Page 307 - Mechanics Analysis Composite Materials

292 Mechanics and analysis of composite materials

(6.42)

Now assume (and this is what should not be done for approximation criteria) that
criterion in Eqs. (6.15) is valid in coordinates (l’, 2’) (see Fig. 6.18) as well. Because
material is orthotropic in these coordinates, we have the following equation similar
to Eq. (6.33):

(6.43)

Matching this result with Eq. (6.30) in which coefficients R are specified by Eqs.
(6.42) we arrive at the following conditions:

which can be reduced to

a45 = a0 . (6.44)

These conditions can hardly be met for cross-ply or fabric composites as in Fig.
6.18.
For an isotropic material, Eqs. (6.25) for the tensor-polynomial criterion and Eq.
(6.43) for the approximation criterion yield, with due regard to Eqs. (6.44) one and
the same strength criterion that can be written as

0: - 0162 + 0;+ 32;, = a’ , (6.45)

where 0 = Oo = 845. Introducing stress intensity c specified by Eqs. (4.19) or
Eq. (4.24) we arrive at the following result:

a=o (6.46)

known in the theory of plasticity as Huber-Mises plasticity criterion. It should be
emphasized that stress intensity, a, is the invariant characteristic of the stress state
(see Section 4.1.2), so for an isotropic material, the approximation criterion in Eq.
(6.45) or Eq. (6.46) is also a tensor-polynomial criterion. In this latter form it was