Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Some Elements of Continuum Mechanics
- 3 Motivation for Seeking a Molecular Scale-Dependent Perspective on Continuum Modelling
- 4 Spatial Localisation, Mass Conservation, and Boundaries
- 5 Motions, Material Points, and Linear Momentum Balance
- 6 Balance of Energy
- 7 Fine-Scale Considerations: Moments, Couple Stress, Inhomogeneity, and Energetics
- 8 Time Averaging and Systems with Changing Material Content
- 9 Elements of Mixture Theory
- 10 Fluid Flow through Porous Media
- 11 Linkage of Microscopic and Macroscopic Descriptions of Material Behaviour via Cellular Averaging
- 12 Modelling the Behaviour of Specific Materials: Constitutive Relations and Objectivity
- 13 Comments on Non-Local Balance Relations
- 14 Elements of Classical Statistical Mechanics
- 15 Summary and Suggestions for Further Study
- Appendix A Vectors, Vector Spaces, and Linear Algebra
- Appendix B Calculus in Euclidean Point Space ℰ
- References
- Index
11 - Linkage of Microscopic and Macroscopic Descriptions of Material Behaviour via Cellular Averaging
Published online by Cambridge University Press: 05 November 2012
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Some Elements of Continuum Mechanics
- 3 Motivation for Seeking a Molecular Scale-Dependent Perspective on Continuum Modelling
- 4 Spatial Localisation, Mass Conservation, and Boundaries
- 5 Motions, Material Points, and Linear Momentum Balance
- 6 Balance of Energy
- 7 Fine-Scale Considerations: Moments, Couple Stress, Inhomogeneity, and Energetics
- 8 Time Averaging and Systems with Changing Material Content
- 9 Elements of Mixture Theory
- 10 Fluid Flow through Porous Media
- 11 Linkage of Microscopic and Macroscopic Descriptions of Material Behaviour via Cellular Averaging
- 12 Modelling the Behaviour of Specific Materials: Constitutive Relations and Objectivity
- 13 Comments on Non-Local Balance Relations
- 14 Elements of Classical Statistical Mechanics
- 15 Summary and Suggestions for Further Study
- Appendix A Vectors, Vector Spaces, and Linear Algebra
- Appendix B Calculus in Euclidean Point Space ℰ
- References
- Index
Summary
Preamble
Analternative strategy for relating continuum field values to their microscopic origins is outlined. Spatial averaging of additive molecular quantities is effected in terms of ‘cells’. In contrast to weighting function methodology, linear momentum balance for macroscopic regions is established before deducing the local form. The existence of a traction field on the boundary of such regions is derived via the assumption of short-range molecular interactions. The corresponding interaction stress tensor is obtained in the standard manner of continuum mechanics. Unlike balances obtained in terms of weighting functions, which hold for any given pair (∈, Δ) of length–time scales, fields obtained as cellular averages exist only if their values are somewhat insensitive to changes in ∈, Δ, and cell shape.
Cellular Averaging
Recall Subsection 4.4.1 in which selection of an appropriate weighting function [namely a mollified version of relations (4.4.4)] delivered spatial cellular averages. The analyses of Sections 4.2, 5.2 through 5.6, 6.2 and 6.3, 7.2 through 7.5, 8.4 through 8.6, 8.9 and 9.2 through 9.7 apply to such a choice. Thus, in particular, the standard relations [which express mass conservation (4.2.16), momentum balance (2.7.30) with T = Tw, where Tw is given by (5.5.20), and energy balance (6.2.75)] hold with field values defined in terms of cellular molecular averages. The associated (i.e., cell-based) notion of material point, and motion thereof, derives from the corresponding velocity field vw defined in Section 5.2.
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- Physical Foundations of Continuum Mechanics , pp. 209 - 224Publisher: Cambridge University PressPrint publication year: 2012