306
S. Ghosh et al.
The perturbed strain in an inclusion is influenced by its interactions with other
inclusions in the MVE. For a population of inclusions represented by the 2-point
probability distribution function S 2 (r) in equation (9), the perturbed strain in the
fiber F I , (I = 1 · · · n p ) due to the interactions of fibers dispersed in mve is
expressed as:
∗
ij
x
F I
= S ij kl (x
I ))
ij
x
I
+
mve \ F I
S 2 (r) ˆ
G ij kl (r)
ij (r) dd
(21)
where S 2 (r) is the 2-point correlation function defined in equation (10). The second
integral term represents the interaction effect of all fibers with the I th fiber, and the
integrand may be denoted as a statistically informed Green’s function or SIGF.
The eigenstrains with n p interacting inclusions are evaluated by applying the
Eshelby’s stress consistency condition, which requires the total stress inside the
fiber F I to be equal to the total stress in the equivalent matrix domain. For the
domain mve consisting of interacting fibers with a distribution represented by the
2-point correlation function S 2 (r), the eigenstrain
ij in a reference fiber occupying
a domain F may be derived using equation (13) as:
ij (x) =
ι F (x)
S ij ab + M ij ab
−
mve \ F
S 2 (r) ˆ
G ij mn (r)
S mnpq + M mnpq
−1 ˆ
G pqab (r) dd
−1
(S abmn + M abmn )
−1
mve \ F
S 2 (r) ˆ
G mnkl (r) dd
−
1
2 (δ ak δ bl + δ al δ bk )
M
kl
= A ij kl (x)) M
kl ∀x ∈ mve
(22)
where M ij kl =
C
F I
ijpq − C M
ijpq
−1
C M
pqkl , C
F I
ij kl is the elastic stiffness of the
inclusion material and r is the distance between a source and field point. The
perturbed displacements at an observation point O in Fig. 2b can be obtained in
terms of the matrix strain M
ij by substituting equation (22) into equation (19) as:
u
∗
i (x) =
⎛
⎜
⎝
mve \ F
S 2
r
L imn
r
A mnkl
r
dd
⎞
⎟
⎠
M
kl
(23)
Finally, using equation (8), the affine transformation-based displacement fields
can be superposed on the above perturbed displacements to prescribe the exterior
statistics-based boundary conditions (ESBCs).
S. Ghosh et al.
The perturbed strain in an inclusion is influenced by its interactions with other
inclusions in the MVE. For a population of inclusions represented by the 2-point
probability distribution function S 2 (r) in equation (9), the perturbed strain in the
fiber F I , (I = 1 · · · n p ) due to the interactions of fibers dispersed in mve is
expressed as:
∗
ij
x
F I
= S ij kl (x
I ))
ij
x
I
+
mve \ F I
S 2 (r) ˆ
G ij kl (r)
ij (r) dd
(21)
where S 2 (r) is the 2-point correlation function defined in equation (10). The second
integral term represents the interaction effect of all fibers with the I th fiber, and the
integrand may be denoted as a statistically informed Green’s function or SIGF.
The eigenstrains with n p interacting inclusions are evaluated by applying the
Eshelby’s stress consistency condition, which requires the total stress inside the
fiber F I to be equal to the total stress in the equivalent matrix domain. For the
domain mve consisting of interacting fibers with a distribution represented by the
2-point correlation function S 2 (r), the eigenstrain
ij in a reference fiber occupying
a domain F may be derived using equation (13) as:
ij (x) =
ι F (x)
S ij ab + M ij ab
−
mve \ F
S 2 (r) ˆ
G ij mn (r)
S mnpq + M mnpq
−1 ˆ
G pqab (r) dd
−1
(S abmn + M abmn )
−1
mve \ F
S 2 (r) ˆ
G mnkl (r) dd
−
1
2 (δ ak δ bl + δ al δ bk )
M
kl
= A ij kl (x)) M
kl ∀x ∈ mve
(22)
where M ij kl =
C
F I
ijpq − C M
ijpq
−1
C M
pqkl , C
F I
ij kl is the elastic stiffness of the
inclusion material and r is the distance between a source and field point. The
perturbed displacements at an observation point O in Fig. 2b can be obtained in
terms of the matrix strain M
ij by substituting equation (22) into equation (19) as:
u
∗
i (x) =
⎛
⎜
⎝
mve \ F
S 2
r
L imn
r
A mnkl
r
dd
⎞
⎟
⎠
M
kl
(23)
Finally, using equation (8), the affine transformation-based displacement fields
can be superposed on the above perturbed displacements to prescribe the exterior
statistics-based boundary conditions (ESBCs).