Here we introduce the notion of a continuum body and discuss how to describe its mass properties, and the various types of forces that may act on it. As we will see, the discussion of internal forces in a continuum body will lead to the notion of a stress tensor field – our first example of a second-order tensor field arising in a physical context. We introduce the basic conditions necessary for the mechanical equilibrium of a continuum body and then derive a corresponding statement in terms of differential equations.

The important ideas in this chapter are: (i) the notion of a mass density field, which enables us to define the mass of an arbitrary part of a body; (ii) the notion of body and surface force fields, which enable us to define the resultant force and torque on an arbitrary part of a body; (iii) the notion of a Cauchy stress field and its relation to surface force fields; (iv) the equations of equilibrium for a body.

Continuum Bodies

The most basic assumption we make in our study of any material body, whether it be a solid, liquid or gas, is that the material involved can be modeled as a continuum: we ignore the atomic nature of the material and assume it is infinitely divisible. This assumption can lead to very effective material models at length scales much longer than typical interatomic spacings. However, at length scales comparable to or smaller than this, it is no longer expected to be valid.