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  3. Introduction to the Network Approximation Method for Materials Modeling
  • Cited by 12
Publisher:
Cambridge University Press
Online publication date:
February 2013
Print publication year:
2012
Online ISBN:
9781139235952
DOI:
https://doi.org/10.1017/CBO9781139235952
Subjects:
Mathematics (general), Mathematics, Engineering, Mathematical Modeling and Methods, Materials Science
Series:
Encyclopedia of Mathematics and its Applications (148)
Information

Book description

In recent years the traditional subject of continuum mechanics has grown rapidly and many new techniques have emerged. This text provides a rigorous, yet accessible introduction to the basic concepts of the network approximation method and provides a unified approach for solving a wide variety of applied problems. As a unifying theme, the authors discuss in detail the transport problem in a system of bodies. They solve the problem of closely placed bodies using the new method of network approximation for PDE with discontinuous coefficients, developed in the 2000s by applied mathematicians in the USA and Russia. Intended for graduate students in applied mathematics and related fields such as physics, chemistry and engineering, the book is also a useful overview of the topic for researchers in these areas.

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Contents

Contents
References
References
References
Abbot, 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 Google Scholar:773–795.
Aboudi, J. (1991). Mechanics of Composite Material: A Unified Micromechanics Approach. Elsevier Science, Amsterdam Google Scholar.
Acrivos, A. and Chang, E.. (1986). A model for estimating transport quantities in two-phase materials. Phys. Fluids, 29 CrossRef | Google Scholar(3):3–4.
Adams, R.A. (1975). Sobolev Spaces. Academic Press, New York Google Scholar.
Ahlfors, L. (1979). Complex Analysis, 3rd ed. McGraw-Hill, New York Google Scholar.
Akhiezer, N.I. (1990). Elements of the Theory of Elliptic Functions.American Mathematical Society, Providence, RI CrossRef | Google Scholar.
Almgren, R.F. (1985). An isotropic three-dimensional structure with Poisson's ratio = −1. J. Elasticity, 15 Google Scholar:427–430.
Ambegaokar, V., Halperin, B.I. and Langer, J.S.. (1971). Hopping conductivity in disordered systems. Phys. Rev., B, 4 CrossRef | Google Scholar(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 CrossRef | Google Scholar(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 CrossRef | Google Scholar(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 CrossRef | Google Scholar: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 Google Scholar.
Aurenhammer, F. and Klein, R.. (1999). Voronoi diagrams. In: Handbook of Computational Geometry (Sack, J.R. and Urrutia, J., eds.), North Holland, Amsterdam Google Scholar, pp. 201–291.
Avellaneda, M. (1987). Optimal bounds and microgeometries for an elastic two-phase composite. SIAM J. Appl. Math., 47 CrossRef | Google Scholar(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 CrossRef | Google Scholar:27–77.
Bakhvalov, N.S. and Panasenko, G.P.. (1989). Homogenization: Averaging Processes in Periodic Media.Kluwer Academic Publishers, Dordrecht CrossRef | Google Scholar.
Balberg, I. (1987). Recent developments in continuum percolation. Phil. Mag., B 30 CrossRef | Google Scholar: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 CrossRef | Google Scholar:313–333.
Batchelor, G.K. and Wen, C.S.. (1972). Sedimentation in a dilute dispersion of spheres. J. Fluid Mech., 52 CrossRef | Google Scholar: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 CrossRef | Google Scholar:95–112.
Bendsøe, M.P. and Sigmund, O.. (2004). Topology Optimization.Springer-Verlag, Berlin CrossRef | Google Scholar.
Bensoussan, A., Lions, J.-L. and Papanicolaou, G.. (1975). Sur quelques phénomènes asymptotiques d'évolution. Compt. Rend. Acad. Set Paris, Ser. A-B, 281 Google Scholar(10):A317–A322.
Bensoussan, A., Lions, J.-L. and Papanicolaou, G.. (1978). Asymptotic Analysis for Periodic Structures.North Holland, Amsterdam Google Scholar.
Beran, M.J. (1968). Statistical Continuum Theories.John Wiley, New York Google Scholar.
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 CrossRef | Google Scholar:107–118.
Berdichevsky, V.L. (2009). Variational Principles of Continuum Mechanics.Springer-Verlag, Berlin Google Scholar.
Bergman, D.J. (1983). The dielectric constant of a composite material – a problem in classical physics. Phys. Reports, C43 Google Scholar: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 CrossRef | Google Scholar(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 CrossRef | Google Scholar: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 CrossRef | Google Scholar(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 CrossRef | Google Scholar, 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 CrossRef | Google Scholar(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 CrossRef | Google Scholar(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 CrossRef | Google Scholar(2):95–112.
