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
Appendix B - Calculus in Euclidean Point Space ℰ
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
Here geometric and analytical pre-requisites for continuum modelling are developed and linked to the algebraic considerations of Appendix A. This material has been included in order to emphasise the direct (i.e., co-ordinate-free) approach employed, which may not be familiar to the reader. The aim has been to provide a reasonably self-contained basis for understanding the notation and methodology used in the main body of the work.
Continuum modelling of material behaviour requires, among other things, mathematical prescriptions of
(i) the location and distribution of matter for the physical system (or body) of interest at a given time,
(ii) changes in location of a body and any associated distortion,
(iii) spatial and temporal variation of local system descriptors (e.g., mass density or velocity), and
(iv) physical descriptors such as mass, momentum, and kinetic energy, which are additive over disjoint regions. (Such descriptors are termed extensive.)
Central to such prescriptions are the notions of point and space, here formalised in terms of Euclidean space ℰ. Distortion is described via one-to-one mappings of points into points (deformations), and local spatial variation of descriptors is treated in terms of generalisations of the derivative of a function of a real variable. Analysis of point (iv) involves relating values of extensive descriptors associated with finite regions to their local densities, and is accomplished via volume integration.
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- Chapter
- Information
- Physical Foundations of Continuum Mechanics , pp. 356 - 406Publisher: Cambridge University PressPrint publication year: 2012