# American Institute of Mathematical Sciences

March  2011, 6(1): 89-109. doi: 10.3934/nhm.2011.6.89

## Non-standard dynamics of elastic composites

 1 Institute for Low Temperature Physics and Engineering, Ukrainian Academy of Sciences, Lenin Ave 47, Kharkiv 61164, Ukraine

Received  April 2010 Revised  November 2010 Published  March 2011

An elastic medium with a large number of small axially symmetric solid particles is considered. It is assumed that the particles are identically oriented and under the influence of elastic medium they move translationally or rotate around symmetry axis but the direction of their symmetry axes does not change. The asymptotic behavior of small oscillations of the system is studied, when the diameters of particles and distances between the nearest particles are decreased. The equations, describing the homogenized model of the system, are derived. It is shown that the homogenized equations correspond to a non-standard dynamics of elastic medium. Namely, the homogenized stress tensor linearly depends not only on the strain tensor but also on the rotation tensor.
Citation: Maksym Berezhnyi, Evgen Khruslov. Non-standard dynamics of elastic composites. Networks and Heterogeneous Media, 2011, 6 (1) : 89-109. doi: 10.3934/nhm.2011.6.89
##### References:
 [1] M. A. Berezhnyy and L. V. Berlyand, Continuum limit for three-dimensional mass-spring networks and discrete Korn's inequality, Journal of the Mechanics and Physics of Solids, 54 (2006), 635-669. doi: 10.1016/j.jmps.2005.09.006. [2] M. A. Berezhnyi, The asymptotic bahavior of viscous incompressible fluid small oscillations with solid interacting particles, Journal of Mathematical Physics, Analysis, Geometry, 3 (2007), 135-156. [3] M. Berezhnyi, L. Berlyand and E. Khruslov, The homogenized model of small oscillations of complex fluids, Networks and Heterogeneous Media, 3 (2008), 835-869. [4] M. Berezhnyi, "Homogenized Models of Complex Fluids," PhD Thesis, ILTPE, 2009 (in ukrainian), 159 p. URL: http://www.dlib.com.ua/useredneni-modeli-strukturovanykh-ridyn.html. [5] L. V. Berlyand and A. D. Okhotsimskii, Averaged description of an elastic medium with a large number of small absolutely rigid inclusions, Dokl. Akad. Nauk SSSR, 268 (1983), 317-320 (in Russian). [6] L. Berlyand and E. Khruslov, Homogenized non-Newtonian viscoelastic rheology of a suspension of interacting particles in a viscous Newtonian fluid, SIAM, Journal of Applied Mathematics, 64 (2004), 1002-1034. doi: i:10.1137/S0036139902403913. [7] E. Cosserat et F. Cosserat, "Théorie des Corps Deformables," Hermann, Paris, 1909. [8] V. A. Ditkin and A. P. Prudnikov, "Integral Transforms and Operational Calculus," Oxford; New York: Pergamon, 1965, 529p. [9] G. Grioli, Ellasticá asymmetrica, Annali di matematica pura ed applicata, 4 (1960), 389-418. doi: 10.1007/BF02414525. [10] T. Kato, "Perturbation Theory for Linear Operators," Springer, 1995, 652 p. [11] L. D. Landau and E. M. Lifshitz, "Course of Theoretical Physics. Quantum Mechanics. Non-relativistic Theory," London: Pergamon, 1958, 515 p. [12] A. I. Leonov, Algebraic theory of linear viscoelastic nematodynamics, Mathematical Physics, Analysis and Geometry, 11 (2008), 87-116. doi: 10.1007/s11040-008-9041-z. [13] V. Marchenko and E. Khruslov, "Homogenization of Partial Differential Equations," Birkhäuser, Boston, 2006, 401 p. [14] A. I. Marcushevich, "Theory of Analytic Functions: Brief Course," Mir, Moscow, 1983. [15] R. D. Mindlin and H. F. Tiersten, Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and Analysis, 11 (1962), 415-448. doi: 10.1007/BF00253946. [16] O. A. Oleinic, A. S. Shamaev and G. A. Iosif'yan, "Mathematical Problems in Elasticity and Homogenization," in "Studies in Mathematics and its Applications," 26, North-Holland Publishing Co., Amsterdam, 1992, 398 p. [17] I. Y. Smolin, P. V. Makarov, D. V. Shmick and I. V. Savlevich, A micropolar model of plastic deformation of polycrystals at the mesolevel, Computational Materials Science, 19 (2000), 133-142. doi: 10.1016/S0927-0256(00)00148-8. [18] X. Zhang and P. Sharma, Inclusions and inhomogeneities in strain gradient elasticity with couple stresses and related problems, International Journal of Solids and Structures, 42 (2005), 3833-3851. doi: 10.1016/j.ijsolstr.2004.12.005.

