15 Homogenization-Based Mechanical Behavior Modeling of Composites …
271
ω3 = cosθ, with θ = 0 (resp. φ = 0) in the x3 (resp. x1) direction of a (x1, x2, x3)
frame):
t
e
pq jn (ω) =
(M
−1
) pj (ω)ω q ω n
( pq),( jn)
,
(15.12)
are built on the acoustic 3 × 3 tensor M (M mp (ω) = C mnpq ω n ω q ) which defines
the Fourier transform of the (static) strain GO as k
2 G
pj (k) = M
−1
pj (ω) (see
recalls in Spagnuolo et al. 2020); on the other hand, the shape function form
in the mGOs involves explicitly the plane wave decomposition (Gel’fand and
Shilov 1964) of the delta function δ(r − r
) = (−1/8 π
2
)
δ
(z − z
,ω)dω,
in comparing with Eq. (15.3) where s
V (z,ω) =
∞
−∞ s V (z
,ω)δ
(z −
z
,ω)dz
in
ψ V (ω,r)dω = 1 = (−1/8
π
2
)
s
V (z,ω)dω
=
(−1/8 π
2
)
s
(V ) (z,ω)dω = X V (r) =
V δ(r − r
)dr
, and X V (r) being the
indicator function at point r in V. Thanks to this latter property, the specific use
of the RT-IRT method in dynamic elasticity of homogeneous media appears in
many works having proposed a dynamic analogous to a static approach, with
also extensions to various coupled property types entering the same formalism as
elasticity (Burridge 1967; Norris 1994; Khutorianski and Sosa 1995; Chen et al.
2007; Zielinski 2010).
The method has been defended as especially efficient for numerical calculations in
cases of anisotropic symmetries (Wang and Achenbach 1993), and recent applications
can be found in (Tavaf et al. 2018; Shrestha and Banerjee 2018) for example. It was
already applied analytically in (Willis 1980a, b), using results from (Korringa 1973),
for studying waves through a matrix reinforced with isolated or aligned spheroids.
These Willis’ works proposed a GO formulation where “static and dynamic terms
uncouple completely, to allow a very direct low-frequency perturbation solution,
starting from the solution of an associated static problem.”
From the presently used framework, in solving the motion equation Divσ (r, t) =
ρ(r)∂
2 u(r, t)/∂t
2 (with as for C(r) writing ρ(r) = ρ
M
+ ρ(r) for the density
field), instead of the stress equilibrium Divσ (r) = 0 and with a dynamic Green
function of the form u p (r, t) = G pj
r − r
, t − t
f j
r
, t
at place of the static
one, one easily verifies (following the static calculations in Spagnuolo et al. 2020)
that the dynamical effects on the GOs under the form of Eqs. (15.2a, b) do not alter, as
expected, the static shape function characteristic of a given inclusion pattern and that
they essentially modify the elementary static GO operators, through a frequencydependent contribution. The latter quite simply enters M in Eq. (15.12) for the
isotropic elastic cases with also extensions to thermo- and poro-elastic domains
which remain of isotropic nature, while piezoelectric and magnetic coupling do not
for they require at most a TI symmetry (Norris 1994; Chen et al. 2007).
It is then permitted to expect that in long wavelength ranges where a composite
can be considered to dynamically behave as a homogeneous equivalent (effective)
medium, although calculations may remain far from simple, applications of wave
propagation analyses to the here considered structures (also allowing to account
for some dispersive features as is pointed in (Christensen 1979b) are possible. Yet,
271
ω3 = cosθ, with θ = 0 (resp. φ = 0) in the x3 (resp. x1) direction of a (x1, x2, x3)
frame):
t
e
pq jn (ω) =
(M
−1
) pj (ω)ω q ω n
( pq),( jn)
,
(15.12)
are built on the acoustic 3 × 3 tensor M (M mp (ω) = C mnpq ω n ω q ) which defines
the Fourier transform of the (static) strain GO as k
2 G
pj (k) = M
−1
pj (ω) (see
recalls in Spagnuolo et al. 2020); on the other hand, the shape function form
in the mGOs involves explicitly the plane wave decomposition (Gel’fand and
Shilov 1964) of the delta function δ(r − r
) = (−1/8 π
2
)
δ
(z − z
,ω)dω,
in comparing with Eq. (15.3) where s
V (z,ω) =
∞
−∞ s V (z
,ω)δ
(z −
z
,ω)dz
in
ψ V (ω,r)dω = 1 = (−1/8
π
2
)
s
V (z,ω)dω
=
(−1/8 π
2
)
s
(V ) (z,ω)dω = X V (r) =
V δ(r − r
)dr
, and X V (r) being the
indicator function at point r in V. Thanks to this latter property, the specific use
of the RT-IRT method in dynamic elasticity of homogeneous media appears in
many works having proposed a dynamic analogous to a static approach, with
also extensions to various coupled property types entering the same formalism as
elasticity (Burridge 1967; Norris 1994; Khutorianski and Sosa 1995; Chen et al.
2007; Zielinski 2010).
The method has been defended as especially efficient for numerical calculations in
cases of anisotropic symmetries (Wang and Achenbach 1993), and recent applications
can be found in (Tavaf et al. 2018; Shrestha and Banerjee 2018) for example. It was
already applied analytically in (Willis 1980a, b), using results from (Korringa 1973),
for studying waves through a matrix reinforced with isolated or aligned spheroids.
These Willis’ works proposed a GO formulation where “static and dynamic terms
uncouple completely, to allow a very direct low-frequency perturbation solution,
starting from the solution of an associated static problem.”
From the presently used framework, in solving the motion equation Divσ (r, t) =
ρ(r)∂
2 u(r, t)/∂t
2 (with as for C(r) writing ρ(r) = ρ
M
+ ρ(r) for the density
field), instead of the stress equilibrium Divσ (r) = 0 and with a dynamic Green
function of the form u p (r, t) = G pj
r − r
, t − t
f j
r
, t
at place of the static
one, one easily verifies (following the static calculations in Spagnuolo et al. 2020)
that the dynamical effects on the GOs under the form of Eqs. (15.2a, b) do not alter, as
expected, the static shape function characteristic of a given inclusion pattern and that
they essentially modify the elementary static GO operators, through a frequencydependent contribution. The latter quite simply enters M in Eq. (15.12) for the
isotropic elastic cases with also extensions to thermo- and poro-elastic domains
which remain of isotropic nature, while piezoelectric and magnetic coupling do not
for they require at most a TI symmetry (Norris 1994; Chen et al. 2007).
It is then permitted to expect that in long wavelength ranges where a composite
can be considered to dynamically behave as a homogeneous equivalent (effective)
medium, although calculations may remain far from simple, applications of wave
propagation analyses to the here considered structures (also allowing to account
for some dispersive features as is pointed in (Christensen 1979b) are possible. Yet,