4 I Introduction

From a mathematical point of view, the hypothesis of continuity may beexpressed by stating that the functions which describe the forces inside thematerial, the displacements, the deformations, etc., are continuous functionsof space and time.

From a physical point of view, this hypothesis corresponds to assumingthat the macroscopically observed material behaviour does not change withthe dimensions of the piece of material considered, especially when they tendto zero. This is equivalent to accepting that the material is a mass of pointswith zero dimensions and all with the same properties.

The validity of this hypothesis is fundamentally related to the size of thesmallest geometrical dimension that must be analysed, as compared with themaximum dimension of the discontinuities actually present in the material.

Thus, in a liquid, the maximum dimension of the discontinuities is thesize of a molecule, which is almost always much smaller than the smallestgeometrical dimension that must be analysed. This is why, in liquids, thehypothesis of continuity may almost always be used without restrictions.

On the other side, in solid materials, the validity of this hypothesis mustbe analysed more carefully. In fact, although in a metal the size of the crystalsis usually much smaller than the smallest geometrical dimension that mustbe analysed, in other materials like concrete, for example, the minimum di-mension that must be analysed is often of the same order of magnitude asthe maximum size of the discontinuities, which may be represented by themaximum dimension of the aggregates or by the distance between cracks.

In gases, the maximum dimension of the discontinuities may be representedby the distance between molecules. Thus, in very rarefied gases the hypothesisof continuity may not be acceptable.

In the theory expounded in the first part of this book the validity of thehypothesis of continuity is always accepted. This allows the material behaviourto be defined independently of the geometrical dimensions of the solid bodyof the liquid mass under consideration. For this reason, the matters studiedhere are integrated into Continuum Mechanics.

I.2 Fundamental Definitions

In the Theory of Structures, actions on the structural elements are defined aseverything which may cause forces inside the material, deformations, acceler-ations, etc., or change its mechanical properties or its internal structure. Inaccordance with this definition, examples of actions are the forces acting ona body, the imposed displacements, the temperature variations, the chemicalaggressions, the time (in the sense that is causes aging and that it is involved inviscous deformations), etc. In the theory expounded here we consider mainlythe effects of applied forces, imposed displacements and temperature.

Some basic concepts are used frequently throughout this book, so it isworthwhile defining them at the beginning. Thus, we define: