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IssuesArchive of Issues2013-5pp.553-560

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S.A. Lychev and A.V. Manzhirov, "Reference Configurations of Growing Bodies," Mech. Solids. 48 (5), 553-560 (2013)
Year 2013 Volume 48 Number 5 Pages 553-560
DOI 10.3103/S0025654413050117
Title Reference Configurations of Growing Bodies
Author(s) S.A. Lychev (Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, pr-tVernadskogo101, str.1, Moscow, 119526 Russia, lychevsa@mail.ru)
A.V. Manzhirov (Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, pr-tVernadskogo101, str.1, Moscow, 119526 Russia, manzh@ipmnet.ru)
Abstract The growing bodies are considered as bodies with induced inhomogeneity caused by junction of inconsistently deformed parts. The body is formalized as an abstract smooth manifold, and all possible affine connections on it are classified. A method is shown for introducing a special connection-material connection-for which neighborhoods of all material points of a growing body pass into a stress-free state. A method for constructing a global stress-free reference configuration of a growing body as an embedding in a space with absolute parallelism is proposed. It is shown that in the case of layered accretion, the material connection corresponding to the stress-free embedding is determined by three independent functions and is in general non-Euclidean. The property of being non-Euclidean is determined by the fact that the torsion of the material connection is nonzero. We suggest to formalize the growing body as a fibration of a three-dimensional smooth manifold over a one-dimensional base, and this formalization characterizes the structure of the material connection.
Keywords growing body, deformation inconsistency, inhomogeneity, natural configuration, internal geometry, absolute parallelism
References
1.  T. Frankel, The Geometry of Physics. An Introduction (Cambridge Univ. Press, Cambridge, 2004).
2.  M. Epstein, The Geometrical Language of Continuum Mechanics (Cambridge Univ. Press, Cambridge, 2010).
3.  G. A. Maugin, Material Inhomogeneities in Elasticity (Chapman and Hall, London, 1993).
4.  A. Kadich and D. Edelen, The Gauge Theory of Dislocations and Disclinations (Springer, Berlin, 1983; Mir, Moscow, 1987).
5.  M. Epstein, and M. Elsanowski, Material Inhomogeneities and Their Evolution: A Geometric Approach (Springer, 2007).
6.  A. Yavari, "A Geometric Theory of Growth Mechanics," J. Nonlin. Sci. 20 (6), 781-830 (2010).
7.  W. Noll, "Materially Uniform Simple Bodies with Inhomogeneities," Arch. Rat. Mech. Anal. 27 (1), 1-32 (1967).
8.  C.-C. Wang, "On the Geometric Structures of Simple Bodies, a Mathematical Foundation for the Theory of Continuous Distributions of Dislocations," Arch. Rat. Mech. Anal. 27 (1), 33-94 (1967).
9.  M. M. Postnikov, Smooth Manifolds (Nauka, Moscow, 1987) [in Russian].
10.  S. Kobayashi and K. Nomidzu, Foundations of Differential Geometry, Vol. 1 (Interscience, New York, 1963; Nauka, Moscow, 1981).
11.  Y. Choquet-Bruhat, C. Dewitt-Morette, and M. Dillard-Bleick, Analysis, Manifolds and Physics, Vol. 1: Basics (North-Holland, Amsterdam, 1982).
12.  A. I. Lurie, Nonlinear Theory of Elasticity (Nauka, Moscow, 1980) [in Russian].
13.  I. A. Kunin, "Theory of Dislacations," in J. A. Schouten, Tensor Analysis for Physicists (Nauka, Moscow, 1965), pp. 373-443 [in Russian].
14.  A. Ozakin and A. Yavari, "A Geometric Theory of Thermal Stresses," J. Math. Phys. 51, 032902 (2010).
15.  A. Yavari and A. Goriely, "Riemann-Cartan Geometry of Nonlinear Dislocation Mechanics," Arch. Rat. Mech. Anal. 205 (1), 59-118 (2012).
16.  C. A. Truesdell, A First Course in Rational Continuum Mechanics (The Johns Hopkins University Press, Baltimore, Maryland, 1972; Mir, Moscow, 1975).
17.  A. V. Manzhirov and S. A. Lychev, "The Mathematical Theory of Growing Solids: Finite Deformations," Dokl. Ross. Akad. Nauk 443 (4), 438-441 (2012) [Dokl. Phys. (Engl. Transl.) 57 (4), 160-163 (2012)].
18.  A. V. Manzhirov and S. A. Lychev, "Residual Stresses in Growing Bodies," in Topical Problems in Solid and Fluid Mechanics (Elite Pub. House, Delhi, 2011), pp. 66-79.
19.  I. A. Miklashevich, Micromechanics of Fracture in Generalized Spaces (Logvinov, Minsk, 2003) [in Russian].
Received 26 June 2013
Link to Fulltext http://link.springer.com/article/10.3103/S0025654413050117
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