Page 323 - Mechanics Analysis Composite Materials

308 Mechanics and analysis of composite materials

These equations generalize Eqs. (3.76)-(3.79) for the case of anisotropic fibers and
specify apparent CTE of a unidirectional ply.
As an example, consider a high-modulus carbon-epoxy composite tested by
Rogers et al. (I 977). Microstructural parameters of the material are as follows
(2' = 27°C):
Efl = 41 1 GPa, Eo = 6.6 GPa, vfl = 0.06,
V~L= 0.35, Efl = -1.2 x 1/"C, CC~L 27.3 x lo-' 1/"C,
Em= 5.7 GPa, vm = 0.316, am= 45 x IO-' l/"C, of = om = 0.5.
For these properties, Eqs. (7.16) yield
El = 208.3 GPa, E2 = 6.5 GPa, V~I= 0.33,
tll -0.57 x IOp6 1/"C, CI?= 43.4 x I/"C,
while experimental results are
El = 208.6 GPa, E2 = 6.3 GPa, v21 = 0.33
CII = -0.5 x I/"C, M? = 29.3 x IO-' 1/"C.
Thus, it can be concluded that the first-order microstructural model providing
proper results for longitudinal material characteristics fails to predict a2 with
required accuracy. Discussion and conclusions concerning this problem and
presented in Section 3.3 for elastic constants are valid for thermal expansion
coefficients as well. For practical applications a1 and CI? are normally determined
by experimental methods. However, in contrast to the elasticity problem, for
which the knowledge of experimental elastic constants and material strength
excludes the micromechanical models from consideration, for the thermoelasticity
problems, these models provide us with useful information even if we know
experimental thermal expansion coefficients. Indeed, consider a unidirectional ply
that is subjected to uniform heating which induces only thermal strains, i.e.,
EIT = E:, E~T= E:, Y~~~= 0. Then, Eqs. (7.14) yield GI = 0, Q = 0, 212 = 0. For
homogeneous materials, this means, that no stresses occur under uniform
heating. However, this is not the case for a composite ply. Generalizing
Eqs. (3.74) that specify longitudinal stresses in the fibers and in the matrix we
get

where aI and a2 are specified by Eqs. (7.16). Thus, because thermal expansion
coefficients of the fibers and the matrix are different from those of the material, there
exist microstructural thermal stresses in the composite structural elements. These
stresses are self-balanced. Indeed