Page 297 - Mechanics Analysis Composite Materials

282 Mechanics and analysis of composite materials

(6.11)

It looks like this criterion yields the same result for tension and compression.
However, it can be readily specified for tension or compression. It is important to
realize that evaluating material strength we usually know the stresses acting in this
material. Thus, we can take in Eq. (6.10)

(6.12)

thus describing the cases of tension and compression. Failure criterion in Eqs. (6.11)
and (6.12) is demonstrated in Fig. 6.9 in application to a fabric composite loaded
with stresses 01 and (r2(712 = 0). Naturally, this criterion is specified by different
equations for different quadrants in Fig. 6.9.
For some problems, e.g., for the problem of design, for which we usually do not
know the signs of stresses, we may need to use a universal form of the polynomial
criterion valid both for tension and compression. In this case, we should apply the
approximation of the type shown in Fig. 6.14(b) and generalize Eq. (6.9) as

Using conditions similar to Eqs. (6. lo), i.e.
F(o, = at, o2= 0, 212 = 0) = 1 if 01 > 0,
F(al = -a;, o2 = 0, 712 = 0) = 1 if o~< 0,

F(ol = 0, o2 = at, 711 = 0) = 1 if a2 > 0, (6.13)
F(ol = 0, a2 = -a;, TIZ = 0) = I if a2 < 0,
F(o1 = 0, 62 = 0, 212 = 212) = 1 ,

we arrive at

(6.14)

Comparison of this criterion with the criteria discussed above and with experimental
results is presented in Fig. 6.9. As can be seen, criteria specified by Eqs. (6.1 l), (6.12)
and (6.14) provide close results which are in fair agreement with experimental data
for all the stress states except, may be, biaxial compression for which there are
practically no experimental results shown in Fig. 6.9. Such results are presented in
Fig. 6.8 and allow us to conclude that the failure envelope can be approximated in
this case by the polynomial of the type shown in Fig. 6.14(c), i.e.