Page 108 - Mechanics Analysis Composite Materials

Chapter 3. Mechanics of a unidirectional ply 93
Then, Eqs. (3.92H3.94) yield

(3.97)

Substituting Eqs. (3.96) and (3.97) into inequality (3.91) we arrive at

where in accordance with Eqs. (3.62) and Fig. 3.34

This result specifying the lower bound on the apparent transverse modulus follows
from Eq. (3.78) if we put Er -+ m. Thus, the lower (solid) line in Fig. 3.36 represents
actually the lower bound on E2.
To derive the expression for the upper bound, we should use the principle of
minimum total potential energy (see Section 2.1 1.1) according to which (we again
assume that the admissible field does not include the actual state)
Tadm > Tact 1 (3.98)

where T = W, -A. Here, &: is determined with Eq. (3.92) in which stresses are
expressed in terms of strains with the aid of Eqs. (3.94) and A, for the problem
under study, is the product of the force acting on the ply by the ply extension
induced by this force. Because the force is the resultant of stress 02 (see Fig. 3.29)
which induces strain E:! the same for actual and admissible states, A is also the same
for both states, and we can present inequality (3.98) as

FFym > . (3.99)

For the actual state, we can write equations similar to Eqs. (3.96), i.e.,

where V = 2Ra in accordance with Fig. 3.38. For the admissible state, we use the
second-order model (see Fig. 3.38) and assume that

where E, is the matrix strain specified by Eq. (3.86). Then, Eqs. (3.94) yield

(3.101)