Composite Materials Research Progress

An Experimental and Analytical Study of Unidirectional Carbon Fiber… 153
For fiber element:
For matrix element
U i+
1,
j −U
i,
j
σ i,
j = E f
(16a)
X
U i+
1,
j −U
i,
j
σ i,
j = Em
(16b)
X
The strain and stress of composites were calculated from average stress of all elements.
c
n m n m
1 ⎡
⎤
= ×
( )
( ) ⎢∑∑σ
2i+
1,
j 1−
V f + ∑∑σ
2i,
jV
f
2N
+ 1 M
⎥
⎣ i−0
1 1 1 ⎦
σ (17)
Strength of Fiber Element and the Failure Criterion
VKΔt
ε c =
(18)
L
Strength assignment to the fiber elements
In simulating, the strength of the fiber elements should be predetermined. According to
the Weibull statistical constitutive model the strength of the fiber follow Equation (4). If we
assume L in Equation (4) equal to mesh length Δx, here σΔx can be obtained from the scale
parameter σ0 at experimental length Lo. n×m random array ηi,j, equally distributed in the
range of (0,1), are produced by the computer, and we let
η
i,
j
β
⎡ Δx
⎛ S ⎤
i,
j ⎞
= P ( Δx,
S ) = ⎢−
⎜
⎟
i,
j exp
⎥
(19)
⎢ L0
⎣ ⎝ σ 0 ⎠ ⎥
⎦
From the Equation (19), we can get the strength of fiber element
The failure criterion
The failure criterion of fiber is
i,
j ≥ S i,
j
1
i,
j
⎡ L0
= − ln j σ
⎤ β
( ηi,
) 0
S ⎢ ⎥
(20)
⎣ Δx
⎦
σ fiber element broken (21a)