  We derive exact probability distributions for the strain ($\epsilon$) at
which the first stress drop event occurs in uniformly strained disordered
crystals, with quenched disorder introduced through polydispersity in particle
sizes. We characterize these first stress drop events numerically as well as
theoretically, and identify them with the first contact breaking event in the
system. Our theoretical results are corroborated with numerical simulations of
quasistatic volumetric strain applied to disordered near-crystalline
configurations of athermal soft particles. We develop a general technique to
determine the $distribution$ of strains at which the first stress drop events
occur, through an exact mapping between the cumulative distribution of first
contact breaking events and the volume of a convex polytope whose dimension is
determined by the number of defects $N_d$ in the system. An exact numerical
computation of this polytope volume for systems with small numbers of defects
displays a remarkable match with the distribution of strains generated through
direct numerical simulations. Finally, we derive the distribution of strains at
which the first stress drop occurs, assuming that individual contact breaking
events are uncorrelated, which accurately reproduces distributions obtained
from direct numerical simulations.
