Materials for engineering, 3rd Edition - (Malestrom)

Composite materials 195
x
dx
F
F
δ
Unit cube
containing a
single inclusion
dδ = ε dx
A 2
A 1
dx
6.4 Unit cube containing a single inclusion.
The total force on the cross-section must equal F, so,
F = A 1 E 1 ε + A 2 E 2 ε
and since A 1 + A 2 = 1, we can write
F = (1 – A 2 ) E 1 ε + A 2 E 2 ε
= ε [E 1 + (E 2 – E 1 )A 2 ]
i.e.
ε =
F
E + ( E – E ) A
1 2 1 2
The total elongation of the cube δ can be expressed:
1
δ = ε( x)dx
∫
= F
0
∫
1
0
dx
E + ( E – E ) A x
1 2 1 2( )