New Approach to Mathematical Model of Elastic in Two-Dimensional Composites

  • Piotr DrygaśEmail author
Conference paper
Part of the Trends in Mathematics book series (TM)


This paper is devoted to boundary value problems for elastic problems modelled by the biharmonic equation in two-dimensional composites. All the problems are studied via the method of complex potentials. The considered boundary value problems for analytic functions are reduced to functional-differential equations. Applications to calculation of the effective properties tensor are discussed.


Functional equation Two-dimensional elastic composite Eisenstein and Natanzon series Effective stress properties 

Mathematics Subject Classification (2010)

Primary 30E25; Secondary 74Q15 



The research has been partially supported by the Centre for Innovation and Transfer of Natural Science and Engineering Knowledge of University of Rzeszów (grant No. WMP/GD-09/2016).


  1. 1.
    J. Byström, N. Jekabsons, J. Varna, An evaluation of different models for prediction of elastic properties of woven composites. Compos. Part B 31, 7–20 (2000)CrossRefGoogle Scholar
  2. 2.
    S. Berggren, D. Lukkassen, A. Meidell, L. Simula, Some methods for calculating stiffness properties of periodic structure. Appl. Math. 48(2), 97–110 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    J. Franců, Homogenization of linear elasticity equations. Apl. Mat. 27, 96–117 (1982)MathSciNetzbMATHGoogle Scholar
  4. 4.
    L. Greengard, J. Helsing, On the numerical evaluation of elastostatic fields in locally isotropic two-dimensional composites. J. Mech. Phys. Solids 46, 1441–1462 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, Reprint of the 2nd English edn. (Springer-Science + Business Media, Dordrecht, 1977)Google Scholar
  6. 6.
    I. Jasiuk, J. Chen, M.F. Thorpe, Elastic Properties of Two-Dimensional Composites Containing Polygonal Holes. Materials Division, vol. 35 (American Society of Mechanical Engineers, New York, 1992), pp. 61–73Google Scholar
  7. 7.
    A.M. Linkov, Boundary Integral Equations in Elasticity Theory (Kluwer Academic Publishers, Dordrecht, 2002)CrossRefzbMATHGoogle Scholar
  8. 8.
    J. Wang, S.L. Crouch, S.G. Mogilevskaya, A complex boundary integral method for multiple circular holes in an infinite plane. Eng. Anal. Bound. Elem. 27(8), 789–802 (2003)CrossRefzbMATHGoogle Scholar
  9. 9.
    S.G. Mogilevskaya, V.I. Kushch, H.K. Stolarski, S.L. Crouch, Evaluation of the effective elastic moduli of tetragonal fiber-reinforced composites based on Maxwell’s concept of equivalent inhomogeneity. Int. J. Solids Struct. 50(25–26), 4161–4172 (2013)CrossRefGoogle Scholar
  10. 10.
    V. Mityushev, S.V. Rogosin, Constructive Methods for Linear and Nonlinear Boundary Value Problems for Analytic Functions. Theory and Applications (Chapman and Hall/CRC, London, 1999)Google Scholar
  11. 11.
    P. Drygaś, Generalized Eisenstein functions. J. Math. Anal. Appl. 444(2), 1321–1331 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    V.Ya. Natanson, On the stresses in a stretched plate weakened by identical holes located in chessboard arrangement. Mat. Sb. 42(5), 616–636 (1935)Google Scholar
  13. 13.
    L.A. Filshtinsky, V. Mityushev, Mathematical Models of Elastic and Piezoelectric Fields in Two-Dimensional Composites, ed. by P.M. Pardalos, T.M. Rassias. Mathematics Without Boundaries (Springer, New York, 2014), pp. 217–262Google Scholar
  14. 14.
    E.I. Grigolyuk, L.A. Filshtinsky, Perforated Plates and Shells (Nauka, Moscow, 1970); [in Russian]Google Scholar
