Mathematical Modeling and Effective Properties in Dispersed Composites

Vladimir  Mityushev

doi:10.32523/bulmathenu.2026/1.3

Authors

Vladimir Mityushev Cracow University of Technology

DOI:

https://doi.org/10.32523/bulmathenu.2026/1.3

Keywords:

Homogenization theory, dispersed composites, self-consistent method, asymptotic analysis, symbolic computation

Abstract

Determining the effective properties of dispersed composites, such as conductivity and permittivity, remains a central challenge in homogenization theory. This paper presents a comprehensive revision of the Self-Consistent Method and its modern variants, demonstrating that many so-called “new models” are, in fact, merely first-order approximations in concentration that reproduce the classical results of Maxwell, Clausius, Mossotti, and others. We establish a rigorous framework for assessing accuracy with respect to both inclusion concentration and contrast parameters. A particular role of simulations addresses to standard packages, when the opacity of numerical calculations often obscures the underlying probability distributions of inclusions. We conclude that the concept of a mathematical model is frequently misapplied in the study of dispersed random composites, especially when the fundamental principles of homogenization and asymptotic analysis are overlooked. Finally, we demonstrate how symbolic computation can be effectively employed to derive analytically correct formulas with precisely quantified accuracy. Special attention is given to the relationship between analytical approaches and numerical simulations in the evaluation of effective material parameters. The paper emphasizes that reliable predictions require consistency with the fundamental principles of homogenization theory and the correct treatment of stochastic microstructures. By comparing classical and modern approaches, we identify the limits of applicability of commonly used approximations and clarify their asymptotic accuracy. The study also highlights the importance of transparent computational procedures, allowing the influence of inclusion geometry, concentration, and spatial distribution to be properly interpreted. The proposed methodology provides a unified perspective on dispersed composites and offers practical guidance for researchers working with effective medium theories, computational homogenization, and multiphase materials.

References

I. Andrianov and V. Mityushev, Exact and “exact” formulae in the theory of composites, Modern Problems in Applied Analysis, Springer International Publishing, Cham, 2018, 15–34.

N.S. Bakhvalov and G. Panasenko, Homogenisation: averaging processes in periodic media: mathematical problems in the mechanics of composite materials. Springer Science and Business Media, 2012.

W. Bielski and R. Wojnar, Plane flow through the porous medium with chessboard-like distribution of permeability, 2nd Workshop on Porous Media, 2018, 7.

B. Bojarski and V. Mityushev, R-linear problem for multiply connected domains and alternating method of Schwarz, Journal of Mathematical Sciences 189 (2013), no 1, 68–77.

D.A.G. Bruggeman, Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitatskonstanten und Leitfahigkeiten der Mischkorper aus isotropen Substanzen, Annalen der Physik 416 (1935), no 7, 636–664.

A. Cherkaev, Optimal three-material wheel assemblage of conducting and elastic composites, International Journal of Engineering Science 59 (2012), 27–39.

A. Cherkaev and A.D. Pruss, Effective conductivity of spiral and other radial symmetric assemblages, Mechanics of Materials 65 (2013), 103–109. https://doi.org/10.1016/j.mechmat.2013.05.004

A. Cherkaev, V. Mityushev, N. Rylko, and P. Kurtyka, The generalized Hashin–Shtrikman approach to Al/nano-TiC composite, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 478:2263 (2022). https://doi.org/10.1098/rspa.2022.0164

R. Czapla, P. Drygas, S. Gluzman, T. Ligocki, and V. Mityushev, Computational Strategy for Analyzing Effective Properties of Random Composites. Part II. Elasticity, Materials 18 (2025), 5041.

A.M. Dykhne, Conductivity of a two-dimensional two-phase system, Soviet Physics JETP 32 (1971), no 1, 63–65.

S. Gluzman, V. Mityushev, and W. Nawalaniec, Computational Analysis of Structured Media. Academic Press, Elsevier, Amsterdam, 2017.

S. Gluzman, Z. Zhunussova, A. Sarvarov, and V. Mityushev, Protocol-Dependent Critical Exponents in Random Composites: Beyond Universality, Symmetry 18 (2026), no 4, 700.

