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Nanoscience and Technology: An International Journal

ISSN Print: 2572-4258
ISSN Online: 2572-4266

Nanoscience and Technology: An International Journal

Formerly Known as Nanomechanics Science and Technology: An International Journal

DOI: 10.1615/NanomechanicsSciTechnolIntJ.v7.i3.10
pages 177-233


S. A. Lychev
A. Yu. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, 101-1 Vernadsky Ave., Moscow, 119526, Russia
K. G. Koifman
A. Yu. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, 101-1 Vernadsky Ave., Moscow, 119526, Russia


In the present paper, modern differential-geometrical methods for modeling the incompatible finite deformations in solids are developed. The incompatibility of deformations may be caused by a variety of physical phenomena, e.g., distributed dislocations and disclinations, point defects, nonuniform thermal fields, shrinkage, growth, etc. Incompatible deformations result in residual stresses and distortion of the geometric shape of a body. These factors determine the critical parameters of modern high-precision technologies, particularly, of additive manufacturing, and are considered to be Ipso Facto essential constituents in corresponding mathematical models. In this context, the development of methods for their quantitative description is an urgent problem of modern solid mechanics. The methods in question are based on the representation of a body and physical space in terms of differentiable manifolds, namely, material manifold and physical manifold. These manifolds are equipped with specific metrics and connections, non-Euclidian in general. All the work as a whole gives a systematic presentation for the geometric aspects of the theory of finite incompatible deformations and contains partial survey of related papers. It is divided into three parts. The present paper represents the first part. It focuses on the physical interpretation of the non-Euclidean structure of the material and physical manifolds. Affine connection on the physical manifold is defined a priori by considerations which are independent of the properties of the deformable body. It is shown that a two-dimensional rigid surface, which formalizes curved substrate used in the deposition process, may serve as an example of non-Euclidean physical manifold. Affine connection on the material manifold represents the intrinsic properties (inner geometry) of the body and is determined by the field of local uniform configurations which performing its "assembly" of identical and uniform infinitesimal "bricks". Uniformity means that the response functional gives for them the same response on all admissible smooth deformations. As a result of assembling, one obtains body, which cannot be immersed in undistorted state into physical manifold. It is an essential feature of residual stressed bodies produced by additive processes. For this reason, it is convenient to use the immersion into a non-Euclidean space (material manifold with non-Euclidean material connection). To this end it is convenient to formalize the body and physical space are in terms of the theory of smooth manifolds. The deformation is formalized as embedding (or, in special case, as immersion) former manifold into the latter one.