ReferencesAbbot, J.R., Tetlow, N., Graham, A.L., Altobell, S.A., Fukushima, E., Mondy, L.A. and Stephens, T.A.. (1991). Experimental observations of particle migration in concentrated suspensions: Couette flow. J. Rheol., 35:773–795.
Aboudi, J. (1991). Mechanics of Composite Material: A Unified Micromechanics Approach. Elsevier Science, Amsterdam.
Acrivos, A. and Chang, E.. (1986). A model for estimating transport quantities in two-phase materials. Phys. Fluids, 29(3):3–4.
Adams, R.A. (1975). Sobolev Spaces. Academic Press, New York.
Ahlfors, L. (1979). Complex Analysis, 3rd ed. McGraw-Hill, New York.
Akhiezer, N.I. (1990). Elements of the Theory of Elliptic Functions.American Mathematical Society, Providence, RI.
Almgren, R.F. (1985). An isotropic three-dimensional structure with Poisson's ratio = −1. J. Elasticity, 15:427–430.
Ambegaokar, V., Halperin, B.I. and Langer, J.S.. (1971). Hopping conductivity in disordered systems. Phys. Rev., B, 4(8):2612–2620.
Andrianov, I.V., Danishevs'kyy, V.V. and Kalamkarov, A.L.. (2002). Asymptotic analysis of effective conductivity of composite materials with large rhombic fibres. Composite Struct., 56(33):229–234.
Andrianov, I.V., Danishevs'kyy, V.V. and Tokarzewski, S.. (1996). Two-point quasifractional approximants for effective conductivity of a simple cubic lattice of spheres. Int. J. Heat Mass Transfer, 39(11):2349–2352.
Andrianov, I.V., Starushenko, G.A., Danishevskiy, V.V. and Tokarzewski, S.. (1999). Homogenization procedure and Padé approximants for effective heat conductivity of a composite material with cylindrical inclusions having square cross-section. Proc. R. Soc. London, A, 455:3401–3413.
Annin, B.D., Kalamkarov, A.L., Kolpakov, A.G. and Parton, V.Z.. (1993). Computation and Design of Composite Material and Structural Elements (in Russian).Nauka, Novosibirsk.
Aurenhammer, F. and Klein, R.. (1999). Voronoi diagrams. In: Handbook of Computational Geometry (Sack, J.R. and Urrutia, J., eds.), North Holland, Amsterdam, pp. 201–291.
Avellaneda, M. (1987). Optimal bounds and microgeometries for an elastic two-phase composite. SIAM J. Appl. Math., 47(6):1216–1228.
Babǔshka, I., Anderson, B., Smith, P. and Levin, K.. (1999). Damage analysis of fiber composites. Part I. Statistical analysis on fiber scale. Comput. Methods Appl. Mech. Engng., 172:27–77.
Bakhvalov, N.S. and Panasenko, G.P.. (1989). Homogenization: Averaging Processes in Periodic Media.Kluwer Academic Publishers, Dordrecht.
Balberg, I. (1987). Recent developments in continuum percolation. Phil. Mag., B 30:991–1003.
Batchelor, G.K. and O'Brien, R.W.. (1977). Thermal or electrical conduction through a granular material. Proc. R. Soc. London, A, 335:313–333.
Batchelor, G.K. and Wen, C.S.. (1972). Sedimentation in a dilute dispersion of spheres. J. Fluid Mech., 52:245–268.
Bendsøe, M.P. and Kikuchi, N.. (1988). Generating optimal topologies in structural design using a homogenization method. Comp. Meth. Appl. Mech. Engng, 71:95–112.
Bendsøe, M.P. and Sigmund, O.. (2004). Topology Optimization.Springer-Verlag, Berlin.
Bensoussan, A., Lions, J.-L. and Papanicolaou, G.. (1975). Sur quelques phénomènes asymptotiques d'évolution. Compt. Rend. Acad. Set Paris, Ser. A-B, 281(10):A317–A322.
Bensoussan, A., Lions, J.-L. and Papanicolaou, G.. (1978). Asymptotic Analysis for Periodic Structures.North Holland, Amsterdam.
Beran, M.J. (1968). Statistical Continuum Theories.John Wiley, New York.
Beran, M.J. and Molyneux, J.. (1966). Use of classical variational principles to determine bounds for the effective bulk modulus in heterogeneous media. Quart. Appl. Math., 24:107–118.
Berdichevsky, V.L. (2009). Variational Principles of Continuum Mechanics.Springer-Verlag, Berlin.
Bergman, D.J. (1983). The dielectric constant of a composite material – a problem in classical physics. Phys. Reports, C43:378–407.
Bergman, D.J., Duering, E. and Murat, M.. (1990). Discrete network models for the low-field Hall effect near a percolation threshold: Theory and simulation. J. Stat. Phys., 1(58):1–43.
Bergman, D.J. and Dunn, K.J.. (1992). Bulk effective dielectric constant of a composite with periodic micro-geometry. Phys. Rev. B, 45:13262–13271.
Berlyand, L., Borcea, L. and Panchenko, A.. (2005). Network approximation for effective viscosity of concentrated suspensions with complex geometries. SIAM J. Math. Anal., 36(5):1580–1628.
Berlyand, L., Gorb, Y. and Novikov, A.. (2005). Discrete network approximation for highly-packed composites with irregular geometry in three dimensions. In: Multiscale Methods in Science and Engineering (Engquist, B., Lotstedt, P. and Runborg, O., eds.), Springer-Verlag, Berlin, pp. 21–58.
Berlyand, L., Gorb, Y. and Novikov, A.. (2009). Fictitious fluid approach and anomalous blow-up of the dissipation rate in a two-dimensional model of concentrated suspensions. Arch. Rational Mech. Anal., 193(3):585–622.
