Page 107 - Mechanics Analysis Composite Materials

92 Mechanics and analysis of composite materials

statically admissible stress field. Equality takes place only if the admissible stress
state coincides with the actual one. Excluding this case, i.e., assuming that the class
of admissible fields under study does not contain the actual field we can write the
following strict inequality
Wadm, > w,""' . (3.91)

For the problem of transverse tension, the fibers can be treated as absolutely rigid,
and only the matrix strain energy can be taken into account. We also can neglect the
energy of shear strain and consider the energy corresponding to normal strains only.
With due regard to these assumptions, we use Eqs. (2.51) and (2.52) to get

(3.92)

vm
where Vm is the volume of the matrix and
u =4(O;"&f + OFEy + OF&?) . (3.93)

To find energy W, entering inequality (3.91), we should express strains in terms of
stresses with the aid of constitutive equations, i.e.,

1
Em
&2" = -(cy - v,oy - vmo$), (3.94)
1
Em - -(OY - VmO, - VmO?) .
m
3-
Em
Consider first the actual stress state. Let the ply in Fig. 3.29 be loaded with stress 02
inducing apparent strain ~2 such that
(3.95)

Here, EYt is the actual apparent modulus, which is not known. With due regard to
Eqs. (3.92) and (3.93) we get

(3.96)

where V is the volume of the material. As an admissible field, we can take any state
of stress that satisfies the equilibrium equations and force boundary conditions.
Using the simplest first-order model shown in Fig. 3.34 we assume that

01"= op = 0, oy = 62 .