Examining the Mechanical Equilibrium of Microscopic Stresses in Molecular Simulations

Abstract

The microscopic stress field provides a unique connection between atomistic simulations and mechanics at the nanoscale. However, its definition remains ambiguous. Rather than a mere theoretical preoccupation, we show that this fact acutely manifests itself in local stress calculations of defective graphene, lipid bilayers, and fibrous proteins. We find that popular definitions of the microscopic stress violate the continuum statements of mechanical equilibrium, and we propose an unambiguous and physically sound definition.

The origin of the distinction between microscopic formulas for stress and Cauchy stress

“…The critical issue is that no correspondence has been established between atomistic formulas for local stress and the fundamental concept of Cauchy stress. [8][9][10][11] Cauchy stress, also known as the true mechanical stress, is defined as the force that the material on one side of a surface element exerts on the material on the other side, divided by the area of the surface. 12 It is a measure of the intensity of internal forces, has a clear physical origin, is the actual physical quantity measured in experiments, and is applicable on all scales.…”

“…The critical issue is that no correspondence has been established between atomistic formulas for local stress and the fundamental concept of Cauchy stress. [8][9][10][11] Cauchy stress, also known as the true mechanical stress, is defined as the force that the material on one side of a surface element exerts on the material on the other side, divided by the area of the surface.12 It is a measure of the intensity of internal forces, has a clear physical origin, is the actual physical quantity measured in experiments, and is applicable on all scales. Atomistic stress formulas, on the other hand, were derived from classical or quantum mechanics as a function of the forces and positions of atoms, and is what have been used in ab initio calculations 13 as an intrinsic property of the quantummechanical ground state of matter, or in classical molecular dynamics or coarse-grained atomistic simulations to predict strength 14 , fracture toughness 15, 16 , hardness 17 , or to quantify the effect of local stress on ferroelectricity 18 , thermal conductivity 19,20 , phase transition 21, 22 , etc.…”

Pressure Profile Calculation with Mesh Ewald Methods

“…3 we apply the method to the case of the water/vapour interface, and in Sec. 4 we describe the scaling properties of the algorithm. Finally, in Sec.…”

“…While some of these profiles have been routinely calculated in the past decades, the importance of determining pressure profiles is only recently being recognized, for example, for the calculation of the mechanical properties of macromolecules. [1][2][3][4] Conjectures related to pressure profiles also play a key role in the possible explanations of several phenomena. Thus, for instance, it was proposed by Cantor that the molecular mechanism of anesthesia is related to the alteration of the lateral pressure profile inside the cell membrane; 5 Imre et al claimed that the spinodal pressure of a liquid phase can be extracted from the lateral pressure profile obtained at the liquid-vapour interface of the same system at the same temperature.…”

A direct two-dimensional pressure formulation in molecular dynamics

“…A generalized method for 3D stress calculations which included temporal averaging weight functions was derived by Yang [8]. Recently, Vanegas [7] and Sanchez et al [9] applied the modified Hardy versions of IK stress to lipid bilayers, coiled coil protein and graphene sheet to determine continuum level properties from atomistic simulations. Further, there exist a few IrvingKirkwood versions [10][11][12] of 1D pressure calculations for 1D inhomogeneous system.…”