GEOMETRIC ASPECTS OF THE THEORY OF INCOMPATIBLE DEFORMATIONS. PART Ⅱ. STRAIN AND STRESS MEASURES

摘要

The present paper is a continuation of an earlier one (Lychev and Koifman, 2016). It introduces non-Euclidean representations for stress and strain distributions on smooth manifolds endowedwith Riemannian metrics in terms of smooth sections and covector-valued forms. The application of non-Euclidean geometry makes it possible to formalize incompatible local deformations in the form similar to the conventional deformation gradient. The only difference is that deformation has to be understood in the generalized sense as embedding of a manifold with non-Euclidean (material) connection into Euclidean one.Material connection characterizes the measure of incompatibility of local deformations and plays the role of a material function that characterizes the body as a "construction" assembled from self-stressed elementary parts. Such bodies are the subject of the paper, which willbe referred to as structurally inhomogeneous bodies. The latter are the archetypal objects of study in modeling and optimization for additive manufacturing. Two classes of structurally inhomogeneous bodies are considered. The first class includes bodies with discrete inhomogeneity, and the second class with a continuous one. The first class represents compound bodies whose finite parts are composed with a preliminary deformation. The stress−strain state of such bodies is determined from the equilibrium conditions for the layers and the ideal contact between them. Modeling of the assembly process is reduced to a recurrent sequence of such problems. To find the stress−strain state of bodieswith a continuous inhomogeneity, the stresses and strains in which are represented by sections of bundles, an evolutionary problem is formulated. In a particular case, this problem reduces to nonlinear integral equation.General constructions are illustrated by discrete and continuous assembling problems for a finite cylinder, whose structural inhomogeneity is a consequence of the layer-by-layer successive shrinkage of the material during manufacturing. It is shown that modeling for discrete process tends to acontinuous one, while the number of layers tends to infinity. The geometric approach developed in the present work may be used in modeling for residual stress distributions and geometric form distortions that appear due to the specific features of additive technological process, such as lithography, LbL processing, etc.