Stochastic Peierls--Nabarro Model for Dislocations in High Entropy Alloys

High entropy alloys (HEAs) are single phase crystals that consist of random solid solutions of multiple elements in approximately equal proportions. This class of novel materials have exhibited superb mechanical properties, such as high strength combined with other desired features. The strength of crystalline materials is associated with the motion of dislocations. In this paper, we derive a stochastic continuum model based on the Peierls--Nabarro framework for interlayer dislocations in a bilayer HEA from an atomistic model that incorporates the atomic level randomness. We use asymptotic analysis and limit theorem in the convergence from the atomistic model to the continuum model. The total energy in the continuum model consists of a stochastic elastic energy in the two layers, and a stochastic misfit energy that accounts for the interlayer nonlinear interaction. The obtained continuum model can be considered as a stochastic generalization of the classical, deterministic Peierls--Nabarro model for the dislocation core and related properties. This derivation also validates the stochastic model adopted by Zhang et al. [Acta Mater., 166 (2019), pp. 424--434].