63
3
The value of V crit is given as
V crit
mu
m
fu
m
=
−
−
∗
∗
σ
σ
σ
σ
(3.7)
where
5 σ mu  = Ultimate tensile strength of the matrix
5 σ m ∗ = Tensile strength of the matrix at fracture
5 σ fu  = Ultimate tensile strength of fibre
The ultimate tensile strength of a composite with V crit may be given
as
σ
σ
σ
σ
cu
mu
fu crit
m
c rit
=
=
+
−
(
)
∗
V
V
1
(3.8)
At V min , the composite will not fail at a stress predicted at Eq. (3.7).
At V min , the volume fraction of fibres is ineffective in restraining the
elongation of the matrix; therefore, the fibres are strained to fracture. So,
V min =
−
+
−
∗
∗
σ
σ
σ
σ
σ
mu
m
fu
mu
m
(3.9)
The ultimate tensile strength of composite may be given as
σ
σ
cu
mu
=
−
(
)
1 V min
(3.10)
=
+
−
(
)
∗
σ
σ
fu
m
V
V
min
m in
1
(3.11)
Volume fraction of fibre
Strength, s
V min
V crit
s f
v f
s min
. Fig. 3.1 Critical volume fraction of fibre and corresponding strength
3.1 · Micromechanics of Polymeric Composites