Some Primitive Concepts in Continuum Mechanics Regarded in Terms of Objective Space-Time Molecular Averaging: The Key Rôle Played by Inertial Observers

Murdoch, A. Ian

doi:10.1007/s10659-005-9048-7

Cited by 15 publications

(10 citation statements)

References 18 publications

(43 reference statements)

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“…The foregoing motivates regarding the contribution −D w to the Cauchy stress T w in (3.23) as thermal stress. It is pressure-like in that D w n · n > 0 for all n ∈ V. 10 In kinetic theory v is defined in terms of expectations; that is, ensemble averages. 11 See, for example, Brush [17].…”

Section: Thermal Stressmentioning

confidence: 99%

On Molecular Modelling and Continuum Concepts

Murdoch

2010

J Elast

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Continuum concepts and field values are related to local (scale-dependent) spacetime atomistic averages. Spatial averaging is effected by both weighting function and cellular localisation procedures, and resulting forms of linear momentum balance are compared. The former yields a local balance directly, with several candidate interaction stress fields. The latter results in a global balance involving a traction field expressible in terms of an interaction stress tensor field. In both approaches the Cauchy stress incorporates distinct interaction and thermokinetic contributions. Inter alia are addressed physically-distinguished choices of weighting function; the scale-dependence of the boundary of a body, its motion and material points thereof; physical interpretations of various candidate interaction stress tensors; temporal averaging and material systems whose content changes with time; and the possible relevance of the latter to investigating a molecular context for configurational forces.

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Section: Thermal Stressmentioning

confidence: 99%

On Molecular Modelling and Continuum Concepts

Murdoch

2010

J Elast

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“…(See [17], p.74 for more explicit details.) Whenever and wherever ρ w = 0, the corresponding velocity field…”

Section: General Relationsmentioning

confidence: 99%

“…For any given scale the fields which appear in these relations are clearly defined in terms of local spatial molecular averages computed at the chosen scale. Choice (4.25) of weighting function (see also Remark 9) is here favoured because of its particularly simple interpretation of velocity field value v w (x, t) as the velocity of the mass centre of molecules which lie within a distance of x at time t. Knowledge of v w means the macroscopic motion of the system can be visualised at scale and the notion of material point (at this scale) motivated and defined (see [17]). This reverses the viewpoint of modern continuum mechanics (e.g.…”

Section: )mentioning

confidence: 99%

“…For example, in Subsection 3.2 neighbouring time-averaged locations x i could be assumed to be governed by an affine deformation field consistent with the macroscopic motion. It is beyond our purposes here to pursue constitutive considerations or, indeed, to discuss energy balance or the rôle of time-averaging (see [4,10] and [17]). 7.5 Current interest in small-scale systems has given rise to extensive studies of how specific atomistic knowledge can be incorporated into scale-dependent simplified descriptions.…”

Section: Remark 18mentioning

confidence: 99%

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A Critique of Atomistic Definitions of the Stress Tensor

Murdoch

2007

J Elasticity

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102

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Current interest in nanoscale systems and molecular dynamical simulations has focussed attention on the extent to which continuum concepts and relations may be utilised meaningfully at small length scales. In particular, the notion of the Cauchy stress tensor has been examined from a number of perspectives. These include motivation from a virial-based argument, and from scale-dependent localisation procedures involving the use of weighting functions. Here different definitions and derivations of the stress tensor in terms of atoms/molecules, modelled as interacting point masses, are compared. The aim is to elucidate assumptions inherent in different approaches, and to clarify associated physical interpretations of stress. Following a critical analysis and extension of the virial approach, a method of spatial atomistic averaging (at any prescribed length scale) is presented and a balance of linear momentum is derived. The contribution of corpuscular interactions is represented by a force density field f. The balance relation reduces to standard form when f is expressed as the divergence of an interaction stress tensor field, T − . The manner in which T − can be defined is studied, since T − is unique only to within a divergencefree field. Three distinct possibilities are discussed and critically compared. An approach to nanoscale systems is suggested in which f is employed directly, so obviating separate modelling of interfacial and edge effects.

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“…3 The point of view has been also discussed in [29] for di¤erent purposes. It is also used in [15] with strict reference to the derivation of configurational balances.…”

mentioning

confidence: 99%