272
P. Franciosi and M. Spagnuolo
from these different cited works among many, it appears that even if the dynamic
problems would in general be solved by a substantial amount when the effective
elastic static behavior for the medium, in terms of effective properties, is known first,
overall calculations to estimate specific wave features, as propagation or dispersion
(scattering cross section) resonance, remain difficult, even for simple cases as aligned
spheroids in an isotropic matrix, as shown in details in (Willis 1980b). Prior to safely
extend from static to dynamic problems the here presented results on mGOs for
“pattern complexity direction,” a major issue to examine is to what extent considering
a global pattern mGO in a single equation does not lose too much of local effects
in wave characteristic concerns, compared with the coupled equation system that
usually describes an inclusion arrangement. Difficulties are expectable too for the
bi- and multi-continuous composite structures which are still not that much examined
as here done in static property contexts. To the authors knowledge, more efficient
frame in terms of microstructure simplifying is still to be proposed.
15.6 Conclusion
The presented results establish that considering infinite patterns of inclusions
embedded in a matrix provides relevant estimates of static linear properties within
a classical first gradient homogenization framework for such composite materials,
using a mean Green operator (mGO) for these patterns at the place of single inclusions or of finite groups of them as is the most frequently done, with a single equation
to solve instead of a system (the substitution is not permitted if the inclusions do not
all have same properties). One shown consequence to referring to infinite patterns
is to modify the notion of spatial distribution of elements, whether them be single
elements or groups. It has been proposed and checked efficient that the relevant
representative domain for evolving infinite patterns was the shape of the influence
(interaction) zone for its elements, the variation of which was firstly related to the
embedding medium spatial anisotropy symmetry and the shape anisotropy characteristics of the elements, both possibly modified but in second row, by an applied
strain. These mGOs are of interest only if they account correctly for the pair interaction between the elements and are not only a mean of the element mGOs. This type
of mGO is unfortunately not at easy analytical hand in all situations of interest and
especially when the matrix properties are far from being isotropic. Even when at hand
analytically, its major interest is to possibly evolve easily in following the current
deformed state of the composite it represents. In that respect, it has been shown
through the several results from the authors and co-workers here synthesized that the
geometrical nature of the RT-IRT method, what has additionally provided the quite
simple so-called decomposition method for determining mGOs of various inclusion
patterns, is a powerful route toward describing complex arrangements of infinite and
possibly networked patterns as this work has hopefully shown. Applications toward
coupled properties, linearizable ones and dynamic ones are discussed.
P. Franciosi and M. Spagnuolo
from these different cited works among many, it appears that even if the dynamic
problems would in general be solved by a substantial amount when the effective
elastic static behavior for the medium, in terms of effective properties, is known first,
overall calculations to estimate specific wave features, as propagation or dispersion
(scattering cross section) resonance, remain difficult, even for simple cases as aligned
spheroids in an isotropic matrix, as shown in details in (Willis 1980b). Prior to safely
extend from static to dynamic problems the here presented results on mGOs for
“pattern complexity direction,” a major issue to examine is to what extent considering
a global pattern mGO in a single equation does not lose too much of local effects
in wave characteristic concerns, compared with the coupled equation system that
usually describes an inclusion arrangement. Difficulties are expectable too for the
bi- and multi-continuous composite structures which are still not that much examined
as here done in static property contexts. To the authors knowledge, more efficient
frame in terms of microstructure simplifying is still to be proposed.
15.6 Conclusion
The presented results establish that considering infinite patterns of inclusions
embedded in a matrix provides relevant estimates of static linear properties within
a classical first gradient homogenization framework for such composite materials,
using a mean Green operator (mGO) for these patterns at the place of single inclusions or of finite groups of them as is the most frequently done, with a single equation
to solve instead of a system (the substitution is not permitted if the inclusions do not
all have same properties). One shown consequence to referring to infinite patterns
is to modify the notion of spatial distribution of elements, whether them be single
elements or groups. It has been proposed and checked efficient that the relevant
representative domain for evolving infinite patterns was the shape of the influence
(interaction) zone for its elements, the variation of which was firstly related to the
embedding medium spatial anisotropy symmetry and the shape anisotropy characteristics of the elements, both possibly modified but in second row, by an applied
strain. These mGOs are of interest only if they account correctly for the pair interaction between the elements and are not only a mean of the element mGOs. This type
of mGO is unfortunately not at easy analytical hand in all situations of interest and
especially when the matrix properties are far from being isotropic. Even when at hand
analytically, its major interest is to possibly evolve easily in following the current
deformed state of the composite it represents. In that respect, it has been shown
through the several results from the authors and co-workers here synthesized that the
geometrical nature of the RT-IRT method, what has additionally provided the quite
simple so-called decomposition method for determining mGOs of various inclusion
patterns, is a powerful route toward describing complex arrangements of infinite and
possibly networked patterns as this work has hopefully shown. Applications toward
coupled properties, linearizable ones and dynamic ones are discussed.