270
P. Franciosi and M. Spagnuolo
0
0
pi/6
pi/3
pi/2
t
i j i j
0
0.05
0.1
0.15
0.2
0.25
T3131 =1/50
T3131 =1/5
T3131 =1
T3131 =5
T3131 =50
T1212 =1/50
T1212 =1/5
T1212 =1
T1212 =5
T1212 =50
T2323 =1/50
T2323 =1/5
T2323 =1
T2323 =5
T2323 =50
0
0
pi/6
pi/3
pi/2
t
i i
i i
0
0.1
0.2
0.3
0.4
0.5
T1111 =1/50
T1111 =1/5
T1111 =1
T1111 =5
T1111 =50
T2222 =1/50
T2222 =1/5
T2222 =1
T2222 =5
T2222 =50
T3333 =1/50
T3333 =1/5
T3333 =1
T3333 =5
T3333 =50
Fig. 15.10 Off-symmetry variations of effective elastic GO terms for a spheroid from flat (1/50) to
long (50) aspect ratios, in a TI symmetric matrix stiffer in the symmetry axis direction x3 (Barboura
and Franciosi 2016)
key information to follow, and in that respect, the results shown in Fig. 15.10 are
important. In order to go further from oblique spheroids toward obliquely embedded
patterns in a TI matrix with any properties of the TMEE type, a still missing essential
analytical information is the calculation of pair interaction operators between two
oblique same inclusions with general same orientation, as the fiber pair of Fig. 15.9
right. Since these calculations do not look to be at easy hand analytically, it is a case
where approximating these interactions with the forms obtained in an isotropic matrix
could be a “better than nothing” option (these isotropic forms are those exemplified
in Fig. 15.2 and reported in Eqs. 15.7a, b).
15.5.3 From Static to Dynamical Problems
Dynamic problems in mechanics and in other fields of physics also are per se at
the origin of a since long huge literature concerned with propagation/dispersion
of various wave types in increasingly many composite structures and metamaterials (Kennett et al. 1978; Grossa et al. 2007; Willis 2019). It is consequently
difficult to here do more than pointing some features in relation with our studies
of global mGOs for large to infinite patterns and networks of inclusions and giving
useful references. It is worth to first point that regarding the use of the PCW (1995)
framework in the presented studies, a result of (Weng 2010) “validated” this framework dedicated to static elasticity of composites as consistent with a dynamical
foundation, a plus to that author owing to other micromechanics models which do
not have a dynamic counterpart.
6 Next considering the foundations of the RT-IRT
method which also are at the basis of the presented results, those look a priori pretty
well adapted to elastodynamic-like problems: On the one hand, the elementary operators involved in the GOs (Eqs. 15.2a, b) which read (for an isotropic elastic-like
matrix of moduli C, using the spherical coordinates ω1 = cosφ sinθ, ω2 = sinφ sinθ,
6 The Mori-Tanaka (MT) model is also validated as having a dynamical foundation, yet it is in
fact only in its restricted validation domain of all congruent ellipsoids in homothetic ellipsoidal
distribution, which coincides with both the HS estimate and the PCW one in this case.
P. Franciosi and M. Spagnuolo
0
0
pi/6
pi/3
pi/2
t
i j i j
0
0.05
0.1
0.15
0.2
0.25
T3131 =1/50
T3131 =1/5
T3131 =1
T3131 =5
T3131 =50
T1212 =1/50
T1212 =1/5
T1212 =1
T1212 =5
T1212 =50
T2323 =1/50
T2323 =1/5
T2323 =1
T2323 =5
T2323 =50
0
0
pi/6
pi/3
pi/2
t
i i
i i
0
0.1
0.2
0.3
0.4
0.5
T1111 =1/50
T1111 =1/5
T1111 =1
T1111 =5
T1111 =50
T2222 =1/50
T2222 =1/5
T2222 =1
T2222 =5
T2222 =50
T3333 =1/50
T3333 =1/5
T3333 =1
T3333 =5
T3333 =50
Fig. 15.10 Off-symmetry variations of effective elastic GO terms for a spheroid from flat (1/50) to
long (50) aspect ratios, in a TI symmetric matrix stiffer in the symmetry axis direction x3 (Barboura
and Franciosi 2016)
key information to follow, and in that respect, the results shown in Fig. 15.10 are
important. In order to go further from oblique spheroids toward obliquely embedded
patterns in a TI matrix with any properties of the TMEE type, a still missing essential
analytical information is the calculation of pair interaction operators between two
oblique same inclusions with general same orientation, as the fiber pair of Fig. 15.9
right. Since these calculations do not look to be at easy hand analytically, it is a case
where approximating these interactions with the forms obtained in an isotropic matrix
could be a “better than nothing” option (these isotropic forms are those exemplified
in Fig. 15.2 and reported in Eqs. 15.7a, b).
15.5.3 From Static to Dynamical Problems
Dynamic problems in mechanics and in other fields of physics also are per se at
the origin of a since long huge literature concerned with propagation/dispersion
of various wave types in increasingly many composite structures and metamaterials (Kennett et al. 1978; Grossa et al. 2007; Willis 2019). It is consequently
difficult to here do more than pointing some features in relation with our studies
of global mGOs for large to infinite patterns and networks of inclusions and giving
useful references. It is worth to first point that regarding the use of the PCW (1995)
framework in the presented studies, a result of (Weng 2010) “validated” this framework dedicated to static elasticity of composites as consistent with a dynamical
foundation, a plus to that author owing to other micromechanics models which do
not have a dynamic counterpart.
6 Next considering the foundations of the RT-IRT
method which also are at the basis of the presented results, those look a priori pretty
well adapted to elastodynamic-like problems: On the one hand, the elementary operators involved in the GOs (Eqs. 15.2a, b) which read (for an isotropic elastic-like
matrix of moduli C, using the spherical coordinates ω1 = cosφ sinθ, ω2 = sinφ sinθ,
6 The Mori-Tanaka (MT) model is also validated as having a dynamical foundation, yet it is in
fact only in its restricted validation domain of all congruent ellipsoids in homothetic ellipsoidal
distribution, which coincides with both the HS estimate and the PCW one in this case.