Page 393 - Mechanics Analysis Composite Materials

378 Mechanics and analysis of composite materials

material constants, the optimal laminates demonstrate characteristics shown in
Table 8.3 only if the orientation angles of the fibers are found from Eqs. (8.35) or
(8.38) and correspond to a particular distribution of stresses ox,cy,and zxy.
As follows from Table 8.3, the modulus of a carbon-epoxy laminate is close to the
modulus of aluminum, while the density of the composite material is less by the
factor of 1.7. This is the theoretical weight-saving factor that can be expected if we
change aluminum for carbon+poxy composite in a thin-walled structure. Because
the stiffness of both materials is approximately the same, to find the optimal
orientation angles of the structure elements, we can substitute in Eq. (8.38) the
stresses acting in the aluminum prototype structure. Thus designed composite
structure will have approximately the same stiffness as the prototype structure and,
as a rule, higher strength because carbon composites are stronger than aluminum
alloys.
To evaluate the strength of the optimal laminate, we should substitute strains
from Eqs. (8.36) into Eqs. (4.69) and thus found strains in the principal material
coordinates of the layers - into constitutive equations, Eqs. (4.56), that specify
stresses ol and o~(z12 = 0)acting in the layers. Applying the proper failure criterion
(see Chapter 6) we can evaluate the laminate strength.
Comparing Tables 1.1 and 8.3 we can see that boron-epoxy optimal laminates
have approximately the same stiffness that titanium (but is lighter by the factor of
about 2) and boron-aluminum can be used to substitute steel with a weight-saving
factor of about 3.
For preliminary evaluation, we can use a monotropic model of unidirectional
plies neglecting stiffness and load-carrying capacity of the matrix. Then, Eqs. (8.37)
acquire the following simple forms:

(8.39)

As an example, consider an aluminum shear web with thickness h = 2 mm, elastic
constants E, = 72 GPa, v, = 0.3 and density pa = 2.7 g/cm3. The panel is loaded
with shear stress z. Its shear stiffness is Bg,= 57.6 GPa mm and the mass of a unit
surface is ma = 5.4 kg/m2. For the composite panel, taking ox = = 0 in Eq. (8.38)
we get 4 = 45". Thus, the composite panel consists of +45" and -45" unidirectional
layers of the same thickness. The total thickness of the laminate is h = 2 mm, i.e.,
the same as for an aluminum panel. Substituting El = 140 GPa and taking into
account that p = 1.55 g/cm3 for a carbon-epoxy composite that is chosen to
substitute aluminum we get B& = 70 GPa mm and m, = 3.1 kg/m3. Stresses acting
in the fiber directions of the composite plies are o; = f2z. Thus, the composite
panel has 21.5% higher stiffness and its mass makes only 57.4% of the mass of a
metal panel. Composite panel has also higher strength because the longitudinal
strength of unidirectional carbon-poxy composite under tension and compression
is more than twice higher than the shear strength of aluminum.
Possibilities of the composite structure under discussion can be enhanced if we use
different materials in the layers with angles @, and @? specified by Eqs. (8.35).