256
P. Franciosi and M. Spagnuolo
(right), up to possibly infinite element number and at all possible compactness up to
contact. We pose ρ 0 = R/L or a/L.
The nonzero components of the GO of a sphere and of a circular cylinder are,
respectively:
t
Sph
iiii = (3A + 5B)/15; t
Sph
(i,j),(i,j) = (2A + 5B)/30;
t
Sph
ii,jj = t
Sph
iiii − 2t
Sph
(i,j),(i,j) , i, j = 1, 2, 3,
(15.6a)
t
Cyl
iiii = (3A + 4B)/8; t
Cyl
(i,3),(i,3) = B/8; t
Cyl
(1,2),(1,2) = (A + 2B)/8,
t
Cyl
11,22 = t
Cyl
iiii − 2t
Cyl
(1,2),(1,2) , i = 1, 2.
(15.6b)
For their infinite (respectively axial and planar) alignments, the mGOs take very
comparable forms:
t
∞Sph
ρ 0
= t
Sph
+
v
(∞)Sph
0
ρ
3
0 + w
(∞)Sph
0
ρ
5
0
= t
Sph
+ 2
Z (3)v
2Sph
0
ρ
3
0 + Z (5)w
2Sph
0
ρ
5
0
,
(15.7a)
t
∞Cyl
ρ 0
= t
Cyl
+
v
(∞)Cyl
0
ρ
2
0 + w
(∞)Cyl
0
ρ
4
0
= t
cyl
+ 2
Z (2)v
2Cyl
0
ρ
2
0 + Z (4)w
2Cyl
0
ρ
4
0
,
(15.7b)
where Z (k) is the Zeta function for k = (2, 3, 4, 5), ρ 0 = R/L and the respective
tensor pairs
v 0 , w 0
are defined from the pair interaction mGOs, all to be found in
(Franciosi et al. 2019).
Among the various obtained results concerning these possibly infinite patterns,
Fig. 15.2 compares the variation with the alignment compactness of these exact
mGOs obtained for the sphere axial infinite alignment case and the cylinder planar
one,
3 borrowed from (Franciosi 2010) and (Franciosi et al. 2019). Although the plots
are in inverted and not identical presentations (respectively versus R/2L = ρ 0 /2
plots for aligned spheres and L/R = 1/ρ 0 plots for cylinders, both of radius R
and 2L axis to axis inter-distance, the contact limit being ρ 0 = 1 in both cases at
2L = 2R), the two major insights are obvious in both cases: The first one is that
the interaction effects in an infinite row are cumulative (in the sphere case, the plot
also reports in grey the sphere pair mGO), and the second is that this cumulating is
bounded by above since beyond a critical distance the pair interactions vanish.
Also, the influence zone extension is clearly less for the sphere axial alignment
(typically five times the sphere size) than for the planar alignment of cylinders
(between ten to twenty times the cylinder diameter). This example corresponds to
an incompressible isotropic matrix, but the corresponding cases with a Poisson ratio
of 0.3, to be found in the sources, give the same dimension of “influence zone.” The
3 The plots for the sphere pair mGO terms also appear in gray on Fig. 15.2 left, not for the cylinder
pair at right.
P. Franciosi and M. Spagnuolo
(right), up to possibly infinite element number and at all possible compactness up to
contact. We pose ρ 0 = R/L or a/L.
The nonzero components of the GO of a sphere and of a circular cylinder are,
respectively:
t
Sph
iiii = (3A + 5B)/15; t
Sph
(i,j),(i,j) = (2A + 5B)/30;
t
Sph
ii,jj = t
Sph
iiii − 2t
Sph
(i,j),(i,j) , i, j = 1, 2, 3,
(15.6a)
t
Cyl
iiii = (3A + 4B)/8; t
Cyl
(i,3),(i,3) = B/8; t
Cyl
(1,2),(1,2) = (A + 2B)/8,
t
Cyl
11,22 = t
Cyl
iiii − 2t
Cyl
(1,2),(1,2) , i = 1, 2.
(15.6b)
For their infinite (respectively axial and planar) alignments, the mGOs take very
comparable forms:
t
∞Sph
ρ 0
= t
Sph
+
v
(∞)Sph
0
ρ
3
0 + w
(∞)Sph
0
ρ
5
0
= t
Sph
+ 2
Z (3)v
2Sph
0
ρ
3
0 + Z (5)w
2Sph
0
ρ
5
0
,
(15.7a)
t
∞Cyl
ρ 0
= t
Cyl
+
v
(∞)Cyl
0
ρ
2
0 + w
(∞)Cyl
0
ρ
4
0
= t
cyl
+ 2
Z (2)v
2Cyl
0
ρ
2
0 + Z (4)w
2Cyl
0
ρ
4
0
,
(15.7b)
where Z (k) is the Zeta function for k = (2, 3, 4, 5), ρ 0 = R/L and the respective
tensor pairs
v 0 , w 0
are defined from the pair interaction mGOs, all to be found in
(Franciosi et al. 2019).
Among the various obtained results concerning these possibly infinite patterns,
Fig. 15.2 compares the variation with the alignment compactness of these exact
mGOs obtained for the sphere axial infinite alignment case and the cylinder planar
one,
3 borrowed from (Franciosi 2010) and (Franciosi et al. 2019). Although the plots
are in inverted and not identical presentations (respectively versus R/2L = ρ 0 /2
plots for aligned spheres and L/R = 1/ρ 0 plots for cylinders, both of radius R
and 2L axis to axis inter-distance, the contact limit being ρ 0 = 1 in both cases at
2L = 2R), the two major insights are obvious in both cases: The first one is that
the interaction effects in an infinite row are cumulative (in the sphere case, the plot
also reports in grey the sphere pair mGO), and the second is that this cumulating is
bounded by above since beyond a critical distance the pair interactions vanish.
Also, the influence zone extension is clearly less for the sphere axial alignment
(typically five times the sphere size) than for the planar alignment of cylinders
(between ten to twenty times the cylinder diameter). This example corresponds to
an incompressible isotropic matrix, but the corresponding cases with a Poisson ratio
of 0.3, to be found in the sources, give the same dimension of “influence zone.” The
3 The plots for the sphere pair mGO terms also appear in gray on Fig. 15.2 left, not for the cylinder
pair at right.