  The current understanding of the earthquake interevent times distribution
(ITD) is incomplete. The Weibull distribution is often used to model the
earthquake ITD. We link the earthquake ITD on single faults with the Earth's
crustal shear strength distribution by means of a phenomenological stick - slip
model. We obtain Weibull ITD for power-law stress accumulation, i.e.,
$\sigma(t) = \alpha t^{\beta}$, where $\beta >0$ for single faults or systems
with homogeneous strength statistics. We show that logarithmic stress
accumulation leads to the log-Weibull ITD. For the Weibull ITD, we prove that
(i) $m= \beta m_s$, where $m$ and $m_s$ are, respectively, the ITD and crustal
shear strength Weibull moduli and (ii) the time scale $\tau_s =
(S_s/\alpha)^{1/\beta}$ where $S_s$ is the scale of crustal shear strength. We
generalize the ITD model for fault systems. We investigate deviations of the
ITD tails from the Weibull due to sampling bias, magnitude selection, and
non-homogeneous strength parameters. Assuming the Gutenberg - Richter law and
independence of $m$ on the magnitude threshold, $M_{L,c},$ we deduce that
$\tau_s \propto e^{- \rho_{M} M_{L,c}},$ where $\rho_M \in [1.15, 3.45]$ for
seismically active regions. We demonstrate that a microearthquake sequence
conforms reasonably well to the Weibull model. The stochastic stick - slip
model justifies the Weibull ITD for single faults and homogeneous fault
systems, while it suggests mixtures of Weibull distributions for heterogeneous
fault systems. Non-universal deviations from Weibull statistics are possible,
even for single faults, due to magnitude thresholds and non-uniform parameter
values.
