15 Homogenization-Based Mechanical Behavior Modeling of Composites …
251
patterns of it, or for an (ellipsoidal) distribution symmetry) and a purely matrixproperty-dependent part, say the property type (elastic-like, dielectric-like or else)
and the property anisotropy (through a set of elementary uniform GOs, one for each
direction around the unit sphere in R3 space). According to this RT-IRT form, the
(m)GO operator of a V domain or pattern reads (skipping the dependency on C
M ),
respectively, at each interior point (r) of V and in terms of mean value over V:
t
V
pq jn (r) =
t
e
pq jn (ω)ψ V (ω, r)dω,
(15.2a)
t
V
pq jn =
t
e
pq jn (ω)ψ V (ω)dω.
(15.2b)
The way the elementary GOs t
e
(ω) showing up in Eqs. (15.2a, b) are obtained is
summarized in most of the cited authors and co-worker references, starting with
(Franciosi and Lormand 2004). Their form will be recalled in Eq. (15.12) and
commented there. They identify with the (uniform) GOs of infinitely flat (oblate)
spheroids, sometimes called platelets, or of laminate layers. Their obtaining is
recalled in Appendix in the case of isotropic elastic (or elastic-like, that is rankfour) tensorial properties, represented by two constant A = −B/2(1 − v),B = 1/μ,
and they are recalled in Table 15.1 [this elastic form contains the dielectric-like,
rank-2, operator form in skipping the terms linked to the constant A while B = 1/D,
with D the dielectric constant, as explained in the cited references, starting from
(Franciosi 2005)]. These elementary GOs related to a transversally isotropic matrix
can be found in (Franciosi 2013) with extension to so-called generalized elastic-like
Table 15.1 Nonzero terms of the elementary operator, with “cθ” and “sθ” for “cosθ” and “sinθ”
(resp. φ)
1
22
33
23
31
12
11 As 4 θc 4 φ
Bs 2 θc 2 φ
As 4 θc 2 φs 2 φ
0
As 2 θc 2 θc 2 φ
0
22 As 4 θc 2 φs 2 φ
0
As 4 θs 4 φ
Bs 2 θs 2 φ
As 2 θc 2 θs 2 φ
0
33 As 2 θc 2 θc 2 φ
0
As 2 θc 2 θs 2 φ
0
As 4 θ
Bc 2 θ
23
As 2 θc 2 θs 2 φ
B(s 2 θs 2 φ +
c 2 θ)/4
31
As 2 θc 2 θc 2 φ
B(s 2 θc 2 φ +
c 2 θ)/4
12
As 4 θc 2 φs 2 φ
Bs 2 θ/4
251
patterns of it, or for an (ellipsoidal) distribution symmetry) and a purely matrixproperty-dependent part, say the property type (elastic-like, dielectric-like or else)
and the property anisotropy (through a set of elementary uniform GOs, one for each
direction around the unit sphere in R3 space). According to this RT-IRT form, the
(m)GO operator of a V domain or pattern reads (skipping the dependency on C
M ),
respectively, at each interior point (r) of V and in terms of mean value over V:
t
V
pq jn (r) =
t
e
pq jn (ω)ψ V (ω, r)dω,
(15.2a)
t
V
pq jn =
t
e
pq jn (ω)ψ V (ω)dω.
(15.2b)
The way the elementary GOs t
e
(ω) showing up in Eqs. (15.2a, b) are obtained is
summarized in most of the cited authors and co-worker references, starting with
(Franciosi and Lormand 2004). Their form will be recalled in Eq. (15.12) and
commented there. They identify with the (uniform) GOs of infinitely flat (oblate)
spheroids, sometimes called platelets, or of laminate layers. Their obtaining is
recalled in Appendix in the case of isotropic elastic (or elastic-like, that is rankfour) tensorial properties, represented by two constant A = −B/2(1 − v),B = 1/μ,
and they are recalled in Table 15.1 [this elastic form contains the dielectric-like,
rank-2, operator form in skipping the terms linked to the constant A while B = 1/D,
with D the dielectric constant, as explained in the cited references, starting from
(Franciosi 2005)]. These elementary GOs related to a transversally isotropic matrix
can be found in (Franciosi 2013) with extension to so-called generalized elastic-like
Table 15.1 Nonzero terms of the elementary operator, with “cθ” and “sθ” for “cosθ” and “sinθ”
(resp. φ)
1
22
33
23
31
12
11 As 4 θc 4 φ
Bs 2 θc 2 φ
As 4 θc 2 φs 2 φ
0
As 2 θc 2 θc 2 φ
0
22 As 4 θc 2 φs 2 φ
0
As 4 θs 4 φ
Bs 2 θs 2 φ
As 2 θc 2 θs 2 φ
0
33 As 2 θc 2 θc 2 φ
0
As 2 θc 2 θs 2 φ
0
As 4 θ
Bc 2 θ
23
As 2 θc 2 θs 2 φ
B(s 2 θs 2 φ +
c 2 θ)/4
31
As 2 θc 2 θc 2 φ
B(s 2 θc 2 φ +
c 2 θ)/4
12
As 4 θc 2 φs 2 φ
Bs 2 θ/4