Irwin's 1957 result is the most useful equation in fracture mechanics: the whole elastic stress field around a crack tip is set by ONE number. $K = Y\,\sigma\,\sqrt{\pi a}$ — read aloud, K equals Y sigma root pi a. $K$ is the stress intensity factor, and its unit is worth meeting slowly: MPa·√m, a stress times the square root of a length, which is why every arithmetic slip under that root does half-power damage. $\sigma$ is the remote applied stress in MPa, far from the crack. $Y$ is the geometry factor, a bare number — 1.00 for a centre crack in a wide plate, 1.12 for an edge crack. And $a$ is the crack length, in metres: the HALF-length for a centre crack, the full depth for an edge crack. Reading a centre crack's full width as $a$ is the classic inspection-report error.

Compare $K$ against the material's fracture toughness $K_{IC}$ and you have your answer: below it the crack sits, at or past it the plate breaks. Turn it around and it gives you the largest crack a part may carry — which is what an inspection interval is really built on.

If the load cycles, the crack advances a little each time, and Paris and Erdogan's 1963 straight line on log-log axes says how far: $\dfrac{da}{dN} = C\,(\Delta K)^{m}$. $da/dN$ is the crack advance per cycle — a length, usually micrometres or less. $\Delta K$ is the RANGE of stress intensity over one cycle, maximum minus minimum, in MPa·√m. $m$ is the exponent, around 3 for steel, and $C$ is the coefficient, which is unit-bound and only means anything alongside the convention it was fitted in — here, metres per cycle with $\Delta K$ in MPa·√m. That cubed exponent is the whole lesson: double the stress range and the crack grows EIGHT times as fast. No material substitution on the market offers leverage like halving $\Delta K$.