the stress intensity factor

If a crack is a knife pressing into the material, the stress intensity factor K measures how hard the knife is being pushed — how severe the conditions are right at the crack tip. It rolls two things that both make a crack want to grow — the applied stress and the crack's length — into a single driving-force number.

For a crack of length a in a body under stress sigma, K = Y times sigma times sqrt(pi a). Sigma is the far-away stress, a is the crack length (half-length for an internal crack), and Y is a dimensionless factor near 1 that accounts for geometry (edge crack, surface crack, finite width). Notice the square root: doubling the stress doubles K, but you must quadruple the crack length to double K. The units look odd — MPa times sqrt(metre), written MPa-m^0.5 — but they simply combine a stress and a length. A worked case: sigma = 100 MPa, a = 2 mm = 0.002 m, Y = 1 gives K = 100 times sqrt(3.14 times 0.002) = about 7.9 MPa-m^0.5.

K is powerful because it captures the whole crack-tip situation in one value, so two different parts with the same K have the same crack-tip severity. Failure is then dead simple to state: the crack runs when K reaches the material's critical value, the fracture toughness K_IC. Do not confuse K (the loading condition, which you can raise by pulling harder or growing the crack) with K_IC (a fixed material property). K also drives fatigue crack growth, where the cyclic swing delta-K sets how fast a crack advances per cycle.