Material Affine Connections for Growing Solids

Abstract

The present paper aims to develop geometrical approach for finite incompatible deformations arising in growing solids. The phenomena of incompatibility is modeled by specific affine connection on material manifold, referred to as material connection. It provides complete description of local incompatible deformations for simple materials. Meanwhile, the differential-geometric representation of such connection is not unique. It means that one can choose different ways for analytical definition of connection for single given physical problem. This shows that, in general, affine connection formalism provides greater potential than is required to the theory of simple materials (first gradient theory). For better understanding of this inconsistence it is advisable to study different ways for material connection formalization in details. It is the subject of present paper. Affine connection endows manifolds with geometric properties, in particular, with parallel transport on them. For simple materials the parallel transport is elegant mathematical formalization of the concept of a materially uniform (in particular, a stress-free) non-Euclidean reference shape. In fact, one can obtain a connection of physical space by determining the parallel transport as a transformation of the tangent vector, which corresponds to the structure of the physical space containing shapes of the body. One can alternatively construct affine connection of material manifold by defining parallel transport as the transformation of the tangent vector, in which its inverse image with respect to locally uniform embeddings does not change. Utilizing of the conception of material connections and the corresponding methods of non-Euclidean geometry may significantly simplify formulation of the initial-boundary value problems of the theory of incompatible deformations. Connection on the physical manifold is compatible with metric and Levi-Civita relations holds for it. Connection on the material manifold is considered in three alternative variants. The first leads to Weitzenbock space (the space of absolute parallelism or teleparallelism, i.e., space with zero curvature and nonmetricity, but with non-zero torsion) and gives a clear interpretation of the material connection in terms of the local linear transformations which transform an elementary volume of simple material into uniform state. The second one allows to choose the Riemannian space structure (with zero torsion and nonmetricity, but nonzero curvature) in material manifold and it is the most convenient way for deriving of field equations. The third variant is based on Weyl manifold with specified volume form and non-vanishing nonmetricity.