Halpin-Tsai Equations

The Halpin-Tsai equations are a set of empirical relationships that enable the property of a composite material to be expressed in terms of the properties of the matrix and reinforcing phases together with their proportions and geometry. These equations were curve fitted to exact elasticity solutions and confirmed by experimental measurements - they work well but the parameter has no scientific basis nor is it related to any material or geometric property. Halpin and Tsai showed that the property of a composite Pc could be expressed in terms of the corresponding property of the matrix Pm and the reinforcing phase (or fibre) Pf using the following relationships:-

The factor is used to describe the influence of geometry of the reinforcing phase on a particular property. This factor is different for different properties in the same composite. The table below summarizes this factor for many typical geometry's.

Aligned continuous fibres


Spherical particles
Oriented short fibres
Oriented plates
Oriented whiskers

In all composite systems the equations are not valid above f=0.9 since these volume fractions of fibres are impossible geometrically.