Page 106 - Mechanics Analysis Composite Materials

Chapter 3. Mechanics of a unidirectional ply 91

numerical (finite element, finite difference methods) stress analysis of the matrix in
the vicinity of fibers,
0 averaging of stress and strain fields for a media filled in with regularly or
randomly distributed fibers,
0 asymptotic solutions of elasticity equations for inhomogeneous solids characteri-
zed with a small microstructural parameter (fiber diameter),
0 photoelasticity methods.
Exact elasticity solution for a periodical system of fibers embedded in an isotropic
matrix (Van Fo Fy (Vanin), 1966) is shown in Figs. 3.36 and 3.37. As can be
seen, due to high scatter of experimental data, the higher-order model does not
demonstrate significant advantages with respect to elementary models.
Moreover, all the micromechanical models can hardly be used for practical
analysis of composite materials and structures. The reason for this is that
irrespective of how rigorous the micromechanical model is, it cannot describe quite
adequately real material microstructure governed by a particular manufacturing
process, take into account voids, microcracks, randomly damaged or misaligned
fibers and many other effects that cannot be formally reflected in a mathematical
model. Because of this, micromechanical models are mostly used for qualitative
analysis providing us with understanding of how material microstructural para-
meters affect its mechanical properties rather than with quantitative information
about these properties. Particularly, the foregoing analysis should result in two main
conclusions. First, the ply stiffness along the fibers is governed by the fibers
and linearly depends on the fiber volume fraction. Second, the ply stiffness across
the fibers and in shear is determined not only by the matrix (which is natural), but
by the fibers as well. Though the fibers do not take directly the load applied in
the transverse direction, they significantly increase the ply transverse stiffness (in
comparison with the stiffness of a pure matrix) acting as rigid inclusions in the
matrix. Indeed, as can be seen in Fig. 3.34, the higher the fiber fraction, af, the lower
is the matrix fraction, a,, for the same a, and the higher stress 02 should be applied
to the ply to cause the same transverse strain ~2 because only matrix strips are
deformable in the transverse direction.
Due to the aforementioned limitations of micromechanics, only the basic models
were considered above. Historical overview of micromechanical approaches and
more detail description of the corresponding results can be found elsewhere
(Bogdanovich and Pastore, 1996; Jones, 1999).
To analyze the foregoing micromechanical models we used traditional approach
based on direct derivation and solution of the system of equilibrium, constitutive,
and strain-displacement equations. As known, the same problems can be solved
with the aid of variational principles discussed in Section 2.1 1. In application to
micromechanics, these principles allow us not only to determine apparent stiffnesses
of the ply, but also to establish the upper and the lower bounds on them.
Consider for example the problem of transverse tension of a ply under the action
of some average stress 02 (see Fig. 3.29) and apply the principle of minimum strain
energy (see Section 2.1 1.2). According to this principle, the actual stress field
provides the value of the body strain energy, which is equal or less than that of any