American Institute of Mathematical Sciences

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

Non-standard dynamics of elastic composites

Maksym Berezhnyi 1, and  Evgen Khruslov 1,

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.
Keywords: asymptotic analysis., elastic material, Microstructure, anisotropic material, inhomogeneous material.
Mathematics Subject Classification: Primary: 35B27, 35Q30, 74Q10; Secondary: 76M30, 76M5.
Citation: Maksym Berezhnyi, Evgen Khruslov. Non-standard dynamics of elastic composites. Networks & 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.  doi: 10.1016/j.jmps.2005.09.006.  Google Scholar

[2]

M. A. Berezhnyi, The asymptotic bahavior of viscous incompressible fluid small oscillations with solid interacting particles,, Journal of Mathematical Physics, 3 (2007), 135.   Google Scholar

[3]

M. Berezhnyi, L. Berlyand and E. Khruslov, The homogenized model of small oscillations of complex fluids,, Networks and Heterogeneous Media, 3 (2008), 835.   Google Scholar

[4]

M. Berezhnyi, "Homogenized Models of Complex Fluids,", PhD Thesis, (2009).   Google Scholar

[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.   Google Scholar

[6]

L. Berlyand and E. Khruslov, Homogenized non-Newtonian viscoelastic rheology of a suspension of interacting particles in a viscous Newtonian fluid,, SIAM, 64 (2004), 1002.  doi: i:10.1137/S0036139902403913.  Google Scholar

[7]

E. Cosserat et F. Cosserat, "Théorie des Corps Deformables,", Hermann, (1909).   Google Scholar

[8]

V. A. Ditkin and A. P. Prudnikov, "Integral Transforms and Operational Calculus,", Oxford; New York: Pergamon, (1965).   Google Scholar

[9]

G. Grioli, Ellasticá asymmetrica,, Annali di matematica pura ed applicata, 4 (1960), 389.  doi: 10.1007/BF02414525.  Google Scholar

[10]

T. Kato, "Perturbation Theory for Linear Operators,", Springer, (1995).   Google Scholar

[11]

L. D. Landau and E. M. Lifshitz, "Course of Theoretical Physics. Quantum Mechanics. Non-relativistic Theory,", London: Pergamon, (1958).   Google Scholar

[12]

A. I. Leonov, Algebraic theory of linear viscoelastic nematodynamics,, Mathematical Physics, 11 (2008), 87.  doi: 10.1007/s11040-008-9041-z.  Google Scholar

[13]

V. Marchenko and E. Khruslov, "Homogenization of Partial Differential Equations,", Birkh\, (2006).   Google Scholar

[14]

A. I. Marcushevich, "Theory of Analytic Functions: Brief Course,", Mir, (1983).   Google Scholar

[15]

R. D. Mindlin and H. F. Tiersten, Effects of couple-stresses in linear elasticity,, Archive for Rational Mechanics and Analysis, 11 (1962), 415.  doi: 10.1007/BF00253946.  Google Scholar

[16]

O. A. Oleinic, A. S. Shamaev and G. A. Iosif'yan, "Mathematical Problems in Elasticity and Homogenization,", in, 26 (1992).   Google Scholar

[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.  doi: 10.1016/S0927-0256(00)00148-8.  Google Scholar

[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.  doi: 10.1016/j.ijsolstr.2004.12.005.  Google Scholar

show all references

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.  doi: 10.1016/j.jmps.2005.09.006.  Google Scholar

[2]

M. A. Berezhnyi, The asymptotic bahavior of viscous incompressible fluid small oscillations with solid interacting particles,, Journal of Mathematical Physics, 3 (2007), 135.   Google Scholar

[3]

M. Berezhnyi, L. Berlyand and E. Khruslov, The homogenized model of small oscillations of complex fluids,, Networks and Heterogeneous Media, 3 (2008), 835.   Google Scholar

[4]

M. Berezhnyi, "Homogenized Models of Complex Fluids,", PhD Thesis, (2009).   Google Scholar

[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.   Google Scholar

[6]

L. Berlyand and E. Khruslov, Homogenized non-Newtonian viscoelastic rheology of a suspension of interacting particles in a viscous Newtonian fluid,, SIAM, 64 (2004), 1002.  doi: i:10.1137/S0036139902403913.  Google Scholar

[7]

