Page 91 - Mechanics Analysis Composite Materials

76 Mechanics and analysis of composite materials
crack in the central fiber, the load carried by this fiber is transmitted by shear
through the matrix to adjacent fibers. At a distance from the end of the fiber more
than c, the stress in the broken fiber becomes close to cr, and for 3 > &, the fiber
behaves as if there is no crack. A portion of the broken fiber corresponding to
0 5 x 5 c. is not fully effective in resisting the applied load, and Zi = K.a is referred to
as the fiber ineffective length. Because the fiber defects are randomly distributed
along its length, their influence on the strength of the ply is minimal if there are no
other defects in the central and the adjacent fibers within distance Zi from the crack.
To minimize the probability of such defects, we should minimize li which depends
on fiber and matrix stiffnesses and material microstructure.
To evaluate Zi, consider Eq. (3.47) and assume that cro(X) becomes close to cr if
e-”;“ -
-k 7
where k is some small parameter indicating how close go(%) should be to o to neglect
the difference between them (as a matter of fact this difference vanishes only for
X + 00). Taking approximately Ai = 2p in accordance with Eq. (3.41) and using
Eq. (3.28) specifying ,u we arrive at

For k = 0.01 we get
li = 2.3~/=$ . (3.57)
.

For a typical carbon-epoxy ply (see Fig. 3.19) with a = 0.016 mm and = 0.6,
Eq. (3.57) yields 0.29 mm.
Thus, for real composites, length Zi is very small, and this expIains why a
unidirectional composite demonstrates much higher strength in longitudinal tension
than a dry bundle of fibers (see Table 3.3). Reducing G,, Le., the matrix stiffness,we

1.5
1.25
1
0.75
0.5
0.25
-
0 X
0 5 10 15 20 25 30 35 40 45 50

Fig. 3.21. Distribution of normal stresses along the fibers n = 0, I, 2,3,4 for k = 4, Er = 250 GPa,
G, = 0.125 GPa.