Rule of Mixtures (longitudinal)⚠ unverified

Understanding the Parameters

• Fiber modulus $E_f$ — usually far stiffer than the matrix (carbon $\sim 230$ GPa vs epoxy $\sim 3$ GPa), so even at $V_f = 0.6$ the fibers carry the large majority of the load and set $E_1$.

• Fiber volume fraction $V_f$ — $E_1$ rises linearly with $V_f$; this near-linearity (well confirmed by test) is why the longitudinal rule of mixtures is trusted for design, unlike the transverse case.

• Load sharing — with equal strain, the fiber-to-matrix stress ratio is $E_f/E_m$: the fibers carry a fraction $E_f V_f/E_1$ of the load, typically $>95\%$. The matrix's main job is to transfer load into the fibers and stabilise them, not to carry axial stress.

• Upper bound — the Voigt model is the theoretical upper bound on composite stiffness; the transverse (Reuss) model is the lower bound (Rule of Mixtures transverse). Real longitudinal moduli sit very close to this upper bound.

Derivation (Approaching a Proof)

Load the composite along the fibers. Fibers and matrix are bonded and side by side, so they stretch together — they experience the same strain $\varepsilon$ (the iso-strain, or Voigt, assumption). By Hooke's law each phase carries a stress $\sigma_f = E_f\varepsilon$ and $\sigma_m = E_m\varepsilon$.

The total force is shared in proportion to area, which for aligned fibers equals volume fraction. The average composite stress is

$$\sigma_1 = \sigma_f V_f + \sigma_m (1-V_f) = (E_f V_f + E_m(1-V_f))\,\varepsilon.$$

The composite modulus is $E_1 = \sigma_1/\varepsilon$:

$$E_1 = E_f V_f + E_m(1-V_f).$$

This is the parallel-springs result: elements sharing a common displacement add their stiffnesses.

Dimensional check. $E_1 = E_f V_f + E_m(1-V_f) = \text{Pa}\cdot(\text{–}) = \text{Pa}$ — a modulus, as required ($V_f$ dimensionless).

History and Development

The iso-strain average is Woldemar Voigt's (1889) bound for polycrystal elastic properties, adapted to fiber composites as the field matured in the 1960s. Paired with the Reuss iso-stress bound (Rule of Mixtures transverse), it brackets the possible stiffness (Voigt–Reuss bounds). The longitudinal rule of mixtures is the one micromechanics formula accurate enough to use directly in design, feeding Composite Laminate Theory and specific-stiffness selection (Specific Modulus).

Related Concepts: Rule of Mixtures transverse, Composite Density, Poisson Ratio Nu12, Longitudinal Strength, Halpin-Tsai Equations, Composite Laminate Theory

Notes: Voigt / iso-strain / parallel model → upper bound, fiber-dominated. Linear in $V_f$, accurate in practice. Fibers carry $>95\%$ of axial load. Transverse direction uses the Reuss lower bound (Rule of Mixtures transverse).