Engineering & Science
Composite Lamina Rule of Mixtures (Voigt / Reuss) & Halpin-Tsai Transverse Modulus
Calculate longitudinal stiffness E1 via the Voigt iso-strain rule of mixtures and transverse stiffness E2 via both the Reuss iso-stress lower bound and the semi-empirical Halpin-Tsai micromechanics equation.
Predict orthotropic carbon-epoxy ply stiffness along and across fibers!
Unidirectional fiber-reinforced composites (such as AS4/3501-6 carbon-epoxy with fiber modulus Ef = 230 GPa, epoxy matrix modulus Em = 3.5 GPa, and fiber volume fraction Vf = 0.60) behave very differently parallel versus perpendicular to the fibers! Along the fiber direction (1-axis), fibers and matrix share equal strain (Voigt iso-strain model), giving longitudinal modulus E1 = Vf*Ef + (1 - Vf)*Em = 0.60*230 + 0.40*3.5 = 139.4 GPa!
Across the fibers (2-axis), the pure iso-stress Reuss model 1 / (Vf/Ef + (1 - Vf)/Em) = 8.55 GPa severely underpredicts transverse modulus because cylindrical fibers constrain lateral matrix deformation! The Halpin-Tsai equation introduces fiber geometry reinforcement factor xi (typically xi = 2.0 for circular fibers in E2) and partitioning parameter eta = ((Ef/Em) - 1) / ((Ef/Em) + xi), yielding E2 = Em * (1 + xi*eta*Vf) / (1 - eta*Vf) = 17.61 GPa—matching experimental coupon tests closely!
| Micromechanics Model | Stiffness Direction | Formula Summary | CFRP Result (Vf = 0.60) |
|---|---|---|---|
| Voigt Rule of Mixtures | Longitudinal E1 (Iso-Strain) | Vf*Ef + (1 - Vf)*Em | 139.4 GPa |
| Halpin-Tsai (xi = 2.0) | Transverse E2 (Realistic) | Em*(1 + xi*eta*Vf)/(1 - eta*Vf) | 17.61 GPa |
| Reuss Inverse Rule | Transverse E2 (Lower Bound) | 1 / (Vf/Ef + (1 - Vf)/Em) | 8.55 GPa |
Longitudinal Modulus E1 — Voigt Iso-Strain Rule of Mixtures (GPa)
Validates that moduli are positive and fiber volume fraction Vf lies strictly in [0, 1], then computes E1.
=IF(OR(B2 <= 0, B3 <= 0, B4 < 0, B4 > 1), "Invalid Composite Inputs", ROUND((B4 * B2) + ((1 - B4) * B3), 2))Transverse Modulus E2 — Halpin-Tsai Equation (GPa)
Computes eta = ((Ef/Em) - 1) / ((Ef/Em) + xi) inline and evaluates E2 = Em * (1 + xi*eta*Vf) / (1 - eta*Vf).
=IF(OR(B2 <= 0, B3 <= 0, B4 < 0, B4 > 1, B5 < 0), "Invalid Composite Inputs", ROUND(B3 * (1 + B5 * (((B2 / B3) - 1) / ((B2 / B3) + B5)) * B4) / (1 - (((B2 / B3) - 1) / ((B2 / B3) + B5)) * B4), 2))Try the example
Use the original values or change them to test your understanding.