Nonlocal hcp kernel functions based on ab initio calculations: Pertinent dislocation problems revisited

Abstract

Eringen's nonlocal theory and an accurate determination of the nonlocal kernel functions for hexagonal close-packed (hcp) crystals are of interest. The kernel functions are closely related to the anisotropy as well as any crystalline symmetries. To this end, five new distinct nonlocal kernel functions which have the characteristics of discrete atomistic Green's functions in the stress space are obtained through consideration of the nonlocal dispersion relations associated with certain directions combined with ab initio Density Functional Perturbation Theory (DFPT) calculations of the pertinent phonon frequencies. This is the first work which provides the nonlocal hcp kernel functions pertinent to each component of the elastic moduli tensor of Eringen's nonlocal continuum theory based on quantum mechanical considerations. Moreover, the nonzero components of the classical elastic moduli as well as the equilibrium lattice parameters associated with the hcp crystals of Mg, Cd, Ti, and Zn have been computed using ab initio Density Functional Theory (DFT) and compared with the available experimental data. In order to show the importance of the anisotropy of the kernel functions, the nonlocal stress field of a straight screw dislocation and a straight edge dislocation inside an infinitely extended hcp medium have been examined for the case where the dislocation lies in the basal plane. Within this theory the unrealistic singular behavior of the classical stresses at the dislocation core is eliminated. In contrast to the classical anisotropy theory that predicts singular shear stress on the slip plane at the core of the dislocation, the corresponding nonlocal shear stress component predicted by the present theory is zero at the dislocation core. It is noteworthy to mention that the normalized nonlocal shear stress distribution and its maximum value are greatly influenced by the anisotropy of the crystalline solids.