Berlyand, L. and Kolpakov, A.. (2001). Network approximation in the limit of small interparticle distance of the effective properties of a high-contrast random dispersed composite. Arch. RationalMech. Anal., 159(3):179–227.
Berlyand, L. and Kozlov, S.. (1992). Asymptotics of the homogenized moduli for the elastic chess-board composite. Arch. Rational Mech. Anal., 118(2):95–112.
Berlyand, L. and Mityushev, V.. (2001). Generalized Clausius–Mossotti formula for random composite with circular fibers. J. Stat. Phys., 102(1/2):115–145.
Berlyand, L. and Mityushev, V.. (2005). Increase and decrease of the effective conductivity of a two phase composite due to polydispersity. J. Stat. Phys., 118(3/4):479–507.
Berlyand, L. and Novikov, A.. (2002). Error of the network approximation for densely packed composites with irregular geometry. SIAM J. Math. Anal., 34(2):385–408.
Berlyand, L. and Promislow, K.. (1995). Effective elastic moduli of a soft medium with hard polygonal inclusions and extremal behavior of effective Poisson's ratio. J. Elasticity, 40(1):45–73.
Berlyand, L.V. and Panchenko, A.. (2007). Strong and weak blow up of the viscous dissipation rates for concentrated suspensions. J. Fluid Mech., 578:1–34.
Bhattacharya, K., Kohn, R.V. and Kozlov, S.. (1999). Some examples of nonlinear homogenization involving nearly degenerate energies. Proc. R. Soc. London, A, 455:567–583.
Bollobás, B. (1998). Modern Graph Theory.Springer-Verlag, New York.
Bonnecaze, R.T. and Brady, J.F.. (1991). The effective conductivity of random suspensions of spherical particles. Proc. R. Soc. London, Ser. A, 432:445–465.
Borcea, L. (1998). Asymptotic analysis of quasi-static transport in high contrast conductive media. SIAM J. Appl. Math., 2(59):597–635.
Borcea, L., Berryman, J.G. and Papanicolaou, G.. (1999). Matching pursuit for imaging high-contrast conductivity. Inverse Problems, 15:811–849.
Borcea, L. and Papanicolaou, G.. (1998). Network approximation for transport properties of high contrast conductivity. Inverse Problems, 4(15):501–539.
Born, M. and Huang, K.. (1954). Dynamical Theory of Crystal Lattices.Oxford University Press, Oxford.
Bourgeat, A., Mikelic, A. and Wright, S.. (1994). Stochastic two-scale convergence in the mean and applications. J. Reine Angew. Math., 456:19–51.
Bourgeat, A. and Piatnitski, A.. (2004). Approximations of effective coefficients in stochastic homogenization. Ann. Inst. H. Poincaré, 40:153–165.
Brady, J.F. (1993). The rheo logical behavior of concentrated colloidal suspensions. J. Chem. Phys., 99:567–581.
Brady, J.F. and Bossis, G.. (1985). The rheology of concentrated suspensions of spheres in simple shear flow by numerical simulation. J. Fluid Mech., 155:105–129.
Brodbent, S.R. and Hammerslay, J.M.. (1957). Percolation processes I. Crystals and mazes. Math. Proc. Cambridge Phil. Soc., 53:629–641.
Broutman, L.J. and Krock, R.H., eds. (1974). Composite Materials. Vol. 1-8. Academic Press, New York.
Brown, W.F. (1956). Dielectrics.Springer-Verlag, Berlin.
Bruno, O. (1991). The effective conductivity of strongly heterogeneous composites. Proc. R. Soc. London, A, 433:353–381.
B¨rger, R. and Wendland, W.L.. (2001). Sedimentation and suspension flows: historical perspective and some recent developments. J. Engng. Math., 41(2/3):101–1 16.
Burkill, J.C. (2004). The Lebesgue Integral.Cambridge University Press, Cambridge.
Caillerie, D. (1978). Sur la comportement limite d'une inclusion mince de grande rigidité dans un corps élastique. Compt. Rend. Acad. Set Paris, Ser. A., 287:675–678.
Carreau, P.J. and Cotton, F.. (2002). Rheological properties of concentrated suspensions. In: Transport Processes in Bubbles, Drops and Particles (De Kee, D. and Chhabra, R.P., eds.), Taylor & Francis, London.
Chang, Ch. and Powell, R.L.. (1994). Effect of particle size distribution on the rheology of a concentrated bimodal suspension. J. Rheol., 38:85–98.
Chen, H.-S. and Acrivos, A.. (1978). The effective elastic moduli materials containing spherical inclusions at non-dilute concentration. Int. J. Solids Struct., 14:349–364.
Cheng, H. and Greengard, L.. (1997). On the numerical evaluation of electrostatic fields in a dense random dispersions of cylinders. J. Comput. Phys., 136:626–639.
Cheng, H. and Greengard, L.. (1998). A method of images for the evaluation of electrostatic fields in a system of closely spaced conducting cylinders. SIAM J. Appl. Math., 50:122–141.
Cherkaev, A.V. (2000). Variational Methods for Structural Optimization.Springer-Verlag, Berlin.
Chinh, Ph.D. (1997). Overall properties of planar quasisymmetric randomly inhomogeneous media: Estimates and cell models. Phys. Rev. E, 56:652–660.
Chou, T.-W. and Ko, F.K., eds. (1989). Textile Structural Composites.Elsevier Science, Amsterdam.
Christensen, R.M. (1979). Mechanics of Composite Materials.John Wiley, New York.
Chung, J.W., De Hosson, J.Th.M. and van der Giessen, E.. (1996). Fracture of a disordered 3-D spring network: A computer simulation methodology. Phys. Rev. B, 54:15094–15100.
