Page 305 - Mechanics Analysis Composite Materials

290 Mechanics and analysis of composite materials

(6.34)

is actually an invariant of the strength tensor. With due regard to Eqs. (6.28) and
(6.31) this invariant can be also written as

I, =Ryl -+fl, =R;: -k S$
If the actual material characteristics do not satisfy Eq. (6.34), then the tensor
strength criterion cannot be applied to this material. However, if this equation is
consistent with experimental data, then the tensor criterion offers considerable
possibilities to study material strength. Indeed, restricting ourselves to two terms
presented in Eq. (6.20) let us write this equation in coordinates (l’, 2’) shown in Fig.
6.18 and assume that 4 # 45”. Then

(6.35)
i.k i.k.nr.n

Here, Si and S$mlfare the components of the second and the fourth rank strength
tensors which are transformed in accordance with tensor calculus as

P.4
(6.36)

Here, I are directional cosines of axes 1’ and 2’ on the plane referred to coordinates 1
and 2 (see Fig. 6.18), i.e. ZII = cos$, 112 = sin$, 121 = -sin 4, 122 = cos $. Sub-
stitution of Eqs. (6.36) into Eq. (6.35) yields the strength criterion in coordinates
(I!, 2’) but written in terms of strength components corresponding to coordinates
(1, 2), i.e.

(6.37)

Apply Eq. (6.37) for the special orthotropic material that was studied above (see
Fig. 6.18) and for which

(6.38)