Page 201 - Mechanics Analysis Composite Materials

186 Mechanics and analysis of composite materials

where ATl =AT, =All, AT2 =Ar2 =A12,Ai2 = =A22, AT4 = AT^ =A14, Ai4 =
-AT4 =A24, A:4 =AT4 =A44, where A,,, (mn= 1I, 12,22, 14,24,44) are specified by
Eqs. (4.72). Substituting Eqs. (4.126) into Eqs. (4.125) we arrive at the following
constitutive equations for an angle-ply layer:

(4.127)

The inverse form of these equations is

(4.128)

where

(4.129)

As follows from Eqs. (4.127) and (4.128), the layer under study is orthotropic.
Now derive constitutive equations relating transverse shear stresses zxzand zy and
the corresponding shear strains yxz and yw. Let the angle-ply layer be loaded by
stress z,. Then for all the plies, z: = 7,; = z, and because the layer is orthotropic,
yz = 7,; = yu, y-2 = y-; = yjz = 0. In a similar way, applying stress z, we have
72 = T; = z,,, y-; = y; = y.y., 7: = y,; = 'yxr = 0. Writing two last constitutive
equations of Eqs. (4.71) for these two cases we arrive at

where stiffnesscoefficients A55 and A66 are specified by Eqs. (4.72).
Dependencies of E, and G,,. on 4 plotted with the aid of Eqs. (4.129) are shown in
Fig. 4.57 with solid lines. Theoretical curve for E, is in very good agreement with
experimental data shown with circles (Lagace, 1985). For comparison, the same
moduli are presented for the +4 anisotropic layer considered in Section 4.3.1. As
can be seen, Ex (*4) 3 E,'. To explain this effect, consider a uniaxial tension of both
layers in the x-direction.While tension of +# ply and of -4 individual plies shown
in Fig. 4.58 is accompanied with shear strain, the system of these plies does not
demonstrate shear under tension and, as a result, has higher stiffness. Working as
plies of a symmetric angle-ply layer individual anisotropic +4 and -4 plies are
loaded not only with normal stress a, that is applied to the layer, but also with shear
stress zx-,, that restricts the shear of individual plies (see Fig. 4.58). To find the
reactive and balanced between the plies shear stress, we can use Eqs. (4.75). Taking