Berlyand, L. and Mityushev, V.. (2001). Generalized Clausius–Mossotti formula for random composite with circular fibers. J. Stat. Phys., 102 CrossRef | Google Scholar(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 CrossRef | Google Scholar(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 CrossRef | Google Scholar(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 CrossRef | Google Scholar(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 CrossRef | Google Scholar: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 CrossRef | Google Scholar:567–583.
Bollobás, B. (1998). Modern Graph Theory.Springer-Verlag, New York CrossRef | Google Scholar.
Bonnecaze, R.T. and Brady, J.F.. (1991). The effective conductivity of random suspensions of spherical particles. Proc. R. Soc. London, Ser. A, 432 CrossRef | Google Scholar:445–465.
Borcea, L. (1998). Asymptotic analysis of quasi-static transport in high contrast conductive media. SIAM J. Appl. Math., 2 CrossRef | Google Scholar(59):597–635.
Borcea, L., Berryman, J.G. and Papanicolaou, G.. (1999). Matching pursuit for imaging high-contrast conductivity. Inverse Problems, 15 CrossRef | Google Scholar:811–849.
Borcea, L. and Papanicolaou, G.. (1998). Network approximation for transport properties of high contrast conductivity. Inverse Problems, 4 Google Scholar(15):501–539.
Born, M. and Huang, K.. (1954). Dynamical Theory of Crystal Lattices.Oxford University Press, Oxford Google Scholar.
Bourgeat, A., Mikelic, A. and Wright, S.. (1994). Stochastic two-scale convergence in the mean and applications. J. Reine Angew. Math., 456 Google Scholar:19–51.
Bourgeat, A. and Piatnitski, A.. (2004). Approximations of effective coefficients in stochastic homogenization. Ann. Inst. H. Poincaré, 40 CrossRef | Google Scholar:153–165.
Brady, J.F. (1993). The rheo logical behavior of concentrated colloidal suspensions. J. Chem. Phys., 99 CrossRef | Google Scholar: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 CrossRef | Google Scholar:105–129.
Brodbent, S.R. and Hammerslay, J.M.. (1957). Percolation processes I. Crystals and mazes. Math. Proc. Cambridge Phil. Soc., 53 CrossRef | Google Scholar:629–641.
Broutman, L.J. and Krock, R.H., eds. (1974). Composite Materials. Vol. 1-8. Academic Press, New York Google Scholar.
Brown, W.F. (1956). Dielectrics.Springer-Verlag, Berlin CrossRef | Google Scholar.
Bruno, O. (1991). The effective conductivity of strongly heterogeneous composites. Proc. R. Soc. London, A, 433 CrossRef | Google Scholar:353–381.
B¨rger, R. and Wendland, W.L.. (2001). Sedimentation and suspension flows: historical perspective and some recent developments. J. Engng. Math., 41 CrossRef | Google Scholar(2/3):101–1 16.
Burkill, J.C. (2004). The Lebesgue Integral.Cambridge University Press, Cambridge Google Scholar.
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 Google Scholar: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 Google Scholar.
Chang, Ch. and Powell, R.L.. (1994). Effect of particle size distribution on the rheology of a concentrated bimodal suspension. J. Rheol., 38 CrossRef | Google Scholar: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 Google Scholar: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 CrossRef | Google Scholar: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 CrossRef | Google Scholar:122–141.
Cherkaev, A.V. (2000). Variational Methods for Structural Optimization.Springer-Verlag, Berlin CrossRef | Google Scholar.
Chinh, Ph.D. (1997). Overall properties of planar quasisymmetric randomly inhomogeneous media: Estimates and cell models. Phys. Rev. E, 56 CrossRef | Google Scholar:652–660.
Chou, T.-W. and Ko, F.K., eds. (1989). Textile Structural Composites.Elsevier Science, Amsterdam Google Scholar.
Christensen, R.M. (1979). Mechanics of Composite Materials.John Wiley, New York Google Scholar.
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 CrossRef | Google Scholar: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 CrossRef | Google Scholar(3):191–309.
Courant, R.S. and Hilbert, D.. (1953). Methods of Mathematical Physics.John Wiley, New York Google Scholar.
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 Google Scholar, 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 CrossRef | Google Scholar:1017–1037.
Curtin, W.A. and Scher, H.. (1990a). Brittle fracture in disordered materials: A spring network model. J. Mater. Res., 5 CrossRef | Google Scholar:535–553.
