"Understanding stress at the atomic scale"

Alan Rosenwinkel
Ph.D. Candidate
Advisor: Professor John L. Bassani
Mechanical Engineering and Applied Mechanics
University of Pennsylvania

Abstract

The ability to construct and manipulate nanoscale structures has renewed interest in understanding the concept of stress at the atomic scale.  In particular, in the setting of molecular dynamics, one would like to compute stress given a description of interatomic potentials along with atomic coordinates and momenta as a function of time.  Key concepts leading to the so-called Virial Stress were introduced by Clausius in 1870 in a study of the "effective force of heat" and has been adopted, more or less consistently, since then.  Nevertheless, a rigorous derivation of atomic-level stress and a clear demonstration of its validity has been somewhat elusive.  Of particular interest is the extent to which the atomistic expression can be localized in time and space while maintaining a meaningful connection to the continuum definition of stress.

Using the principle of virtual work we present a derivation of stress in a discrete, dynamical medium that yields an expression comprising three terms: the two well known terms of the Virial Stress and one that vanishes by the virial theorem.  To understand the significance of each term and to establish the validity of this atomic-level stress, we have carried out a series of molecular dynamics simulations for heating of both harmonic and anharmonic crystals, with and without constraint. The calculated stresses are shown to agree both with physical intuition and continuum limits.  Localization of the atomic stress is also investigated, both from a theoretical perspective and by molecular dynamics simulation of both space- and time-varying stress fields.