Differential geometry applied to continuum mechanics

Differential geometry provides the suitable background to present and discuss continuum mechanics with an integrative and mathematically precise terminology. By starting with a review of linear geometry in affine point spaces, the paper introduces modern differential geometry on manifolds including the following topics: topology, tensor algebra, bundles and tensor fields, exterior algebra, differential and integral calculi. The tools worked out are applied subsequently to basic topics of continuum mechanics. In particular, kinematics of a material body and balance of mass are formulated by applying the geometric terminology, the principles of objectivity and material frame indifference of constitutive equations are examined, and a clear distinction of the Lagrangian formulation from the Eulerian formulation is drawn. Moreover, the paper outlines a generalized Arbitrary Lagrangian-Eulerian (ALE) formulation of continuum mechanics on differentiable manifolds. As an essential part, the grid manifold introduced therein facilitates a consistent description of the relations between the material body, the ambient space and the arbitrary reference domain of the ALE formulation. Not least, the objective of the paper is to provide a compilation of important formulae and basic results –some of them with a full proof– frequently used by the community. If practical, point arguments and changes in points within equations will be clearly indicated, and component and direct (or absolute) tensor notation will be applied as needed, avoiding a single-track approach to the subject.