the thermal-shock parameter R
How do you put a single number on a ceramic's ability to survive a sudden temperature jump? The thermal-shock parameter R does exactly that for crack initiation: it is the largest temperature drop a piece can take, all at once, before the surface stress reaches its strength and a crack starts. A bigger R means a tougher material against thermal shock — you can shock it harder before it cracks. R has units of temperature, because it literally is a critical temperature difference.
The formula follows straight from setting thermal stress equal to strength. Recall the surface of an instantly quenched body goes into tension E times alpha times delta-T divided by (1 - nu). Set that equal to the fracture strength sigma_f and solve for delta-T, and you get R = sigma_f times (1 - nu) divided by (E times alpha), where sigma_f is strength, nu the Poisson ratio, E the elastic modulus, and alpha the expansion coefficient. This is the worst-case limit: it assumes the quench is instantaneous and complete, so the surface reaches the bath temperature before any heat conducts inward — in heat-transfer language, an infinite Biot number. R answers the question, how big a temperature difference can this material take under the most severe possible shock?
Reading the formula tells you how to build a shock-resistant ceramic. You want R large, so you want high strength sigma_f on top, and low modulus E and low expansion alpha on the bottom. This is why fused silica, with a tiny alpha and modest E, has an enormous R and shrugs off flame-to-water quenches, while a dense high-modulus alumina has only a modest R despite being strong. Notice what is missing: R contains no thermal conductivity. That is deliberate — R describes the infinitely fast shock where conduction has no time to help. The moment the shock is merely fast rather than instant, conductivity starts to matter, and you need the companion parameter R'.
For fused silica take sigma_f about 100 MPa, E about 72 GPa, alpha about 0.5 x 10^-6 /K, nu about 0.17: R is roughly 100 x 10^6 times 0.83 divided by (72 x 10^9 times 0.5 x 10^-6), near 2300 degrees C. For a dense alumina (sigma_f 350 MPa, E 380 GPa, alpha 8 x 10^-6) R is only about 80 degrees C — a thirty-fold gap dominated by alumina's much larger expansion.
R ranks materials for the most severe instantaneous quench; low expansion and low modulus dominate the ranking.
R deliberately omits thermal conductivity because it models the infinitely severe (infinite-Biot) quench where conduction cannot help. It therefore underestimates the real shock resistance of high-conductivity ceramics like SiC — for those you must use R'.