A pre-existing micro-flaw of length a sits at the edge of a monazite grain under a remote applied stress σ. Linear-elastic fracture mechanics (LEFM) gives the crack-tip stress-intensity factor for an edge crack:
K_I = Y · σ · √(π·a) (Y ≈ 1.12)
Below the fracture toughness KIC, the crack still creeps forward slowly through stress-corrosion / static-fatigue crack growth — a real, well-documented mechanism in brittle ceramics and minerals — at a rate that rises steeply with how close KI is to KIC:
da/dt = rate · v0 · (K_I / K_IC)^n
Because K_I itself grows with √a, this is a positive-feedback loop: the crack accelerates itself. Once a reaches the Griffith–Irwin critical length ac — where K_I(ac) = KIC exactly — the equilibrium becomes mechanically unstable and the crack runs through the rest of the grain in an instant, exactly as brittle ore shatters under a crusher jaw.
- Applied stress / impact rate — how hard and how often the grain is struck (jaw crusher or mill stroke).
- KIC — the grain's intrinsic resistance to fracture; harder gangue needs more energy to liberate.
- a₀ — the size of the worst pre-existing micro-flaw the grain starts with (mining-induced damage).
- n — how sharply crack growth accelerates as KI approaches KIC (material brittleness).
Each full fracture event is one liberation step in comminution: the rare-earth-bearing monazite lattice is progressively exposed as the grain is broken down, which is the real mineral-processing motivation behind analyzing crack propagation in ore grains.