Page 301 - Mechanics Analysis Composite Materials

286 Mechanics and analysis of composite materials

components form a tensor induces some conditions that should be imposed on these
components and not necessarily correlate with experimental data.
To be specific, consider a second-order tensor criterion. Introducing contracted
notations for tensor components and restricting ourselves to the consideration of
orthotropic materials referred to the principal material coordinates 1, 2, 3 (see Fig.
6.1) we can present Eq. (6.20) as

which corresponds to Eq. (6.20) if we put

Superscript “0” indicates that the components of the strength tensors are referred to
the principal materia1 coordinates. Applying strength conditions in Eqs. (6.13) we
can reduce Eq. (6.21) to the following form:

(6.22)

This equation looks similar to Eq. (6.14),but there is a principal difference between
them. While Eq. (6.14)is only an approximation of experimental results, and we can
take any suitable value of coefficient RY2 (in particular, we put R12 = 0), criterion in
Eq. (6.22) has an invariant tensor form, and coefficient R12 should be determined
using this property of the criterion.
Following Gol’denblat and Kopnov (1968) consider two cases of pure shear in
coordinates l’, 2’ shown in Fig. 6.17 and assume that T& = T15 and zqg = Ti5, where

2 2
4

(4 (6)

Fig. 6.17. Pure shear in coordinates (1’,2’) rotated by 45” with respect to the principal material
coordinates (1. 2).