2 - Continuum Mechanics

Summary

This chapter shows how to transition exactly from discrete molecular dynamics to the averaged continuum dynamics controlling the movement of the center of mass of large numbers of molecules. Discrete particle dynamics is described from the classical Newton–Maxwell perspective and from the quantum perspective with an emphasis on how quantum effects control the force interactions between molecules. Representing atoms using the Dirac delta function in three dimensions (a field) is introduced along with the volume-averaging theorem that defines the macroscopic gradient of volume-averaged fields. The continuum statements of the conservation of mass and momentum of large numbers of atoms are derived. It is shown that the forces causing the center of mass of a collection of molecules to move come entirely from molecules that lie outside the collection. In so doing, the stress tensor is obtained as sums over the molecular-force interactions and a sum over the thermal (random) kinetic energy of the molecules. Body forces are defined as the long-range force fields of electromagnetism and gravity acting on each collection of molecules.