April  2022, 17(2): 163-202. doi: 10.3934/nhm.2022002

Homogenization of stiff inclusions through network approximation

Université de Paris, Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586, 75205 Paris CEDEX 13 France

*Corresponding author: Alexandre Girodroux-Lavigne

Received  June 2021 Revised  January 2022 Published  April 2022 Early access  February 2022

We investigate the homogenization of inclusions of infinite conductivity, randomly stationary distributed inside a homogeneous conducting medium. A now classical result by Zhikov shows that, under a logarithmic moment bound on the minimal distance between the inclusions, an effective model with finite homogeneous conductivity exists. Relying on ideas from network approximation, we provide a relaxed criterion ensuring homogenization. Several examples not covered by the previous theory are discussed.

Citation: David Gérard-Varet, Alexandre Girodroux-Lavigne. Homogenization of stiff inclusions through network approximation. Networks and Heterogeneous Media, 2022, 17 (2) : 163-202. doi: 10.3934/nhm.2022002
References:
[1]

M. A. Akcoglu and U. Krengel, Ergodic theorems for superadditive processes, J. Reine Angew. Math., 323 (1981), 53-67.  doi: 10.1515/crll.1981.323.53.

[2]

H. Ammari, P. Garapon, H. Kang and H. Lee, Effective viscosity properties of dilute suspensions of arbitrarily shaped particles, Asymptot. Anal., 80 (2012) 189–211. doi: 10.3233/ASY-2012-1101.

[3]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, American Mathematical Soc., 2013. doi: 10.1090/surv/186.

[4]

L. BerlyandL. Borcea and A. Panchenko, Network approximation for effective viscosity of concentrated suspensions with complex geometry, SIAM J. Math. Anal., 36 (2005), 1580-1628.  doi: 10.1137/S0036141003424708.

[5]

L. BerlyandY. Gorb and A. Novikov, Fictitious fluid approach and anomalous blow-up of the dissipation rate in a two-dimensional model of concentrated suspensions, Arch. Ration. Mech. Anal., 193 (2009), 585-622.  doi: 10.1007/s00205-008-0152-2.

[6]

L. Berlyand and A. Kolpakov, Network approximation in the limit of small interparticle distance of the effective properties of a high-contrast random dispersed composite, Arch. Ration. Mech. Anal., 159 (2001), 179-227.  doi: 10.1007/s002050100142.

[7]

L. Berlyand, A. G Kolpakov and A. Novikov, Introduction to the Network Approximation Method for Materials Modeling, Cambridge University Press, 148, 2013.

[8]

L. Berlyand and V. Mityushev, Increase and decrease of the effective conductivity of two phase composites due to polydispersity, J. Stat. Phys., 118 (2005), 481-509.  doi: 10.1007/s10955-004-8818-0.

[9]

L. Berlyand and A. Novikov, Error of the network approximation for densely packed composites with irregular geometry, SIAM Journal on Mathematical Analysis, 34 (2002), 385-408.  doi: 10.1137/S0036141001397144.

[10]

L. Berlyand and A. Panchenko, Strong and weak blow-up of the viscous dissipation rates for concentrated suspensions, Journal of Fluid Mechanics, 578 (2007), 1-34.  doi: 10.1017/S0022112007004922.

[11]

B. Blaszczyszyn, Lecture notes on random geometric models. random graphs, point processes and stochastic geometry, 2017.

[12]

B. Bollobás, Modern Graph Theory, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0619-4.

[13]

L. Borcea and G. C. Papanicolaou, Network approximation for transport properties of high contrast materials, SIAM J. Appl. Math., 58 (1998), 501-539.  doi: 10.1137/S0036139996301891.

[14]

G. Dal Maso and L. Modica, Nonlinear stochastic homogenization, Ann. Mat. Pura Appl., 144 (1986), 347-389.  doi: 10.1007/BF01760826.

[15]

M. Duerinckx, Effective viscosity of random suspensions without uniform separation, arXiv: 2008.13188.

[16]

M. Duerinckx and A. Gloria, Corrector equations in fluid mechanics: Effective viscosity of colloidal suspensions, Arch. Ration. Mech. Anal., 239 (2021), 1025-1060.  doi: 10.1007/s00205-020-01589-1.

[17]

M. Duerinckx and A. Gloria, On Einstein's effective viscosity formula, arXiv: 2008.03837.

[18]

M. Duerinckx and A. Gloria, Continuum percolation in stochastic homogenization and the effective viscosity problem, arXiv: 2108.09654.

