Identification of gradient elasticity parameters based on interatomic interaction potentials accounting for modified Lorentz-Berthelot rules

Abstract

The identification algorithm developed here for scale parameters of gradient elasticity combines solutions for a deformed heterogeneous composite fragment in a continuous one-dimensional model and for a diatomic chain in a discrete atomistic model. For the identification, the models are taken equivalent and the effective stiffnesses of equivalent composite fragments are compared. In the discrete model, only the nearest neighbor atoms interact and the interaction between dissimilar atoms are determined by a modified Lorentz-Berthelot rule. As a result, the effective stiffness of the discrete composite represented as a nonuniform atomic chain is found. The continuous model is a gradient one and takes into account nonlocal effects in the volume and adhesive properties of phase boundaries. The problem of determining the effective stiffness of a composite fragment is solved analytically in the one-dimensional approximation. The study is aimed to develop a procedure of identifying the scale parameters of gradient theories such that the parameters would be independent of the choice of potentials used in discrete modeling. On the example of modeling using the Morse and Lennard-Jones potentials, we propose an identification methodology invariant with respect to the choice of potentials. It is shown that the invariance is provided if the potentials in discrete modeling are coincided in the vicinity of equilibrium points. It is demonstrated that for unambiguous determination of the scale parameters of gradient elasticity, it suffices to use the simplest two-parameter potentials approximating any other potentials subject to equal equilibrium bond distances and equal second derivatives at the equilibrium point (i.e., force constants). An example of identifying the gradient elasticity parameters is presented for a two-phase W-Si composite.