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P. Franciosi and M. Spagnuolo
it remains not permitted to treat multi-phased inclusion patterns or fiber networks in
the PCW framework of Eqs. (15.1), for a single mGO for elements of different phases
(in the sense of different properties) cannot manage with phase to phase interactions
in a mixed network. The route to follow in such important cases of multi-phased
inclusion patterns is indicated by Eq. (15.5a) which consider two different (E, F)
inclusion types: whether the type differences be shapes or properties, the global
mGO shares into three parts, the mGOs for the sub-patterns E and F and the third
mixed interaction mGO, as here partly explicated for a mixture of spheroids and
of finite cylinder coaxial elements. As far as the E, F elements have same physical
(elastic or else) properties and are only shape-different, a single mGO can be used
as indicated. If the cylinders and the spheroids do not have same properties, then
three distinct mGOs need be separately accounted for, say shortly speaking the E
and the F uniform ones and the E–F interaction one, using the n-site homogenization
method from (Fassi-Fehri et al. 1989) specialized to two elastic phases in (Franciosi
and Charles 2016b). The same three parts would be needed if aligning all shapeidentical inclusions but with two different property types. Equations (15.5a, b) here
clearly show how infinite patterns mixing elements with different properties can also
be treated following the same here proposed method, with this specific variant that
a single equation for a single phase in terms of properties becomes a n-set equation
when the phase number with different properties in the pattern is n.
At last, it cannot be ignored here, as far as possibly large elastic deformations
may be concerned, the as well tremendous developments of second gradient analyses
especially devoted to highly deformable fiber networks also mostly inspired by the
pantograph specific structure type, owing to torsion and bending components of the
deformation (Mindlin 1965; Germain 1973; Sciarra et al. 2007; Pideri and Seppecher
1997; Rahali et al. 2015; dell’Isola et al. 2019) which are often hard to disregard as
here done. However, explicating GOs for second gradient deformation behavior to
extend the here discussed type of homogenization schemes is a highly challenging
domain. Hence, if the here built structures are “pantographic-inspired”, they can
only deform moderately as described. If this constitutes a widely opened field for
further research on innovative composites, some other ones are briefly discussed in
the next section for which, to various extents, applications of the presented results
for describing complicated structures from a global mGO are more at hand. Second
gradient approaches of classical Eshelby/Green problems with more or less implicit
connections to the RT-IRT method can be found in (Drugan and Willis 1996; Gao
and Ma 2009; Wu et al. 2015), with references cited there in.
15.5 Other Potential Application Extension Directions
Not much far after the Eshelby seminal publications in the fifties, a lot of research
started to explore further the potential application domains of the Eshelby results on
embedded ellipsoidal inclusions at various, from nano to mega, scales and “generalized” Eshelby tensors for inclusions have been calculated in various other contexts as,
P. Franciosi and M. Spagnuolo
it remains not permitted to treat multi-phased inclusion patterns or fiber networks in
the PCW framework of Eqs. (15.1), for a single mGO for elements of different phases
(in the sense of different properties) cannot manage with phase to phase interactions
in a mixed network. The route to follow in such important cases of multi-phased
inclusion patterns is indicated by Eq. (15.5a) which consider two different (E, F)
inclusion types: whether the type differences be shapes or properties, the global
mGO shares into three parts, the mGOs for the sub-patterns E and F and the third
mixed interaction mGO, as here partly explicated for a mixture of spheroids and
of finite cylinder coaxial elements. As far as the E, F elements have same physical
(elastic or else) properties and are only shape-different, a single mGO can be used
as indicated. If the cylinders and the spheroids do not have same properties, then
three distinct mGOs need be separately accounted for, say shortly speaking the E
and the F uniform ones and the E–F interaction one, using the n-site homogenization
method from (Fassi-Fehri et al. 1989) specialized to two elastic phases in (Franciosi
and Charles 2016b). The same three parts would be needed if aligning all shapeidentical inclusions but with two different property types. Equations (15.5a, b) here
clearly show how infinite patterns mixing elements with different properties can also
be treated following the same here proposed method, with this specific variant that
a single equation for a single phase in terms of properties becomes a n-set equation
when the phase number with different properties in the pattern is n.
At last, it cannot be ignored here, as far as possibly large elastic deformations
may be concerned, the as well tremendous developments of second gradient analyses
especially devoted to highly deformable fiber networks also mostly inspired by the
pantograph specific structure type, owing to torsion and bending components of the
deformation (Mindlin 1965; Germain 1973; Sciarra et al. 2007; Pideri and Seppecher
1997; Rahali et al. 2015; dell’Isola et al. 2019) which are often hard to disregard as
here done. However, explicating GOs for second gradient deformation behavior to
extend the here discussed type of homogenization schemes is a highly challenging
domain. Hence, if the here built structures are “pantographic-inspired”, they can
only deform moderately as described. If this constitutes a widely opened field for
further research on innovative composites, some other ones are briefly discussed in
the next section for which, to various extents, applications of the presented results
for describing complicated structures from a global mGO are more at hand. Second
gradient approaches of classical Eshelby/Green problems with more or less implicit
connections to the RT-IRT method can be found in (Drugan and Willis 1996; Gao
and Ma 2009; Wu et al. 2015), with references cited there in.
15.5 Other Potential Application Extension Directions
Not much far after the Eshelby seminal publications in the fifties, a lot of research
started to explore further the potential application domains of the Eshelby results on
embedded ellipsoidal inclusions at various, from nano to mega, scales and “generalized” Eshelby tensors for inclusions have been calculated in various other contexts as,