E. Cosserat et F. Cosserat, "Théorie des Corps Deformables,", Hermann, (1909).   Google Scholar

[8]

V. A. Ditkin and A. P. Prudnikov, "Integral Transforms and Operational Calculus,", Oxford; New York: Pergamon, (1965).   Google Scholar

[9]

G. Grioli, Ellasticá asymmetrica,, Annali di matematica pura ed applicata, 4 (1960), 389.  doi: 10.1007/BF02414525.  Google Scholar

[10]

T. Kato, "Perturbation Theory for Linear Operators,", Springer, (1995).   Google Scholar

[11]

L. D. Landau and E. M. Lifshitz, "Course of Theoretical Physics. Quantum Mechanics. Non-relativistic Theory,", London: Pergamon, (1958).   Google Scholar

[12]

A. I. Leonov, Algebraic theory of linear viscoelastic nematodynamics,, Mathematical Physics, 11 (2008), 87.  doi: 10.1007/s11040-008-9041-z.  Google Scholar

[13]

V. Marchenko and E. Khruslov, "Homogenization of Partial Differential Equations,", Birkh\, (2006).   Google Scholar

[14]

A. I. Marcushevich, "Theory of Analytic Functions: Brief Course,", Mir, (1983).   Google Scholar

[15]

R. D. Mindlin and H. F. Tiersten, Effects of couple-stresses in linear elasticity,, Archive for Rational Mechanics and Analysis, 11 (1962), 415.  doi: 10.1007/BF00253946.  Google Scholar

[16]

O. A. Oleinic, A. S. Shamaev and G. A. Iosif'yan, "Mathematical Problems in Elasticity and Homogenization,", in, 26 (1992).   Google Scholar

[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.  doi: 10.1016/S0927-0256(00)00148-8.  Google Scholar

[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.  doi: 10.1016/j.ijsolstr.2004.12.005.  Google Scholar

[1]

Adrian Viorel, Cristian D. Alecsa, Titus O. Pinţa. Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020407
[2]

Vincent Ducrot, Pascal Frey, Alexandra Claisse. Levelsets and anisotropic mesh adaptation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 165-183. doi: 10.3934/dcds.2009.23.165
[3]

Patrick W. Dondl, Martin Jesenko. Threshold phenomenon for homogenized fronts in random elastic media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 353-372. doi: 10.3934/dcdss.2020329
[4]

Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020075
[5]

Andreas Kreuml. The anisotropic fractional isoperimetric problem with respect to unconditional unit balls. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020290
[6]

Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021004
[7]

Josselin Garnier, Knut Sølna. Enhanced Backscattering of a partially coherent field from an anisotropic random lossy medium. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1171-1195. doi: 10.3934/dcdsb.2020158
[8]

Hongbo Guan, Yong Yang, Huiqing Zhu. A nonuniform anisotropic FEM for elliptic boundary layer optimal control problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1711-1722. doi: 10.3934/dcdsb.2020179
[9]

Shao-Xia Qiao, Li-Jun Du. Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity. Electronic Research Archive, , () : -. doi: 10.3934/era.2020116
[10]

Claudia Lederman, Noemi Wolanski. An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020391
[11]

Jia Cai, Guanglong Xu, Zhensheng Hu. Sketch-based image retrieval via CAT loss with elastic net regularization. Mathematical Foundations of Computing, 2020, 3 (4) : 219-227. doi: 10.3934/mfc.2020013
[12]

George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003
[13]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259
[14]

Yunfeng Jia, Yi Li, Jianhua Wu, Hong-Kun Xu. Cauchy problem of semilinear inhomogeneous elliptic equations of Matukuma-type with multiple growth terms. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3485-3507. doi: 10.3934/dcds.2019227
[15]

Dongfen Bian, Yao Xiao. Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1243-1272. doi: 10.3934/dcdsb.2020161
[16]

Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251
[17]

Vivina Barutello, Gian Marco Canneori, Susanna Terracini. Minimal collision arcs asymptotic to central configurations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 61-86. doi: 10.3934/dcds.2020218
[18]

Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301
[19]

José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091
[20]

Yueh-Cheng Kuo, Huey-Er Lin, Shih-Feng Shieh. Asymptotic dynamics of hermitian Riccati difference equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020365

2019 Impact Factor: 1.053

Tools

Metrics

  • PDF downloads (34)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences