Calculation of stress in atomistic simulation

Abstract

Atomistic simulation is a useful method for studying material science phenomena. Examination of the state of a simulated material and the determination of its mechanical properties is accomplished by inspecting the stress field within the material. However, stress is inherently a continuum concept and has been proven difficult to define in a physically reasonable manner at the atomic scale. In this paper, an expression for continuum mechanical stress in atomistic systems derived by Hardy is compared with the expression for atomic stress taken from the virial theorem. Hardy's stress expression is evaluated at a fixed spatial point and uses a localization function to dictate how nearby atoms contribute to the stress at that point; thereby performing a local spatial averaging. For systems subjected to deformation, finite temperature, or both, the Hardy description of stress as a function of increasing characteristic volume displays a quicker convergence to values expected from continuum theory than volume averages of the local virial stress. Results are presented on extending Hardy's spatial averaging technique to include temporal averaging for finite temperature systems. Finally, the behaviour of Hardy's expression near a free surface is examined, and is found to be consistent with the mechanical definition for stress.

Applicability of Continuum Fracture Mechanics in Atomistic Systems

“…This verifies the statement in the previous studies [69,71] that the Hardy stress with the high order localization functions performs a quicker convergence than the local virial stress. Theoretically, the smaller averaging volume where a local stress can converge, the closer this atomistic stress definition can approach the ideal setting of the macroscopic stress.…”

“…Hardy stresses cross and times the size of lattice constant, respectively [69,71]. As for the multi-atom crystal, the advantage of the Hardy's definition diminishes and both stress definitions require a radius of times the size of lattice constant to achieve convergence [70].…”

“Bulk” Nanocrystalline Metals: Review of the Current State of the Art and Future Opportunities for Copper and Copper Alloys

“…89,90 Moreover, simulations show that some solute atoms (dopants) can provide additional strengthening for grain boundary related deformation processes and nanocrystalline materials, in general. 29,92 Furthermore, the development of tools and algorithms that extend both the length and time scales of molecular dynamics [93][94][95] as well as the ability to characterize and visualize atomistic results [96][97][98][99][100] can aid in using molecular dynamics as a tool for insight into nanocrystalline materials design and behavior. While these simulations can give atomistic details of the nature of these solute atoms in thermal stability and deformation, there have been a limited number of atomistic studies of binary and higher-order alloys compared to the large amount of literature devoted to pure nanocrystalline copper, in part due to the limited number of appropriate interatomic potentials.…”

“…Previous studies have used per-atom structure metrics (e.g., centrosymmetry, 206 slip vector, 207 common-neighbor analysis, 208 or Ackland analysis 209 ) for distinguishing and visualizing grain boundaries, triple junctions, dislocations, twins, and point defects in static or dynamic simulations. 96,210 In addition to these structure-based metrics, there have emerged a number of continuum mechanics-motivated metrics that define different strain or stress measures based on the local environment, e.g., the Hardy stress by Zimmerman and colleagues, 97,211 deformation gradient-based metrics by Tucker et al 212,213 and Stukowski and Arsenlis. 214 Moreover, the ability to track, visualize, and characterize dislocations can offer even further insight into deformation in nanocrystalline materials.…”

Central-force decomposition of spline-based modified embedded atom method potential

“…Other approaches to the calculation of local stress have been proposed over time [36-38, 40, 65], with the definition itself often an object of debate [40,48,50,51,[53][54][55]. The features of a number of measures of local stress were examined in detail by Admal and Tadmor [53,54], who showed through numerical experiments [53] that of the three most commonly used approaches, Hardy stress is the most accurate approach, with the most favourable convergence with respect to the averaging volume.…”