Clerc, J.P., Giraud, G., Laugier, J.M. and Luck, J.M.. (1990). The electrical conductivity of binary disordered systems, percolation clusters, fractals and related models. Adv. Phys., 39(3):191–309.
Courant, R.S. and Hilbert, D.. (1953). Methods of Mathematical Physics.John Wiley, New York.
Coussot, P. (2002). Flows of concentrated granular mixtures. In: Transport Processes in Bubbles, Drops and Particles (Chhabra, R.P. and De Kee, D., eds.), Taylor & Francis, London, pp. 291–315.
Craster, R.V. and Obnosov, Yu.V.. (2004). A three-phase tessellation: Solution and effective properties. Proc. R. Soc. London, A, 460:1017–1037.
Curtin, W.A. and Scher, H.. (1990a). Brittle fracture in disordered materials: A spring network model. J. Mater. Res., 5:535–553.
Curtin, W.A. and Scher, H.. (1990b). Mechanical modeling using a spring network. J. Mater. Res., 5:554–562.
Del Maso, G. (1993). An Introduction to Γ-Convergence.Birkhäuser, Boston.
Diaz, A.R. and Kikuchi, N.. (1992). Solutions to shape and topology eigenvalue optimization problems using a homogenization method. Int. J. Num. Meth. Engng, 35:1487–1502.
Dieudonne, J.A. (1969). Treatise on Analysis.Academic Press, New York.
Ding, J., Warriner, H.E. and Zasadzinski, J.A.. (2002). Viscosity of two-dimensional suspensions. Phys. Rev. Lett., 88(16):168102.1-168102.4.
Dobrodumov, A.M. and El'yashevich, A.M.. (1973). Simulation of brittle fracture of polymers by a network model in the Monte Carlo method. Sov. Solid State Phys., 15:1259–1260.
Doyle, W.T. (1978). The Clasius–Mossotti problem for cubic arrays of spheres. J. Appl. Phys., 49:795–797.
Drummon, J.E. and Tahir, M.I.. (1984). Laminar viscous flow through regular arrays of parallel solid cylinders. Int. J. Multiphase Flow, 10:515–540.
Drygaś, P. and Mityushev, V.. (2009). Effective conductivity of unidirectional cylinders with interfacial resistance. Quarterly J. Mech. Appl. Math., 62(3):235–262.
Dykhne, A.M. (1971). Conductivity of a two-dimensional two-phase system. Sov. Phys., 32(63):63–65.
Einstein, A. (1906). Eine neue Bestimmung der Molekuldimensionen. Ann. Phys., 19:289–306.
Ekeland, I. and Temam, R.. (1976). Convex Analysis and Variational Problems.North Holland, Amsterdam.
Evans, L.C. and Gangbo, W.. (1999). Differential equation methods for the Monge–Kantorovich mass transfer problem. Mem. Amer. Math. Soc., 137(653):viii+66.
Evans, L.C. and Gariepy, R.F.. (1992). Measure Theory and Fine Properties of Functions.CRC Press, Boca Raton, FL.
Feng, N.A. (1985). Percolation properties of granular elastic networks in two dimensions. Phys. Rev. B, 32(1):510–513.
Feng, N.A. and Acrivos, A.. (1985). On the viscosity of concentrated suspensions of solid spheres. Chem. Engng Sci., 22:847–853.
Flaherty, J.E. and Keller, J.B.. (1973). Elastic behavior of composite media. Comm. Pure Appl. Math., 26:565–580.
Flory, P.J. (1941). Molecular size distribution in three dimensional polymers. I. Gelation. J. Amer. Chem. Soc., 63:3083–3090.
Fox, L. (1964). An Introduction to Numerical Linear Algebra.Clarendon Press, Oxford.
Francfort, G.A. and Murat, F.. (1986). Homogenization and optimal bounds in linear electricity. Arch. Rational Mech. Anal., 94(4):307–334.
Frenkel, N.A. and Acrivos, A.. (1967). On the viscosity of concentrated suspension of solid spheres. Chem. Engng Sci., 22:847–853.
Friis, E.A., Lakes, R.S. and Park, J.B.. (1988). Negative Poisson's ratio polymeric and metallic foams. J. Mater. Sci., 23:4406–4414.
Gakhov, F.D. (1966). Boundary Value Problems.Pergamon Press, Oxford.
Garboczi, E.J. and Douglas, J.F.. (1996). Intrinsic conductivity of objects having arbitrary shape and conductivity. Phys. Rev. E, 53(6):6169–6180.
Gaudiello, A. and Kolpakov, A.G.. (2011). Influence of non degenerated joint on the global and local behavior of joined rods. Int. J. Engng. Sci., 49(3):295–309.
Good, I.J. (1949). The number of individuals in a cascade process. Math. Proc. Cambridge Phil. Soc., 45:360–363.
Goto, H. and Kuno, H.. (1984). Flow of suspensions containing particles of two different sizes through a capillary tube. II. Effect of the particle size ratio. J. Rheol., 28:197–205.
Graham, A.L. (1981). On the viscosity of a suspension of solid particles. Appl. Sci. Res., 37:275–286.
Greengard, L. and Lee, J.-Y.. (2006). Electrostatics and heat conduction in high contrast composite materials. J. Comput. Phys., 211(1):64–76.
Greengard, L. and Moura, M.. (1994). On the numerical evaluation of electrostatic fields in composite materials. Acta Numerica, 3:379–410.
Grigolyuk, E.I. and Filshtinskij, L.A.. (1972). Periodical Piecewise Homogeneous Elastic Structures (in Russian).Nauka, Moscow.
Grimet, G. (1992). Percolation.Springer-Verlag, Berlin.
Gupta, P.K. and Cooper, A.R.. (1990). Topologically disordered networks of rigid polytopes. J. Non-Crystal. Solids, 123(14):14–21.
Halperin, B.I., Feng, S. and Sen, P.N.. (1985). Difference between lattice and continuum percolation transport exponents. Phys. Rev. Lett., 54:2391–2394.
Happel, J. (1959). Viscous flow relative to arrays of cylinders. AIChE J., 5:174–177.
Hasimoto, H. (1959). On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres. J. Fluid Mech., 5:317–328.
Haug, E.J., Choi, K.K. and Komkov, V.. (1986). Design Sensitivity Analysis of Structural Systems.Academic Press, Orlando, FL.
Herrmann, H.J., Hansen, A. and Roux, S.. (1989). Fracture of disordered, elastic lattices in two dimensions. Phys. Rev. B, 39:637–648.
Hill, R. (1963). Elastic properties of reinforced solids: Some theoretical principles. J. Mech. Phys. Solids, 11:357–372.
Hill, R. (1996). Characterization of thermally conductive epoxy composite fillers. Proc. Technical Program “Emerging Packing Technology”,Surface Mount Tech. Symp., pp. 125–131.
Hill, R.F. and Supancic, P.H.. (2002). Thermal conductivity of platelet-filled polymer composite. J. Am. Cer. Soc., 85:851–857.
Hinsen, K. and Felderhof, B.U.. (1992). Dielectric constant of a suspension of uniform spheres. Phys. Rev. B, 46(20):12955–12963.
Hrennikoff, A. (1941). Solution of a problem of elasticity by the framework method. J. Appl. Mech., 8:169–175.
Jabin, P.-E. and Otto, F.. (2004). Identification of the dilute regime in particle sedimentation. Commun. Math. Phys., 250:415–432.
Jeffrey, D.J. and Acrivos, A.. (1976). The rheological properties of suspensions of rigid particles. AIChE J., 22:417–432.
Jikov, V.V., Kozlov, S.M. and Oleinik, O.A.. (1994). Homogenization of Differential Operators and Integral Functionals.Springer-Verlag, Berlin.
Kalamkarov, A.L. and Kolpakov, A.G.. (1996). On the analysis and design of fiber reinforced composite shells. Trans. ASME. J. Appl. Mech., 63(4):939–945.
Kalamkarov, A.L. and Kolpakov, A.G.. (1997). Analysis, Design and Optimization of Composite Structures.John Wiley, Chichester.
Karal, F.C. Jr. and Keller, J.B.. (1966). Effective dielectric constant, permeability, and conductivity of a random medium and the velocity and attenuation coefficient of coherent waves. J. Math. Phys., 7:661–670.
Kato, T. (1976). Perturbation Theory for Linear Operators.Springer-Verlag, New York.
Keller, J.B. (1963). Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders. J. Appl. Phys., 4(34):991–993.
Keller, J.B. (1964). A theorem on the conductivity of a composite medium. J. Math. Phys., 5:548–549.
Keller, J.B. (1987). Effective conductivity of a periodic composite composed of two very unequal conductors. J. Math. Phys., 10(28):2516–2520.
Keller, J.B. and Sachs, D.. (1964). Calculations of conductivity of a medium containing cylindrical inclusions. J. Appl. Phys., 35:537–538.
Kellomaki, M., Astrom, J. and Timonen, J.. (1996). Rigidity and dynamics of random spring networks. Phys. Rev. Lett., 77:2730–2733.
Kelly, A. and Rabotnov, Yu.N., eds. (1988). Handbook of Composites.North Holland, Amsterdam.
Kesten, H. (1992). Percolation Theory for Mathematicians.Birkhäuser, Boston.
Kolmogorov, A.N. and Fomin, S.V.. (1970). Introductory Real Analysis.Prentice Hall, Englewood Cliffs, NJ.
Kolpakov, A.A. (2007). Numerical verification of existence of the energy-concentration effect in a high-contrast high-filled composite material. J. Engng Phys. Thermophys., 80(4):812–819.
Kolpakov, A.A. and Kolpakov, A.G.. (2007). Asymptotics of the capacity of a system of closely placed bodies. Tamm's shielding effect and network models. Doklady Phys., 415(2):188–192.
Kolpakov, A.A. and Kolpakov, A.G.. (2010). Capacity and Transport in Contrast Composite Structures: Asymptotic Analysis and Applications.CRC Press, Boca Raton, FL.
Kolpakov, A.G. (1987). Averaged characteristics of thermoelastic frames. Izvestiay of the Academy of Science of the USSR. Mechanics of Solids, 22(6):53–61.
Kolpakov, A.G. (1985). Determination of the average characteristics of elastic frameworks. J. Appl. Math. Mech., 49:739–745.
Kolpakov, A.G. (1988). Asymptotics of the first boundary value problem for an elliptic equation in a region with a thin covering. Siberian Math. J., 6:74–84.
Kolpakov, A.G. (1992). Glued bodies. Differential Equations, 28(8):1131–1139.
Kolpakov, A.G. (2004). Stressed Composite Structures: Homogenized Models for Thin-Walled Nonhomogeneous Structures with Initial Stresses.Springer-Verlag, Berlin.
Kolpakov, A.G. (2005). Asymptotic behavior of the conducting properties of high-contrast media. J. Appl. Mech. Tech. Phys., 46(3):412–422.
Kolpakov, A.G. (2006a). The asymptotic screening and network models. J. Engng Phys. Thermophys., 2:39–47.
Kolpakov, A.G. (2006b). Convergence of solutions for a network approximation of the two-dimensional Laplace equation in a domain with a system of absolutely conducting disks. Comp. Math. Math. Phys., 46(9):1682–1691.
Kolpakov, A.G. (2011). Influence of non degenerated joint on the global and local behavior of joined plates. Int. J. Engng. Sci., 49(11):1216–1231.
Koplik, J. (1982). Creeping flow in two-dimensional networks. J. Fluid Mech., 119:219–247.
Kozlov, S.M. (1978). Averaging of random structures (in Russian). Doklady Acad. Nauk SSSR, 241(5):1016–11019.
Kozlov, S.M. (1980). Averaging of random operators. Math. USSR Sbornik, 37:167–180.
Kozlov, S.M. (1989). Geometric aspects of averaging. Russian Math. Surv., 2(44):91–144.
Kozlov, S.M. (1992). On the domain of variations of apparent added masses, polarization and effective characteristics of composites. J. Appl. Math. Mech., 56(1):102–107.
Kuchling, H. (1980). Physics.VEB Fachbuchverlag, Leipzig.
Kun, F. and Herrmann, H.. (1996). A study of fragmentation processes using a discrete element method. Comput. Meth. Appl. Mech. Engng, 138:3–18.
Ladd, A.J.C. (1997). Sedimentation of homogeneous suspensions of non-Brownian spheres. Phys. Fluids, 9(3):491–499.
Ladyzhenskaya, O.A. and Ural'tseva, N.N.. (1968). Linear and Quasilinear Elliptic Equations.Academic Press, New York.
Lakes, R. (1991). Deformation mechanisms of negative Poisson's ratio materials: Structural aspects. J. Mater. Sci., 26:2287–2292.
Lamb, H. (1991). Hydrodynamics.Dover, New York.
Landauer, R. (1978). Electrical conductivity in inhomogeneous media. In: Electrical Transport and Optical Properties of Inhomogeneous Media (Garland, J.C., Tanner, D.B., eds.), American Institute of Physics. Woodbury, New York, pp. 2–43.
Leal, G. (1992). Laminar Flow and Convective Transport Processes: Scaling Principles and Asymptotic Analysis.Butterworth-Heinemann, Amsterdam.
Leighton, D. and Acrivos, A.. (1987). Measurement of shear-induced self-diffusion in concentrated suspensions of spheres. J. FluidMech., 177:109–131.
Lenczner, M. (1997). Homogénéisation d'un circuit électrique. C.R. Acad. Sci. Paris, Série II B, 324(9):537–542.
Lieberman, G.M. (1988). Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal., 12(11):1203–1219.
Limat, L. (1988). Percolation and Cosserat elasticity: Exact results on a deterministic fractal. Phys. Rev., B, 37:672–675.
Lions, J.-L. (1978). Notes on some computational aspects of the homogenization method in composite materials. In: Computational Methods in Mathematics, Geophysics and Optimal Control,Nauka, Novosibirsk, pp. 5–19.
Lions, J.-L. and Magenes, E.. (1972). Non-Homogeneous Boundary Value Problems and Applications, Vol. 1, 2. Springer-Verlag, Berlin.
Lipton, R. (1994). Optimal bounds on the effective elastic tensor for orthotropic composites. Proc. R. Soc. London, A, 444:399–410.
Love, A.E.H. (1929). A Treatise on the Mathematical Theory of Elasticity.Oxford University Press, Oxford.
Lu, J.-K. (1995). Complex Variable Methods in Plane Elasticity.World Scientific, Singapore.
Lévy, T. (1986). Application of homogenization to the study of a suspension of force-free particles. In: Trends in Applications of Pure Mathematics to Mechanics. Lecture Notes in Physics 249, Springer-Verlag, Berlin, pp. 349–353.
Makaruk, S.F., Mityushev, V.V. and Rogosin, S.V.. (2006). An optimal design problem for two-dimensional composite materials. A constructive approach. In: Analytic Methods of Analysis and Differential Equations. AMADE 2003 (Kilbas, A.A. and Rogosin, S.V., eds.). Cambridge Scientific, Cottenham, Cambridge, pp. 153–167.
Markov, K.Z. (2000). Elementary micromechanics of heterogeneous media. In: Heterogeneous Media: Micromechanics Modeling Methods and Simulation (Markov, K. and Preziosi, L., eds.), Birkhauser, Basel, pp. 1–162.
Maury, B. (1999). Direct simulations of 2D fluid-particle flows in biperiodic domains. J. Comput. Phys., 156(2):325–351.
Maxwell, J.C. (1873). Treatise on Electricity and Magnetism.Clarendon Press, Oxford.
McAllister, L.E. and Lachman, W.L.. (1983). Multidirectional carbon-carbon composites. In: Handbook of Composites, Vol. 4. Fabrication of Composites (Kelly, A. and Mileiko, S.T., eds.), North Holland, Amsterdam, pp. 109–176.
McKenzie, D.R., McPhedran, R.C. and Derrik, G.H.. (1978). The conductivity of a lattice of spheres II. The body centered and face centered lattices. Proc. R. Soc. London, A, 362:211–232.
McPhedran, R. (1986). Transport property of cylinder pairs and of the square array of cylinders. Proc. R. Soc. London, A, 408:31–43.
McPhedran, R., Poladian, L. and Milton, G.W.. (1988). Asymptotic studies of closely spaced, highly conducting cylinders. Proc. R. Soc. London, A, 415:195–196.
McPhedran, R.C. and McKenzie, D.R.. (1978). The conductivity of a lattice of spheres I. The simple cubic lattice. Proc. R. Soc. London, A, 359:45–63.
McPhedran, R.C. and Milton, G.W.. (1987). Transport properties of touching cylinder pairs and of a square array of touching cylinders. Proc. R. Soc. London, A411:313–326.
Meester, R. and Roy, R.. (1992). Continuum Percolation.Cambridge University Press, Cambridge.