[19]

D. Gérard-Varet, Derivation of the Batchelor-Green formula for random suspensions, Journal de Mathématiques Pures et Appliquées, 152 (2021), 211–250. doi: 10.1016/j.matpur.2021.05.002.

[20]

D. Gérard-Varet and M. Hillairet, Analysis of the viscosity of dilute suspensions beyond Einstein's formula, Arch. Ration. Mech. Anal., 238 (2020), 1349-1411.  doi: 10.1007/s00205-020-01567-7.

[21]

D. Gérard-Varet and R. M. Höfer, Mild assumptions for the derivation of Einstein's effective viscosity formula, Communications in Partial Differential Equations, 46 (2021), 611-629.  doi: 10.1080/03605302.2020.1850780.

[22]

D. Gérard-Varet and A. Mecherbet, On the correction to Einstein's formula for the effective viscosity, arXiv: 2004.05601.

[23]

L. Giovanni, A First Course in Sobolev Spaces, American Mathematical Society, Providence, RI, 2017. doi: 10.1090/gsm/181.

[24]

Y. Gorb and L. Berlyand, The Effective conductivity of densely packed high contrast composites with inclusions of optimal shape, Continuum Models and Discrete Systems, 158 (2004), 63-74.  doi: 10.1007/978-1-4020-2316-3_11.

[25]

B. M. Haines and A. L. Mazzucato, A proof of Einstein's effective viscosity for a dilute suspension of spheres, SIAM Journal on Mathematical Analysis, 44 (2012), 2120-2145.  doi: 10.1137/100810319.

[26]

M. Hillairet and D. Wu, Effective viscosity of a polydispersed suspension, Journal de Mathématiques Pures et Appliquées. Neuvième Série, 138 (2020), 413-447.  doi: 10.1016/j.matpur.2020.03.001.

[27]

V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators, Springer-Verlag, Berlin Heidelberg, 1994. doi: 10.1007/978-3-642-84659-5.

[28]

J. B. Keller, A theorem on the conductivity of a composite medium, Journal of Mathematical Physics, 5 (1964), 548-549.  doi: 10.1063/1.1704146.

[29]

S. Mischler, An introduction to evolution PDEs, (2017).

[30]

B. Niethammer and R. Schubert, A local version of Einstein's formula for the effective viscosity of suspensions, SIAM J. Math. Anal., 52 (2020), 2561-2591.  doi: 10.1137/19M1251229.

[31]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1994.

show all references

References:
[1]

M. A. Akcoglu and U. Krengel, Ergodic theorems for superadditive processes, J. Reine Angew. Math., 323 (1981), 53-67.  doi: 10.1515/crll.1981.323.53.

[2]

H. Ammari, P. Garapon, H. Kang and H. Lee, Effective viscosity properties of dilute suspensions of arbitrarily shaped particles, Asymptot. Anal., 80 (2012) 189–211. doi: 10.3233/ASY-2012-1101.

[3]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, American Mathematical Soc., 2013. doi: 10.1090/surv/186.

[4]

L. BerlyandL. Borcea and A. Panchenko, Network approximation for effective viscosity of concentrated suspensions with complex geometry, SIAM J. Math. Anal., 36 (2005), 1580-1628.  doi: 10.1137/S0036141003424708.

[5]

L. BerlyandY. Gorb and A. Novikov, Fictitious fluid approach and anomalous blow-up of the dissipation rate in a two-dimensional model of concentrated suspensions, Arch. Ration. Mech. Anal., 193 (2009), 585-622.  doi: 10.1007/s00205-008-0152-2.

[6]

L. Berlyand and A. Kolpakov, Network approximation in the limit of small interparticle distance of the effective properties of a high-contrast random dispersed composite, Arch. Ration. Mech. Anal., 159 (2001), 179-227.  doi: 10.1007/s002050100142.

[7]

L. Berlyand, A. G Kolpakov and A. Novikov, Introduction to the Network Approximation Method for Materials Modeling, Cambridge University Press, 148, 2013.

[8]

L. Berlyand and V. Mityushev, Increase and decrease of the effective conductivity of two phase composites due to polydispersity, J. Stat. Phys., 118 (2005), 481-509.  doi: 10.1007/s10955-004-8818-0.

[9]

L. Berlyand and A. Novikov, Error of the network approximation for densely packed composites with irregular geometry, SIAM Journal on Mathematical Analysis, 34 (2002), 385-408.  doi: 10.1137/S0036141001397144.