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##### References:
 [1] M. A. Berezhnyy and L. V. Berlyand, Continuum limit for three-dimensional mass-spring networks and discrete Korn's inequality, Journal of the Mechanics and Physics of Solids, 54 (2006), 635-669. doi: 10.1016/j.jmps.2005.09.006. [2] M. A. Berezhnyi, The asymptotic bahavior of viscous incompressible fluid small oscillations with solid interacting particles, Journal of Mathematical Physics, Analysis, Geometry, 3 (2007), 135-156. [3] M. Berezhnyi, L. Berlyand and E. Khruslov, The homogenized model of small oscillations of complex fluids, Networks and Heterogeneous Media, 3 (2008), 835-869. [4] M. Berezhnyi, "Homogenized Models of Complex Fluids," PhD Thesis, ILTPE, 2009 (in ukrainian), 159 p. URL: http://www.dlib.com.ua/useredneni-modeli-strukturovanykh-ridyn.html. [5] L. V. Berlyand and A. D. Okhotsimskii, Averaged description of an elastic medium with a large number of small absolutely rigid inclusions, Dokl. Akad. Nauk SSSR, 268 (1983), 317-320 (in Russian). [6] L. Berlyand and E. Khruslov, Homogenized non-Newtonian viscoelastic rheology of a suspension of interacting particles in a viscous Newtonian fluid, SIAM, Journal of Applied Mathematics, 64 (2004), 1002-1034. doi: i:10.1137/S0036139902403913. [7] E. Cosserat et F. Cosserat, "Théorie des Corps Deformables," Hermann, Paris, 1909. [8] V. A. Ditkin and A. P. Prudnikov, "Integral Transforms and Operational Calculus," Oxford; New York: Pergamon, 1965, 529p. [9] G. Grioli, Ellasticá asymmetrica, Annali di matematica pura ed applicata, 4 (1960), 389-418. doi: 10.1007/BF02414525. [10] T. Kato, "Perturbation Theory for Linear Operators," Springer, 1995, 652 p. [11] L. D. Landau and E. M. Lifshitz, "Course of Theoretical Physics. Quantum Mechanics. Non-relativistic Theory," London: Pergamon, 1958, 515 p. [12] A. I. Leonov, Algebraic theory of linear viscoelastic nematodynamics, Mathematical Physics, Analysis and Geometry, 11 (2008), 87-116. doi: 10.1007/s11040-008-9041-z. [13] V. Marchenko and E. Khruslov, "Homogenization of Partial Differential Equations," Birkhäuser, Boston, 2006, 401 p. [14] A. I. Marcushevich, "Theory of Analytic Functions: Brief Course," Mir, Moscow, 1983. [15] R. D. Mindlin and H. F. Tiersten, Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and Analysis, 11 (1962), 415-448. doi: 10.1007/BF00253946. [16] O. A. Oleinic, A. S. Shamaev and G. A. Iosif'yan, "Mathematical Problems in Elasticity and Homogenization," in "Studies in Mathematics and its Applications," 26, North-Holland Publishing Co., Amsterdam, 1992, 398 p. [17] I. Y. Smolin, P. V. Makarov, D. V. Shmick and I. V. Savlevich, A micropolar model of plastic deformation of polycrystals at the mesolevel, Computational Materials Science, 19 (2000), 133-142. doi: 10.1016/S0927-0256(00)00148-8. [18] X. Zhang and P. Sharma, Inclusions and inhomogeneities in strain gradient elasticity with couple stresses and related problems, International Journal of Solids and Structures, 42 (2005), 3833-3851. doi: 10.1016/j.ijsolstr.2004.12.005.
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