Introduction

I.1. Outlining the problem

Let us consider a solid S that is subjected to imposed displacements and external forces.

The aim of the mechanics of deformable solids is to study the internal state of the material (notion of stress) and the way in which it becomes deformed (notion of strain) [FRA 12, SAL 01, LEM 96].

In mechanics, a mechanical piece or system may be designed:

– to prevent it from breaking;
– to prevent it from becoming permanently deformed;
– to prevent it from becoming too deformed, or;
– for any another purposes.

A solid shall be deemed a continuous medium, meaning that it shall be regarded as a continuous set of material points with a mass, representing the state of matter that is surrounded by an infinitesimal volume.

Mechanics of deformable solids enables the study of cohesive forces (notion of stress) at a point M, like the forces exerted on the small volume surrounding it, called a Representative Elementary Volume (REV). For metals, the REV is typically within the range of a tenth of a millimeter.

The matter in this REV must be seen as continuous and homogeneous:

– if it is too small, the matter cannot be as seen homogeneous: atomic piling, inclusion within matter, grains, etc. (for example: for concrete, the REV is within the range of 10 cm);
– if it is too big, the state of the cohesive forces in its center will no longer represent the REV state.