Composite Materials Research Progress

An Experimental and Analytical Study of Unidirectional Carbon Fiber… 151
Finite Difference Method
We define the non-dimensional displacement coordinate as
where
U = u
i,
j i,
j
/ ξ
, X x
ξ =
The Equation 9 can be rewriten as
A
i
d
2
U
dX
= /ξ ,
The second-order derivative at point (i,j) is:
2 k
dU i,j
d X
no element break
element (i,j)~(i,j+1) break
element (i,j-1)~(i,j) break
2
τ i / i+
1,
j = τ i / i+
1,
j
( + )
dfE f G fdm G mdf
2G
G
f m
( U − U + U ) = 0
i,
j
+ 1,
2 2 i+
j i,
j i−1,
j
Substituting Equation (13) into (12), one can obtain:
UL =
⎧
⎨
⎩
⎪⎧
UR
= ⎨
⎪⎩
U
0
ξ /
E f
df
(10)
(11)
(12)
⎧ 1 k
k k
⎪ 2 ( U i,j − 1 − 2U i,j + U i,j + 1)
( )
⎪
Δx
⎪ 4 k k
= ⎨ 2 ( U i,j − U i,j −1)
⎪ 3(
Δx)
⎪ 4 k k
2 ( U i,j + 1 − U i,j )
⎩⎪
3(
Δx
)
(13)
( UL + UR)
+ ( ΔX
)
C
C
2
k Ai
(
2
4
U = i,
j
C1Ai
+ C2C
3 Δ
k
i,
j−1
U , + 1
k
i j
0
element (i,j-1)~(i,j) unbroken
element (i,j-1)~(i,j) broken
( ) 2
X
UD +
C
5
UU )
(14)