Non-Euclidean Geometry and Defected Structure for Bodies with Variable Material Composition

Abstract

In the paper the relationship between pure geometrical concepts of the theory of affine connections, physical concepts related with non-linear theory of distributed defects and concepts of non-linear continuum mechanics for bodies with variable material composition is discussed. Distinguishing feature of the bodies with variable material composition is that their global reference shapes can not be embedded into Euclidean space and have to be represented as smooth manifolds with specific (material) connection and metric. The method for their synthesis based on the modeling of additive process are proposed. It involves specific boundary problem referred to as evolutionary problem. The statement of such problem as well as illustrative exact solutions for it are obtained. Because non-Euclidean connection is rarely used in continuum mechanics, it is illustrated from the perspective of differential geometry as well as from the point of view, adopted in the theory of finite incompatible deformations. In order to compare formal structures defined within the models of solids with variable material composition with their counterpart in non-linear theory of distributed defects, a brief sketch for latter is given. The examples for cylindrical and spherical non-linear problems are presented. The correspondences between geometrical structures that defines material connection, fields of related defect densities and evolutionary problems for bodies with variable material composition are shown.