Linear elasticity represents only a simplified and very particular behavior of solids. The purpose of this chapter is to present some simple examples of problems encountered when the constitutive laws are nonlinear, like some of the laws described in Chapter 5, when the stress tensor σ is a nonlinear function of the deformation tensor ε(u) in the framework of nonlinear elasticity in small displacements (see Chapter 5). As a result, the corresponding equilibrium equations are nonlinear, contrary to the equations encountered in the previous chapters in Part 3 of this book. Nonlinear mechanical phenomena are at this time a very active domain of solid mechanics in connection with the research of new materials and with the study of their mechanical properties (polymers, composite materials, etc.).

This chapter will be rather short; we only consider stationary problems and sometimes limit ourselves to problems of mechanics that involve only one space variable.

In the first three sections of this chapter, the presentation is based on energy theorems similar to those of Chapter 15 and, as indicated in Remark 15.1, we consider energy functionals w(ε) that are no longer quadratic functions of ε. In Section 16.1, in connection with nonlinear elasticity, we consider cases in which w is a strictly convex function of ε. In Section 16.2, in connection with plasticity, we consider energy functionals possessing some degeneracies: typically, w(ε) is convex but not strictly convex, which may lead to discontinuities (cracks, sliding lines).

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.

Hypotheses on the cohesion forces

We are given a material system S. Let S = S1 ∪ S2 be a partition of S, Ω1 and Ω2 being the domains occupied by S1 and S2 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 Σ of Ω1 and Ω2 (see Figure 3.1) is sufficiently regular.

Concerning the actions exerted by S2 on S1, we make the following assumptions introduced by Cauchy:

1. (H1) The forces exerted by S2on S1are contact forces, which means that they can be represented by a vector measure dφ concentrated on Σ = ∂ω1 ∩ ∂Ω2.

2. (H2) The measure dφ is absolutely continuous with respect to the surface measure dΣ, that is,

3. where T is the (vector) surface density of the forces.

4. (H3) The function T depends only on the point x of Σ and on the unit normal n to Σ at point x: