Page 79 - 3D Fibre Reinforced Polymer Composites

68 30 Fibre Reinforced Polymer Composites
conditions given in equations (4.8) and (4.9) are expected to be identical, Le., the
effective compliance matrix, [SI, is the inverse matrix of the effective elastic stiffness
matrix [c] . When applying the boundary conditions in equations (4.8) or (4.9) to a
representative volume, solutions for the true stresses or true strains can be obtained
either analytically or numerically. In the analytical approaches, various assumptions are
introduced to simplify the solutions, which in turn yield simple and closed-form
expressions for the effective properties of the representative volume. However, the
selected assumptions may not allow consideration of certain characteristics and their
corresponding parameters. In the numerical analysis approach, finite element methods
may be used and lesser number of assumptions is required in the analysis, which allows
consideration of more characteristics of a representative volume element. With the
advance of computing techniques, numerical simulation can be achieved at multi-scales,
which allows modelling of more features. However, numerical analysis approaches can
be expensive and require more microstructural information of a representative volume.

4.2.3 Rules of Mixtures and Mori-Tanaka Theory
As an illustrative example, consider a two-phase composite consisting of an elastic
matrix reinforced by randomly dispersed spherical elastic inclusions. The average stress
and strain are given by:

{ F} = c, { F'I' } + c2 { 32) }
[ F} = c, [ 8"' } + c2 { 82) } (4.14)

where cl, c2 are the volume fraction of each phase with cl+c2=l, ?) and Z(') (i=1,2)
are the average stress and strain vectors in phase 1 and 2 respectively.
Using the relations between stresses and strains at any point in the phase as given in
equation (4.10) and (4.1 l), the above equations can be written as:

(4.15)

The average strains and stresses in each phase are uniquely dependent on the average
strains and stresses in representative volume element, namely,

where Ai and Bi (i=1,2) are referred to as concentration matrices, and
c,[A,I+c2[A21 =[I1 and cl[~,l+c2[B21=[~l.
Substituting equations (4.16) into (4.15) yields the following expressions for the
effective stiffness and compliance matrices of the composite material:

=
[c] C,[C"'][A,]+C,[C'~'][A~] (4.17a)