Griffith’s theory of brittle fracture (1920)

Glass should be as strong as its atomic bonds, about \(E/10\), yet it breaks at a few hundredths of that. Alan Arnold Griffith (paper read in 1920, published 1921) blamed small cracks. A crack of half-length \(a\) in a plate under stress \(\sigma\) releases elastic energy \(\pi\sigma^2a^2/E\) per unit thickness and costs surface energy \(4a\gamma_s\). It grows when the release outpaces the cost, at \(\sigma_f = \sqrt{2E\gamma_s/\pi a}\) (griffith_fracture_stress()). Griffith confirmed it on glass bulbs and tubes with scratches of known size, and drawing fresh glass fibres showed strengths approaching the theoretical value.

import matplotlib.pyplot as plt
import numpy as np

from engineeringkit.materials import griffith_fracture_stress

The energy balance

At fixed stress the total energy \(U(a) = 4a\gamma_s - \pi\sigma^2a^2/E\) has a maximum. A crack shorter than the maximum would need energy to grow; a longer one runs. The maximum lies exactly where Griffith’s stress equals the applied stress.

E, gamma = 70e9, 1.0
sigma = 100e6
a = np.linspace(1e-7, 2e-5, 4001)
U = 4 * a * gamma - np.pi * sigma**2 * a**2 / E
a_peak = a[np.argmax(U)]
print(f"at {sigma / 1e6:.0f} MPa the energy peaks at a = {1e6 * a_peak:.2f} um")
assert np.isclose(griffith_fracture_stress(E, gamma, a_peak), sigma, rtol=1e-3)

at 100 MPa the energy peaks at a = 4.46 um

Strength against flaw size

Strength falls as \(1/\sqrt a\). A flaw of a few nanometres is needed to reach the ideal strength; scratches of a few micrometres bring glass down to the 100 MPa range measured in practice.

flaws = np.geomspace(1e-9, 1e-3, 200)
strength = griffith_fracture_stress(E, gamma, flaws)
ideal = E / 10
print(f"1 um flaw: {griffith_fracture_stress(E, gamma, 1e-6) / 1e6:.0f} MPa; ideal strength {ideal / 1e9:.0f} GPa")
assert np.allclose(strength * np.sqrt(flaws), strength[0] * np.sqrt(flaws[0]))

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(9.5, 3.8))
ax1.plot(1e6 * a, U * 1e6)
ax1.axvline(1e6 * a_peak, color="0.6", ls=":")
ax1.set(xlabel="crack half-length [um]", ylabel="energy [uJ/m]", title=f"Energy balance at {sigma / 1e6:.0f} MPa")
ax2.loglog(1e6 * flaws, strength / 1e6, label="Griffith")
ax2.axhline(ideal / 1e6, color="k", ls="--", label="ideal, E/10")
ax2.set(xlabel="crack half-length [um]", ylabel="fracture stress [MPa]", title="Glass")
ax2.legend()
fig.tight_layout()

plt.show()

1 um flaw: 211 MPa; ideal strength 7 GPa