The Halpin-Tsai equations (1969)

The Voigt and Reuss bounds are far apart for fibre composites, and the exact elasticity solutions for fibre arrays were too complicated for design. John Halpin and Stephen Tsai, at the US Air Force Materials Laboratory, fitted those solutions with one interpolating formula,

\[\frac{P}{P_m} = \frac{1 + \xi\eta V_f}{1 - \eta V_f}, \qquad \eta = \frac{P_f/P_m - 1}{P_f/P_m + \xi}\]

(halpin_tsai()), in which a single parameter \(\xi\) measures how well the reinforcement carries load: \(\xi = 0\) is the Reuss bound, \(\xi \to \infty\) the Voigt bound, and \(\xi = 2\) fits the transverse modulus of round fibres. The example estimates the transverse and shear moduli of glass-epoxy and the stiffness of short fibres.

import matplotlib.pyplot as plt
import numpy as np

from engineeringkit.materials import halpin_tsai, rule_of_mixtures_E_longitudinal, rule_of_mixtures_E_transverse

Transverse and shear moduli

For E-glass (\(E_f = 72\) GPa, \(G_f = 30\) GPa) in epoxy (\(E_m = 3.5\) GPa, \(G_m = 1.3\) GPa) at \(V_f = 0.6\), Halpin-Tsai gives about 15 GPa across the fibres, well above the Reuss value of 8 GPa that springs in series predict.

Ef, Em, Gf, Gm, Vf = 72e9, 3.5e9, 30e9, 1.3e9, 0.6
E2 = halpin_tsai(Ef, Em, Vf, 2.0)
G12 = halpin_tsai(Gf, Gm, Vf, 1.0)
reuss = rule_of_mixtures_E_transverse(Ef, Em, Vf)
print(f"E2 = {E2 / 1e9:.1f} GPa (Reuss {reuss / 1e9:.1f}), G12 = {G12 / 1e9:.2f} GPa")
assert reuss < E2 < rule_of_mixtures_E_longitudinal(Ef, Em, Vf)
assert np.isclose(halpin_tsai(Ef, Em, Vf, 0.0), reuss)

E2 = 14.9 GPa (Reuss 8.2), G12 = 4.48 GPa

One parameter between the bounds

For aligned short fibres of aspect ratio \(\ell/d\), the longitudinal modulus uses \(\xi = 2\ell/d\): short stubby fibres sit near the lower bound, and long ones approach the Voigt rule of mixtures.

fractions = np.linspace(0, 0.7, 100)
fig, ax = plt.subplots(figsize=(6, 3.8))
ax.plot(fractions, rule_of_mixtures_E_longitudinal(Ef, Em, fractions) / 1e9, "k--", label="Voigt")
for aspect in (1, 10, 100):
    ax.plot(fractions, halpin_tsai(Ef, Em, fractions, 2 * aspect) / 1e9, label=f"l/d = {aspect}")
ax.plot(fractions, rule_of_mixtures_E_transverse(Ef, Em, fractions) / 1e9, "k:", label="Reuss")
ax.set(xlabel="fibre volume fraction", ylabel="modulus [GPa]", title="Halpin-Tsai for glass-epoxy")
ax.legend()
fig.tight_layout()
short, long_ = halpin_tsai(Ef, Em, 0.5, 2.0), halpin_tsai(Ef, Em, 0.5, 200.0)
assert short < long_ < rule_of_mixtures_E_longitudinal(Ef, Em, 0.5)

plt.show()