  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
inter-layer 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
inter-layer 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, 424-434, 2019).
