Page 101 - Mechanics Analysis Composite Materials

86 Mechanics and analysis of eomposite marerials

Substituting Eqs. (3.74) into this equation we can express e1 in terms of 01and 62.
Matching this result to the first constitutive equation in Eqs. (3.58) we arrive at

(3.77)

The first of these equations specifies the apparent longitudinal modulus of the ply
and corresponds to the so-called rule of mixtures according to which the property of
a composition can be calculated as the sum of its constituent material properties
multiplied by the corresponding volume fractions.
Now consider Eq. (3.68) that can be written as
~2 = EZUf + .
f

Substituting strains E: and E? from Eqs. (3.72), stresses cf and 0;" from
Eqs. (3.74) and E] from Eqs. (3.58) with due regard to Eqs. (3.76) and (3.77) we
can express ~2 in terms of 01 and 62.Comparing this expression with the second
constitutive equation in Eqs. (3.58) we get

(3.78)

Using Eqs. (3.76) and (3.79) we have

V2I = vfq+ v,u, . (3.80)
This result corresponds to the rule of mixtures. Another Poisson's ratio can be
found from Eqs. (3.77) and (3.78). Finally, Eqs. (3.58), (3.70), and (3.73) yield the
apparent shear modulus

(3.81)

This expression can be derived from the rule of mixtures if we use compliance
coefficients instead of stiffnesses as in Eq. (3.76).
Since the fiber modulus is typically many times greater than the matrix modulus,
Eqs. (3.76), (3.78), and (3.81) can be simplified neglecting small terms and presented
in the following approximate form