A generalized Irving–Kirkwood formula for the calculation of stress in molecular dynamics models

Abstract

In non-equilibrium molecular dynamics simulations, continuum mechanics quantities can be computed from the position and momentum of the particles based on the classical Irving-Kirkwood formalism. For practical purposes, the implementations of Irving-Kirkwood formulas often involve a spatial averaging using a smooth kernel function. The resulting formula for the stress has been known as Hardy stress. Usually results obtained this way still need to be further processed to reduce the fluctuation, e.g., by ensemble or time averaging. In this paper we extend Hardy's formulas by systematically incorporating both spatial and temporal averaging into the expression of continuum quantities. The derivation follows the Irving-Kirkwood formalism, and the average quantities still satisfy conservation laws in continuum mechanics. We will discuss the selection of kernel functions and present several numerical tests.

Continuum balances from extended Hamiltonian dynamics

“…In recent years, there has been a renewed interest in these matters; with no pretense to exhaustivity, we mention the works of Murdoch 9 (who proposes an identification procedure alternative to that used in the IKN procedure along lines anticipated by Hardy 10 ), Admal & Tadmor, 11 Seguin & Fried, 12 and others. [13][14][15][16][17][18][19][20] We remark that in the case of microstructured continua a microscopic definition of angular momentum is often necessary and would significantly affect the continuum balance of the density of angular momentum, as recently discussed by Seguin & Fried 12 and Davydov & Steinmann. 20 Nevertheless, in the present treatment, we restrict attention to continua associated with systems of material points that do not possess rotational degrees of freedom.…”

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

“…The expressions (52) and (53) define a central-force decomposition of the s-MEAM potential and constitute the main result of this work.…”

“…Murdoch [44] introduced spatial and temporal averaging to the IK procedure, which was another way of circumventing the use of Dirac delta distributions and obviated the need to calculate ensemble averages. The approach of [43,44] (termed the Murdoch-Hardy procedure, with the corresponding quantity known as Hardy stress) was later further generalised [45][46][47][48][49][50][51][52][53][54]. Owing to its many advantages, Hardy stress has since found widespread use in atomistic simulations [55][56][57][58][59][60][61][62][63][64].…”

Extracting continuum-like deformation and stress from molecular dynamics simulations

“…Yang et al [20] extended Hardy's formulas by systematically incorporating both spatial and temporal averaging into the expression of continuum quantities. The derivation followed the Irving-Kirkwood formalism, and the average quantities still satisfied conservation laws in continuum mechanics.…”

“…This interest has also resulted in a number of works that have revisited the original works [1][2][3] to find new ways to extract continuum parameters from MD simulations [13][14][15][16][17][18][19][20][21][22].…”