobile dislocation segments and collinear superjogs.
In addition, assuming that beyond the multiplication peak, the dislocation
density storage rate in ice single crystals is reasonably well estimated from a classical
Kocks–Mecking storage equation [30], the strain hardening coefficient associated
to collinear superjog strengthening on basal slip is expected to be of the form:
pffiffiffiffi0ffi
a0
a0
¼
ð3Þ
pffiffiffi ¼ 0
2 2K
pffiffiffiffiffiffiffi
where K ¼ a00 is, in single slip condition, a constant proportional to the
dislocation mean free path [7,27].
Considering the same fraction of collinear superjogs in ice as in FCC materials
(k 10%) and the same dislocation mean free path coefficient (K ¼ 180), simple
predictions on basal slip system properties can be drawn from Equations (2) and (3).
The results of such calculations at two different dislocation densities and assuming
244
B. Devincre
Table 4. DD
pffiffiffiffiffisimulation predictions for the basal slip self-interaction
coefficient a00 , the plastic flow stresses 1 and 2 for basal dislocation
densities equal to 1 ¼ 1011 m2 and 2 ¼ 1013 m2 and the strain hardening
coefficient . Such calculations are made for a density of collinear superjogs
of 0.1 and a very large mean free path for mobile dislocations, i.e. K ¼ 180.
The occurrence of collinear superjogs inherited from cross-slip in either
prismatic or pyramidal glide planes is evaluated.
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P superjogs
1 superjogs
pffiffiffiffi0ffi
a0
1 (MPa)
2 (MPa)
/
0.25
0.22
0.15
0.13
1.5
1.3
1/5500
1/7200
either the existence of collinear superjogs in the prismatic or in the pyramidal glide
planes are listed in Table 4.
Whatever the slip system considered for the collinear superjogs, DD simulations
predict a self-interaction coefficient for the basal slip larger that the Taylor
coefficient, ¼ 0.1, usually considered in the literature. Little is know about the real
density of collinear superjogs existing in the ice dislocation microstructure, but it is
reasonable to guess that such density is not negligible as the cross-slip process is
commonly observed experimentally. Here, assuming that the dislocation dynamics in
basal slip is characterized
by the same rate of collinear superjog storage as in FCC
pffiffiffiffiffi
0:22.
crystals we find a00 4 p
ffiffiffiffiffi
Considering such a a00 value for the basal self-interaction, Equation (2) gives a
flow stress amplitude in the order of 0.15 and 1.5 MPa with basal dislocation
densities in the order of 1011 and 1013 m2, respectively. Such a flow stress level
represents a significant fraction of the actual experimental values. Hence, it is
reasonable to assume that forest strengthening in ice may be larger than commonly
assumed.
Finally, it is important to realize that the latter suggestion is still compatible with
the small strain hardening rate experimentally observed in ice. Indeed, the results
reported in Table 4 always give a tiny strain hardening rate whatever the hypothesis
made on the dislocation microstructure.
5. Conclusion
Three-dimensional DD simulations were performed to investigate forest strengthening in ice single crystals deformed in basal slip. The following conclusions were
obtained:
(i) The first example of asymmetric interaction between slip systems is found in
ice. This result should apply to all HCP crystals with dislocations in basal
and prismatic slips systems.
(ii) Slip system interactions involving collinear annihilations potentially introduce in ice crystals a strong forest strengthening. Conversely, slip system
interactions involving junctions are weak and cannot contribute significantly
to critical stresses and strain hardening.
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(iii) A simple model, initially proposed for single slip in FCC crystals, is applied
to basal slip and yields results consistent with experimental data for the
plastic flow stress and strain hardening rate for ice single crystals. This
suggests that the interactions between segments gliding in basal planes and a
small density of short collinear segments (superjogs) inherited from
relaxation processes in the prismatic or pyramidal glide planes may be
essential in ice to quantitatively predict the latter plastic properties.
(iv) The difference between the two strain hardening rates estimated for basal
slip when considering the existence of collinear superjogs either in the
prismatic or in the pyramidal planes is small. For this reason, no
information can be extracted from the present analysis regarding a possible
preference for basal dislocation to cross-slip either in the prismatic or in the
pyramidal slip planes.
Acknowledgments
This paper is dedicated to the memory of Patrick Veyssière. The author is indebted to
G. Monnet for having developed the HCP crystal module in the DD simulation code
microMegas. The support of the research project ElasticGB, through Grant ANR-09-BLAN0079, is acknowledged.
Note
1. See the mM home page at: http://zig.onera.fr/mm_home_page
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