Statistical Modeling of the Orowan Bypass Mechanism for Randomly Distributed Obstacles

Mesoscopic obstacles (e.g., precipitates) serve as an important strengthening and hardening mechanism in metallic alloy systems. Several example material systems include Mg and Al alloys as well as those found in nuclear reactors. Material strength in these systems originates from dislocation–obstacle interactions, in which the obstacles impede the motion of dislocations. The spatial distribution of obstacles in alloy systems is known to be stochastic in nature. Since alloy strengthening scales inversely with obstacle spacing, a spatial distribution of obstacles yields a statistical distribution of strengths. To elucidate relations between planar obstacle density (\(\rho \)), size (D), and the statistical distribution of alloy strength (\(\tau_{crit}\)), we conduct a large number of simulations, yielding statistically representative samples of \(\tau _{crit}\) for randomly generated spatial distributions of obstacles. We interpret the simulation results probabilistically, treating \(\tau _{crit}\) as a random variable with an associated distribution function. To rationalize our results, we develop a weakest link type statistical analysis in which the key quantity is the spatial distribution of link lengths between obstacles. Our analysis yields an expectation and standard deviation of \(\tau _{crit}\) which approximately scale with \(\sim \sqrt{\rho } \ln (D)\). Our analysis corroborates our simulations and unifies similar studies found in the literature.