Non-Local Elasticity Kernels Extracted from Atomistic Simulations
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Non-Local Elasticity Kernels Extracted from Atomistic Simulations R.C. Picu Department of Mechanical, Aerospace and Nuclear Engineering Rensselaer Polytechnic Institute, Troy, NY 12180 ABSTRACT Non-local elasticity kernels are calibrated based on the atomic scale structure of glasses obtained from atomistic simulations. A model Morse material, with interatomic interactions described by a pair potential, and Al, characterized by an embedded-atom potential are considered. The study is limited to linear isotropic non-local elasticity. The functional form of the derived kernels is significantly different than that usually assumed in non-local constitutive models (Gaussian). They have a range that extends up to the cut-off radius of the interatomic potential, are positive at the origin, and become negative approximately one atomic distance away. These kernels involve 2 internal length scales that are both derived from atomistics for the materials mentioned above. The kernel for Al is tensorial, a different function weighting each entry of the stiffness tensor. Model materials interacting by pair potentials may be described by a single function that weights the whole stiffness tensor. Two applications are briefly described in closure. The new non-local model improves upon the Gaussian one by predicting a more realistic wave dispersion relationship, with essentially zero group velocity at the boundary of the Brillouin zone. Next, the role of non-locality in defining the Peierls stress of a dislocation is studied within a Peierls-Nabarro model and it is shown that the predictions of the critical stress improve upon consideration of non-locality. INTRODUCTION The formulation of elasticity used at the macroscopic scale is local; stresses at a point depend on strains at the same location. When used at the micro and nanoscale, local elasticity predicts infinite stresses at the crack tip, at the core of dislocations and fails to accurately represent the energetics of point defects such as interstitials and vacancies. On the atomic scale, the magnitude of stress at the location of an atom is determined by the details of the interatomic interactions and the distribution of interacting neighbors about that atom. Here we refer to a “representative atom” (RA) and the atomic level stress is thought of in the ensemble average sense. Displacing a neighbor located at some distance from the RA, leads to a perturbation of the stress state at the location of the central atom. The size of this influence zone is determined by the range of the interatomic potential (the cut-off radius). Hence, the atomic scale stress is intrinsically non-local. This problem was recognized and addressed many years ago. The proposed non-local theories designed to avoid the effects mentioned above may be classified as integral [e.g. 1-3] and gradient [4-6]. In the integral formulations, the stress at a point depends on the deformation within a surrounding region of volume V according to the equation σ ij (x) = ∫ α(| x − y |)t ij (y )dV (y ) , where tij(y) are the
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