Each small sphere is one grain (or, if dark, a pore) inside a cross-section of a fired ceramic block. Raising the firing temperature and sintering rate drives two competing processes at once: densification, where pores shrink and disappear as atoms diffuse to grain boundaries, and grain growth, where the grains themselves keep coarsening the longer the part stays hot. A little coarsening is normal and even helps close the last pores — but push temperature and time too far and grains keep growing well past the point of full density, which is exactly what "overfiring" means in a real kiln.
a_flaw ∝ grain size · (1 + porosity·5)
σ_fracture = K_IC / √(π·a_flaw) (Griffith criterion)
The estimated fracture strength box applies that Griffith relation using the material's own fracture toughness (K_IC), treating the largest grain/pore as the critical flaw. Slide "Applied stress" past that strength and the block cracks along the weakest row of grain boundaries — shown in red, with the two halves pulling apart — because ceramics have essentially no ductility to blunt a crack tip the way metals do.
- Material — alumina and silicon carbide are hard and stiff but relatively brittle; silicon nitride and zirconia trade some hardness for much higher fracture toughness (zirconia via transformation toughening), so they tolerate larger flaws before failing.
- Firing temperature / sintering rate — together set how fast pores vanish and grains coarsen; watch the strength number rise as pores close, then fall again if you keep firing hot for too long.
- Applied stress — the mechanical load on the part; cross the estimated strength and fracture triggers immediately, since ceramics fail with almost no warning.
Real-world relevance: this is why ceramic firing schedules are tightly controlled — underfired parts stay porous and weak, overfired parts grow oversized grains and get weak again, and the strongest part sits at a narrow "just right" window in between.