Griffith Critical Stress

A. A. Griffith was working at the Royal Aircraft Establishment on why materials are so much weaker than theory says they should be. The cohesive strength of a solid — the stress it takes to pull two planes of atoms apart — is roughly $E/10$, which for glass is about 7000 MPa. Real glass breaks around 50 MPa. The gap is a factor of more than a hundred, and no amount of care in making the glass closed it.

His 1921 paper in the Philosophical Transactions solved it with an energy argument, and the argument is worth following because everything since is a refinement of it. Take a plate under tension and put a crack in it. Two things happen. The material around the crack unloads, releasing stored elastic energy — and that release grows with the square of the crack length, because both the length of the unloaded region and its depth scale with $a$. Meanwhile the crack has created new surface, which costs energy — and that cost grows only linearly with $a$, because two faces of length $a$ each need $2\gamma a$ of energy per unit thickness.

One term goes as $a^2$ and the other as $a$, so their difference has a maximum. Below that crack length the energy cost of extending exceeds the release, and the crack is stable. Above it, every increment of growth releases more than it costs, the surplus goes into kinetic energy, and the crack accelerates. There is no equilibrium past the peak: brittle fracture is unstable by construction, and that is why glass does not crack a little and stop. Setting the derivative to zero gives $\sigma_c = \sqrt{2E\gamma/\pi a}$.

The square root is the physics. Strength falls as the square root of flaw size, so a flaw four times larger halves the strength and a flaw a hundred times larger divides it by ten. That single relation explains almost everything odd about brittle materials. A freshly drawn glass fibre is extraordinarily strong — several GPa — and the same fibre after being touched is weak, because handling put surface flaws on it and nothing else changed. Glass is stronger in compression because compression closes cracks instead of opening them. Tempered glass is strong because the surface is left in residual compression, so an applied tension must first cancel that before any surface flaw feels anything. And ceramics scatter enormously in strength while metals do not, because a ceramic's strength is set by its single largest flaw and flaw populations vary from piece to piece — which is why ceramic design uses Weibull statistics rather than a single allowable.

Where it fails, and why the failure was productive. Apply Griffith's equation to a structural steel with a true surface energy of about 1 J/m² and it predicts a fracture stress one to three orders of magnitude below what steel actually achieves. The reason is that a metal does not cleave cleanly. It yields at the crack tip, and the plastic work done in that zone dwarfs the cost of the two new surfaces — 10 000 to 100 000 J/m² against roughly 1. Orowan and Irwin independently patched this in the late 1940s by replacing $\gamma$ with an effective fracture energy covering surface plus plastic work, and the patch is where modern fracture mechanics began. You may use this page that way, and it will give sensible answers for metals — but write in your notes that the number is an effective fracture energy, because a $\gamma$ of 30 000 J/m² filed as a surface energy will mislead whoever reads the file next.

Two practical notes. The factor of 2 in the numerator is there because a growing crack creates two new surfaces, one on each face; dropping it halves the answer and is a common slip. And Griffith's $a$ is the HALF length of an interior crack: the flaw in his plate is $2a$ long. His own experiment ran the equation backwards — he broke glass rods, computed the flaw size implied by the strength, and found figures far larger than anything visible, which is how the idea that strength is governed by invisible defects entered engineering at all.