Microelasticity model of random alloys. Part I: mean square displacements and stresses

Abstract

In concentrated solid solutions, the random distribution of elements of different sizes induces characteristic displacement and stress fields at the root of solid solution strengthening. The aim of this two-part article is to derive the statistical properties of these elastic fields. The present Part I focuses on the variance of the elastic fields, while Part II addresses their spatial correlations. In this first part, we develop two elastic models of random solid solutions, based respectively on realspace and spectral methods, to derive the mean square displacement and shear stress. Both approaches hold advantages and drawbacks that are discussed here. We show in particular that both the mean square displacement and shear stress are directly proportional to the variance of the atomic eigenstrains, which embodies the atomic size differences and simplifies to the classical lattice mismatch parameter if the alloy satisfies Vegard's law. This allows to clarify the scaling relations between various quantities (mismatch parameter, mean square displacement and yield stress) and to bridge energy-based and stress-based models of concentrated solid solution strengthening proposed in the literature. The elastic predictions for the mean-square displacement and stresses are also successfully compared with atomistic results obtained for a model Lennard-Jones system and with a more complex Al-Mg interatomic potential.

Influence of stress correlations on dislocation glide in random alloys

“…To assess the capability of the elastic model to describe dislocations in concentrated alloys, we first compare its predictions with atomistic simulations performed in a binary Al-Mg system modeled with the interatomic potential of Ref. [52], which was already employed in our previous work [41,42].…”

“…Based on the microelastic model developed in Refs. [41,42], the autocorrelation of a shear stress component τ mn (m = n) is expressed in Fourier space for a system of dimensions L x , L y and L z as:…”

“…The spreading parameter a and noise variance τ 2 p were determined for Al 50 Mg 50 in Ref. [41]. random alloy.…”

“…In previous contributions [41,42], we investigated the statistical properties of the stress field emerging from concentrated random solid solutions. Atoms were represented as dilatation and compression centers in an average isotropic elastic medium.…”

“…We point out that this elastic model only accounts for atomic size mismatch and does not incorporate the potential role of spatially varying elastic constants or chemical interactions between species that may also contribute to solute strengthening. The atomic size differences were modeled with eigenstrains distributed around the lattice positions with a smallscale regularization parameter a, which was found close to 1 Å to match atomistic calculations [41]. Within the linear elasticity theory, the variance of tion, τ 2 p , and of the atomic displacements, u 2 (the so-called lattice distortion [43]), were shown proportional to the variance of the eigenstrains defined as ε 2 = i c i ε 2 i (with c i and ε i , respectively the concentration and eigenstrain of element i).…”