248
P. Franciosi and M. Spagnuolo
denoted as “PCW”, it points the essential role in it of the (rapidly reintroduced)
GOs, which are an advantageous equivalent to the Eshelby tensors, and it presents
the calculation of these operators from the Radon and inverse (RT-IRT ) transform
method which was used in all the cited works by the authors team on infinite inclusion
patterns. Section 15.3 reports and discusses some of the obtained results, in terms of
their representative mean GOs (mGOs) which determine the effective properties of
the composites embedding them, as well as the (mean) stress and strain fields in the
pattern in comparison with those applied to the composite: from the cases of axial
infinite alignments of spheres and spheroids (Franciosi 2010) and of finite, flat or
long, cylinders (Franciosi et al. 2015) and then from the cases of planar alignments
of infinite parallel cylinders (Franciosi et al. 2019) and rectangular and square beams
(Franciosi 2018; Spagnuolo et al. 2020). Based on the results of these works on
“elementary” alignments, it is next in Sect. 15.4 discussed and exemplified how mD
networks can be built to be then embedded in a matrix and possibly be evolving under
straining of the composite structure (the new case of inhomogeneous axial alignment
of inclusions is introduced there), as was successfully attempted in building 1D fiber
bundles and 3D fiber networks from planar fiber arrays. The last example of the
building of a 3D fiber network from assembling planar arrays of beams, reported
from (Spagnuolo et al. 2020) with the array interconnections being inspired by those
of pantographic 2D fiber architectures (Alibert et al. 2003; Andreaus et al. 2018;
Barchiesi and Placidi 2017; Barchiesi et al. 2019; Boutin et al. 2017; Cuomo et al.
2017; dell’Isola et al. 2015, 2016, 2017; Eremeyev et al. 2016, 2019; Eremeyev
2018; Placidi et al. 2016, 2017, 2018, 2019), points that mathematical constraints
describing the physical interconnection mechanisms can substitute them efficiently if
hard to account for in the pattern description. Section 15.5 evokes different potential
other application fields of the presented results beyond the one of linear static elasticlike (and dielectric-like) ones, in pointing the remaining difficulties to overcome for
progressing further. Section 15.6 concludes.
15.2 Effective (Piece-Wise) Linear Elastic-like Properties
of Composites of the Matrix-Reinforced Type
We here select the two-point statistic framework of (Ponte-Castaneda and Willis
1995), which is an extension of the well-defined variational approach from Hashin
and Shtrikman (1963), to be referred to as “PCW” and “HS”, respectively, for a
uniform medium with (stiffness) properties C
M in which are embedded elements of
one or n phases (V j), with properties C
V j and volume fractions f V j , with an a priori
single request of not overpassing space filling at
n
j=1 f V j = 1 (that is a null matrix
phase fraction f M = 0). For the following, it is enough to recall that it takes the
generic form:
P. Franciosi and M. Spagnuolo
denoted as “PCW”, it points the essential role in it of the (rapidly reintroduced)
GOs, which are an advantageous equivalent to the Eshelby tensors, and it presents
the calculation of these operators from the Radon and inverse (RT-IRT ) transform
method which was used in all the cited works by the authors team on infinite inclusion
patterns. Section 15.3 reports and discusses some of the obtained results, in terms of
their representative mean GOs (mGOs) which determine the effective properties of
the composites embedding them, as well as the (mean) stress and strain fields in the
pattern in comparison with those applied to the composite: from the cases of axial
infinite alignments of spheres and spheroids (Franciosi 2010) and of finite, flat or
long, cylinders (Franciosi et al. 2015) and then from the cases of planar alignments
of infinite parallel cylinders (Franciosi et al. 2019) and rectangular and square beams
(Franciosi 2018; Spagnuolo et al. 2020). Based on the results of these works on
“elementary” alignments, it is next in Sect. 15.4 discussed and exemplified how mD
networks can be built to be then embedded in a matrix and possibly be evolving under
straining of the composite structure (the new case of inhomogeneous axial alignment
of inclusions is introduced there), as was successfully attempted in building 1D fiber
bundles and 3D fiber networks from planar fiber arrays. The last example of the
building of a 3D fiber network from assembling planar arrays of beams, reported
from (Spagnuolo et al. 2020) with the array interconnections being inspired by those
of pantographic 2D fiber architectures (Alibert et al. 2003; Andreaus et al. 2018;
Barchiesi and Placidi 2017; Barchiesi et al. 2019; Boutin et al. 2017; Cuomo et al.
2017; dell’Isola et al. 2015, 2016, 2017; Eremeyev et al. 2016, 2019; Eremeyev
2018; Placidi et al. 2016, 2017, 2018, 2019), points that mathematical constraints
describing the physical interconnection mechanisms can substitute them efficiently if
hard to account for in the pattern description. Section 15.5 evokes different potential
other application fields of the presented results beyond the one of linear static elasticlike (and dielectric-like) ones, in pointing the remaining difficulties to overcome for
progressing further. Section 15.6 concludes.
15.2 Effective (Piece-Wise) Linear Elastic-like Properties
of Composites of the Matrix-Reinforced Type
We here select the two-point statistic framework of (Ponte-Castaneda and Willis
1995), which is an extension of the well-defined variational approach from Hashin
and Shtrikman (1963), to be referred to as “PCW” and “HS”, respectively, for a
uniform medium with (stiffness) properties C
M in which are embedded elements of
one or n phases (V j), with properties C
V j and volume fractions f V j , with an a priori
single request of not overpassing space filling at
n
j=1 f V j = 1 (that is a null matrix
phase fraction f M = 0). For the following, it is enough to recall that it takes the
generic form: