Page 94 - 3D Fibre Reinforced Polymer Composites

Micromechanics Models for Mechanical Properties 83

(4.39)

where E,, G, and v, are the matrix elastic properties, Efl, Eft, Gflz, Gf~3 and vf12 are the
fibre elastic properties, and V, represents the fibre volume fraction of the yarn. In terms
of the engineering constants, the corresponding stiffness matrix [a and the compliance
matrix [SI can be determined following the standard procedure (Christensen, 1979) in
the yarn-related local coordinate system. From the yarn undulation function, the off-
axis angle can be determined and then used to compute the off-axis elastic stiffness
matrix, which can be a function of x or y. With reference to Figure 4.6, the off-axis
stiffness and compliance matrix for the warp are [C"(y)] and [S w(y)] , and those for
the weft are [C'(x)] and [S' (x)] , by employing the transformation matrix [q defined
between the yarn-based local coordinate system and the global coordinate system for the
unit cell.
In the 3D model proposed by Hahn and Pandey (1994) the representative volume
element as shown in Figure 4.6 is considered but without modelling the upper and lower
layer of pure matrix. Yarn undulations and geometry are described by using the
sinusoidal functions similar to those given in equations (4.35), (4.37) and (4.38) with
h,=O, h, and h+=O being exchanged, Lf replacing a,, and aw+gw, and ayt and L, replacing
afigf. Average stresses and strains are defined as those in equation (4.7). To simplify
the analysis, it is assumed that the strains are uniform throughout the unit cell when it is
subject to homogenous displacement boundary conditions similar to the definition in
equation (4.8). This is a key assumption in this model as it introduces the
approximation. Under this iso-strain assumption, the effective elastic properties [c] as
defined in equation (4.10) can be determined by:

(4.40)

where the subscripts and superscripts w, f and nz represent the warp, weft and pure
matrix, respectively, V is the total volume of the unit cell. Closed form expressions for
the effective elastic constant matrix [e] were given by Hahn and Pandey (1994). The
iso-strain assumption offers a significant simplification in evaluating effective elastic
stiffness matrix [c], but it also at the same time creates an opportunity for future
research to enhance the predicted results by removing the iso-strain assumption.