Journal of Applied Mathematics and Mechanics (about journal) Journal of Applied
Mathematics and Mechanics

Russian Academy of Sciences
in January 1936
(Translated from 1958)
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ISSN 0021-8928
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IssuesArchive of Issues2013-4pp.421-432

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S.A. Lychev and A.V. Manzhirov, "The mathematical theory of growing bodies. Finite deformations," J. Appl. Math. Mech. 77 (4), 421-432 (2013)
Year 2013 Volume 77 Issue 4 Pages 421-432
Title The mathematical theory of growing bodies. Finite deformations
Author(s) S.A. Lychev (Moscow, Russia,
A.V. Manzhirov (Moscow, Russia)
Abstract The fundamentals of the mathematical theory of accreting bodies for finite deformations are explained using the concept of the bundle of a differentiable manifold that enables one to construct a clear classification of the accretion processes. One of the possible types of accretion, as due to the continuous addition of stressed material surfaces to a three-dimensional body, is considered. The complete system of equations of the mechanics of accreting bodies is presented. Unlike in problems for bodies of constant composition, the tensor field of the incompatible distortion, which can be found from the equilibrium condition for the boundary of growth, that is, a material surface in contact with a deformable three-dimensional body, enters into these equations. Generally speaking, a growing body does not have a stress-free configuration in three-dimensional Euclidean space. However, there is such a configuration on a certain three-dimensional manifold with a non-Euclidean affine connectedness caused by a non-zero torsion tensor that is a measure of the incompatibility of the deformation of the growing body. Mathematical models of the stress-strain state of a growing body are therefore found to be equivalent to the models of bodies with a continuous distribution of the dislocations.
Received 03 February 2013
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