SERVE-Boundary Conditions
321
5.4 Micromechanical Analysis of the Polydispersed SERVE
with ESBCs
The ESBCs for the statistically nonhomogeneous microstructures in Sect. 2 are
extended for polydispersed microstructures in this section. For polydispersed fibers,
the distribution of fiber radius a is represented by the probability density function
of the fiber size P DF (a). A relative position vector r = x − x I , where x is a
generic spatial point and x I is a reference point, is used to represent the 2-point
kernel functions. The eigenstrain
ij in a reference fiber occupying a domain F
given in equation (22) is correspondingly modified as:
ij (x) = [ ι F (x)
S ij ab + M ij ab
−
mve \ F
P DF (a)S 2 (r) ˆ
G ij mn (r, a)
S mnpq + M mnpq
−1
ˆ
G pqab (r, a)dd ] −1 [ ( (S abmn + M abmn ) −1
mve \ F
P DF (a)S 2 (r) ˆ
G mnkl (r, a)dd ) −
1
2
(δ ak δ bl + δ al δ bk ) ] M
kl = A ij kl (x)) M
kl
∀x ∈ mve
(29)
S ij kl are the spatially invariant interior, ˆ
G ij kl (r, a) are the position- dependent exterior Eshelby tensors, respectively, and ι is a phase-based unit function. Furthermore,
M ij kl =
C F
ijpq − C M
ijpq
−1
C M
pqkl , with C M
ij kl and C F
ij kl being the elastic stiffness
of the matrix and inclusion materials, respectively. The perturbed displacements at
an observation point O in Fig. 2b are written in terms of the matrix strain as:
u
∗
i (x) =
⎡
⎢
⎣
mve \ F
P DF (a) S 2
r
L imn
r
, a
A mnkl
r
dd
⎤
⎥
⎦
M
kl
(30)
where L ikl (r) is a unified displacement-transfer tensor. Finally, using equation (8),
the affine transformation-based displacement fields are superposed on the perturbed
displacements to prescribe the exterior statistics-based boundary conditions. The
ESBCs are applied on the boundary serve of a SERVE domain serve of size L
using the steps given in Sect. 2.2 with a few modifications.
In step 5, the perturbed displacements u ∗
i are computed using equation (30)
incorporating P DF (a) and S 2 (r, θ ) for all the boundary nodes, using the following
steps:
• Each radial orientation is discretized into N r number of equally spaced segments
with increment r =
R−a
N r
, where a is the fiber radius, R is the radius of horizon
(extent of the MVE), and the lower limit of the integration is r = a. The αth
radial point is given as r α = α
R−a
N r
.
321
5.4 Micromechanical Analysis of the Polydispersed SERVE
with ESBCs
The ESBCs for the statistically nonhomogeneous microstructures in Sect. 2 are
extended for polydispersed microstructures in this section. For polydispersed fibers,
the distribution of fiber radius a is represented by the probability density function
of the fiber size P DF (a). A relative position vector r = x − x I , where x is a
generic spatial point and x I is a reference point, is used to represent the 2-point
kernel functions. The eigenstrain
ij in a reference fiber occupying a domain F
given in equation (22) is correspondingly modified as:
ij (x) = [ ι F (x)
S ij ab + M ij ab
−
mve \ F
P DF (a)S 2 (r) ˆ
G ij mn (r, a)
S mnpq + M mnpq
−1
ˆ
G pqab (r, a)dd ] −1 [ ( (S abmn + M abmn ) −1
mve \ F
P DF (a)S 2 (r) ˆ
G mnkl (r, a)dd ) −
1
2
(δ ak δ bl + δ al δ bk ) ] M
kl = A ij kl (x)) M
kl
∀x ∈ mve
(29)
S ij kl are the spatially invariant interior, ˆ
G ij kl (r, a) are the position- dependent exterior Eshelby tensors, respectively, and ι is a phase-based unit function. Furthermore,
M ij kl =
C F
ijpq − C M
ijpq
−1
C M
pqkl , with C M
ij kl and C F
ij kl being the elastic stiffness
of the matrix and inclusion materials, respectively. The perturbed displacements at
an observation point O in Fig. 2b are written in terms of the matrix strain as:
u
∗
i (x) =
⎡
⎢
⎣
mve \ F
P DF (a) S 2
r
L imn
r
, a
A mnkl
r
dd
⎤
⎥
⎦
M
kl
(30)
where L ikl (r) is a unified displacement-transfer tensor. Finally, using equation (8),
the affine transformation-based displacement fields are superposed on the perturbed
displacements to prescribe the exterior statistics-based boundary conditions. The
ESBCs are applied on the boundary serve of a SERVE domain serve of size L
using the steps given in Sect. 2.2 with a few modifications.
In step 5, the perturbed displacements u ∗
i are computed using equation (30)
incorporating P DF (a) and S 2 (r, θ ) for all the boundary nodes, using the following
steps:
• Each radial orientation is discretized into N r number of equally spaced segments
with increment r =
R−a
N r
, where a is the fiber radius, R is the radius of horizon
(extent of the MVE), and the lower limit of the integration is r = a. The αth
radial point is given as r α = α
R−a
N r
.