Kinematics

Abstract

In this chapter we discuss the admissible kinematics of a continuous body in the physical space from a differential geometric point of view, as it is proposed by Epstein and Segev [1, 2]. A major part of the chapter deals with the introduction of the necessary differential geometric concepts. These geometric concepts are then directly applied to the description of a first gradient continuum as a model of a deformable body. Section 2.1 introduces the objects of continuum mechanics, the body and the physical space as manifolds. The idea to regard a body as a smooth manifold originates from Noll [3] and is applied explicitly in [1]. In Sect. 2.2, tangent bundles, vector fields and global flows are defined to formulate the idea of a smooth spatial virtual displacement field. In Sect. 2.3, we introduce the configuration as a mapping between manifolds and discuss the infinite dimensional manifold structure of the set of all differentiable mappings. Furthermore, we introduce pullback tangent bundles which are required to represent elements of the tangent space of the configuration manifold, i.e. virtual displacement fields. In Sect. 2.4, we give a brief introduction to affine connections.