Page 220 - Mechanical Behavior of Materials

Section 5.4 Anisotropic Materials 221

Substituting for each L in this equation from Eq. 5.59 yields

ε r L r + ε m L m
ε Y = (5.61)
L
Next, substitute for the strains, using Eq. 5.57, and also note that all of the stresses are equal, to
obtain

1 1 L r 1 L m
= + (5.62)
E Y E r L E m L

The length ratios are equivalent to volume fractions:

L r L m
V r = , V m = 1 − V r = (5.63)
L L
Thus, we finally obtain

1 V r V m E r E m
= + , E Y = (a, b) (5.64)
E Y E r E m V r E m + V m E r
where (b) is obtained from (a) by solving for E Y .

5.4.6 Other Elastic Constants, and Discussion

Similar logic also leads to an estimate of ν XY , the larger of the two Poisson’s ratios, called the major
Poisson’s ratio, and also an estimate of the shear modulus:

ν XY = V r ν r + V m ν m (5.65)

G r G m
G XY = (5.66)
V r G m + V m G r

The estimates of composite elastic constants just described are all approximations. Actual
values of E X are usually reasonably close to the estimate. Since E Y from Eq. 5.64 is a lower bound
for the case of fibers, actual values are somewhat higher. Books on composite materials contain
more accurate, but considerably more complex, derivations and equations. In addition, fibers may
occur in two directions, and laminated materials are often employed that consist of several layers of
unidirectional or woven composite. Estimates for these more complex cases can also be made.
◦
◦
◦
In a laminate, if equal numbers of fibers occur in several directions, such as the 0 ,90 , +45 ,
and −45 directions, the elastic constants may be approximately the same for any direction in the
◦
X-Y plane, but different in the Z-direction. Such a material is said to be quasi-isotropic, and it may
be approximated as an isotropic material for in-plane loading, or as a transversely isotropic material
for general three-dimensional analysis.