Foundations of Large Deformations

Abstract

In these lecture notes we will recapitulate some basic facts from the mechanics of a continuous medium at finite deformations, which are needed for the development of thermomechanical constitutive theory, presented in separate articles of this course on thermoplasticity. Our topic are thus material independent quantities like strain, strain-rate, stress or mechanical work. Whereas in the older literature index notation of tensorial quantities is almost exclusively used, either in spatial coordinate systems, e.g. ERINGEN (1962), or in convected coordinate systems, e.g. SEDOV (1965), GREEN and ZERNA (1954), there is a tendency towards a direct or coordinate-free notation, e.g. TRUESDELL and NOLL (1965). Since in textbooks on continuum mechanics usually the one or the other approach is preferred, it is difficult for the student to interpret results stated in a “language” he has not learned. The main purpose of these notes is therefore to develop the various concepts partly parallel, and to work out in some detail the existing relations. Special emphasis is laid on the transition from a description in convected coordinates to the coordinate-free notation and vice versa. To introduce the notation used here, we list a few defining relations on vectors and second order tensors.