9.1 Basic Equations

In this chapter, we introduce the concept of effective elastic constants for the laminate. These constants are the effective extensional modulus in the x direction E _x, the effective extensional modulus in the y direction E _y, the effective Poisson's ratios v _xy and v _yx, and the effective shear modulus in the x- y plane G _xy.

The effective elastic constants are usually defined when considering the inplane loading of symmetric balanced laminates. In the following equations, we consider only symmetric balanced or symmetric cross-ply laminates. We therefore define the following three average laminate stresses [1]:

(9.1)

(9.2)

(9.3)

where H is the thickness of the laminate. Comparing (9.1), (9.2), and (9.3) with (7.13), we obtain the following relations between the average stresses and the force resultants:

(9.4)

(9.5)

(9.6)

Solving (9.4), (9.5), and (9.6) for N _x, N _y, and N _xy, and substituting the results into (8.11) and (8.12) for symmetric balanced laminates, we obtain:

(9.7)

The above 3 3 matrix is defined as the laminate compliance matrix for symmetric balanced laminates. Therefore, by analogy with (4.5), we obtain the following effective elastic constants for the laminate:

(9.8a)

(9.8b)

(9.8c)

(9.8d)

(9.8e)

It is clear from the above equations that v _xy and v _yx are not independent and are related by the following reciprocity relation:

(9.9)

Finally, we note that the expressions of the effective...