Geometric aspects of the theory of incompatible deformations for simple structurally inhomogeneous solids with variable composition

The present paper is intended to formalize strain measures on non-Euclidean spaces for simple body. Use of non-Euclidean geometry methods allows one: i) to identify a global materially uniform reference shape for bodies with structural inhomogeneity, which caused by layer-by-layer formation of a solid during an additive manufacturing process; ii) to identify a global actual shape for bodies immersed into non-Euclidean physical space, in particular, for 2-dimensional solids on material surfaces. In present paper the expressions for strain measures are derived. The latter are generated by embeddings of Riemannian manifold, which represents a simple body, into Riemannian manifold, which represents a physical space. A method for description of solids with variable composition is suggested. Such a solid is considered as a family of Riemannian manifolds. Operations of partitioning and joining are defined over them. These operations characterize structural features of inhomogeneities, which are defined by a scenario of an additive manufacturing process. Specific cases for discrete and continuous structural inhomogeneity are considered in detail. A procedure for material metric synthesizing is suggested. Inclusion map is introduced. It allows one to establish relationship between the classical deformation gradient and tangent map, which is defined over smooth manifold representing a shape of the body. Essential features of suggested method for description of deformation incompatibility are demonstrated by example of hollow structurally inhomogeneous cylinder with incompressible material.