Curtin, W.A. and Scher, H.. (1990b). Mechanical modeling using a spring network. J. Mater. Res., 5 CrossRef | Google Scholar:554–562.
Del Maso, G. (1993). An Introduction to Γ-Convergence.Birkhäuser, Boston CrossRef | Google Scholar.
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 CrossRef | Google Scholar:1487–1502.
Dieudonne, J.A. (1969). Treatise on Analysis.Academic Press, New York Google Scholar.
Ding, J., Warriner, H.E. and Zasadzinski, J.A.. (2002). Viscosity of two-dimensional suspensions. Phys. Rev. Lett., 88 CrossRef | Google Scholar | PubMed(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 Google Scholar:1259–1260.
Doyle, W.T. (1978). The Clasius–Mossotti problem for cubic arrays of spheres. J. Appl. Phys., 49 CrossRef | Google Scholar: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 CrossRef | Google Scholar:515–540.
Drygaś, P. and Mityushev, V.. (2009). Effective conductivity of unidirectional cylinders with interfacial resistance. Quarterly J. Mech. Appl. Math., 62 CrossRef | Google Scholar(3):235–262.
Dykhne, A.M. (1971). Conductivity of a two-dimensional two-phase system. Sov. Phys., 32 Google Scholar(63):63–65.
Einstein, A. (1906). Eine neue Bestimmung der Molekuldimensionen. Ann. Phys., 19 CrossRef | Google Scholar:289–306.
Ekeland, I. and Temam, R.. (1976). Convex Analysis and Variational Problems.North Holland, Amsterdam Google Scholar.
Evans, L.C. and Gangbo, W.. (1999). Differential equation methods for the Monge–Kantorovich mass transfer problem. Mem. Amer. Math. Soc., 137 Google Scholar(653):viii+66.
Evans, L.C. and Gariepy, R.F.. (1992). Measure Theory and Fine Properties of Functions.CRC Press, Boca Raton, FL Google Scholar.
Feng, N.A. (1985). Percolation properties of granular elastic networks in two dimensions. Phys. Rev. B, 32 CrossRef | Google Scholar | PubMed(1):510–513.
Feng, N.A. and Acrivos, A.. (1985). On the viscosity of concentrated suspensions of solid spheres. Chem. Engng Sci., 22 Google Scholar:847–853.
Flaherty, J.E. and Keller, J.B.. (1973). Elastic behavior of composite media. Comm. Pure Appl. Math., 26 CrossRef | Google Scholar:565–580.
Flory, P.J. (1941). Molecular size distribution in three dimensional polymers. I. Gelation. J. Amer. Chem. Soc., 63 CrossRef | Google Scholar:3083–3090.
Fox, L. (1964). An Introduction to Numerical Linear Algebra.Clarendon Press, Oxford Google Scholar.
Francfort, G.A. and Murat, F.. (1986). Homogenization and optimal bounds in linear electricity. Arch. Rational Mech. Anal., 94 CrossRef | Google Scholar(4):307–334.
Frenkel, N.A. and Acrivos, A.. (1967). On the viscosity of concentrated suspension of solid spheres. Chem. Engng Sci., 22 CrossRef | Google Scholar: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 CrossRef | Google Scholar:4406–4414.
Gakhov, F.D. (1966). Boundary Value Problems.Pergamon Press, Oxford Google Scholar.
Garboczi, E.J. and Douglas, J.F.. (1996). Intrinsic conductivity of objects having arbitrary shape and conductivity. Phys. Rev. E, 53 CrossRef | Google Scholar | PubMed(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 CrossRef | Google Scholar(3):295–309.
Good, I.J. (1949). The number of individuals in a cascade process. Math. Proc. Cambridge Phil. Soc., 45 CrossRef | Google Scholar: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 CrossRef | Google Scholar:197–205.
Graham, A.L. (1981). On the viscosity of a suspension of solid particles. Appl. Sci. Res., 37 CrossRef | Google Scholar:275–286.
Greengard, L. and Lee, J.-Y.. (2006). Electrostatics and heat conduction in high contrast composite materials. J. Comput. Phys., 211 CrossRef | Google Scholar(1):64–76.
Greengard, L. and Moura, M.. (1994). On the numerical evaluation of electrostatic fields in composite materials. Acta Numerica, 3 CrossRef | Google Scholar:379–410.
Grigolyuk, E.I. and Filshtinskij, L.A.. (1972). Periodical Piecewise Homogeneous Elastic Structures (in Russian).Nauka, Moscow Google Scholar.
Grimet, G. (1992). Percolation.Springer-Verlag, Berlin Google Scholar.
