Materials for engineering, 3rd Edition - (Malestrom)

196
Materials for engineering
where A 2(x) describes the variation of A 2 as a function of x.
The value of Young’s modulus for the composite, E c , will be, for the unit
cube, the ratio F/δ, i.e.
1
1 dx
= [6.1]
Ec ∫
0
E1 + ( E2 – E1 ) A2(x)
We may thus derive an expression for E c for various volume fractions of
any particular distribution of the embedded material for which A 2(x) is a welldefined
function of x. We will consider three examples.
Cubic inclusions
Consider a composite material consisting of a volume fraction f of identical
cubic inclusions of Young’s modulus E 2 in a matrix of modulus E 1 . In our
unit cube of Fig. 6.4, one inclusion will be of volume f, so the edge length
of the inclusion will be f 1/3 , and the area of one face, A 2 , will be f 2/3 . Thus,
A 2(x) will be of the form shown in Fig. 6.5.
Substituting in equation [6.1], we obtain:
∫
1 d x
dx
= +
E
E
1/3 E + ( E – E ) f
c 0
(1– f 1/3 )
∫
1
1 (1– f )
1 2 1
2/3
= 1 – 1/3
f
+
E
1
1/3
f
E + ( E – E ) f
1 2 1
2/3
=
2/3
E1 + ( E2 – E1) f – ( E2 – E1f
2/3
E [ E + ( E – E ) f ]
1 1 2 1
f 2/3
A 2
1 – f 1/3 f 1/3 x
6.5 Inclusion area vs. distance graph for cubic inclusion.