Chapter 15
Homogenization-Based Mechanical
Behavior Modeling of Composites Using
Mean Green Operators for Infinite
Inclusion Patterns or Networks Possibly
Co-continuous with a Matrix
Patrick Franciosi and Mario Spagnuolo
Abstract The Fourier–Green homogenization method for estimating the behavior
of composites was first developed for aggregates and inhomogeneity-reinforced (weakened) matrices, based on Eshelby (Proc R Soc Lond, A 421:379–396, 1957, Proc
R Soc Lond, A 252:561–569, 1959) solution of the isolated inclusion problem. The
need to address increasingly complex structures opened fruitful development routes,
firstly solving the inhomogeneity pair interaction problem and the one of heterogeneous (double or multilayered) inhomogeneities, in order to account for inclusion
dense concentrations and patterns. This work reports recent developments from the
authors and co-workers which examined in that framework possibly infinite inclusion patterns, possibly arranged into an infinite network possibly co-continuous with
the embedding matrix and possibly evolving under strain. The proposed modeling
amounts to determining the representative mean Green operator (mGO) for the
infinite pattern or network in its current (strain evolving) state. Once the method
foundations being summarized, previously solved “elementary” cases are recalled,
concerning infinite coaxial alignments of spheres, spheroids or finite cylinders and
planar alignments of infinite parallel cylinders or rectangular beams. It is next shown
how other complex patterns or networks could be represented in combining such
elementary ones. The mGO solution for a new family of inhomogeneous axial inclusion alignments is reported to support the discussion. Potential other application
fields are evoked.
Keywords Composites · Inclusion patterns · Phase co-continuity · Green
operators · Homogenization
P. Franciosi (B)
CNRS, UPR3407 LSPM, Université Sorbonne Paris-Nord, 93430 Villetaneuse, France
e-mail: patrick.franciosi@univ-paris13.fr
M. Spagnuolo
International Research Center M&MoCS, Universitàdegli Studi dell’Aquila, L’Aquila, Italy
© Springer Nature Switzerland AG 2021
F. dell’Isola and L. Igumnov (eds.), Dynamics, Strength of Materials and Durability
in Multiscale Mechanics, Advanced Structured Materials 137,
https://doi.org/10.1007/978-3-030-53755-5_15
245
Homogenization-Based Mechanical
Behavior Modeling of Composites Using
Mean Green Operators for Infinite
Inclusion Patterns or Networks Possibly
Co-continuous with a Matrix
Patrick Franciosi and Mario Spagnuolo
Abstract The Fourier–Green homogenization method for estimating the behavior
of composites was first developed for aggregates and inhomogeneity-reinforced (weakened) matrices, based on Eshelby (Proc R Soc Lond, A 421:379–396, 1957, Proc
R Soc Lond, A 252:561–569, 1959) solution of the isolated inclusion problem. The
need to address increasingly complex structures opened fruitful development routes,
firstly solving the inhomogeneity pair interaction problem and the one of heterogeneous (double or multilayered) inhomogeneities, in order to account for inclusion
dense concentrations and patterns. This work reports recent developments from the
authors and co-workers which examined in that framework possibly infinite inclusion patterns, possibly arranged into an infinite network possibly co-continuous with
the embedding matrix and possibly evolving under strain. The proposed modeling
amounts to determining the representative mean Green operator (mGO) for the
infinite pattern or network in its current (strain evolving) state. Once the method
foundations being summarized, previously solved “elementary” cases are recalled,
concerning infinite coaxial alignments of spheres, spheroids or finite cylinders and
planar alignments of infinite parallel cylinders or rectangular beams. It is next shown
how other complex patterns or networks could be represented in combining such
elementary ones. The mGO solution for a new family of inhomogeneous axial inclusion alignments is reported to support the discussion. Potential other application
fields are evoked.
Keywords Composites · Inclusion patterns · Phase co-continuity · Green
operators · Homogenization
P. Franciosi (B)
CNRS, UPR3407 LSPM, Université Sorbonne Paris-Nord, 93430 Villetaneuse, France
e-mail: patrick.franciosi@univ-paris13.fr
M. Spagnuolo
International Research Center M&MoCS, Universitàdegli Studi dell’Aquila, L’Aquila, Italy
© Springer Nature Switzerland AG 2021
F. dell’Isola and L. Igumnov (eds.), Dynamics, Strength of Materials and Durability
in Multiscale Mechanics, Advanced Structured Materials 137,
https://doi.org/10.1007/978-3-030-53755-5_15
245