We study amplify-and-forward (AF) relay networks operating with source and relay amplifier distortion, where the distortion dominates the noise power. The diversity order is shown to be $0$ for fixed-gain (FG) and $1$ for variable-gain (VG) if distortion occurs at the relay; if distortion occurs only at the source, the diversity order will be $1$ for both. With $ε_β=N_0/η_β$ ($N_0$ the noise power, $η_β$ the distortion power at node $β\in\{S,R\}$, the source or relay), we demonstrate the emergence of what we call an $ε$-critical signal-to-noise plus distortion ratio (SNDR) threshold (a threshold that emerges when $\min\{ε_β\}$ becomes small) for both forwarding protocols. We show that crossing this threshold in distortion limited regions will cause a phase transition (a dramatic drop) in the network's outage probability. Thus, small reductions in the required end-to-end transmission rate can have significant reductions in the network's outage probability.