Isostress rule of mixtures ($E^{-1}=\varphi E_1^{-1}+(1-\varphi )E_2^{-1}$)

The isostress rule of mixtures estimates a composite material’s Young's modulus by assuming that its constituent phases carry the same axial stress. For two phases with positive Young's moduli $E_1,E_2$ and volume fractions $\varphi ,1-\varphi$, the effective modulus is their weighted harmonic mean:This is the one-dimensional equal-stress specialization of the Reuss elastic approximation, rather than the equal-strain assumption of the longitudinal rule of mixtures.^[ref]

To derive it, put two ideal elastic segments in series, with the same sectional area $A$, lengths $L_1,L_2$, and a common axial force $F$. Force balance gives $\sigma =F/A$ in each segment. Under one-dimensional linear elasticity, ${\epsilon }_i=\sigma /E_i$. Since the extensions add,Equal sectional areas make the length fraction $L_1/(L_1+L_2)$ equal to the volume fraction $\varphi$. Dividing by the common stress proves the formula.

For an illustrative pair $E_1=100\mathrm{GPa}$, $E_2=2\mathrm{GPa}$ and $\varphi =0.8$,At $\sigma =10\mathrm{MPa}$, the segment strains are $0.0001$ and $0.005$, so the mean strain is $0.00108$. The softer phase supplies most of the extension despite occupying only one fifth of the volume. Placing the same proportions in ideal parallel load paths with equal strain instead gives $0.8E_1+0.2E_2=80.4\mathrm{GPa}$. Geometry has changed the result substantially.

The series derivation is exact within its one-dimensional model, with bonded segments and no bending or additional transverse constraints. It is not an exact prediction for every real fibre-reinforced composite material loaded across its fibres: stress and strain vary around the actual reinforcement geometry. The primary teaching resource explicitly distinguishes the equal-stress slab estimate from the more complicated transverse response.^[ref]