D. Grzonka, The New Engineer: Psychology, Systems, Open Source. Talk at the WordCamp Europe, Krakow, Poland, 4–6 June, 2026.

Z. Hashin and S. Shtrikman, A variational approach to the theory of the effective magnetic permeability of multiphase materials, Journal of Applied Physics 33 (1962), 3125–3131. https://doi.org/10.1063/1.1728579

R. Hill, Elastic properties of reinforced solids: Some theoretical principles, Journal of the Mechanics and Physics of Solids 11 (1963), 357–372.

R. Hill, A self-consistent mechanics of composite materials, Journal of the Mechanics and Physics of Solids 13 (1965), 213–222.

V.V. Jikov, S.M. Kozlov, and O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals. Springer Science & Business Media, 2012.

J.B. Keller, A theorem on the conductivity of a composite medium, Journal of Mathematical Physics 5 (1964), no 4, 548–549.

R. Landauer, Electrical conductivity in inhomogeneous media, AIP Conference Proceedings 40 (1978), no 1, 2–45.

G. Matheron, Eléments pour une théorie des milieux poreux. Masson, Paris, 1967.

J.C. Maxwell, A Treatise on Electricity and Magnetism, Vol. 1. Clarendon Press, 1873.

J.C.M. Maxwell-Garnett, Colours in Metal Glasses and in Metallic Films, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 203 (1904), no 359–371, 385–420.

S.G. Mikhlin, Integral Equations: and Their Applications to Certain Problems in Mechanics, Mathematical Physics and Technology, Vol. 4. Elsevier, 2014.

V. Mityushev, Hilbert boundary value problem for multiply connected domains, Complex Variables 35 (1998), 283–295.

V. Mityushev, R-linear problem on torus and its application to composites, Complex Variables 50 (2005), no 7–10, 621–630.

V.V. Mityushev, R-linear and Riemann–Hilbert Problems for Multiply Connected Domains, Advances in Applied Analysis. Springer Basel, Basel, 2012, 147–176.

V. Mityushev and P. Dryga'{s}, Effective Properties of Fibrous Composites and Cluster Convergence, SIAM Journal on Multiscale Modeling and Simulation 17 (2019), no 2, 696–715. https://doi.org/10.1137/18M1184278

V. Mityushev and N. Rylko, Effective properties of two-dimensional dispersed composites. Part I. Schwarz's alternating method, Computers and Mathematics with Applications 111 (2022), 50–60.

V. Mityushev, Effective properties of two-dimensional dispersed composites. Part II. Revision of self-consistent methods, Computers and Mathematics with Applications 121 (2022), 74–84.

V. Mityushev, R.A. Kycia, W. Nawalaniec, and N. Rylko, Introduction to Mathematical Modeling and Computer Simulations. Chapman and Hall/CRC, 2025.

T. Mori and K. Tanaka, Average stress in matrix and average elastic energy of materials with misfitting inclusions, Acta Metallurgica 21 (1973), no 5, 571–574.

A.N. Norris, A differential scheme for the effective moduli of composites, Mechanics of Materials 4 (1985), no 1, 1–16.

Y.V. Obnosov, Exact solution of a boundary-value problem for a rectangular checkerboard field, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 452 (1996), no 1954, 2423–2442.

A. Bensoussan, J.L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Vol. 374. American Mathematical Society, 2011.

L. Rayleigh, On the influence of obstacles arranged in rectangular order upon the properties of a medium, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 34 (1892), no 211, 481–502.

K. Schulgasser, Sphere assemblage model for polycrystals and symmetric materials, Journal of Applied Physics 54 (1983), 1380–1382. https://doi.org/10.1063/1.332161

S. Yakubovich, P. Drygas, and V. Mityushev, Closed-form evaluation of two-dimensional static lattice sums, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472 (2016), no 2195.

A.Z. Zhubanysheva, N. Temirgaliev, and Z.N. Temirgalieva, Application of divisor theory to the construction of tables of optimal coefficients for quadrature formulas, Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki 49 (2009), no 1, 14–25.