[10]

L. Berlyand and A. Panchenko, Strong and weak blow-up of the viscous dissipation rates for concentrated suspensions, Journal of Fluid Mechanics, 578 (2007), 1-34.  doi: 10.1017/S0022112007004922.

[11]

B. Blaszczyszyn, Lecture notes on random geometric models. random graphs, point processes and stochastic geometry, 2017.

[12]

B. Bollobás, Modern Graph Theory, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0619-4.

[13]

L. Borcea and G. C. Papanicolaou, Network approximation for transport properties of high contrast materials, SIAM J. Appl. Math., 58 (1998), 501-539.  doi: 10.1137/S0036139996301891.

[14]

G. Dal Maso and L. Modica, Nonlinear stochastic homogenization, Ann. Mat. Pura Appl., 144 (1986), 347-389.  doi: 10.1007/BF01760826.

[15]

M. Duerinckx, Effective viscosity of random suspensions without uniform separation, arXiv: 2008.13188.

[16]

M. Duerinckx and A. Gloria, Corrector equations in fluid mechanics: Effective viscosity of colloidal suspensions, Arch. Ration. Mech. Anal., 239 (2021), 1025-1060.  doi: 10.1007/s00205-020-01589-1.

[17]

M. Duerinckx and A. Gloria, On Einstein's effective viscosity formula, arXiv: 2008.03837.

[18]

M. Duerinckx and A. Gloria, Continuum percolation in stochastic homogenization and the effective viscosity problem, arXiv: 2108.09654.

[19]

D. Gérard-Varet, Derivation of the Batchelor-Green formula for random suspensions, Journal de Mathématiques Pures et Appliquées, 152 (2021), 211–250. doi: 10.1016/j.matpur.2021.05.002.

[20]

D. Gérard-Varet and M. Hillairet, Analysis of the viscosity of dilute suspensions beyond Einstein's formula, Arch. Ration. Mech. Anal., 238 (2020), 1349-1411.  doi: 10.1007/s00205-020-01567-7.

[21]

D. Gérard-Varet and R. M. Höfer, Mild assumptions for the derivation of Einstein's effective viscosity formula, Communications in Partial Differential Equations, 46 (2021), 611-629.  doi: 10.1080/03605302.2020.1850780.

[22]

D. Gérard-Varet and A. Mecherbet, On the correction to Einstein's formula for the effective viscosity, arXiv: 2004.05601.

[23]

L. Giovanni, A First Course in Sobolev Spaces, American Mathematical Society, Providence, RI, 2017. doi: 10.1090/gsm/181.

[24]

Y. Gorb and L. Berlyand, The Effective conductivity of densely packed high contrast composites with inclusions of optimal shape, Continuum Models and Discrete Systems, 158 (2004), 63-74.  doi: 10.1007/978-1-4020-2316-3_11.

[25]

B. M. Haines and A. L. Mazzucato, A proof of Einstein's effective viscosity for a dilute suspension of spheres, SIAM Journal on Mathematical Analysis, 44 (2012), 2120-2145.  doi: 10.1137/100810319.

[26]

M. Hillairet and D. Wu, Effective viscosity of a polydispersed suspension, Journal de Mathématiques Pures et Appliquées. Neuvième Série, 138 (2020), 413-447.  doi: 10.1016/j.matpur.2020.03.001.

[27]

V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators, Springer-Verlag, Berlin Heidelberg, 1994. doi: 10.1007/978-3-642-84659-5.

[28]

J. B. Keller, A theorem on the conductivity of a composite medium, Journal of Mathematical Physics, 5 (1964), 548-549.  doi: 10.1063/1.1704146.

[29]

S. Mischler, An introduction to evolution PDEs, (2017).

[30]

B. Niethammer and R. Schubert, A local version of Einstein's formula for the effective viscosity of suspensions, SIAM J. Math. Anal., 52 (2020), 2561-2591.  doi: 10.1137/19M1251229.

[31]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1994.

Figure 1.  Geometry of the inclusions with a close-up on a gap
Figure 2.  Spherical set up. The graph obtained with the whites lines is isomorphic to the multigraph of inclusions
Figure 3.  Two inclusions configuration, shorted on the right
Figure 4.  Cycle-free configuration set up
Figure 5.  On the left, all clusters are far away from the others. On the right, groups of inclusions joined by a grey line form a short $ F' $ that verifies $ \mathrm{Ed}(F') = \emptyset $
Figure 6.  Separation of the domain $ \mathcal{P} = \mathcal{P}_1 \cup \mathcal{P}_2 $ with the boundary $ \Sigma $
Figure 7.  Geometry of the gap
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