Real solids break far below their ideal bond strength because a crack pays for its own growth: it trades stored elastic energy for the surface energy of two new faces. The fracture stress is σ_f = √(2Eγ/πa) — it falls as the crack half-length a grows, scaling as 1/√a. Rendered, not quoted.
source Griffith, A. A., The Phenomena of Rupture and Flow in Solids (1921), Phil. Trans. R. Soc. A 221, 163–198 · doi:10.1098/rsta.1921.0006
An infinite plate under remote tension σ holds a central through-crack of half-length a. Two competing energies, per unit thickness:
Elastic energy released by opening the crack: U_el = −πσ²a²/E (grows as a²).
Surface energy spent making two new faces: U_s = 4γa (grows as a).
Total U(a) = 4γa − πσ²a²/E. The crack runs when growing it lowers U — i.e. past the peak where dU/da = 0, which pins the fracture stress at σ_f = √(2Eγ/πa).
The-stress-strain law gives the ideal: linear σ=Eε up to a theoretical strength near E/10. Griffith 1921 explains the gap — every real solid carries flaws, and the largest flaw a drops its strength to √(2Eγ/πa), a tiny fraction of ideal.
Neighbour: the-stress-strain (the ideal elastic slope). Here that slope stores the energy a crack then spends. The material constant that survives is the toughness K_c = σ_f√(πa) = √(2Eγ).
Live re-check of the engine's laws — 1/√a scaling, quadruple-halves, larger-is-weaker, and constant K_c. Flips red the instant window 6 tampers the formula.
Modulus E = 70 GPa (aluminium), surface energy γ = 1.0 J/m², crack half-length a swept from 0.1–4 mm. All exact inputs to a closed-form law — no fitting.
Proven: quadrupling a halves σ_f exactly; longer cracks are weaker; toughness K_c = √(2Eγ) is constant across crack lengths.
“Griffith's balance is an energy criterion — it gives no stress field, and for ductile metals the surface term is tiny next to plastic work at the tip.”
True. Irwin & Orowan (1948–49) fixed this: replace 2γ with an effective 2γ+G_p (plastic work), giving G_c and K_c=√(EG_c). The 1/√a scaling this engine proves survives intact — only the constant changes.
“Strength is a fixed material number; a scratch is cosmetic.”
→ Strength depends on the largest flaw. A 4× longer crack halves it.
“Bigger part, same alloy → same failure stress.”
→ Bigger parts hide bigger flaws, so they fail sooner. Only K_c is size-free.
“σ_f should rise with crack length (more material to tear).”
→ Backwards. Longer crack releases more stored energy per step, so it needs less stress. That inversion is exactly window 6's tamper.
Plant the disclosed void: move a into the numerator (no square root) so σ_f rises with crack length. The WITNESS in 7 catches it, the panel flags, K_c stops being constant.