Page 99 - Mechanics Analysis Composite Materials

84 Mechanics and analysis of composite materials
=EIEI, 02 = 0, ti2 = 0 , (3.61)

where El =Epf. Being very simple and too approximate to be used in stress-strain
analysis of composite structures, Eqs. (3.61) are extremely efficient for design of
optimal composite structures in which the loads are carried mainly by fibers (see
Chapter 8).
First-order models allow for the matrix stiffness but require only one structural
parameter to be specified-fiber volume fraction, uf. Because the fiber distribution in
the ply is not important for these models, the ply can be presented as a system of
strips shown in Fig. 3.34 and simulating fibers (shadowed areas) and matrix (light
areas). Structural parameters of the model can be expressed in terms of fiber and
matrix volume fractions only, i.e.,

(3.62)

Assume that the model ply is under in-plane loading with some effective stresses 01,
02, and ti2 as in Fig. 3.34 and find the corresponding effective elastic constants El,
E2, G12, v12, and v21 entering Eqs. (3.58). Constitutive equations for isotropic fiber
and matrix strips can be written as

(3.63)

Here f and m indices correspond, as earlier, to fibers and matrix, respectively.
Let us make some assumptions concerning the model behavior. First, it is natural
to assume that effective stress resultant ala is distributed between fiber and matrix
strips and that the longitudinal strains of these strips are the same that the effective
(apparent) longitudinal strain of the ply, 81, i.e.,

(3.64)
(3.65)

Second, as can be seen in Fig. 3.34, under transverse tension the stresses in the strips
are the same and are equal to the effective stress 02, while the ply elongation in the
transverse direction is the sum of the fiber and matrix strips elongations, i.e.,

a2= Oy = a2 , (3.66)
f
Aa = Aaf + Aam (3.67)