Melrose, D.B. and McPhedran, R.C.. (1991). Electromagnetic Processes in Dispersive Media.Cambridge University Press, Cambridge.
Meredith, R.E. and Tobias, C.W.. (1960). Resistance to potential flow through a cubical array of spheres. J. Appl. Physics, 31:1270–1273.
Mertensson, E. and Gafvert, U.. (2003). Three-dimensional impedance networks for modeling frequency dependent electrical properties of composite materials. J. Phys. D: Appl. Phys., 36:1864–1872.
Mertensson, E. and Gafvert, U.. (2004). A three-dimensional network model describing a non-linear composite material. J. Phys. D: Appl. Phys., 37:112–119.
Michel, J.C., Moulinec, H. and Suquet, P.. (2000). A computational method based on augmented Lagrangians and fast Fourier transforms for composites with high contrast. Comp. Model. Engng Sci., 1(2):79–88.
Michel, J.C., Moulinec, H. and Suquet, P.. (2002). A computational scheme for linear and non-linear composites with arbitrary phase contrast. Int. J. Numer. Meth. Engng., 52:139–160.
Milton, G.M. (1992). Composite materials with Poisson's ratios close to —1. J. Mech. Phys. Solids, 40:1105–1137.
Milton, G.W. (2002). The Theory of Composites.Cambridge University Press, Cambridge.
Mityushev, V. (1993). Plane problem for the steady heat conduction of a material with circular inclusions. Arch. Mech., 45(2):211–215.
Mityushev, V. (1994). Solution of the Hilbert boundary value problem for a multiply connected domain. Slupskie Prace Mat.-Przyr., 9a:33–67.
Mityushev, V. (1997a). A functional equation in a class of analytic functions and composite materials. Demostratio Math., 30:63–70.
Mityushev, V. (1997b). Functional equations and their applications in the mechanics of composites. Demonstratio Math., 30(1):64–70.
Mityushev, V. (1998). Hilbert boundary value problem for multiply connected domains. Complex Variables, 35:283–295.
Mityushev, V. (1999). Transport properties of two-dimensional composite materials with circular inclusions. Proc. R. Soc. London, A455:2513–2528.
Mityushev, V. (2001). Transport properties of doubly periodic arrays of circular cylinders and optimal design problems. Appl. Math. Optim., 44:17–31.
Mityushev, V. (2005). R-linear problem on the torus and its application to composites. Complex Variables, 50(7–10):621–630.
Mityushev, V. (2009). Conductivity of a two-dimensional composite containing elliptical inclusions. Proc. R. Soc. A, 465:2991–3010.
Mityushev, V. and Adler, P.M.. (2002a). Longitudinal permeability of a doubly periodic rectangular array of cylinders. I. Z. Angew. Math. Mech., 82:335–345.
Mityushev, V. and Adler, P.M.. (2002b). Longitudinal permeability of a doubly periodic rectangular array of cylinders. II. An arbitrary distribution of cylinders inside the unit cell. Z. Angew. Math. Phys., 53:486–517.
Mityushev, V., Pesetskaya, E. and Rogosin, S.. (2008). Analytical methods for heat conduction in composites and porous media in cellular and porous materials. In: Cellular and Porous Materials: Thermal Properties Simulation and Prediction (Ochsner, A., Murch, G. and de Lemos, M., eds.), Wiley-VCH, Weinheim.
Mityushev, V. and Rogozin, S.V.. (2000). Constructive Methods for Linear and Nonlinear Boundary Value Problems of Analytic Function Theory.Chapman & Hall/CRC, Boca Raton, FL.
Mityushev, V.V. (1997). Transport properties of doubly-periodic arrays of circular cylinders. Z Angew. Math. Mech., 77:115–120.
Mizohata, S. (1973). The Theory of Partial Differential Equations.Cambridge University Press, Cambridge.
Molchanov, S. (1994). Lectures on random media. In: Lectures on Probability Theory (Bakry, D., Gill, R.D. and Molchanov, S.A., eds.), Springer-Verlag, Berlin, pp. 242–411.
Molyneux, J. (1970). Effective permittivity of a polycrystalline dielectric. J. Math. Phys., 11(4):1172–1184.
Movchan, A.B., Movchan, N.V. and Poulton, C.G.. (2002). Asymptotic Models of Fields in Dilute and Densely Packed Composites.Imperial College Press, London.
Nemat-Nasser, S. and Hori, M.. (1993). Micromechanics. Elsevier Science, Amsterdam.
Nettelblad, B., Mårtensson, E., Önneby, C., Gäfvert, U. and Gustafsson, A.. (2003). Two percolation thresholds due to geometrical effects: Experimental and simulated results. J. Phys. D:Appl. Phys., 36(4):399–405.
Newman, M.E.J. (2003). The structure and functions of complex networks. SIAM Rev., 45(2): 167-256.
Nicorovici, N.A. and McPhedran, R.C.. (1996). Transport properties of arrays of elliptical cylinders. Phys. Rev. E, 54:1945–1957.
Noor, A.K. (1988). Continuum modeling for repetitive structures. Appl. Mech. Rev., 41(7): 285–296.
Nott, P.R. and Brady, J.F.. (1994). Pressure-driven flow of suspensions: Simulation and theory. J. Fluid Mech., 275:157–199.
Novikov, A. (2009). A discrete network approximation for effective conductivity of non-ohmic high-contrast composites. Commun. Math. Sci., 7(3):719–740.
Novikov, V.V. and Friedrich, Chr.. (2005). Viscoelastic properties of composite materials with random structure. Phys. Rev. E, 72:021506-1-021506-9.
