the Griffith criterion

In 1920 A. A. Griffith was puzzled by a strange fact: thin glass fibres were enormously stronger than thick rods of the same glass, and none reached the strength their atomic bonds promised. His insight was to stop asking whether the stress at a crack tip was high enough to snap a bond, and instead to do energy bookkeeping. A crack can only grow if growing it pays for itself: the elastic energy the material releases as the crack opens must be at least enough to pay for the fresh surface the crack creates. If it does not pay, the crack sits still; the moment it does, the crack runs.

Picture a plate stretched by a stress sigma with a crack of length 2c through it. Opening the crack relaxes the material around it, releasing stored elastic energy that grows with the area unzipped, roughly as c squared. But every bit of crack advance makes two new surfaces, and creating surface costs energy in proportion to the crack length, as c. Because the release grows as c squared while the cost grows only as c, there is a critical crack length beyond which release always wins. Balancing the two gives the Griffith equation for the fracture stress: sigma_f = sqrt(2 times E times gamma_s / (pi times c)), where E is Young's modulus, gamma_s the surface energy, and c the half-length of the flaw. The strength falls off as one over the square root of the flaw size, exactly explaining why the thinner, less flawed fibre was stronger.

The Griffith criterion is the foundation of all ceramic fracture mechanics. It says a ceramic's measured strength is controlled by the size of its worst crack, not by the strength of its bonds, and it explains why real ceramics fail at roughly one part in a hundred of their theoretical strength. Later work by Irwin repackaged the same energy balance into the stress-intensity form K = Y times sigma times sqrt(pi times c) reaching a critical value K_IC. An honest caveat: Griffith's original balance counted only the thermodynamic surface energy, which is right for a perfectly brittle glass but far too small for a tough polycrystalline ceramic, where extra energy is eaten by processes like grain bridging and microcracking, so in practice one uses the measured fracture energy or toughness rather than 2 gamma_s.