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

1. (i) the location and distribution of matter for the physical system (or body) of interest at a given time,

2. (ii) changes in location of a body and any associated distortion,

3. (iii) spatial and temporal variation of local system descriptors (e.g., mass density or velocity), and

4. (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.