Gupta, P.K. and Cooper, A.R.. (1990). Topologically disordered networks of rigid polytopes. J. Non-Crystal. Solids, 123 CrossRef | Google Scholar(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 CrossRef | Google Scholar:2391–2394.
Happel, J. (1959). Viscous flow relative to arrays of cylinders. AIChE J., 5 CrossRef | Google Scholar: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 CrossRef | Google Scholar:317–328.
Haug, E.J., Choi, K.K. and Komkov, V.. (1986). Design Sensitivity Analysis of Structural Systems.Academic Press, Orlando, FL Google Scholar.
Herrmann, H.J., Hansen, A. and Roux, S.. (1989). Fracture of disordered, elastic lattices in two dimensions. Phys. Rev. B, 39 CrossRef | Google Scholar | PubMed:637–648.
Hill, R. (1963). Elastic properties of reinforced solids: Some theoretical principles. J. Mech. Phys. Solids, 11 CrossRef | Google Scholar:357–372.
Hill, R. (1996). Characterization of thermally conductive epoxy composite fillers. Proc. Technical Program “Emerging Packing Technology”, Google ScholarSurface 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 CrossRef | Google Scholar:851–857.
Hinsen, K. and Felderhof, B.U.. (1992). Dielectric constant of a suspension of uniform spheres. Phys. Rev. B, 46 CrossRef | Google Scholar | PubMed(20):12955–12963.
Hrennikoff, A. (1941). Solution of a problem of elasticity by the framework method. J. Appl. Mech., 8 Google Scholar:169–175.
Jabin, P.-E. and Otto, F.. (2004). Identification of the dilute regime in particle sedimentation. Commun. Math. Phys., 250 CrossRef | Google Scholar:415–432.
Jeffrey, D.J. and Acrivos, A.. (1976). The rheological properties of suspensions of rigid particles. AIChE J., 22 CrossRef | Google Scholar:417–432.
Jikov, V.V., Kozlov, S.M. and Oleinik, O.A.. (1994). Homogenization of Differential Operators and Integral Functionals.Springer-Verlag, Berlin CrossRef | Google Scholar.
Kalamkarov, A.L. and Kolpakov, A.G.. (1996). On the analysis and design of fiber reinforced composite shells. Trans. ASME. J. Appl. Mech., 63 CrossRef | Google Scholar(4):939–945.
Kalamkarov, A.L. and Kolpakov, A.G.. (1997). Analysis, Design and Optimization of Composite Structures.John Wiley, Chichester Google Scholar.
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 Google Scholar:661–670.
Kato, T. (1976). Perturbation Theory for Linear Operators.Springer-Verlag, New York Google Scholar.
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 CrossRef | Google Scholar(34):991–993.
Keller, J.B. (1964). A theorem on the conductivity of a composite medium. J. Math. Phys., 5 CrossRef | Google Scholar:548–549.
Keller, J.B. (1987). Effective conductivity of a periodic composite composed of two very unequal conductors. J. Math. Phys., 10 CrossRef | Google Scholar(28):2516–2520.
Keller, J.B. and Sachs, D.. (1964). Calculations of conductivity of a medium containing cylindrical inclusions. J. Appl. Phys., 35 CrossRef | Google Scholar:537–538.
Kellomaki, M., Astrom, J. and Timonen, J.. (1996). Rigidity and dynamics of random spring networks. Phys. Rev. Lett., 77 CrossRef | Google Scholar | PubMed:2730–2733.
Kelly, A. and Rabotnov, Yu.N., eds. (1988). Handbook of Composites.North Holland, Amsterdam Google Scholar.
Kesten, H. (1992). Percolation Theory for Mathematicians.Birkhäuser, Boston Google Scholar.
Kolmogorov, A.N. and Fomin, S.V.. (1970). Introductory Real Analysis.Prentice Hall, Englewood Cliffs, NJ Google Scholar.
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 CrossRef | Google Scholar(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 Google Scholar(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 Google Scholar.
Kolpakov, A.G. (1987). Averaged characteristics of thermoelastic frames. Izvestiay of the Academy of Science of the USSR. Mechanics of Solids, 22 Google Scholar(6):53–61.
Kolpakov, A.G. (1985). Determination of the average characteristics of elastic frameworks. J. Appl. Math. Mech., 49 CrossRef | Google Scholar: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 Google Scholar:74–84.
Kolpakov, A.G. (1992). Glued bodies. Differential Equations, 28 Google Scholar(8):1131–1139.
Kolpakov, A.G. (2004). Stressed Composite Structures: Homogenized Models for Thin-Walled Nonhomogeneous Structures with Initial Stresses.Springer-Verlag, Berlin CrossRef | Google Scholar.
