= Longitudinal rule of mixtures
{title2=$E_L$}
The <longitudinal rule of mixtures> estimates the axial <Young's modulus> of continuous aligned fibres in a <composite matrix>:
$$
E_L=V_fE_f+(1-V_f)E_m.
$$
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> $\varepsilon$ in both phases. Adding their <forces> gives the average axial <stress> $\sigma=V_fE_f\varepsilon+(1-V_f)E_m\varepsilon$, 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.
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