15 Homogenization-Based Mechanical Behavior Modeling of Composites …
247
2006; McCue et al. 2016) or ceramics (Pavese et al. 2007) in passing by woven
textile composites (Udhayaraman and Mulay 2017), bones (Kinney et al. 2005),
metal matrix (Peng et al. 2001), compacted powders (Poquillon et al. 2002) or metamaterials (Yang et al. 2018), existing three-phase structures with phase concentration
domains exhibiting tri-continuity (Han et al. 2009; Ryoo 2009) as well as assemblages
where phases duplicate in partly continuous and partly discontinuous (Veenstra et al.
2000; Limodin et al. 2007; Torres et al. 2012) made this extension to multiple cocontinuity of phases worthy of interest as applications show in (Franciosi 2020a).
In this enormous flux of developments for analysing the overall behavior of
increasingly varied composite structures several issues are still scarcely investigated:
non-ellipsoidal inhomogeneity shapes (Walpole 1967; Kinoshita and Mura 1971;
Chiu 1977; Rodin 1996; Franciosi 2005; Traxl and Lackner 2018), which are still so
far abusively approximated by ellipsoids [a sphere would then represent both a cube
and an octahedron which are opposite polyhedrons (Franciosi and Lormand 2004)
and a finite cylinder is badly represented by a spheroid of same aspect ratio (Franciosi et al. 2016)]; aside of inclusions distributions (Ponte-Castaneda and Willis 1995;
Buryachenko 2001; Franciosi and Lebail 2004), many specific inclusion patterns with
potentially significant interactions when the group density becomes high and for any
pattern size, possibly up to infinite elements, still need improved descriptions. Limit
cases of interest are when the elements in the group become at contact, what may
possibly yield to connectivity inversions (Franciosi and Gaertner 1998) and to the
transition toward interconnected pattern elements. This last problem is the one to
solve for describing an inclusion pattern turning from an embedded (discontinuous)
phase into a continuous (networked) one or a multi-continuous state change in a
n-phase composite.
The authors and co-workers have addressed several of these questions in recent
works (Franciosi 2010; Franciosi et al. 2015, 2019; Spagnuolo et al. 2020; Franciosi
2020b), offering some results which may serve further to estimate effective properties
for several composite structures comprising a uniform matrix and a second phase (or
more): A second phase can be described by several infinite arrangements (as linear
or planar arrays) of elements which can then in turn be spatially organized into a
single pattern having one- or multi-directional (mD) co-continuity with a reference
matrix phase (the volume fraction of which possibly going down to zero, that is
infinitesimal). Such arrangements can thus realize a mD-networked pattern of second
phase (m = 1, 2, 3, …, omni), the element interconnections being either physically
present or mathematically described, as will be commented and exemplified. We here
summarize, with a minimum of equations to be complemented by the referenced
papers, these previous results with this particular viewpoint in mind, and we aim
at projecting them toward potential diversified applications on complex composite
structures and architectures at different (from micro to macro) scales. A new type
of inhomogeneous axial alignment of inclusions, built from a mixed alignment of
several inclusion shapes, is presented and solved (not in the details) for spheroids
and finite cylinders axial associations, serving as support to the discussion.
Section 15.2 briefly recalls the foundations of the homogenization-based model
we make use, namely the (Ponte-Castaneda and Willis 1995) framework, to be
247
2006; McCue et al. 2016) or ceramics (Pavese et al. 2007) in passing by woven
textile composites (Udhayaraman and Mulay 2017), bones (Kinney et al. 2005),
metal matrix (Peng et al. 2001), compacted powders (Poquillon et al. 2002) or metamaterials (Yang et al. 2018), existing three-phase structures with phase concentration
domains exhibiting tri-continuity (Han et al. 2009; Ryoo 2009) as well as assemblages
where phases duplicate in partly continuous and partly discontinuous (Veenstra et al.
2000; Limodin et al. 2007; Torres et al. 2012) made this extension to multiple cocontinuity of phases worthy of interest as applications show in (Franciosi 2020a).
In this enormous flux of developments for analysing the overall behavior of
increasingly varied composite structures several issues are still scarcely investigated:
non-ellipsoidal inhomogeneity shapes (Walpole 1967; Kinoshita and Mura 1971;
Chiu 1977; Rodin 1996; Franciosi 2005; Traxl and Lackner 2018), which are still so
far abusively approximated by ellipsoids [a sphere would then represent both a cube
and an octahedron which are opposite polyhedrons (Franciosi and Lormand 2004)
and a finite cylinder is badly represented by a spheroid of same aspect ratio (Franciosi et al. 2016)]; aside of inclusions distributions (Ponte-Castaneda and Willis 1995;
Buryachenko 2001; Franciosi and Lebail 2004), many specific inclusion patterns with
potentially significant interactions when the group density becomes high and for any
pattern size, possibly up to infinite elements, still need improved descriptions. Limit
cases of interest are when the elements in the group become at contact, what may
possibly yield to connectivity inversions (Franciosi and Gaertner 1998) and to the
transition toward interconnected pattern elements. This last problem is the one to
solve for describing an inclusion pattern turning from an embedded (discontinuous)
phase into a continuous (networked) one or a multi-continuous state change in a
n-phase composite.
The authors and co-workers have addressed several of these questions in recent
works (Franciosi 2010; Franciosi et al. 2015, 2019; Spagnuolo et al. 2020; Franciosi
2020b), offering some results which may serve further to estimate effective properties
for several composite structures comprising a uniform matrix and a second phase (or
more): A second phase can be described by several infinite arrangements (as linear
or planar arrays) of elements which can then in turn be spatially organized into a
single pattern having one- or multi-directional (mD) co-continuity with a reference
matrix phase (the volume fraction of which possibly going down to zero, that is
infinitesimal). Such arrangements can thus realize a mD-networked pattern of second
phase (m = 1, 2, 3, …, omni), the element interconnections being either physically
present or mathematically described, as will be commented and exemplified. We here
summarize, with a minimum of equations to be complemented by the referenced
papers, these previous results with this particular viewpoint in mind, and we aim
at projecting them toward potential diversified applications on complex composite
structures and architectures at different (from micro to macro) scales. A new type
of inhomogeneous axial alignment of inclusions, built from a mixed alignment of
several inclusion shapes, is presented and solved (not in the details) for spheroids
and finite cylinders axial associations, serving as support to the discussion.
Section 15.2 briefly recalls the foundations of the homogenization-based model
we make use, namely the (Ponte-Castaneda and Willis 1995) framework, to be