Composite Materials Research Progress

An Experimental and Analytical Study of Unidirectional Carbon Fiber… 149
β
⎡ L ⎛ σ ⎞ ⎤
P ( σ ) = exp⎢−
⎜
⎟ ⎥
(4)
⎢ L0
⎣ ⎝σ
0 ⎠ ⎥⎦
where, P is the survive probability of fiber at stress σ , and the σ 0 and β are Weibull scale
L length and reference length of fiber.
In addition, it is assumed that epoxy matrix is homogeneous and linear elastic.
parameter and Weibull shape parameter, L and 0
⎧σ
m = Emε
⎨
⎩τ
m = Gmγ
where, E m and Gm are tensile modulus and shear modulus.
Shear Stress on the Interface
The shear stress at i−1/i,j interface (shown in Figure 21) τ i−1/i,j can be expressed as a function
of the fiber displacement ui,j and the interface displacement ui−1/i,j:
( u − u ) /( df / 2)
τ =
(6a)
i−1 / i,
j G f i,
j i−1
/ i,
j
where G f is the shear modulus of the fiber. τ i−1/i,j can be also expressed as :
( u − u ) / ( dm / 2)
τ i−1 / i,
j = Gm i−1
/ i,
j i−1,
j
(6b)
If the interface does not break, combine (6A) and (6B) to eliminate ui−1/i,j, then we get
when the interface breaks, we have
2GmG
f
τ i−1
/ i,
j =
( ui,
j − ui−1,
j )
(7)
G df + G dm
m
τ = τ
i−
1 / i,
j
where, τ c is the friction between the fiber and the matrix when the matrix cracks or the
interface debonds.
f
c
(5)