What does this equation mean?

The governing correlation is empirical. Paris and Erdogan proposed that crack growth per cycle depends on the range of the stress intensity factor over the cycle: $\frac{\mathrm{d}a}{\mathrm{d}N} = C\,(\Delta K)^{m}$. The parameters C and m are fitted, not derived, and depend on environment, frequency, temperature and stress ratio. A mechanics review that proposes a generalised form of the law opens by conceding that “fatigue life prediction is still very much an empirical art rather than a science”, and notes that the power law holds only in an intermediate regime, deviating near the threshold below which cracks do not propagate and again in the fast-growth stage approaching final fracture [ 6 ] . The same review sets out the live disagreement in the field: short cracks, comparable in size to the microstructure or to the local plastic zone, grow at rates that the long-crack law does not describe, and there is no consensus on whether a single stress-intensity-based similitude parameter can be repaired to cover them or whether short-crack behaviour requires a separate model with its own length scale [ 6 ] . Practitioners who need a number today generally handle this by treating the threshold conservatively rather than by resolving the physics.