TABLE OF CONTENTS This chapter is central to continuum mechanics. Our aim is to model and study the cohesion forces (or internal forces) of a system, that is to say the actions exerted by part of a system S on another part of S. Our study leads to the definition of the Cauchy stress tensor and to the equations of statics and dynamics that then follow by application of the fundamental law of dynamics.
The Cauchy stress tensor is expressed in the Eulerian variable; its analogue in the Lagrangian variable is the Piola-Kirchhoff tensor introduced in the last section of this chapter.
We are given a material system S. Let S = S 1 ? S 2 be a partition of S, ? 1 and ? 2 being the domains occupied by S 1 and S 2 at a given time. In Sections 3.1 and 3.2, the time will be fixed and will not appear explicitly. We assume that the common boundary ? of ? 1 and ? 2 (see Figure 3.1) is sufficiently regular.
Concerning the actions exerted by S 2 on S 1, we make the following assumptions introduced by Cauchy:
(H1) The forces exerted by S 2 on S 1 are contact forces, which means that they can be represented by a vector measure d ? concentrated on ? = ?
