Page 266 - Mechanics Analysis Composite Materials

Chapter 5. Mechanics of laminates 25 1

where Ai:) = G.rz and A;:) = C;, are shear moduli of the core.

5.9. Coordinate of the reference plane

Stiffness coefficients specified by Eqs. (5.28) include coordinate of the reference
plane e (see Fig. 5.1) that, being properly pre-assigned, allows us to simplify
constitutive equations for the laminate. As was shown in Sections 5.2 and 5.6,
taking the middle plane as the reference plane, i.e., putting e = h/2, we sometimes
get C,,,,,= 0, and constitutive equations acquire the simplest form without
membrane-bending coupling terms.
Now, a natural question as to whether it is possible to reduce Eqs. (5.5) to this
form in the general case arises. Taking C,,l,l= 0 in Eqs. (5.28) we get

(5.63)

It is important, that the reference plane should be one and the same for all mn = 11,
12, 22, 14, 24, 44, and these six equations should give the same value of e. In the
general case, this is not possible, so the universal reference plane providing Cm,= 0
cannot exist.
However, there are some other (in addition to the homogeneous and symmetric
structures) particular laminates for which this condition can be met. For example,
consider a laminate composed of isotropic layers (see Sections 4.1 and 5.2). For such
laminates

and in accordance with Eqs. (5.42)

As can be seen, these parameters, being substituted into Eqs. (5.63) do not provide
one and the same value of e. But if Poisson's ratio is the same for all the layers, Le.,
vi = v (i = I, 2, 3,...,k) we get