In this last chapter of Part 1, we return to the fundamental concepts of continuum mechanics and develop two new independent subjects.

On the one hand, we introduce some thermodynamical concepts, namely, internal energy, heat, and temperature to express the energy conservation principle, which leads to a new equation.

On the other hand, we study shock waves: contrary to the regularity assumptions consistently made until now, we consider here the case in which some physical and mechanical quantities are piecewise regular, that is, everywhere regular except at the crossing of some surfaces. It is this framework that is used, for instance, in perfect fluid mechanics, to model the shock waves produced by planes flying at transsonic or supersonic speeds.

Heat and energy

We consider a material system S that fills the domain Ωt at time t.

Definition 6.1.For every material system S and at each time t, there exists a measure carried by Ωtof the form e(x, t) dx, where e is nonnegative. By definition

is the internal energy of S at time t, e(x, t) is the mass density of specific internal energy of S at time t, and ρe is the volume density of internal energy.

Definition 6.2.The energy of the system S at time t is the sum of its kinetic energy and of its internal energy:

The energy ε is sometimes called the total energy of the system and is thus defined by its volume density

Remark 6.1: For fluids, thermodynamics yields relations between ρ, p, and e (p is the pressure). In particular, it postulates the existence of a relation, called the equation of state, of the form e = g(p, ρ).