Chapter 10: Deformation and Rotation, Deformation Rate and Spin

10.1 Introduction

Consider the geometrical change of a coherent amount of material or material fraction, for which modelling as a continuum is assumed to be permitted. The first part of the present chapter is focussed on the description of the local deformation (generally coupled with rotations of the material) and along with that, the introduction of a number of different strain measures.

Only after choosing a reference configuration is it possible to define deformation in a meaningful way (deformation is a relative concept). This implies that initially the theory and the accompanying application area are related to solids. When the material is a mixture of several material fractions, each fraction can, with regard to local geometrical changes, in principle, be isolated from the other fractions.

The second part of this chapter discusses geometrical changes in time. Central concepts in this part are deformation rate and rotation velocity (spin). The derivations in this part are not only relevant for solids, but even more important for applications including fluids.

10.2 A Material Line Segment in the Reference and Current Configuration

Consider a coherent amount of material in a fully defined state (the reference configuration). In a material point P, with position vector in the reference configuration, we focus our attention on an arbitrary infinitesimally small material line segment , see Fig. 10.1.