Page 157 - Mechanics Analysis Composite Materials

142 Mechanics and analysis of composite materials

usually found testing the layer under compression in the z-direction. Transverse
shear moduli G13 and G23 can be obtained by different methods, e.g., by inducing
pure shear in two symmetric specimens shown in Fig. 4.15 and calculating shear
modulus as G13 =P/(2Ay),where A is the in-plane area of the specimen.
For unidirectional composites, G13 = Gl2 (see Table 3.5) while typical values of
G23 are listed in Table 4.1 (Herakovich , 1998).
Poisson’s ratios v31 and ~32can be determined measuring the change of the layer
thickness under in-plane tension in directions 1 and 2.

4.2.2. Nonlinear models

Consider Figs. 3.40-3.43 showing typical stress-strain diagrams for unidirection-
al advanced composites. As can be seen, materials demonstrate linear behavior only
under tension. The curves corresponding to compression are slightly nonlinear,
while the shear curves are definitely nonlinear. It should be emphasized that this
does not mean that linear constitutive equations presented in Section 4.2.1 are
not valid for these materials. First, it should be taken into account that the
deformations of properly designed composite materials are controlled by the fibers,
and they do not allow the shear strain to reach the values at which the shear stress-
strain curve is strongly nonlinear. Second, the shear stiffness is usually very small in
comparison with the longitudinal one, and such is its contribution to the apparent
material stiffness. Material behavior is usually close to linear even if the shear
deformation is nonlinear. Thus, a linear elastic model provides, as a rule, a
reasonable approximation to the actual material behavior. However, there exist
problems, to solve which we should allow for material nonlinearity and apply one
of nonlinear constitutive theories discussed below.
First, note that material behavior under elementary loading (pure tension,
compression, and shear) is specified by experimental stress-strain diagrams of the
type shown in Figs. 3.40-3.43, and we do not need any theory. The necessity of the

Fig. 4.15. A test to determine transverse shear modulus.

Table 4.1
Transverse shear moduli of unidirectional composites (Herakovich, 1998).
Material Glass-epoxy Carbon-poxy Aramid-epoxy Boron-AI

G23 GPa) 4.1 3.2 1.4 49.1