Geometric Aspects of the Theory of Incompatible Deformations in Growing Solids

Abstract

Differential-geometrical methods for modeling the incompatible finite deformations in growing solids are developed. 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 and are considered to be essential constituents in corresponding mathematical models. Affine connection on the material manifold represents the intrinsic properties (proper 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 embedded 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 embedding into a non–Euclidean space (material manifold with non–Euclidean material connection). To this end one can formalize the body and physical space in terms of the theory of smooth manifolds. The deformation is formalized as embedding (or, in special case, as immersion) of former manifold into the latter one.