Novozilov, V.V. (1970). On the relationship between average values of the stress tensor and strain tensor in statistically isotropic elastic bodies. Appl. Math. Mech., 34(1):67–74.
Nunan, K.C. and Keller, J.B.. (1984a). Effective elasticity tensor for a periodic composite. J. Mech. Phys. Solids, 32:259–280.
Nunan, K.C. and Keller, J.B.. (1984b). Effective velocity of aperiodic suspension. J. Fluid Mech., 142:269–287.
Oleinik, O.A., Shamaev, A.S. and Yosifian, G.A.. (1962). Mathematical Problems in Elasticity and Homogenization.North Holland, Amsterdam.
Ostoja-Starzewski, M. (2006). Material spatial randomness – from statistical to representative volume element. Probab. Eng. Mech., 21(2):112–132.
Panasenko, G.P. (2005). Multi-Scale Modeling for Structures and Composites.Springer-Verlag, Berlin.
Panasenko, G.P. and Virnovsky, G.. (2003). Homogenization of two-phase flow: high contrast of phase permeability. C.R. Mecanique, 331:9–15.
Papanicolaou, G.C. (1995). Diffusion in random media. In: Surveys in Applied Mathematics (Keller, J.B., McLaughlin, D. and Papanicolaou, G., eds.), Plenum Press, New York, pp. 205–255.
Papanicolaou, G.C. and Varadhan, S.R.S.. (1981). Boundary value problems with rapidly oscillating random coefficients. Seria Coll. Janos Bolyai, 27:835–873.
Perrins, W.T., McPhedran, R.C. and McKenzie, D.R.. (1979). Transport properties of regular arrays of cylinders. Proc. R. Soc. London, A, 369:207–225.
Pesetskaya, E.V (2005). Effective conductivity of composite materials with random positions of cylindrical inclusions: Finite number inclusions in the cell. Applic. Anal., 84(8):843–865.
Peterseim, D. (2010). Triangulating a system of disks. In: Proc. 26th Eur. Workshop Comp. Geometry, pp. 241–244.
Peterseim, D. (2012). Robustness of finite elements simulation in densely packed random particle composites. Networks Meter. Media, 7(1):113–126.
Pham Huy, H. and Sanchez-Palencia, E.. (1974). Phénomènes de transmission à travers des couches minces de conductivité élevée. J. Math. Anal. Appl., 47:284—309.
Phillips, R.J., Armstrong, R.C., Brown, R.A., Graham, A.L. and Abbot, J.R.. (1992). A constitutive equation for concentrated suspensions that accounts for shear-induced particle migration. Phys. Fluids, A, 4:30–40.
Poincaré, H. (1886). Sur les integrals irregulieres des equations lineaires. Ada Math., 8:295–344.
Poslinski, A.J., Ryan, M.E., Gupta, R.K., Seshadri, S.G. and Frechette, F.J.. (1988). Rheological behavior of filled polymeric systems II. The effect of bimodal size distribution of particulates. J. Rheol., 32:751–771.
Prager, S. (1963). Diffusion and viscous flow in concentrated suspension. Physica, 29:129–139.
Pshenichnov, G.I. (1993). A Theory of Latticed Plates and Shells.World Scientific, Singapore.
Rayleigh, Lord (Strutt, J.W.). (1892). On the influence of obstacles arranged in rectangular order upon the properties of the medium. Phil. Mag., 34(241):481—491.
Reuss, A. (1929). Berechnung der Flieβgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle. Z. Angew. Math. Mech., 9:49–58.
Robinson, D.A. and Friedman, S.F. (2001). Effect of particle size distribution on the effective dielectric permittivity of saturated granular media. Water Resour. Res., 37(1):33–40.
Rockafellar, R.T. (1969). Convex Functions and Duality in Optimization Problems and Dynamics.Springer-Verlag, Berlin.
Rockafellar, R.T. (1970). Convex Analysis.Princeton University Press, Princeton, NJ.
Roux, S. and Guyon, E.. (1985). Mechanical percolation: A small beam lattice study. J. Physique Lett., 46:999–1004.
Rudin, W. (1964). Principles of Mathematical Analysis.McGraw-Hill, New York.
Rudin, W. (1992). Functional Analysis.McGraw-Hill, New York.
Runge, I. (1925). Zur elektrischen Leitfahigkeit metallischer Aggregate. Z. Tech. Physic, 61(6):61–68.
Rylko, N. (2000). Transport properties of a rectangular array of highly conducting cylinders. J. Engng. Math., 38:1–12.
Rylko, N. (2008a). Effect of polydispersity in conductivity of unidirectional cylinders. Arch. Mater. Sci. Engng, 29:45–52.
Rylko, N. (2008b). Structure of the scalar field around unidirectional circular cylinders. Proc. R. Soc. London, A, 464:391–407.
Sab, K. (1992). On the homogenization and the simulation of random materials. Eur. J. Mech., A/Solids, 11(5):585–607.
Sahimi, M. (2003). Heterogeneous Materials, Vol. 1, 2. Springer-Verlag, New York.
Sanchez-Palencia, E. (1974). Problemes de perturbations liés aux phénomènes de conduction à travers des couches minces de grande résistivité. J. Math. Pure Appl., 53:251–270.
Sanchez-Palencia, E. (1980). Non-Homogeneous Media and Vibration Theory.Springer-Verlag, Berlin.
Sangani, A.S. and Acrivos, A.. (1983). The effective conductivity of a periodic array of spheres. Proc. R. Soc. London, A, 386:263–275.
Schwartz, L. (1966). Theorie des Distributions.Hermann, Paris.
Schwartz, L.M., Johnson, D.L. and Feng, S.. (1984). Vibration modes in granular materials. Phys. Rev. Lett., 52(831):831–834.