Kolpakov, A.G. (2005). Asymptotic behavior of the conducting properties of high-contrast media. J. Appl. Mech. Tech. Phys., 46 CrossRef | Google Scholar(3):412–422.
Kolpakov, A.G. (2006a). The asymptotic screening and network models. J. Engng Phys. Thermophys., 2 Google Scholar: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 CrossRef | Google Scholar(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 CrossRef | Google Scholar(11):1216–1231.
Koplik, J. (1982). Creeping flow in two-dimensional networks. J. Fluid Mech., 119 CrossRef | Google Scholar:219–247.
Kozlov, S.M. (1978). Averaging of random structures (in Russian). Doklady Acad. Nauk SSSR, 241 Google Scholar(5):1016–11019.
Kozlov, S.M. (1980). Averaging of random operators. Math. USSR Sbornik, 37 CrossRef | Google Scholar:167–180.
Kozlov, S.M. (1989). Geometric aspects of averaging. Russian Math. Surv., 2 CrossRef | Google Scholar(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 CrossRef | Google Scholar(1):102–107.
Kuchling, H. (1980). Physics.VEB Fachbuchverlag, Leipzig Google Scholar.
Kun, F. and Herrmann, H.. (1996). A study of fragmentation processes using a discrete element method. Comput. Meth. Appl. Mech. Engng, 138 CrossRef | Google Scholar:3–18.
Ladd, A.J.C. (1997). Sedimentation of homogeneous suspensions of non-Brownian spheres. Phys. Fluids, 9 CrossRef | Google Scholar(3):491–499.
Ladyzhenskaya, O.A. and Ural'tseva, N.N.. (1968). Linear and Quasilinear Elliptic Equations.Academic Press, New York Google Scholar.
Lakes, R. (1991). Deformation mechanisms of negative Poisson's ratio materials: Structural aspects. J. Mater. Sci., 26 CrossRef | Google Scholar:2287–2292.
Lamb, H. (1991). Hydrodynamics.Dover, New York Google Scholar.
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 Google Scholar, pp. 2–43.
Leal, G. (1992). Laminar Flow and Convective Transport Processes: Scaling Principles and Asymptotic Analysis.Butterworth-Heinemann, Amsterdam Google Scholar.
Leighton, D. and Acrivos, A.. (1987). Measurement of shear-induced self-diffusion in concentrated suspensions of spheres. J. FluidMech., 177 CrossRef | Google Scholar:109–131.
Lenczner, M. (1997). Homogénéisation d'un circuit électrique. C.R. Acad. Sci. Paris, Série II B, 324 Google Scholar(9):537–542.
Lieberman, G.M. (1988). Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal., 12 CrossRef | Google Scholar(11):1203–1219.
Limat, L. (1988). Percolation and Cosserat elasticity: Exact results on a deterministic fractal. Phys. Rev., B, 37 CrossRef | Google Scholar | PubMed: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 Google Scholar, pp. 5–19.
Lions, J.-L. and Magenes, E.. (1972). Non-Homogeneous Boundary Value Problems and Applications, Vol. 1, 2. Springer-Verlag, Berlin Google Scholar.
Lipton, R. (1994). Optimal bounds on the effective elastic tensor for orthotropic composites. Proc. R. Soc. London, A, 444 CrossRef | Google Scholar:399–410.
Love, A.E.H. (1929). A Treatise on the Mathematical Theory of Elasticity.Oxford University Press, Oxford Google Scholar.
Lu, J.-K. (1995). Complex Variable Methods in Plane Elasticity.World Scientific, Singapore CrossRef | Google Scholar.
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 CrossRef | Google Scholar, 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 Google Scholar, 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 CrossRef | Google Scholar, pp. 1–162.
Maury, B. (1999). Direct simulations of 2D fluid-particle flows in biperiodic domains. J. Comput. Phys., 156 CrossRef | Google Scholar(2):325–351.
Maxwell, J.C. (1873). Treatise on Electricity and Magnetism.Clarendon Press, Oxford Google Scholar.
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 Google Scholar, 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 CrossRef | Google Scholar:211–232.
McPhedran, R. (1986). Transport property of cylinder pairs and of the square array of cylinders. Proc. R. Soc. London, A, 408 CrossRef | Google Scholar: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 CrossRef | Google Scholar: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 CrossRef | Google Scholar: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 CrossRef | Google Scholar:313–326.
Meester, R. and Roy, R.. (1992). Continuum Percolation.Cambridge University Press, Cambridge Google Scholar.
