SERVE-Boundary Conditions
325
{α} =
1
8
1 − ν M
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
4ν M − 1
3 − 4ν M
0
0
0
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎭
, {β}(r) =
ρ 2
8
1 − ν M
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
−2
1 + 2ν M + 9ρ 2
2 − 3ρ 2
4(1 + 2ν M ) − 12ρ 2
4 − 12ρ 2
16 − 24ρ 2
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎭
The parameter ν M is the Poisson’s ratio of the matrix material. The circumference
basis tensor is given as:
{ ij kl }(θ ) =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
δ ij δ kl
δ ik δ jl + δ il δ jk
δ ij n k n l
n i n j δ kl
n i n j n k n l
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎭
, where
⎧
⎨
⎩
n 1
n 2
n 3
⎫
⎬
⎭
=
⎧
⎨
⎩
cos θ
sin θ
1
⎫
⎬
⎭
For the cylindrical fiber of circular cross-section, the interior and exterior
displacement-transfer tensors are given by:
T ij k
x, x
I
= {η}
T (r){ ij k }(θ ) and D ij k
x, x
I
= {γ }
T (r){ ij k }(θ )
(33)
where
{η}(r) = a
ρ
8
1 − ν M
⎧
⎪ ⎨
⎪ ⎩
4ν M − 1
3 − 4ν M
0
⎫
⎪ ⎬
⎪ ⎭
, {γ }(r) = a
ρ
8
1 − ν M
⎧
⎪ ⎨
⎪ ⎩
−2
1 − 2ν M + ρ 2
2
1 − 2ν M + ρ 2
4
1 − ρ 2
⎫
⎪ ⎬
⎪ ⎭
and
{ ij k }(θ ) =
⎧
⎨
⎩
n i δ jk
n j δ ik + n k δ ij
n i n j n k
⎫
⎬
⎭
Acknowledgements This work has been supported through a grant No. FA9550-12-1-0445
to the Center of Excellence on Integrated Materials Modeling (CEIMM) at Johns Hopkins
University awarded by the AFOSR/RSL Computational Mathematics Program (Manager Dr. A.
Sayir) and AFRL/RX (Monitors Drs. C. Woodward and C. Przybyla). These sponsorships are
gratefully acknowledged. Computing support by the Homewood High-Performance Compute
Cluster (HHPC) and Maryland Advanced Research Computing Center (MARCC) is gratefully
acknowledged.