Page 391 - Mechanics Analysis Composite Materials

376 Mechanics and analysis of composite materials

Table 8.2
Parameters of typical advanced composites.

Parameter Fabric-epoxy composites Unidirectional-epoxy composites Boron-AI
Glass Carbon Aramid Glass Carbon Aramid Boron
-
0~/8l 0.99 0.99 0.83 0.022 0.025 0.012 0.054 0.108
n 0.85 1.0 1.O 0.28 0.I 0.072 0.11 0.7

with Eqs. (8.32), this means that hl -Kh2, and the ratio h2/h1 varies from 12.7 for
glass-epoxy to 2.04 for boron+poxy composites. Now, return to the discussion
presented in section 4.4.2 from which it follows that in laminated composites
transverse stresses 02 reaching their ultimate value, &, cause cracks in the matrix
which do not result in the failure of the laminate whose strength is controlled by
fibers. To describe the laminate with cracks in the matrix (naturally, if the cracks are
admitted for the structure under design), we can use the monotropic model of the
ply and, hence, results of optimization presented in Section 8.1.
Consider again the optimality condition Eq. (8.25). As can be seen, this equation
can be satisfied not only by strains in Eqs. (8.26), but also if we take

Y.VJ
tan24i = -. (8.34)
Ex - E,.
Because the left-hand side of this equation is a periodic function with period 7c,
Eq. (8.34) determines two angles, Le.

(8.35)

Thus, the optimal laminate consists of two layers, and the fibers in both layers are
directed along the lines of principal stresses. Assume that the layers are made of the
same composite material and have the same thickness, i.e. hl = h2 = h/2, where h is
the thickness of the laminate. Then, using Eqs. (8.24) and (8.35) we can show that
BII= B22 and B24 = -BI~for this laminate. After some transformation involving
elimination of y.!,,. from the first two equations of Eqs. (8.23) with the aid of
Eq. (8.34) and similar transformation of the third equation from which and E: are
eliminated using again Eq. (8.34) we get

Nx = (BII + 814 tan 24)~:+ (B12 - 814tan %)E!,
N,,= (~12- B14tan 2414 + (BI 1 +B14 tan 24)~:~
Nx?; = (B44 f B14 cot 24)$,. .

Upon substitution of coefficients B,,, from Eqs. (8.24) we arrive at