15 Homogenization-Based Mechanical Behavior Modeling of Composites …
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has one property type among the TMEE sub-cases what needs at most transversal
isotropy (TI) symmetry. Owing to the vast literature in these cases, an important
classification to be done is whether the works address the specific determination
of GOs in a uniform infinite medium of most general anisotropy, or directly face
estimating overall properties of composite structures. In the former group, a very
instructive work of that type is the one of (Buroni and Saez 2010) which partly uses
the RT-IRT method plus the Stroh formalism to explicit all the necessary integral
roots in the general anisotropic elasticity case. The obtained complicated analytical
form can be used in further numerical calculations. In the second group, when the
matrix has isotropic, hence uncoupled, properties (Poizat and Sester 1999; Clyne
et al. 2005; Liu 2011), it can be considered that the problem of an infinite inclusion
pattern or fiber network having any TMEE properties of any anisotropic symmetry
can be treated as a generalized thermo-elastic problem with an isotropic matrix and an
embedded phase of the here discussed morphology, possibly with TMEE properties
(remind that as pointed in the previous section problems of a multi-phased network
in a matrix cannot be treated by a single equation as for a single-phase pattern or
network). These isotropic matrix cases already represent quite a lot of structures of
interest.
When not isotropic, the matrix phase properties are hardly found in the literature
to be less than transversally isotropic, what is already so challenging that numerical
calculations are the most often used. And even so, most results remain restricted to
axially symmetric problems that is reinforcing phase properties and shapes axially
symmetric with the matrix TI symmetry. See (Huang et al. 1998; Mikata 2001; Hou
and Leung 2004; Chen et al. 2004; Lee et al. 2005) to cite a few. For example, to
the authors knowledge, there are not even Eshelby-based approaches proposing fully
explicit analytical solutions for off-symmetric structures in a TI matrix, although the
formalism for them is frequently presented. In the already cited application of the
present RT-IRT method in (Franciosi 2013) for MEE axially symmetric (spheroids,
laminates and fibers) inclusions in a MEE matrix, a newly obtained result toward
the follow-up of evolving patterns was the analytical mGO calculations for oblique
orientations of the inclusion axis with regard to the symmetry axis of the matrix,
which to the authors knowledge has no anterior proposal. The resulting effective
properties do not have TI symmetry anymore, yet the result for spheroids is obtained
from implementing analytical explicit integrals (oblique infinite fibers and laminates
in a TI matrix can be treated in full analytical manner). Not all published results on
the GO variations with orientations for spheroids were presented in (Barboura and
Franciosi 2016) in the TI elastic case. In addition to the few presented graphs in the
cited short reference, Fig. 15.10 illustrates the case of stiffer axial than transverse
properties, the (not shown) converse being indeed very different and not just reversed.
Applications to piezo-elastic problems are directly at hand, just needing numerical
implementation. Studying this inclusion orientation effect in not isotropic (here TI)
matrices shows that “tuning” a property does not necessarily correspond to in-axes
orientation choices but can be (case per case) in between.
When an inclusion pattern has no specific orientation parameter as a 1D fiber
bundle, the orientation changes with some matrix property anisotropy becomes a
269
has one property type among the TMEE sub-cases what needs at most transversal
isotropy (TI) symmetry. Owing to the vast literature in these cases, an important
classification to be done is whether the works address the specific determination
of GOs in a uniform infinite medium of most general anisotropy, or directly face
estimating overall properties of composite structures. In the former group, a very
instructive work of that type is the one of (Buroni and Saez 2010) which partly uses
the RT-IRT method plus the Stroh formalism to explicit all the necessary integral
roots in the general anisotropic elasticity case. The obtained complicated analytical
form can be used in further numerical calculations. In the second group, when the
matrix has isotropic, hence uncoupled, properties (Poizat and Sester 1999; Clyne
et al. 2005; Liu 2011), it can be considered that the problem of an infinite inclusion
pattern or fiber network having any TMEE properties of any anisotropic symmetry
can be treated as a generalized thermo-elastic problem with an isotropic matrix and an
embedded phase of the here discussed morphology, possibly with TMEE properties
(remind that as pointed in the previous section problems of a multi-phased network
in a matrix cannot be treated by a single equation as for a single-phase pattern or
network). These isotropic matrix cases already represent quite a lot of structures of
interest.
When not isotropic, the matrix phase properties are hardly found in the literature
to be less than transversally isotropic, what is already so challenging that numerical
calculations are the most often used. And even so, most results remain restricted to
axially symmetric problems that is reinforcing phase properties and shapes axially
symmetric with the matrix TI symmetry. See (Huang et al. 1998; Mikata 2001; Hou
and Leung 2004; Chen et al. 2004; Lee et al. 2005) to cite a few. For example, to
the authors knowledge, there are not even Eshelby-based approaches proposing fully
explicit analytical solutions for off-symmetric structures in a TI matrix, although the
formalism for them is frequently presented. In the already cited application of the
present RT-IRT method in (Franciosi 2013) for MEE axially symmetric (spheroids,
laminates and fibers) inclusions in a MEE matrix, a newly obtained result toward
the follow-up of evolving patterns was the analytical mGO calculations for oblique
orientations of the inclusion axis with regard to the symmetry axis of the matrix,
which to the authors knowledge has no anterior proposal. The resulting effective
properties do not have TI symmetry anymore, yet the result for spheroids is obtained
from implementing analytical explicit integrals (oblique infinite fibers and laminates
in a TI matrix can be treated in full analytical manner). Not all published results on
the GO variations with orientations for spheroids were presented in (Barboura and
Franciosi 2016) in the TI elastic case. In addition to the few presented graphs in the
cited short reference, Fig. 15.10 illustrates the case of stiffer axial than transverse
properties, the (not shown) converse being indeed very different and not just reversed.
Applications to piezo-elastic problems are directly at hand, just needing numerical
implementation. Studying this inclusion orientation effect in not isotropic (here TI)
matrices shows that “tuning” a property does not necessarily correspond to in-axes
orientation choices but can be (case per case) in between.
When an inclusion pattern has no specific orientation parameter as a 1D fiber
bundle, the orientation changes with some matrix property anisotropy becomes a