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.

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), 635-669. 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.

[1]

Mustapha El Jarroudi, Youness Filali, Aadil Lahrouz, Mustapha Er-Riani, Adel Settati. Asymptotic analysis of an elastic material reinforced with thin fractal strips. Networks and Heterogeneous Media, 2022, 17 (1) : 47-72. doi: 10.3934/nhm.2021023

[2]

Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10

[3]

Rejeb Hadiji, Ken Shirakawa. Asymptotic analysis for micromagnetics of thin films governed by indefinite material coefficients. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1345-1361. doi: 10.3934/cpaa.2010.9.1345

[4]

Francesco Maddalena, Danilo Percivale, Franco Tomarelli. Adhesive flexible material structures. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 553-574. doi: 10.3934/dcdsb.2012.17.553

[5]

Huicong Li, Jingyu Li. Asymptotic behavior of Dirichlet eigenvalues on a body coated by functionally graded material. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1493-1516. doi: 10.3934/cpaa.2017071

[6]

Simone Göttlich, Sebastian Kühn, Jan Peter Ohst, Stefan Ruzika, Markus Thiemann. Evacuation dynamics influenced by spreading hazardous material. Networks and Heterogeneous Media, 2011, 6 (3) : 443-464. doi: 10.3934/nhm.2011.6.443

[7]

Víctor Manuel Jiménez, Manuel de León. The evolution equation: An application of groupoids to material evolution. Journal of Geometric Mechanics, 2022  doi: 10.3934/jgm.2022001

[8]

Toyohiko Aiki, Joost Hulshof, Nobuyuki Kenmochi, Adrian Muntean. Analysis of non-equilibrium evolution problems: Selected topics in material and life sciences. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : i-iii. doi: 10.3934/dcdss.2014.7.1i

[9]

Víctor Manuel Jiménez Morales, Manuel De León, Marcelo Epstein. Lie groupoids and algebroids applied to the study of uniformity and homogeneity of material bodies. Journal of Geometric Mechanics, 2019, 11 (3) : 301-324. doi: 10.3934/jgm.2019017

[10]

Agnes Lamacz, Ben Schweizer. Effective acoustic properties of a meta-material consisting of small Helmholtz resonators. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 815-835. doi: 10.3934/dcdss.2017041

[11]

Claude Stolz. On estimation of internal state by an optimal control approach for elastoplastic material. Discrete and Continuous Dynamical Systems - S, 2016, 9 (2) : 599-611. doi: 10.3934/dcdss.2016014

[12]

Rainer Picard. On a comprehensive class of linear material laws in classical mathematical physics. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 339-349. doi: 10.3934/dcdss.2010.3.339

[13]

Michela Eleuteri, Jana Kopfová, Pavel Krejčí. A new phase field model for material fatigue in an oscillating elastoplastic beam. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2465-2495. doi: 10.3934/dcds.2015.35.2465

[14]

Huicong Li. Effective boundary conditions of the heat equation on a body coated by functionally graded material. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1415-1430. doi: 10.3934/dcds.2016.36.1415

[15]

Manuel Friedrich, Bernd Schmidt. On a discrete-to-continuum convergence result for a two dimensional brittle material in the small displacement regime. Networks and Heterogeneous Media, 2015, 10 (2) : 321-342. doi: 10.3934/nhm.2015.10.321

[16]

Qiong Liu, Ahmad Reza Rezaei, Kuan Yew Wong, Mohammad Mahdi Azami. Integrated modeling and optimization of material flow and financial flow of supply chain network considering financial ratios. Numerical Algebra, Control and Optimization, 2019, 9 (2) : 113-132. doi: 10.3934/naco.2019009

[17]

Caiyan Li, Dongsheng Li. $ W^{1,p} $ estimates for elliptic systems on composite material with almost partially BMO coefficients. Communications on Pure and Applied Analysis, 2021, 20 (9) : 3143-3159. doi: 10.3934/cpaa.2021100

[18]

Patrick Ballard, Bernadette Miara. Formal asymptotic analysis of elastic beams and thin-walled beams: A derivation of the Vlassov equations and their generalization to the anisotropic heterogeneous case. Discrete and Continuous Dynamical Systems - S, 2019, 12 (6) : 1547-1588. doi: 10.3934/dcdss.2019107

[19]

Grigory Panasenko, Ruxandra Stavre. Asymptotic analysis of the Stokes flow with variable viscosity in a thin elastic channel. Networks and Heterogeneous Media, 2010, 5 (4) : 783-812. doi: 10.3934/nhm.2010.5.783

[20]

Grigory Panasenko, Ruxandra Stavre. Asymptotic analysis of a non-periodic flow in a thin channel with visco-elastic wall. Networks and Heterogeneous Media, 2008, 3 (3) : 651-673. doi: 10.3934/nhm.2008.3.651

2020 Impact Factor: 1.213

Metrics

  • PDF downloads (70)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]