Fibre Packing

In all systems the equations which predict the properties of a composite breakdown at high volume fractions of reinforcement because of geometric packing limitations and the necessity for the reinforcing phase to be surrounded by the matrix in order that load can be transferred to it. There are two simple packing models which we can use to establish an upper bound for the volume fraction, a square array and an hexagonal array with circular section reinforcement. For the hexagonal close packing



the maximum volume fraction of fibres,fmax occurs when the fibres are just touching, i.e.,

so,
For the simple cubic packing

the maximum volume fraction of fibres occurs when the fibres are just touching, i.e.,

so,

From the two figures it is readily apparent that volume fractions higher then 90% are impossible and that even 78% fibre loading would be very difficult to achieve. In practice, the maximum volume fraction is around 60% in unidirectional aligned fibre composites. In woven materials, the total volume fraction rarely exceeds 40% in a given layer of cloth and so the effective fibre fraction in either the warp or weft directions is unlikely to exceed 20% for a plain weave fabric. For loosely packed fabrics such as chopped strand mat, the total volume fraction of fibres is unlikely to exceed 10% and are normally used to provide filler layers between the outer load bearing layers in a multilayer laminate.

Density, Weight and Volume

The density of the composite is easily calculated by adding up the mass of each component



To convert from volume fraction of fibres, f, to weight fraction of fibres, fw, we just need to establish the ratio of the mass of the fibres to the total mass, this is simply

To convert from weight fraction, fw, to volume fraction, f, we need to establish the ratio of the volume of reinforcement to the total volume of the composite. Again this is simply

Now we need to implement the design rules in MATHCAD so that we can quickly find the required volume or weight fractions of fibres to produce the desired elastic properties. The following exercise is relatively trivial but, it demonstrates a number of important techniques, such as
  • How to set up a design model so that it is easy to change the properties of the materials used.
  • How to write equations in terms of the design variable.
  • How to solve for the design variable.
  • How to plot property variations as a function of the design variable.
  • How to simplify and re-use equations to solve more complex problems.
If you want to skip the exercise, then move on to the next section which examines how to determine the strength of aligned continuous fibre composites.