15 Homogenization-Based Mechanical Behavior Modeling of Composites …
257
Fig. 15.2 Evolution of the mGO terms for sphere (left) and cylinder (right) alignments versus
inter-distance
one for planar cylinder alignments is confirmed for similar arrays of square beams
in (Franciosi 2018; Spagnuolo et al. 2020).
A net consequence of this influence zone finiteness is on the second GO showing
up in the PCW estimate (Eq. 15.1b) which represents some spatial distribution of
the inclusions or patterns of them: If the considered pattern is infinite, considering
distributions of infinite patterns does not make sense anymore, and this distribution
operator more likely corresponds to a finite “window” on the infinite matrix substructure related to a representative zone in terms of interactions. It then well corresponds
to that influence zone out of which the rest of the pattern does not contribute to the
overall properties of the composite. Under this viewpoint, the zone is more characterized by the (possibly evolving) matrix properties and by the details of the infinite
pattern structure. This assumption has been successfully tested in (Franciosi et al.
2019) for transversally compressed 1D bundles of parallel fibers embedded in an
isotropic matrix: Estimates from the PCW modeling (Eq. 15.1b) where the distribution GO was kept invariant (cylindrical symmetry) while the fiber bundle mGO was
evolving (such as to change the number and position of the fibers in the influence
zone) gave consistent results (in terms of effective elastic modulus evolutions) with
numerically calculated ones for comparisons. In contrast, a distribution symmetry
taken to evolve homothetically to the fiber arrangement in a finite pattern did not
yield relevant estimates.
In comparison with the simple forms of Eqs. (15.7a, b) for spheres and infinite
cylinders, the mGOs obtained for coaxial finite cylinders and for rectangular or
257
Fig. 15.2 Evolution of the mGO terms for sphere (left) and cylinder (right) alignments versus
inter-distance
one for planar cylinder alignments is confirmed for similar arrays of square beams
in (Franciosi 2018; Spagnuolo et al. 2020).
A net consequence of this influence zone finiteness is on the second GO showing
up in the PCW estimate (Eq. 15.1b) which represents some spatial distribution of
the inclusions or patterns of them: If the considered pattern is infinite, considering
distributions of infinite patterns does not make sense anymore, and this distribution
operator more likely corresponds to a finite “window” on the infinite matrix substructure related to a representative zone in terms of interactions. It then well corresponds
to that influence zone out of which the rest of the pattern does not contribute to the
overall properties of the composite. Under this viewpoint, the zone is more characterized by the (possibly evolving) matrix properties and by the details of the infinite
pattern structure. This assumption has been successfully tested in (Franciosi et al.
2019) for transversally compressed 1D bundles of parallel fibers embedded in an
isotropic matrix: Estimates from the PCW modeling (Eq. 15.1b) where the distribution GO was kept invariant (cylindrical symmetry) while the fiber bundle mGO was
evolving (such as to change the number and position of the fibers in the influence
zone) gave consistent results (in terms of effective elastic modulus evolutions) with
numerically calculated ones for comparisons. In contrast, a distribution symmetry
taken to evolve homothetically to the fiber arrangement in a finite pattern did not
yield relevant estimates.
In comparison with the simple forms of Eqs. (15.7a, b) for spheres and infinite
cylinders, the mGOs obtained for coaxial finite cylinders and for rectangular or