2.1. The concept of mass

The aim of this chapter is to allow to express and to generalize to continuum media the well-known law in mechanics (for one material point) f = m ?. Having introduced the acceleration, we now need to introduce the concepts of mass and force.

In this section, we consider a material system S in motion. This system fills the domain ? _{t 0} at time t ₀ and the domain ? _t at time t. We define the concept of mass through the following hypothesis:

For every material system S and at each time t, there exists a positive measure _t carried by ? _t and called the mass distribution.

From here on, we will, most of the time, assume that _t is regular (with respect to the Lebesgue measure dx), that is to say, there exists a function ? = ?( x, t) such that

? is called the volumic mass (or density) of the system at point x and at time t (we use here the Eulerian description of the motion). As usual, we assume that the function ? is as smooth as necessary at least of class ¹ in x and t.

Other cases, which we will not consider here, are important in mechanics; for plates and shells the measure _t is carried by a surface ?