Page 258 - Mechanics Analysis Composite Materials

Chapter 5. Mechanics of laminates 243

5.5. Quasi-isotropic laminates

The layers of the laminate can be arranged in such a way that the laminate
will behave as an isotropic layer under in-plane loading. Actually, the laminate is
not isotropic (that is why it is called a quasi-isotropic laminate) because under
transverse (normal to the laminate plane) loading and under interlaminar shear its
behavior is different from that of an isotropic (e.g., metal) layer.
To derive the conditions that should be met by the structure of a quasi-isotropic
laminate consider in-plane loading with stresses o.~,o,., and z.~,. that are shown in
Fig. 5.1 and induce only in-plane strains E:, E;, and Y:,~. Taking IC, = IC,. = K.~~,= 0 in
Eqs. (5.5) and introducing average (through the laminate thickness 6) stresses as

we can write the first three equations of Eqs. (5.5) in the following form:

(5.51)

where in accordance with Eqs. (5.28) and (5.42)

k
B,,,= EAikA;, hi = hi/h , (5.52)
i= I
where, hi is the thickness of the ith layer normalized to the laminate thickness and
A,,, are the stiffness coefficients specified by Eqs. (4.72). For an isotropic layer,
constitutive equations analogous to Eqs. (5.51) are

0, = E(8: + v&;), IT! = E(&:+ V&!), Z.,. = Gy:-v , (5.53)

where

- E E =?(l-V)E .
1
E=- G=- (5.54)
1 - v2' 2(1 + v)
Matching Eqs. (5.51) and (5.53) we can see that shear stretching coefficients of the
laminate, Le., = B41 and 1%4= B42 should be equal to zero. As follows from
Eqs. (4.72) and Section 5.4.3, this means that the laminate should be balanced, Le.,
it should be composed of O", &4i(or di and IT - 4i),and 90"layers only. Because
the laminate stiffness in the x- and the y-directions must be the same, we require
that Bll = B22. Using Eqs. (4.72), taking hi= h for all i, and performing some
transformation we arrive at the following condition: