Reconsideration of Continuum Thermomechanical Quantities in Atomic Scale Simulations

Webb, Edmund B.; Zimmerman, Jonathan A.; Seel, Steven C.

doi:10.1177/1081286507086899

Cited by 68 publications

(74 citation statements)

References 59 publications

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“…Numerous works have appeared that have attempted the reconciliation of classical moleculardynamics stress and the continuum definition of stress. The reader is referred to the following representative (non-exhaustive) list of papers and the references therein: Murdoch & Bedeaux (1994); Cormier et al (2001);Murdoch (2003aMurdoch ( ,b, 2007; Zhou (2003); Zimmerman et al (2004);Chen (2006); Silling & Lehoucq (2007);Webb et al (2008), among others. In particular, we highlight the classic paper by Noll (2009) (originally published in 1955 in German), which was recently updated (and translated by Lehoucq and Lilienfeld-Toal).…”

Section: Introductionmentioning

confidence: 99%

Revisiting quantum notions of stress

Maranganti

Sharma

2010

Proc. R. Soc. A.

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An important aspect of multi-scale modelling of materials is to link continuum concepts, such as fields, to the underlying discrete microscopic behaviour in a seamless manner. With the growing importance of atomistic calculations to understand material behaviour, reconciling continuum and discrete concepts is necessary to interpret molecular and quantum-mechanical simulations. In this work, we provide a quantum-mechanical framework to a distinctly continuum quantity: mechanical stress. While the concept of the global macroscopic stress tensor in quantum mechanics has been well established, there still exist open issues when it comes to a spatially varying local quantum stress tensor. We attempt to shed some light on this topic by establishing a general quantummechanical operator-based approach to continuity equations and from those, introduce a local quantum-mechanical stress tensor. Further, we elucidate the analogies that exist between the (classical) molecular-dynamics-based stress definition and the quantum stress. Our derivations appear to suggest that the local quantum-mechanical stress may not be an observable in quantum mechanics and therefore traces the non-uniqueness of the atomistic stress tensor to the gauge arbitrariness of the quantum-mechanical state function. Lastly, the virial stress theorem (of empirical molecular dynamics) is re-derived in a transparent manner that elucidates the analogy between quantum-mechanical global stresses.

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Section: Introductionmentioning

confidence: 99%

Revisiting quantum notions of stress

Maranganti

Sharma

2010

Proc. R. Soc. A.

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“…In contrast to previous definitions of fluxes (e.g. equations (10) and (12)) the RHS of equation (15), as it will be shown later, can not be expressed only as a divergence of q i ; therefore, a scalar term q has to be introduced. Note that in this approach the internal potential energy is introduced implicitly (thus the name for this formulation) via…”

Section: Fluxes In Space-probability Averagingmentioning

confidence: 93%

“…Considering the harmonic potential and the long wavelength approximation the macroscopic elasticity tensor corresponding to the Wigner-Seitz cell of periodic crystal could be obtained [7]. Relationships between atomistic and continuum worlds have been also considered in the context of thermal properties of matter [8][9][10][11] as well as dissipation effects in solids [12]. Note that some processes, such as electron-phonon interaction, can not be explained within the molecular mechanics formulation and require the quantum-mechanical approach.…”

Section: Introductionmentioning

confidence: 99%

Reviewing the roots of continuum formulations in molecular systems. Part III: Stresses, couple stresses, heat fluxes

Davydov

Steinmann

2013

Mathematics and Mechanics of Solids

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This contribution is the third part in a series devoted to the fundamental link between discrete particle systems and continuum descriptions. The basis for such a link is the postulation of the primary continuum fields such as density and kinetic energy in terms of atomistic quantities using space and probability averaging.In this part, solutions to the flux quantities (stress, couple stress, and heat flux), which arise in the balance laws of linear and angular momentum, and energy are discussed based on the Noll's lemma. We show especially that the expression for the stress is not unique. Integrals of all the fluxes over space are derived. It is shown that the integral of both the microscopic Noll-Murdoch and Hardy couple stresses (more precisely their potential part) equates to zero. Space integrals of the Hardy and the Noll-Murdoch Cauchy stress are equal and symmetric even though the local NollMurdoch Cauchy stress is not symmetric. Integral expression for the linear momentum flux and the explicit heat flux are compared to the virial pressure and the Green-Kubo expression for the heat flux, respectively.It is proven that in the case when the Dirac delta distribution is used as kernel for spatial averaging, the Hardy and the Noll-Murdoch solution for all fluxes coincide.The heat fluxes resulting from both the so-called explicit and implicit approaches are obtained and compared for the localized case. We demonstrate that the spatial averaging of the localized heat flux obtained from the implicit approach does not equate to the expression obtained using a general averaging kernel. In contrast this happens to be true for the linear momentum flux, i.e. the Cauchy stress.

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“…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].…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Winczewski

Dziedzic

2016

Modelling Simul. Mater. Sci. Eng.

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Abstract. Central-force decompositions are fundamental to the calculation of stress fields in atomic systems by means of Hardy stress. We derive expressions for a central-force decomposition of the spline-based modified embedded atom method (s-MEAM) potential. The expressions are subsequently simplified to a form that can be readily used in molecular-dynamics simulations, enabling the calculation of the spatial distribution of stress in systems treated with this novel class of empirical potentials. We briefly discuss the properties of the obtained decomposition and highlight further computational techniques that can be expected to benefit from the results of this work. To demonstrate the practicability of the derived expressions, we apply them to calculate stress fields due to an edge dislocation in bcc Mo, comparing their predictions to those of linear elasticity theory.

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Reconsideration of Continuum Thermomechanical Quantities in Atomic Scale Simulations

Cited by 68 publications

References 59 publications

Revisiting quantum notions of stress

Revisiting quantum notions of stress

Reviewing the roots of continuum formulations in molecular systems. Part III: Stresses, couple stresses, heat fluxes

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

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