the critical flaw size
At any given stress there is a threshold crack size that separates safe from catastrophic. A crack shorter than that threshold sits quietly, doing nothing; a crack even a hair longer suddenly runs and the part shatters. That threshold is the critical flaw size. It is the practical meeting point of the two big ideas in ceramic fracture: the stress you apply, and the toughness the material offers. Below the threshold the material wins; above it the crack wins.
It comes straight from rearranging the fracture condition K_IC = Y times sigma times sqrt(pi times c). Solving for the flaw size at which a crack becomes critical gives c_crit = (1/pi) times (K_IC / (Y times sigma))^2. Put in numbers: with a toughness K_IC = 3 MPa sqrt(m), an applied stress sigma = 300 MPa, and geometry Y near 1, you get c_crit = (1/3.14) times (3/300)^2 = (1/3.14) times (0.01)^2, about 3.2 times 10^-5 metre, or roughly 32 microns. So at 300 MPa any flaw larger than about 30 microns is fatal, while anything smaller survives. Notice the two levers: raising toughness enlarges the tolerable flaw as the square of K_IC, while raising the applied stress shrinks it as the inverse square of sigma.
The critical flaw size is what makes non-destructive inspection so demanding for ceramics. To certify a part you must be able to detect any flaw bigger than c_crit, yet that size is often only tens of microns, right at or beyond the limit of ultrasonics and X-ray imaging. This is precisely why proof testing exists: loading every part once to a proof stress guarantees that no flaw larger than the critical size for that stress has survived, giving a certified minimum strength without having to find the flaw at all. It is also why the flaw's shape and location matter: a surface flaw with an unfavourable geometry factor Y becomes critical at a smaller size than a rounded internal one.
An alumina component with K_IC = 4 MPa sqrt(m) is designed to work at 200 MPa. Its critical flaw size is (1/pi)(4/200)^2, about 130 microns. Machining scratches must be kept well below that; a single 150 micron chip from careless handling would push a flaw past critical and the part would fail at its rated load.
Set the operating stress and toughness, and the critical flaw size follows; every flaw must be kept smaller than it.
The critical flaw is often only tens of microns, smaller than routine inspection can reliably find, so non-destructive testing alone cannot guarantee a brittle part. That gap is exactly why proof testing, which weeds out any part whose flaw already exceeds the critical size, is used for safety-critical ceramics.