15 Homogenization-Based Mechanical Behavior Modeling of Composites …
261
D ell (θ ) = R
1 +
ζ 2 − 1
cos 2 θ and ζ = a/R < 1 with regard to the spheroid
direction along the x3 alignment axis such that ˜
s
V E
(z, θ) contributes to the θ integrals.
This is likely at hand for the integrals related to spheroid shapes have been solved
already, but the total resolution is let to further works, together with finalizing the
sphere case.
An important point here is, as stressed already, that such an alignment with
elements at contact is not equivalent to connected elements although they have
the same (undifferentiated) mGOs: Depending on the case, it would be necessary
to additionally assign the elements in the alignments to remain at contact or not.
What means substituting mathematical interconnection (connectedness) conditions
to supply for missing physical ones. If the exemplified mixed alignment is kept
undissociated by appropriately chosen conditions, the axial stiffness for the spheroid
array will be enhanced by the added cylinders in relevant manner with regard to the
structure. In comparison, averaging the spheroid alignment mGO with the GO of an
infinite cylinder to similarly give axial stiffness to such alignments as if the array was
connected is an efficient assumption that saves more needs of interface conditions,
but it is arbitrary. Being necessary to finalize the mGO calculation for such mixed
alignments in order to comparing and discussing further, this is let to forthcoming
papers.
15.4 From Elementary Alignments to Bundles or Networks
The second pointed fact in this work is that determining mGOs for various patterned
of networked substructures is simplified when fully using the geometrical nature of
the RT-IRT method. This section therefore focuses on the mGOs obtained that way
in the case of substructures that can be represented by axial and planar alignments of
similar (possibly being composite) elements, possibly up to infinite numbers, from
which more complicated patterns or networked could be represented, as is exemplified. This section addresses the more complex structures one can represent/describe
from the treated elementary ones recalled in Sect. 15.3.
Using axial alignments of spheres, spheroids, finite cylinders or mixtures of them
allows to building more complex patterns or networks as 1D bundles of such alignments, or multi-directional (mD) ones which can also be omni-directional as an
isotropic structure for example would be. Thanks to the knowledge of the mGOs for
these axial (axially symmetric) arrays, this is as easily at hand than creating bundles
of fibers, with only the knowledge of the rotation tensor to apply on the 1D mGO
and averaging over directional mGOs is a relevant approximate for the mGO of such
a, pretty much networked already, pattern. It is noteworthy that depending on how
the averaging is weighted, it is possible to build isotropic or anisotropic patterns, the
use of ellipsoidal spatial distributions being still an efficient weighting method. In
similar way, planar arrays of parallel fibers can be used to build 1D fiber bundles,
with a representative mGO obtained in averaging the mGOs of the planar fiber arrays
261
D ell (θ ) = R
1 +
ζ 2 − 1
cos 2 θ and ζ = a/R < 1 with regard to the spheroid
direction along the x3 alignment axis such that ˜
s
V E
(z, θ) contributes to the θ integrals.
This is likely at hand for the integrals related to spheroid shapes have been solved
already, but the total resolution is let to further works, together with finalizing the
sphere case.
An important point here is, as stressed already, that such an alignment with
elements at contact is not equivalent to connected elements although they have
the same (undifferentiated) mGOs: Depending on the case, it would be necessary
to additionally assign the elements in the alignments to remain at contact or not.
What means substituting mathematical interconnection (connectedness) conditions
to supply for missing physical ones. If the exemplified mixed alignment is kept
undissociated by appropriately chosen conditions, the axial stiffness for the spheroid
array will be enhanced by the added cylinders in relevant manner with regard to the
structure. In comparison, averaging the spheroid alignment mGO with the GO of an
infinite cylinder to similarly give axial stiffness to such alignments as if the array was
connected is an efficient assumption that saves more needs of interface conditions,
but it is arbitrary. Being necessary to finalize the mGO calculation for such mixed
alignments in order to comparing and discussing further, this is let to forthcoming
papers.
15.4 From Elementary Alignments to Bundles or Networks
The second pointed fact in this work is that determining mGOs for various patterned
of networked substructures is simplified when fully using the geometrical nature of
the RT-IRT method. This section therefore focuses on the mGOs obtained that way
in the case of substructures that can be represented by axial and planar alignments of
similar (possibly being composite) elements, possibly up to infinite numbers, from
which more complicated patterns or networked could be represented, as is exemplified. This section addresses the more complex structures one can represent/describe
from the treated elementary ones recalled in Sect. 15.3.
Using axial alignments of spheres, spheroids, finite cylinders or mixtures of them
allows to building more complex patterns or networks as 1D bundles of such alignments, or multi-directional (mD) ones which can also be omni-directional as an
isotropic structure for example would be. Thanks to the knowledge of the mGOs for
these axial (axially symmetric) arrays, this is as easily at hand than creating bundles
of fibers, with only the knowledge of the rotation tensor to apply on the 1D mGO
and averaging over directional mGOs is a relevant approximate for the mGO of such
a, pretty much networked already, pattern. It is noteworthy that depending on how
the averaging is weighted, it is possible to build isotropic or anisotropic patterns, the
use of ellipsoidal spatial distributions being still an efficient weighting method. In
similar way, planar arrays of parallel fibers can be used to build 1D fiber bundles,
with a representative mGO obtained in averaging the mGOs of the planar fiber arrays