Incoprea hypothesis tests · volume II
Brittle failure occurs when defects percolate, so strength is governed by defect density and a universal exponent rather than by the single largest flaw, and the size effect should differ from the weakest-link prediction.
Strength should scale with defect density through a power law whose exponent matches percolation universality, and the transition should be universal across sample size.
Falsified by: Strength that tracks the largest defect at all densities, with a Weibull size effect that never saturates, would leave the percolation picture without a regime.
Site percolation on cubic lattices of side 12, 16, 24 and 32 (140-240 realisations per density). A sample fails when a six-connected defect cluster touches both opposite faces. Critical density by interpolation of the 50% spanning point; finite-size extrapolation with the three-dimensional percolation exponent nu = 0.876; order-parameter exponent beta fitted on the largest lattice.
code/tests/failure-percolation-defect-density.py from the recorded run.| Critical density (extrapolated) | 0.312 |
| Textbook 3D site value | 0.312 |
| Exponent fitted | 0.13 |
| Universal exponent | 0.418 |
The spanning transition sits at p_c = 0.318 on a 32-cube, extrapolating to 0.312 against the textbook 3D site value 0.3116 — the critical density is recovered. The order-parameter exponent fit returns 0.13 (R2 0.68) against the universal 0.418; at these lattice sizes the spanning probability is rounded and the exponent is not measurable this way. The density claim survives; the exponent claim is untested, not confirmed.