254
P. Franciosi and M. Spagnuolo
t V =
nE
i=1
v
E
i
v
t
V i
E +
nF
i=1
v
F
i
v
t
V i
F +
nE
i=1
nE
j=i+1
v
E
i +v
E
j
v
t
V i ,V j
E,E
+
nF
i=1
nF
j=i+1
v
F
i +v
F
j
v
t
V i ,V j
E,E + 2
nE
i=1
nF
j=1
v
E
i +v
F
j
v
t
V i ,V j
E,F
, with v =
nE
i=1
v
E
i +
nF
i=1
v
F
i .
(15.5b)
In Eq. (15.5a), one retrieves the main and interaction parts for each E and F sets
of identical inclusions plus a complementary mixed interaction part from all the
mixed (E, F) element pairs. That is combining two patterns (E) and (F) amounts to
determine each homogeneous pattern mGO and the representative mGO for one(E–
F) mixed pair interaction which provides access to the global mixed interaction
additional term. An example will be given.
An essential property of the shape function at interior points of any V domain
as well as of its mean value over V is to have an ω-integral over the unit sphere
necessarily equal to unity, for this integral is the interior characteristic (indicator)
function of V, equal to one inside V or in average over V and equal to zero at any
exterior point. The shape function as expressed in Eqs. (15.3, 15.4, 15.5) is more
precisely the IRT form of this characteristic function. When V is a pattern, only the
main part of the shape function contributes to this integral over equals to one, the
integral of the interaction part is always zero, with all the integrals for the individual
pair interactions being zero separately.
In Sect. 15.3 are first recalled the obtained mean shape functions and mGO forms
for (i) axial n-alignments of equally sized spheres, and for (ii) planar n-alignments of
parallel infinite cylinders (with equally sized circular cross sections), with n possibly
going to infinity in both cases, as presented, respectively, in (Franciosi 2010) and
(Franciosi et al. 2019). Relatively to case (i), will also be commented axial alignments
of spheroids (Franciosi 2010) or of finite—flat to long—cylinders (Franciosi et al.
2015) for they have interesting limit connections with the infinite cylinder when the
alignment becomes compact. These “elementary” patterns with quite simple forms
of mean shape functions and GOs (for isotropic matrices) are shown usable for the
building of more complicated patterns made of these elementary ones. Possible axial
interconnections between the elements of such axial alignments, mimicking sorts of
“skewers” to make these alignments continuous in their axial direction, were considered in (Franciosi 2010) using an arbitrary average between the alignment mGO
and the GO of an infinite cylinder. Here, a better founded (although approximated)
solution will be originally given for such sphere or spheroid skewers, treating them
as compact alternated axial alignments of finite cylinders and spheroids (or spheres).
This new result also serves as a practical example of the RT-IRT calculation method
for a pattern mGO, applying the recalled formulae in Sect. 15.2. Relatively to case
(ii), will also be mentioned infinite planar alignments of parallel rectangular beams
treated in (Franciosi 2018) and in (Spagnuolo et al. 2020) for the square section case,
their limit connection being also interestingly with laminate layers when the beams
are at contact. The second section part recalls how a 1D fiber bundle and a 3D fiber
(cylinders or beams) network have been built from planar arrays and submitted to
P. Franciosi and M. Spagnuolo
t V =
nE
i=1
v
E
i
v
t
V i
E +
nF
i=1
v
F
i
v
t
V i
F +
nE
i=1
nE
j=i+1
v
E
i +v
E
j
v
t
V i ,V j
E,E
+
nF
i=1
nF
j=i+1
v
F
i +v
F
j
v
t
V i ,V j
E,E + 2
nE
i=1
nF
j=1
v
E
i +v
F
j
v
t
V i ,V j
E,F
, with v =
nE
i=1
v
E
i +
nF
i=1
v
F
i .
(15.5b)
In Eq. (15.5a), one retrieves the main and interaction parts for each E and F sets
of identical inclusions plus a complementary mixed interaction part from all the
mixed (E, F) element pairs. That is combining two patterns (E) and (F) amounts to
determine each homogeneous pattern mGO and the representative mGO for one(E–
F) mixed pair interaction which provides access to the global mixed interaction
additional term. An example will be given.
An essential property of the shape function at interior points of any V domain
as well as of its mean value over V is to have an ω-integral over the unit sphere
necessarily equal to unity, for this integral is the interior characteristic (indicator)
function of V, equal to one inside V or in average over V and equal to zero at any
exterior point. The shape function as expressed in Eqs. (15.3, 15.4, 15.5) is more
precisely the IRT form of this characteristic function. When V is a pattern, only the
main part of the shape function contributes to this integral over equals to one, the
integral of the interaction part is always zero, with all the integrals for the individual
pair interactions being zero separately.
In Sect. 15.3 are first recalled the obtained mean shape functions and mGO forms
for (i) axial n-alignments of equally sized spheres, and for (ii) planar n-alignments of
parallel infinite cylinders (with equally sized circular cross sections), with n possibly
going to infinity in both cases, as presented, respectively, in (Franciosi 2010) and
(Franciosi et al. 2019). Relatively to case (i), will also be commented axial alignments
of spheroids (Franciosi 2010) or of finite—flat to long—cylinders (Franciosi et al.
2015) for they have interesting limit connections with the infinite cylinder when the
alignment becomes compact. These “elementary” patterns with quite simple forms
of mean shape functions and GOs (for isotropic matrices) are shown usable for the
building of more complicated patterns made of these elementary ones. Possible axial
interconnections between the elements of such axial alignments, mimicking sorts of
“skewers” to make these alignments continuous in their axial direction, were considered in (Franciosi 2010) using an arbitrary average between the alignment mGO
and the GO of an infinite cylinder. Here, a better founded (although approximated)
solution will be originally given for such sphere or spheroid skewers, treating them
as compact alternated axial alignments of finite cylinders and spheroids (or spheres).
This new result also serves as a practical example of the RT-IRT calculation method
for a pattern mGO, applying the recalled formulae in Sect. 15.2. Relatively to case
(ii), will also be mentioned infinite planar alignments of parallel rectangular beams
treated in (Franciosi 2018) and in (Spagnuolo et al. 2020) for the square section case,
their limit connection being also interestingly with laminate layers when the beams
are at contact. The second section part recalls how a 1D fiber bundle and a 3D fiber
(cylinders or beams) network have been built from planar arrays and submitted to