Melrose, D.B. and McPhedran, R.C.. (1991). Electromagnetic Processes in Dispersive Media.Cambridge University Press, Cambridge CrossRef | Google Scholar.
Meredith, R.E. and Tobias, C.W.. (1960). Resistance to potential flow through a cubical array of spheres. J. Appl. Physics, 31 CrossRef | Google Scholar: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 CrossRef | Google Scholar: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 CrossRef | Google Scholar: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 Google Scholar(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 CrossRef | Google Scholar:139–160.
Milton, G.M. (1992). Composite materials with Poisson's ratios close to —1. J. Mech. Phys. Solids, 40 CrossRef | Google Scholar:1105–1137.
Milton, G.W. (2002). The Theory of Composites.Cambridge University Press, Cambridge CrossRef | Google Scholar.
Mityushev, V. (1993). Plane problem for the steady heat conduction of a material with circular inclusions. Arch. Mech., 45 Google Scholar(2):211–215.
Mityushev, V. (1994). Solution of the Hilbert boundary value problem for a multiply connected domain. Slupskie Prace Mat.-Przyr., 9a Google Scholar:33–67.
Mityushev, V. (1997a). A functional equation in a class of analytic functions and composite materials. Demostratio Math., 30 Google Scholar:63–70.
Mityushev, V. (1997b). Functional equations and their applications in the mechanics of composites. Demonstratio Math., 30 Google Scholar(1):64–70.
Mityushev, V. (1998). Hilbert boundary value problem for multiply connected domains. Complex Variables, 35 Google Scholar:283–295.
Mityushev, V. (1999). Transport properties of two-dimensional composite materials with circular inclusions. Proc. R. Soc. London, A455 CrossRef | Google Scholar:2513–2528.
Mityushev, V. (2001). Transport properties of doubly periodic arrays of circular cylinders and optimal design problems. Appl. Math. Optim., 44 CrossRef | Google Scholar:17–31.
Mityushev, V. (2005). R-linear problem on the torus and its application to composites. Complex Variables, 50 Google Scholar(7–10):621–630.
Mityushev, V. (2009). Conductivity of a two-dimensional composite containing elliptical inclusions. Proc. R. Soc. A, 465 CrossRef | Google Scholar: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 CrossRef | Google Scholar: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 CrossRef | Google Scholar: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 Google Scholar.
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 Google Scholar.
Mityushev, V.V. (1997). Transport properties of doubly-periodic arrays of circular cylinders. Z Angew. Math. Mech., 77 CrossRef | Google Scholar:115–120.
Mizohata, S. (1973). The Theory of Partial Differential Equations.Cambridge University Press, Cambridge Google Scholar.
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 Google Scholar, pp. 242–411.
Molyneux, J. (1970). Effective permittivity of a polycrystalline dielectric. J. Math. Phys., 11 CrossRef | Google Scholar(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 CrossRef | Google Scholar.
Nemat-Nasser, S. and Hori, M.. (1993). Micromechanics. Elsevier Science, Amsterdam Google Scholar.
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 CrossRef | Google Scholar(4):399–405.
Newman, M.E.J. (2003). The structure and functions of complex networks. SIAM Rev., 45 CrossRef | Google Scholar(2): 167-256.
Nicorovici, N.A. and McPhedran, R.C.. (1996). Transport properties of arrays of elliptical cylinders. Phys. Rev. E, 54 Google Scholar | PubMed:1945–1957.
Noor, A.K. (1988). Continuum modeling for repetitive structures. Appl. Mech. Rev., 41 CrossRef | Google Scholar(7): 285–296.
Nott, P.R. and Brady, J.F.. (1994). Pressure-driven flow of suspensions: Simulation and theory. J. Fluid Mech., 275 CrossRef | Google Scholar:157–199.
Novikov, A. (2009). A discrete network approximation for effective conductivity of non-ohmic high-contrast composites. Commun. Math. Sci., 7 CrossRef | Google Scholar(3):719–740.
Novikov, V.V. and Friedrich, Chr.. (2005). Viscoelastic properties of composite materials with random structure. Phys. Rev. E, 72 CrossRef | Google Scholar | PubMed: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 Google Scholar(1):67–74.
Nunan, K.C. and Keller, J.B.. (1984a). Effective elasticity tensor for a periodic composite. J. Mech. Phys. Solids, 32 CrossRef | Google Scholar:259–280.
Nunan, K.C. and Keller, J.B.. (1984b). Effective velocity of aperiodic suspension. J. Fluid Mech., 142 CrossRef | Google Scholar:269–287.
