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IssuesArchive of Issues2011-3pp.348-356

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L.M. Zubov, "Continuum Theory of Dislocations and Disclinations in Nonlinearly Elastic Micropolar Media," Mech. Solids. 46 (3), 348-356 (2011)
Year 2011 Volume 46 Number 3 Pages 348-356
DOI 10.3103/S0025654411030022
Title Continuum Theory of Dislocations and Disclinations in Nonlinearly Elastic Micropolar Media
Author(s) L.M. Zubov (South Federal University, Mil'chikova 8a, Rostov-on-Don, 344090 Russia, zubov@math.rsu.ru)
Abstract A nonlinear theory of continuously distributed dislocation and disclination type defects in elastic media with intrinsic rotational degrees of freedom and couple stresses is proposed. The medium strains are assumed to be finite. The solving equations of the continuum theory of defects are obtained by passing to the limit from a discrete set of isolated dislocations and disclinations to their continuous distribution. The notions of dislocation and disclination densities in a micropolar body under large deformations are introduced. Incompatibility equations are obtained and a boundary-value problem of equilibrium is posed for an elastic micropolar body with a given density of distributed defects. A nonlinear problem of determining the intrinsic stresses in a hollow circular cylinder due to a given distribution of disclinations is solved.

A mathematical model of moment (micropolar) media can be used to describe the deformations of structurally inhomogeneous bodies [1], liquid crystals, composites, nano-structural and magnetic materials.

Earlier [2], continuously distributed dislocations were studied in the framework of the model of a simple nonlinear elastic medium without the material microstructure taken into account. A nonlinear theory of isolated dislocations and disclinations in micropolar media is presented in [3].
Keywords couple stress, large deformations, dislocation and disclination densities, incompatibility equation
References
1.  V. E. Panin, V. A. Likhachev, and Yu. V. Grinyaev, Structure Levels of Deformation of Solids (Nauka, Novosibirsk, 1985) [in Russian].
2.  S. V. Derezin and L. M. Zubov, "Equations of a Nonlinear Elastic Medium with Continuously Distributed Dislocations and Disclinations," Dokl. Ross. Akad. Nauk 366 (6), 762-765 (1999) [Dokl. Phys. (Engl. Transl.) 44 (6), 391-394 (1999)].
3.  L. M. Zubov, Nonlinear Theory of Dislocations and Disclinations in Elastic Bodies (Springer, Berlin-Heidelberg-New York, 1997).
4.  E. Nikitin and L. M. Zubov, "Conservation Laws and Conjugate Solutions in the Elasticity of Simple Materials and Materials with Couple Stress," J. Elasticity 51 (1), 1-22 (1998).
5.  E. Kröner, Continuum Theory of Dislocations and Self Stresses (Springer, Berlin, 1958; Mir, Moscow, 1965) [in German, in Russian].
6.  A. A. Vakulenko, "Relation between Micro- and Macroproperties in Elastoplastic Media," in Advances in Science and Technology: Mechanics of Solids, Vol. 22 (VINITI, Moscow, 1991), pp. 3-54 [in Russian].
7.  L. M. Zubov, "Variational Principles and Invariant Integrals for Nonlinearly Elastic Bodies with Couple Stresses," Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, No. 6, 10-16 (1990) [Mech. Solids (Engl. Transl.)].
8.  E. Cartan, Riemannian Geometry in an Orthogonal Frame (Izd-vo MGU, Moscow, 1960) [in Russian].
9.  L. M. Zubov, "Theory of Volterra Dislocations in Nonlinearly Elastic Bodies," Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, No. 5, 140-147 (1987) [Mech. Solids (Engl. Transl.)].
10.  L. I. Shkutin, Mechanics of Deformation of Flexible Bodies (Nauka, Novosibirsk, 1988) [in Russian].
Received 18 January 2008
Link to Fulltext http://www.springerlink.com/content/d1366h8812374174/
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