  We investigate a model for fatigue crack growth in which damage accumulation
is assumed to follow a power law of the local stress amplitude, a form which
can be generically justified on the grounds of the approximately self-similar
aspect of microcrack distributions. Our aim is to determine the relation
between model ingredients and the Paris exponent governing subcritical
crack-growth dynamics at the macroscopic scale, starting from a single small
notch propagating along a fixed line. By a series of analytical and numerical
calculations, we show that, in the absence of disorder, there is a critical
damage-accumulation exponent $\gamma$, namely $\gamma_c=2$, separating two
distinct regimes of behavior for the Paris exponent $m$. For $\gamma>\gamma_c$,
the Paris exponent is shown to assume the value $m=\gamma$, a result which
proves robust against the separate introduction of various modifying
ingredients. Explicitly, we deal here with (i) the requirement of a minimum
stress for damage to occur; (ii) the presence of disorder in local damage
thresholds; (iii) the possibility of crack healing. On the other hand, in the
regime $\gamma<\gamma_c$ the Paris exponent is seen to be sensitive to the
different ingredients added to the model, with rapid healing or a high minimum
stress for damage leading to $m=2$ for all $\gamma<\gamma_c$, in contrast with
the linear dependence $m=6-2\gamma$ observed for very long characteristic
healing times in the absence of a minimum stress for damage. Upon the
introduction of disorder on the local fatigue thresholds, which leads to the
possible appearance of multiple cracks along the propagation line, the Paris
exponent tends to $m\approx 4$ for $\gamma\lesssim 2$, while retaining the
behavior $m=\gamma$ for $\gamma\gtrsim 4$.
