Page 248 - Mechanics Analysis Composite Materials

Chapter 5. Mechanics of laminates 233

To compare two possible forms of constitutive equations for transverse shear,
consider for the sake of brevity an orthotropic layer for which

For transverse shear in the xz-plane Eqs. (5.15) yield

(5.20)
where

(5.21)
- 1'
in accordance with Eq. (5.17), while Eq. (5.19) yields

(5.22)

If the shear modulus does not depend on z, both equations, Eqs. (5.21) and (5.22),
give the same result S55 = G,h.
Using the energy method applied in Section 3.3 we can show that the Eqs. (5.21)
and (5.22) provide the upper and the lower bounds for the actual transverse shear
stiffness. Indeed, consider a strip with unit width experiencing transverse shear
induced by force Y, as in Fig. 5.7, Assume that Eq. (5.20) links the actual force K.
with the actual angle yx = A/l through the actual shear stiffness SSSwhich we do not
know and which we would like to evaluate. To do this, we can use two variational
principles described in Section 2.11. According to the principle of minimum total
potential energy

(5.23)
Tact < rtdm 7

Y

Fig. 5.7. Transverse shear of a strip with unit width.