= Isostress rule of mixtures
{title2=$E^{-1}=\phi E_1^{-1}+(1-\phi)E_2^{-1}$}
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> $\phi,1-\phi$, the effective modulus is their <weighted harmonic mean>:
$$
\frac1{E_{\mathrm{eff}}}=\frac{\phi}{E_1}+\frac{1-\phi}{E_2}.
$$
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>.<Mechanics of Composite Materials by Bill Clyne and Boban Tanovic>[Load Transfer, What about the Transverse Stiffness?, Figure 2]{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>, $\varepsilon_i=\sigma/E_i$. Since the extensions add,
$$
\varepsilon=\frac{\Delta L_1+\Delta L_2}{L_1+L_2}
=\frac{L_1}{L_1+L_2}\frac{\sigma}{E_1}
+\frac{L_2}{L_1+L_2}\frac{\sigma}{E_2}.
$$
Equal sectional areas make the length fraction $L_1/(L_1+L_2)$ equal to the <volume fraction> $\phi$. Dividing by the common <stress> proves the formula.
For an illustrative pair $E_1=100\,\mathrm{GPa}$, $E_2=2\,\mathrm{GPa}$ and $\phi=0.8$,
$$
E_{\mathrm{eff}}=\left(\frac{0.8}{100}+\frac{0.2}{2}\right)^{-1}\mathrm{GPa}
\approx9.26\,\mathrm{GPa}.
$$
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.<Mechanics of Composite Materials by Bill Clyne and Boban Tanovic>[What about the Transverse Stiffness?, limitations of the slab model]{ref}
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