 Longitudinal rule of mixtures

The longitudinal rule of mixtures estimates the axial Young's modulus of continuous aligned fibres in a composite matrix:The model assumes perfect bonding, small elastic strain, no voids and loading parallel to the fibres. Equal extension over a common initial length gives equal axial strain $\epsilon$ in both phases. Adding their forces gives the average axial stress $\sigma =V_f{E}_f\epsilon +(1-V_f)E_m\epsilon$, which proves the formula. It is the axial Voigt elastic approximation.

For example, $V_f=0.6$, $E_f=70\mathrm{GPa}$ and $E_m=3\mathrm{GPa}$ give $E_L=43.2\mathrm{GPa}$. These are illustrative inputs, not universal constituent values. The formula makes reinforcement efficient because the stiffer constituent carries the larger stress at the common strain. Loading across the fibres changes the compatibility and traction conditions, so the same formula is not a general transverse-modulus law. Fibre waviness or poor fibre-matrix load transfer can also invalidate the ideal axial estimate.