15 Homogenization-Based Mechanical Behavior Modeling of Composites …
259
axial array. This corresponds to the particular case of Fig. 15.3 middle left with
not contacting alternated spheroids and finite cylinder elements (at the contact limit,
the spheroid large section must be assumed larger enough than the cylinder one to
consider the surface contact to be flat). For such mixed alignments, a solution is at
hand from calculating the interaction between a mixed (finite cylinder with spheroid
or sphere) element pair, according to Fig. 15.3 middle right and right. Resolving
this pair interaction problem provides the mGO that remains needed according to
Eq. (15.5a) in the case of two (E, F) element shapes present in a same alignment, for
which the respective alignment mGOs (main plus interaction parts) are known. Note
that in terms of potential applications, the structures in Fig. 15.3 left correspond,
among other possibilities, and when n is a finite number, to dendritic-like ones in
metallic materials.
Determining a pair interaction mGO from the RT-IRT method can be exemplified
on this new case: One calculates as recalled the section areas and derivative by planes
(of θ normal here since the problem is axially symmetric) passing simultaneously
through both elements. One element belonging to the ellipsoid family is a typical
case where the first form of the mean interaction weight function in Eq. (15.3b) is of
easier use, writing, with v = v E + v F the volume of the element pair:
ψ V E ,V F (ω) = −
1
8 π 2 v
D
+
V (ω)
D
−
V (ω)
˜
s
V E
(z,ω)s V F (z,ω)dz = −
˜
s
V E
(z,θ )
4π v
D
+
V (θ)
D
−
V (θ)
s V F (z,θ )dz.
(15.8a)
The area second derivative for the ellipsoidal (or spherical) element (E) being
constant, it simplifies the remaining integral as being all or a part of the volume of
the second element (F) and the φ integral around the alignment (x3) axis contributes
the 2π factor that appeared in Eq. (15.8a).
As shown in Fig. 15.3 middle right, there are five angular θ sectors—defined by
the orientations of the tangent planes to the spheroid passing through the cylinder
vertical section “corners” P, Q, R, S to separately “treat”: The first one (0–θ 1) corresponds to no plane simultaneously cutting both elements, and the last one (θ 4–π /2)
corresponds to directions for which the entire cylinder volume is swept out by the
simultaneously cutting planes (the supplementary (π /2–π ) sector is equivalent by
symmetry). In the three intermediate sectors (θ 1–θ 2, θ 2–θ 3, θ 3–θ 4), only a θ-varying
part of the cylinder volume is swept out by crossing planes. In the central θ-angular
part limited by the cylinder “corners” R and Q, this cylinder volume fraction equal
to
D
+
V (θ)
D
−
V (θ)
s V F (z,θ )dz reads from simple geometry πr
2
(2h − 2r sin θ ), with 2h the
cylinder height and with r the cylinder radius taken smaller enough than the sphere
radius R, for the assumption of a flat contact section to hold up to the limit contact
case. The integrals to solve to fully determine the interaction shape function are
at quite easy hand provided manipulations of little interest here. For room saving
and not developing out of purpose calculations, a simplified solution is sketched
in simplifying the five angular domains to be treated into the remaining three in
259
axial array. This corresponds to the particular case of Fig. 15.3 middle left with
not contacting alternated spheroids and finite cylinder elements (at the contact limit,
the spheroid large section must be assumed larger enough than the cylinder one to
consider the surface contact to be flat). For such mixed alignments, a solution is at
hand from calculating the interaction between a mixed (finite cylinder with spheroid
or sphere) element pair, according to Fig. 15.3 middle right and right. Resolving
this pair interaction problem provides the mGO that remains needed according to
Eq. (15.5a) in the case of two (E, F) element shapes present in a same alignment, for
which the respective alignment mGOs (main plus interaction parts) are known. Note
that in terms of potential applications, the structures in Fig. 15.3 left correspond,
among other possibilities, and when n is a finite number, to dendritic-like ones in
metallic materials.
Determining a pair interaction mGO from the RT-IRT method can be exemplified
on this new case: One calculates as recalled the section areas and derivative by planes
(of θ normal here since the problem is axially symmetric) passing simultaneously
through both elements. One element belonging to the ellipsoid family is a typical
case where the first form of the mean interaction weight function in Eq. (15.3b) is of
easier use, writing, with v = v E + v F the volume of the element pair:
ψ V E ,V F (ω) = −
1
8 π 2 v
D
+
V (ω)
D
−
V (ω)
˜
s
V E
(z,ω)s V F (z,ω)dz = −
˜
s
V E
(z,θ )
4π v
D
+
V (θ)
D
−
V (θ)
s V F (z,θ )dz.
(15.8a)
The area second derivative for the ellipsoidal (or spherical) element (E) being
constant, it simplifies the remaining integral as being all or a part of the volume of
the second element (F) and the φ integral around the alignment (x3) axis contributes
the 2π factor that appeared in Eq. (15.8a).
As shown in Fig. 15.3 middle right, there are five angular θ sectors—defined by
the orientations of the tangent planes to the spheroid passing through the cylinder
vertical section “corners” P, Q, R, S to separately “treat”: The first one (0–θ 1) corresponds to no plane simultaneously cutting both elements, and the last one (θ 4–π /2)
corresponds to directions for which the entire cylinder volume is swept out by the
simultaneously cutting planes (the supplementary (π /2–π ) sector is equivalent by
symmetry). In the three intermediate sectors (θ 1–θ 2, θ 2–θ 3, θ 3–θ 4), only a θ-varying
part of the cylinder volume is swept out by crossing planes. In the central θ-angular
part limited by the cylinder “corners” R and Q, this cylinder volume fraction equal
to
D
+
V (θ)
D
−
V (θ)
s V F (z,θ )dz reads from simple geometry πr
2
(2h − 2r sin θ ), with 2h the
cylinder height and with r the cylinder radius taken smaller enough than the sphere
radius R, for the assumption of a flat contact section to hold up to the limit contact
case. The integrals to solve to fully determine the interaction shape function are
at quite easy hand provided manipulations of little interest here. For room saving
and not developing out of purpose calculations, a simplified solution is sketched
in simplifying the five angular domains to be treated into the remaining three in