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P. Franciosi and M. Spagnuolo
15.1 Introduction
The modeling of the overall behavior of heterogeneous macro-homogeneous materials and the estimate of their effective properties have experienced considerable
developments from the Eshelby solution of the stress and strain fields in and out
a finite ellipsoidal inhomogeneity in an infinite matrix subjected to stress or strain
loading conditions at infinity (Eshelby 1957, 1959). Having first established that
interior fields in an ellipsoid resulting from uniform loading at infinity were also
uniform, a “boulevard” was open for applications to, on the one hand composites
of the reinforced-matrix type, say a uniform matrix embedding particles of one or
several other phases, yielding to Hashin and Shtrikman (1963) variational estimate
type of effective properties (Walpole 1981; Willis 1981), and on the other hand
granular materials, including polycrystals, where each grain type could in turn be
considered as embedded into some equivalent uniform matrix to the aggregate, with
the self-consistent scheme being the leading one for these structures (Hill 1952, 1965;
Kröner 1958, 1990; Zeller and Dederich 1973).
A considerable and still rapidly expanding, more or less all “Eshelby-based”,
literature followed, exploding with the progress in computational means meanwhile,
impossible to summarize here where only a few ones will be cited, with several key
steps being taken in the following decades and up to now: from linear elasticity to
other (as dielectric or thermal) properties and to nonlinear (including plastic) mechanical behavior (Levin 1967; Kröner 1990; Ponte-Castaneda and Suquet 1998; Suquet
1997; Clyne et al. 2005; Buryachenko and Brun 2012), from isolated inclusions to
groups, either in statistical terms (Ponte-Castaneda and Willis 1995; Bornert et al.
1996) or in deterministic manner from the solution of the (ellipsoidal) inhomogeneity
pair interaction problem (Berveiller et al. 1987), also from homogeneous to heterogeneous, multi-phased inclusions, from either the proposal of composite inclusion
models (Christensen and Lo 1979) or the solution of the double inclusion problem,
an inhomogeneity inside an inhomogeneity (Hori and Nemat-Nasser 1993; Hu and
Weng 2000) and for particular (spherical, cylindrical) multilayered structures (Hervé
and Zaoui 1993, 1995) to only mention major milestones.
With regard to this state of the art from the last century, it is noteworthy that, within
the so-called Fourier–Green homogenization framework applications (as recalled
next on, the Eshelby tensor is intimately related to a Green operator with “better”
properties), a missing composite structure was the one where possibly several to all
phases are co-continuous, with still several phases possibly being discontinuous (say
embedded), although several attempts for two-phase co-continuity without follow-up
were presented in (Postma 1955; Boucher 1974; Christensen 1979a). A new start was
proposed in (Franciosi and El Omri 2011) in terms of “laminate system schemes” for
two-phase bi-continuous composites first and then for any p phase number among n
in (Franciosi and Charles 2016a; Franciosi 2020a). Although the bi-continuous twophase composites are the most frequent co-continuous structures with a variety that
goes from sponge-like or foam-like structures (Roberts and Garboczi 2002; Gong
et al. 2005; Bender et al. 2008) to dual phase metallic materials (Delannay et al.
P. Franciosi and M. Spagnuolo
15.1 Introduction
The modeling of the overall behavior of heterogeneous macro-homogeneous materials and the estimate of their effective properties have experienced considerable
developments from the Eshelby solution of the stress and strain fields in and out
a finite ellipsoidal inhomogeneity in an infinite matrix subjected to stress or strain
loading conditions at infinity (Eshelby 1957, 1959). Having first established that
interior fields in an ellipsoid resulting from uniform loading at infinity were also
uniform, a “boulevard” was open for applications to, on the one hand composites
of the reinforced-matrix type, say a uniform matrix embedding particles of one or
several other phases, yielding to Hashin and Shtrikman (1963) variational estimate
type of effective properties (Walpole 1981; Willis 1981), and on the other hand
granular materials, including polycrystals, where each grain type could in turn be
considered as embedded into some equivalent uniform matrix to the aggregate, with
the self-consistent scheme being the leading one for these structures (Hill 1952, 1965;
Kröner 1958, 1990; Zeller and Dederich 1973).
A considerable and still rapidly expanding, more or less all “Eshelby-based”,
literature followed, exploding with the progress in computational means meanwhile,
impossible to summarize here where only a few ones will be cited, with several key
steps being taken in the following decades and up to now: from linear elasticity to
other (as dielectric or thermal) properties and to nonlinear (including plastic) mechanical behavior (Levin 1967; Kröner 1990; Ponte-Castaneda and Suquet 1998; Suquet
1997; Clyne et al. 2005; Buryachenko and Brun 2012), from isolated inclusions to
groups, either in statistical terms (Ponte-Castaneda and Willis 1995; Bornert et al.
1996) or in deterministic manner from the solution of the (ellipsoidal) inhomogeneity
pair interaction problem (Berveiller et al. 1987), also from homogeneous to heterogeneous, multi-phased inclusions, from either the proposal of composite inclusion
models (Christensen and Lo 1979) or the solution of the double inclusion problem,
an inhomogeneity inside an inhomogeneity (Hori and Nemat-Nasser 1993; Hu and
Weng 2000) and for particular (spherical, cylindrical) multilayered structures (Hervé
and Zaoui 1993, 1995) to only mention major milestones.
With regard to this state of the art from the last century, it is noteworthy that, within
the so-called Fourier–Green homogenization framework applications (as recalled
next on, the Eshelby tensor is intimately related to a Green operator with “better”
properties), a missing composite structure was the one where possibly several to all
phases are co-continuous, with still several phases possibly being discontinuous (say
embedded), although several attempts for two-phase co-continuity without follow-up
were presented in (Postma 1955; Boucher 1974; Christensen 1979a). A new start was
proposed in (Franciosi and El Omri 2011) in terms of “laminate system schemes” for
two-phase bi-continuous composites first and then for any p phase number among n
in (Franciosi and Charles 2016a; Franciosi 2020a). Although the bi-continuous twophase composites are the most frequent co-continuous structures with a variety that
goes from sponge-like or foam-like structures (Roberts and Garboczi 2002; Gong
et al. 2005; Bender et al. 2008) to dual phase metallic materials (Delannay et al.