15 Homogenization-Based Mechanical Behavior Modeling of Composites …
267
in addition to already cited thermal or dielectric properties in introduction, to acoustic
or dynamic problems, plasticity and nonlinear properties and behavior, generalized
media,…, showing first that the tensor uniformity property in the linear, elastic-like
or dielectric-like, contexts were in general not preserved, reinforcing the interest for
mean values.
As was synthesized in the abstract, the present work aimed at presenting recent
developments from the authors and collaborators which extended the seminal Eshelby
isolated inclusion problem to large and possibly infinite inclusion patterns with all
pair interactions between elements being accounted for, to patterns possibly turning
into an infinite network arrangement possibly then being co-continuous with the
embedding matrix and with the possibility to account for that arrangement evolving
under strain in the specific field of mechanical behavior. The proposed method which
is based on using the RT-IRT to determine one representative global mGO for the
infinite pattern or network in its current state (such that only having a single inclusion
to solve for obtaining an equivalent homogeneous material say) presents the helpful
advantage for calculations of separating, for linear and static problems, the morphological and the physical features in the mGO expression. Beyond the formally simple
validity extension to other linear static physical (dielectric, magnetic, thermal,…)
application fields as mentioned in introduction, it is expectable that all the performed
work on the morphological mGO parts remains valid and can be used more broadly,
for instance when the behavior of concern is not linear, when properties are coupled
or when dynamical effects cannot be ignored. Yet, even in the simplest of these
problems, say when the reference matrix properties are “not too much anisotropic”,
the full calculation of the mGOs is not necessarily made so simple with using the
RT-IRT, especially when aiming at analytical well enough approximate solutions
of easy use in larger modeling developments. Computational help will frequently
remain necessary to attain a solution. For the three mentioned directions (each of
which is a research field), simple extension possibilities of the here presented work
are briefly exemplified, pointing some already realized and easily realizable openings
and locking points to solve.
15.5.1 From Linear to Nonlinear Behavior
The extension to nonlinear (static) effective behavior of composites calls for an
appropriate development of the homogenization framework to use. Remaining in the
examples of mechanical behavior, the incremental homogenization method of (Hill
1965) using a piece-wise linearization of the phase constitutive behavior to linearly
increment the composite response has been used for years to estimating the plastic
response of aggregates and polycrystals. The step-wise reference to comparison
materials of affine thermo-elastic (i.e., linear elastic with eigenstrain) behavior type
for the constitutive phases provided a rigorous improvement which permits in quite
simple manner the extension of a linear problem to nonlinear ones. That is, using
constitutive laws of the form, at each point r in the medium with behavior σ (r) =
267
in addition to already cited thermal or dielectric properties in introduction, to acoustic
or dynamic problems, plasticity and nonlinear properties and behavior, generalized
media,…, showing first that the tensor uniformity property in the linear, elastic-like
or dielectric-like, contexts were in general not preserved, reinforcing the interest for
mean values.
As was synthesized in the abstract, the present work aimed at presenting recent
developments from the authors and collaborators which extended the seminal Eshelby
isolated inclusion problem to large and possibly infinite inclusion patterns with all
pair interactions between elements being accounted for, to patterns possibly turning
into an infinite network arrangement possibly then being co-continuous with the
embedding matrix and with the possibility to account for that arrangement evolving
under strain in the specific field of mechanical behavior. The proposed method which
is based on using the RT-IRT to determine one representative global mGO for the
infinite pattern or network in its current state (such that only having a single inclusion
to solve for obtaining an equivalent homogeneous material say) presents the helpful
advantage for calculations of separating, for linear and static problems, the morphological and the physical features in the mGO expression. Beyond the formally simple
validity extension to other linear static physical (dielectric, magnetic, thermal,…)
application fields as mentioned in introduction, it is expectable that all the performed
work on the morphological mGO parts remains valid and can be used more broadly,
for instance when the behavior of concern is not linear, when properties are coupled
or when dynamical effects cannot be ignored. Yet, even in the simplest of these
problems, say when the reference matrix properties are “not too much anisotropic”,
the full calculation of the mGOs is not necessarily made so simple with using the
RT-IRT, especially when aiming at analytical well enough approximate solutions
of easy use in larger modeling developments. Computational help will frequently
remain necessary to attain a solution. For the three mentioned directions (each of
which is a research field), simple extension possibilities of the here presented work
are briefly exemplified, pointing some already realized and easily realizable openings
and locking points to solve.
15.5.1 From Linear to Nonlinear Behavior
The extension to nonlinear (static) effective behavior of composites calls for an
appropriate development of the homogenization framework to use. Remaining in the
examples of mechanical behavior, the incremental homogenization method of (Hill
1965) using a piece-wise linearization of the phase constitutive behavior to linearly
increment the composite response has been used for years to estimating the plastic
response of aggregates and polycrystals. The step-wise reference to comparison
materials of affine thermo-elastic (i.e., linear elastic with eigenstrain) behavior type
for the constitutive phases provided a rigorous improvement which permits in quite
simple manner the extension of a linear problem to nonlinear ones. That is, using
constitutive laws of the form, at each point r in the medium with behavior σ (r) =