Past exam of the materials science course of the University of Cambridge / 2017 / Paper 1 / 5 / b

The optimum axial-stiffness arrangement has continuous straight fibres parallel to the beam axis with good fibre–matrix bonding. For a common axial strain $\epsilon$, force balance on a section gives $F=E_f\epsilon A_f+E_m\epsilon A_m$. Divide by total area and by $\epsilon$ to obtain the longitudinal rule of mixtures,This is an isostrain upper idealization, consistent with the Voigt elastic approximation. The glass fraction required by the deflection constraint isAbout 95.8% glass by volume is impractical for an ordinary glass-fibre/polymer board. It leaves little matrix for wetting, load transfer and defect tolerance. Identical circular parallel fibres pack at most $\pi /(2\sqrt{3})=0.9069$ in a regular hexagonal array; even that zero-void ideal gives only $68.3\mathrm{GPa}$. Polydisperse or noncircular fibres can change that geometric limit, so it is not an absolute theorem about every possible composite. Waviness, voids, incomplete bonding and nonaxial fibres further reduce the achievable modulus. A thicker section or a stiffer reinforcement is a more plausible design change.