Oleinik, O.A., Shamaev, A.S. and Yosifian, G.A.. (1962). Mathematical Problems in Elasticity and Homogenization.North Holland, Amsterdam Google Scholar.
Ostoja-Starzewski, M. (2006). Material spatial randomness – from statistical to representative volume element. Probab. Eng. Mech., 21 CrossRef | Google Scholar(2):112–132.
Panasenko, G.P. (2005). Multi-Scale Modeling for Structures and Composites.Springer-Verlag, Berlin Google Scholar.
Panasenko, G.P. and Virnovsky, G.. (2003). Homogenization of two-phase flow: high contrast of phase permeability. C.R. Mecanique, 331 CrossRef | Google Scholar: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 CrossRef | Google Scholar, 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 Google Scholar: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 CrossRef | Google Scholar: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 CrossRef | Google Scholar(8):843–865.
Peterseim, D. (2010). Triangulating a system of disks. In: Proc. 26th Eur. Workshop Comp. Geometry, Google Scholar pp. 241–244.
Peterseim, D. (2012). Robustness of finite elements simulation in densely packed random particle composites. Networks Meter. Media, 7 Google Scholar(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 Google Scholar: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 CrossRef | Google Scholar:30–40.
Poincaré, H. (1886). Sur les integrals irregulieres des equations lineaires. Ada Math., 8 Google Scholar: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 CrossRef | Google Scholar:751–771.
Prager, S. (1963). Diffusion and viscous flow in concentrated suspension. Physica, 29 CrossRef | Google Scholar:129–139.
Pshenichnov, G.I. (1993). A Theory of Latticed Plates and Shells.World Scientific, Singapore CrossRef | Google Scholar.
Rayleigh, Lord (Strutt, J.W.). (1892). On the influence of obstacles arranged in rectangular order upon the properties of the medium. Phil. Mag., 34 CrossRef | Google Scholar(241):481—491.
Reuss, A. (1929). Berechnung der Flieβgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle. Z. Angew. Math. Mech., 9 CrossRef | Google Scholar: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 CrossRef | Google Scholar(1):33–40.
Rockafellar, R.T. (1969). Convex Functions and Duality in Optimization Problems and Dynamics.Springer-Verlag, Berlin CrossRef | Google Scholar.
Rockafellar, R.T. (1970). Convex Analysis.Princeton University Press, Princeton, NJ CrossRef | Google Scholar.
Roux, S. and Guyon, E.. (1985). Mechanical percolation: A small beam lattice study. J. Physique Lett., 46 CrossRef | Google Scholar:999–1004.
Rudin, W. (1964). Principles of Mathematical Analysis.McGraw-Hill, New York Google Scholar.
Rudin, W. (1992). Functional Analysis.McGraw-Hill, New York Google Scholar.
Runge, I. (1925). Zur elektrischen Leitfahigkeit metallischer Aggregate. Z. Tech. Physic, 61 Google Scholar(6):61–68.
Rylko, N. (2000). Transport properties of a rectangular array of highly conducting cylinders. J. Engng. Math., 38 CrossRef | Google Scholar:1–12.
Rylko, N. (2008a). Effect of polydispersity in conductivity of unidirectional cylinders. Arch. Mater. Sci. Engng, 29 Google Scholar:45–52.
Rylko, N. (2008b). Structure of the scalar field around unidirectional circular cylinders. Proc. R. Soc. London, A, 464 CrossRef | Google Scholar:391–407.
Sab, K. (1992). On the homogenization and the simulation of random materials. Eur. J. Mech., A/Solids, 11 Google Scholar(5):585–607.
Sahimi, M. (2003). Heterogeneous Materials, Vol. 1, 2. Springer-Verlag, New York Google Scholar.
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 Google Scholar:251–270.
Sanchez-Palencia, E. (1980). Non-Homogeneous Media and Vibration Theory.Springer-Verlag, Berlin Google Scholar.
Sangani, A.S. and Acrivos, A.. (1983). The effective conductivity of a periodic array of spheres. Proc. R. Soc. London, A, 386 CrossRef | Google Scholar:263–275.
Schwartz, L. (1966). Theorie des Distributions.Hermann, Paris Google Scholar.
Schwartz, L.M., Johnson, D.L. and Feng, S.. (1984). Vibration modes in granular materials. Phys. Rev. Lett., 52 CrossRef | Google Scholar(831):831–834.
Shermergor, T.D. (1977). Elasticity Theory of Micro-Inhomogeneous Media (in Russian).Nauka, Moscow Google Scholar.
