Page 109 - Mechanics Analysis Composite Materials

94 Mechanics and analysis of composite materials

where

Substituting Eqs. (3.101) into Eq. (3.93) and performing integration in accordance
with Eq. (3.92) we get

(3.102)

Here

and r(1) is given in notation to Eq. (3.89). Applying Eqs. (3.100) and (3.102) in
conjunction with inequality (3.99) we arrive at

where

ZEm
E' -
- 20f(1 - 2vmp.J
is the upper bound on E2 shown in Fig. 3.36 with a broken line.
Taking statically and kinematically admissible stress and strain fields that are
more close to the actual state of stress and strain one can increase E: and decrease
E; making the difference between the bounds smaller (Hashin and Rosen, 1964).
It should be emphasized that thus established bounds are not the bounds on the
modulus of a real composite material but on the result of calculation corresponding
to the accepted material model. Indeed, return to the first-order model shown in
Fig. 3.34 and consider in-plane shear with stress TI?.As can be readily proved, the
actual stress-strain state of the matrix in this case is characterized with the following
stresses and strains

(3.103)

Assuming that fibers are absolutely rigid and taking stresses and strains in
Eqs. (3.103) as statically and kinematically admissible we can readily find that