On the Roots of Continuum Mechanics in Differential Geometry

The aim of this contribution is to illustrate the roots of the geometrically nonlinear kinematics of (generalized) continuum mechanics in differential geometry. Firstly several relevant concepts from differential geometry, such as connection, parallel transport, torsion, curvature, and metric (in index notation) for holonomic and anholonomic coordinate transformations are reiterated. The notation and the selection of these topics are essentially motivated by their relation to the geometrically nonlinear kinematics of continuum mechanics. Then, secondly, the kinematics are considered from the point of view of nonlinear coordinate transformations and nonlinear point transformations, respectively. Together with the discussion on the integrability conditions for the (first-order) distortions, the concept of dislocation density tensors is introduced. After touching on the possible interpretations of nonlinear elasticity using concepts from differential geometry, a detailed discussion of the kinematics of multiplicative elastoplasticity is given. The discussion culminates in a comprehensive set of twelve different types of dislocation density tensors. Potentially, these can be used to model densities of geometrically necessary dislocations and the accompanying hardening in crystalline materials. Continuum elastoplasticity formulations of this kind fall into the class of generalized (gradient-type) plasticity models.