Shook, C.A and Rocko, M.C.. (1991). Slurry Flow, Principles and Practice.Butterworth-Heinemann, Boston, MA Google Scholar.
Sierou, A. and Brady, J.F.. (2002). Rheology and micro structure in concentrated noncolloidal suspensions. J. Rheol., 46 CrossRef | Google Scholar(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 Google Scholar: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 Google Scholar:1870–1878.
Simonenko, I.B. (1975b). Limit problem of conductivity in an inhomogeneous medium. Siberian Math. J., 16 Google Scholar:1291–1300.
Smythe, W.R. (1950). Static and Dynamical Electricity, 2nd ed. McGraw-Hill, New York Google Scholar.
Sobolev, S.L. (1937). On the boundary value problem for polyharmonic functions (in Russian). Matem. Zbornik, 2 Google Scholar(3):465–499.
Sobolev, S.L. (1950). Some Applications of Functional Analysis to Mathematical Physics (in Russian). Leningrad State University, Leningrad Google Scholar.
Stauffer, D. and Aharony, A.. (1992). Introduction to Percolation Theory.Taylor & Francis, London Google Scholar.
Stockmayer, W.H. (1943). Theory of molecular size distribution and gel formation in branched-chain polymers. J. Chem. Phys., 11 CrossRef | Google Scholar: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 CrossRef | Google Scholar:193–219.
Szczepkowski, J., Malevich, A.E. and Mityushev, V.. (2003). Macroscopic properties of similar arrays of cylinders. Quart. J. Appl. Math. Mech., 56 CrossRef | Google Scholar(4):617–628.
Tamm, I.E. (1979). Fundamentals of the Theory of Electricity.Mir Publishers, Moscow Google Scholar.
Temam, R. (1979). Navier-Stokes Equations.North Holland, Amsterdam Google Scholar.
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 CrossRef | Google Scholar: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 CrossRef | Google Scholar:6088–6098.
Timoshemko, S. and Goodier, J.N.. (1951). Theory of Elasticity.McGraw-Hill, New York Google Scholar.
Torquato, S. (2002). Random Heterogeneous Materials.Springer-Verlag, Berlin CrossRef | Google Scholar.
van Lint, J.H. and Wilson, R.M.. (2001). A Course in Combinatorics, 2nd ed. Cambridge University Press, Cambridge CrossRef | Google Scholar.
Vinogradov, V. and Milton, G.W.. (2005). An accelerated fast Fourier transform algorithm for nonlinear composites. Advances Comp. Experim. Enging. Set Proc. ICCES'05. Google Scholar Available at www.math.utah.edu/vladim/papers/publications.html.
Voigt, W. (1910). Lehrbuch der Kristallphysik.Teubner, Stuttgart Google Scholar.
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 Google Scholar(198):198–287.
Walpole, L.J. (1966). On bounds for the overall elastic moduli of inhomogeneous systems. J. Mech. Phys. Solids, 14 CrossRef | Google Scholar:151–162.
Weil, A. (1976). Elliptic Functions According to Eisenstein andKronecker.Springer-Verlag, Berlin CrossRef | Google Scholar.
Wermer, J. (1974). Potential Theory.Springer-Verlag, Berlin CrossRef | Google Scholar.
West, B.W. (2000). Introduction to Graph Theory.Prentice Hall, NJ Google Scholar.
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 Google Scholar, 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 CrossRef | Google Scholar:7–13.
Yang, C.S. and Hui, P.M.. (1991). Effective nonlinear response in random nonlinear resistor networks: Numerical studies. Phys. Rev. B, 44 CrossRef | Google Scholar | PubMed: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 CrossRef | Google Scholar:395–423.
Yeh, R.H.T. (1970a). Variational principles of elastic moduli of composite materials. J. Appl. Phys., 41 CrossRef | Google Scholar(8):3353–3356.
Yeh, R.H.T. (1970b). Variational principles of transport properties of composite materials. J. Appl. Phys., 41 CrossRef | Google Scholar(1):224–226.
Yosida, K. (1971). Functional Analysis.Springer-Verlag, Berlin CrossRef | Google Scholar.
Yurinski, V.V. (1980). Average of an elliptic boundary problem with random coefficients. Siberian Math. J., 21 CrossRef | Google Scholar:470–482.
Yurinski, V.V. (1986). Averaging of symmetric diffusion in a random medium. Siberian Math. J., 27 CrossRef | Google Scholar(4):603–613.
Zeidler, E. (1995). Applied Functional Analysis: Applications to Mathematical Physics.Springer-Verlag, Berlin Google Scholar.
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 CrossRef | Google Scholar(6):979–992.

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