Shermergor, T.D. (1977). Elasticity Theory of Micro-Inhomogeneous Media (in Russian).Nauka, Moscow.
Shook, C.A and Rocko, M.C.. (1991). Slurry Flow, Principles and Practice.Butterworth-Heinemann, Boston, MA.
Sierou, A. and Brady, J.F.. (2002). Rheology and micro structure in concentrated noncolloidal suspensions. J. Rheol., 46(5):1031–1056.
Simonenko, I.B. (1974). Electrostatics problems for an inhomogeneous medium: A case of thin dielectric with high dielectric constant: I. Differential Equations, 10:301–309.
Simonenko, I.B. (1975a). Electrostatics problems for an inhomogeneous medium: A case of thin dielectric with high dielectric constant: II. Differential Equations, 11:1870–1878.
Simonenko, I.B. (1975b). Limit problem of conductivity in an inhomogeneous medium. Siberian Math. J., 16:1291–1300.
Smythe, W.R. (1950). Static and Dynamical Electricity, 2nd ed. McGraw-Hill, New York.
Sobolev, S.L. (1937). On the boundary value problem for polyharmonic functions (in Russian). Matem. Zbornik, 2(3):465–499.
Sobolev, S.L. (1950). Some Applications of Functional Analysis to Mathematical Physics (in Russian). Leningrad State University, Leningrad.
Stauffer, D. and Aharony, A.. (1992). Introduction to Percolation Theory.Taylor & Francis, London.
Stockmayer, W.H. (1943). Theory of molecular size distribution and gel formation in branched-chain polymers. J. Chem. Phys., 11:45–55.
Subia, S., Ingber, M.S., Mondy, L.A., Altobelli, S.A. and Graham, A.L.. (1998). Modeling of concentrated suspensions using a continuum constitutive equation. J. Fluid Mech., 373:193–219.
Szczepkowski, J., Malevich, A.E. and Mityushev, V.. (2003). Macroscopic properties of similar arrays of cylinders. Quart. J. Appl. Math. Mech., 56(4):617–628.
Tamm, I.E. (1979). Fundamentals of the Theory of Electricity.Mir Publishers, Moscow.
Temam, R. (1979). Navier-Stokes Equations.North Holland, Amsterdam.
Thovert, J.F. and Acrivos, A.. (1989). The effective thermal conductivity of a random polydispersed suspension of spheres to order c2. Chem. Eng. Comm., 82:177–191.
Thovert, J.F., Kim, I.C., Torquato, S. and Acrivos, A.. (1990). Bounds on the effective properties of polydispersed suspensions of spheres: An evaluation of two relevant morphological parameters. J. Appl. Phys., 67:6088–6098.
Timoshemko, S. and Goodier, J.N.. (1951). Theory of Elasticity.McGraw-Hill, New York.
Torquato, S. (2002). Random Heterogeneous Materials.Springer-Verlag, Berlin.
van Lint, J.H. and Wilson, R.M.. (2001). A Course in Combinatorics, 2nd ed. Cambridge University Press, Cambridge.
Vinogradov, V. and Milton, G.W.. (2005). An accelerated fast Fourier transform algorithm for nonlinear composites. Advances Comp. Experim. Enging. Set Proc. ICCES'05. Available at www.math.utah.edu/vladim/papers/publications.html.
Voigt, W. (1910). Lehrbuch der Kristallphysik.Teubner, Stuttgart.
Voronoi, G. (1908). Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxieme Memoire. Recherches sur les parallelloedres primitifs. J. ReineAngew. Math., 134(198):198–287.
Walpole, L.J. (1966). On bounds for the overall elastic moduli of inhomogeneous systems. J. Mech. Phys. Solids, 14:151–162.
Weil, A. (1976). Elliptic Functions According to Eisenstein andKronecker.Springer-Verlag, Berlin.
Wermer, J. (1974). Potential Theory.Springer-Verlag, Berlin.
West, B.W. (2000). Introduction to Graph Theory.Prentice Hall, NJ.
Willis, J.R. (2002). Lectures on mechanics of random media. In: Mechanics of Random and Multiscale Microstructures, CISM Lecture Notes (Jeulin, D. and Ostoja-Starzewski, M., eds.), Springer-Verlag, Vienne, pp. 221–267.
Yan, Y., Li, J. and Sander, L.M.. (1989). Fracture growth in 2-D elastic networks with the Born model. Europhys. Lett., 10:7–13.
Yang, C.S. and Hui, P.M.. (1991). Effective nonlinear response in random nonlinear resistor networks: Numerical studies. Phys. Rev. B, 44:12559–12561.
Yardley, J.G., Reuben, A.J. and McPhedran, R.C.. (2001). The transport properties of layers of elliptical cylinders. Proc. R. Soc. London, A, 457:395–423.
Yeh, R.H.T. (1970a). Variational principles of elastic moduli of composite materials. J. Appl. Phys., 41(8):3353–3356.
Yeh, R.H.T. (1970b). Variational principles of transport properties of composite materials. J. Appl. Phys., 41(1):224–226.
Yosida, K. (1971). Functional Analysis.Springer-Verlag, Berlin.
Yurinski, V.V. (1980). Average of an elliptic boundary problem with random coefficients. Siberian Math. J., 21:470–482.
Yurinski, V.V. (1986). Averaging of symmetric diffusion in a random medium. Siberian Math. J., 27(4):603–613.
Zeidler, E. (1995). Applied Functional Analysis: Applications to Mathematical Physics.Springer-Verlag, Berlin.
Zuzovsky, M. and Brenner, H.. (1977). Effective conductivities of composite materials composed of a cubic arrangement of spherical particles embedded in an isotropic matrix. Z. Angew. Math. Phys., 28(6):979–992.
