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
4 - Spatial Localisation, Mass Conservation, and Boundaries
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
Upon modelling molecules as point masses, volumetric densities ρw of mass and pw of momentum are defined as local spatial averages of molecular masses and momenta using a weighting function w which, while possessing certain essential features, is otherwise unspecified and general. Partial (time) differentiation of ρw yields the continuity equation (2.5.16) in which the velocity field vw ≔ pw/ρw. The physical interpretations of ρw,pw and vw depend crucially upon the choice of w. Several physically distinguished classes of weighting function are discussed. Emphasis is placed upon a particular class because the corresponding interpretations of the mass density and velocity fields, and of the boundary, associated with any body are particularly simple. The conceptual problems C.P.1, C.P.2, and C.P.3 listed in Section 3.8 are addressed and completely resolved.
Weighted Averages and the Continuity Equation
The mass density ρ(x, t) at a given location x (a geometrical point) and time t is a local measure of ‘mass per unit volume’. The key questions here are ‘What mass?’ and ‘What volume?’
The mass of any given body of matter derives ultimately from that of its constituent fundamental discrete entities (i.e., electrons and atomic nuclei). While any such fundamental entity could be modelled as a point mass whose location is that of its mass centre, for the purposes of this chapter we adopt a molecular viewpoint. Specifically, we choose here to regard a material system (or body) ℳ to be a fixed, identifiable set of (N, say) molecules modelled as point masses.
- Type
- Chapter
- Information
- Physical Foundations of Continuum Mechanics , pp. 44 - 70Publisher: Cambridge University PressPrint publication year: 2012