260
P. Franciosi and M. Spagnuolo
Fig. 15.3 right (small sectors 2 and 4 are shared between the first, central and last
ones) and also considering spheres, from which the spheroid case follows provided
some more manipulations not of much interest either here and the differences being
in a different expression for the sector limiting θ U and θ V orientations (and of a
well-known θ-dependent section area derivative for spheroids). These approximate
median limit orientations are taken to, respectively, correspond to the tangent planes
to the sphere that pass through the middle points w and y of the RS and PQ diameters
of the cylinder, respectively. We thus have the mean interaction shape function for
the (E, F) pair as:
ψ V E ,V F (θ ) = −
˜
s
V E
(z,θ )
4π(v E + v F )
×
⎧
⎨
⎩
0
f o r θ ∈ (0 − θ U )
πr
2
(2h − 2r sin θ) for θ ∈ (θ U − θ V )
v F
for θ ∈ (θ V − π/2),
(15.8b)
with ˜
s
V E
(z,θ ) = −3 v/2 D ell (θ )
3 for spheroids with breadth (thickness) 2D ell (θ )
in each θ direction which becomes uniformly −4π R
3
/2R
3
= −2π for a sphere of
radius R (equal to D). The θ-integral of ψ V E ,V F (θ ) reads, with C = 0.5(v E +v F )
−1
=
0.5(4π R
3
/3 + 2πr
2 h)
−1 :
2
π/2
0
ψ V E ,V F (θ ) sin θ dθ = 2C
⎛
⎝ 2πr
2
θ V
θ U
(h − r sin θ ) sin θ dθ + 2πr
2 h
π/2
θ V
sin θ dθ
⎞
⎠
= 4πr
2 C
h cos θ U −
r
2
(θ V − θ U ) −
sin 2θ V − sin 2θ U
2
.
(15.9)
The point here is not to calculate it but to simply show that the necessary integrals
to obtain the related mixed pair interaction mGO (in an isotropic elastic matrix) are
at hand, say the integrals for p = 1, 2:
I
2 p
= 2
π/2
0
ψ V E ,V F (θ ) cos
2 p
θ sin θ dθ
= 4πr
2 C
⎛
⎝ h
π/2
θ U
cos
2 p
θ sin θ dθ − r
θ V
θ U
cos
2 p
θ sin
2
θ dθ
⎞
⎠ .
(15.10)
This is the case for these two integral pairs have simple primitives which are
not to be explicated further here (the shape function in Eq. (15.9) corresponds to the
integral of Eq. (15.10) when p = 0). These results allow to specifying the mixed series
showing up in Eqs. (15.5a, b) for sphere/cylinder pairs and to obtaining a mGO for
the mixed n (possibly infinite) axial alignment. For the spheroidal shapes, the section
derivatives ˜
s
V E
(z, θ) = −3 v/2 D ell (θ )
3 read −2πζ
1 +
ζ
2
− 1
cos
2
θ
−3/2 from
P. Franciosi and M. Spagnuolo
Fig. 15.3 right (small sectors 2 and 4 are shared between the first, central and last
ones) and also considering spheres, from which the spheroid case follows provided
some more manipulations not of much interest either here and the differences being
in a different expression for the sector limiting θ U and θ V orientations (and of a
well-known θ-dependent section area derivative for spheroids). These approximate
median limit orientations are taken to, respectively, correspond to the tangent planes
to the sphere that pass through the middle points w and y of the RS and PQ diameters
of the cylinder, respectively. We thus have the mean interaction shape function for
the (E, F) pair as:
ψ V E ,V F (θ ) = −
˜
s
V E
(z,θ )
4π(v E + v F )
×
⎧
⎨
⎩
0
f o r θ ∈ (0 − θ U )
πr
2
(2h − 2r sin θ) for θ ∈ (θ U − θ V )
v F
for θ ∈ (θ V − π/2),
(15.8b)
with ˜
s
V E
(z,θ ) = −3 v/2 D ell (θ )
3 for spheroids with breadth (thickness) 2D ell (θ )
in each θ direction which becomes uniformly −4π R
3
/2R
3
= −2π for a sphere of
radius R (equal to D). The θ-integral of ψ V E ,V F (θ ) reads, with C = 0.5(v E +v F )
−1
=
0.5(4π R
3
/3 + 2πr
2 h)
−1 :
2
π/2
0
ψ V E ,V F (θ ) sin θ dθ = 2C
⎛
⎝ 2πr
2
θ V
θ U
(h − r sin θ ) sin θ dθ + 2πr
2 h
π/2
θ V
sin θ dθ
⎞
⎠
= 4πr
2 C
h cos θ U −
r
2
(θ V − θ U ) −
sin 2θ V − sin 2θ U
2
.
(15.9)
The point here is not to calculate it but to simply show that the necessary integrals
to obtain the related mixed pair interaction mGO (in an isotropic elastic matrix) are
at hand, say the integrals for p = 1, 2:
I
2 p
= 2
π/2
0
ψ V E ,V F (θ ) cos
2 p
θ sin θ dθ
= 4πr
2 C
⎛
⎝ h
π/2
θ U
cos
2 p
θ sin θ dθ − r
θ V
θ U
cos
2 p
θ sin
2
θ dθ
⎞
⎠ .
(15.10)
This is the case for these two integral pairs have simple primitives which are
not to be explicated further here (the shape function in Eq. (15.9) corresponds to the
integral of Eq. (15.10) when p = 0). These results allow to specifying the mixed series
showing up in Eqs. (15.5a, b) for sphere/cylinder pairs and to obtaining a mGO for
the mixed n (possibly infinite) axial alignment. For the spheroidal shapes, the section
derivatives ˜
s
V E
(z, θ) = −3 v/2 D ell (θ )
3 read −2πζ
1 +
ζ
2
− 1
cos
2
θ
−3/2 from