  15. 15.
    E.I. Grigolyuk, L.A. Filshtinsky, Periodic Piecewise-Homogeneous Elastic Structures (Nauka, Moscow, 1992); [in Russian]Google Scholar
  16. 16.
    P. Drygaś, V. Mityushev, Effective elastic properties of random two-dimensional composites. Int. J. Solids Struct. 97–98, 543–553 (2016)CrossRefGoogle Scholar
  17. 17.
    P. Drygaś, Functional-differential equations in a class of analytic functions and its application to elastic composites. Complex Variables Elliptic Equ. 61(8), 1145–1156 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    V. Mityushev, Thermoelastic plane problem for material with circular inclusions. Arch. Mech. 52(6), 915–932 (2000)zbMATHGoogle Scholar
  19. 19.
    A. Weil, Elliptic Functions According to Eisenstein and Kronecker (Springer, Berlin, 1976)CrossRefzbMATHGoogle Scholar
  20. 20.
    N.I. Akhiezer, Elements of the Theory of Elliptic Functions (American Mathematical Society, Providence, RI, 1990)CrossRefzbMATHGoogle Scholar
  21. 21.
    V.V. Mityushev, E. Pesetskaya, S.V. Rogosin, Analytical Mathods for Heat Conduction in Composites and Porous Media, ed. by A. Ochsner, G.E. Murch, M.J.S. de Lemos (Wiley, New York, 2008)Google Scholar
  22. 22.
    V.V. Mityushev, Representative cell in mechanics of composites and generalized Eisenstein’s-Rayleigh sums. Complex Variables 51(8–11), 1033–1045 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    V. Mityushev, N. Rylko, Optimal distribution of the nonoverlapping conducting disks. Multiscale Model. Simul. 10(1), 180–190 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    R. Czapla, W. Nawalaniec, V. Mityushev, Effective conductivity of random two-dimensional composites with circular non-overlapping inclusions. Comput. Mater. Sci. 63, 118–126 (2012)CrossRefGoogle Scholar
  25. 25.
    J.W. Eischen, S. Torquato, Determining elastic behavior of composites by the boundary element method. J. Appl. Phys. 74, 159–170 (1993)CrossRefGoogle Scholar
  26. 26.
    P. Drygaś, V. Mityushev, Effective conductivity of unidirectional cylinders with interfacial resistance. Q. J. Mech. Appl. Math. 62, 235–262 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    S. Yakubovich, P. Drygaś, V. Mityushev, Closed-form evaluation of 2D static lattice sums. Proc. R. Soc. A 472, 20160510 (2016); CrossRefzbMATHGoogle Scholar
  28. 28.
    P. Drygaś, Steady heat conduction of material with coated inclusion in the case of imperfect contact. Math. Model. Anal. 12(3), 291–296 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    P. Drygaś, A functional-differential equation in a class of analytic functions and its application. Aequationes Math. 73(3), 22–232 (2007)MathSciNetzbMATHGoogle Scholar
  30. 30.
    P. Drygaś, Functional-differential equations in Hardy-type classes. Tr. Inst. Mat. 15(1), 105–110 (2007)Google Scholar
  31. 31.
    V.V. Mityushev, Exact solution of the R-linear problem for a disc in a class of doubly periodic functions. J. Appl. Funct. Anal. 2(2), 115–127 (2007)MathSciNetzbMATHGoogle Scholar
  32. 32.
    V. Mityushev, Transport properties of two-dimensional composite materials with circular inclusions. Proc. R. Soc. Lond. A 455, 2513–2528 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    V.V. Mityushev, E. Pesetskaya, S.V. Rogosin, Analytical methods for heat conduction in composites and porous media, in Cellular and Porous Materials: Thermal Properties Simulation and Prediction, ed. by A. Öchsner, G.E. Murch, M.J.S. de Lemos (Wiley, Weinheim, 2008)Google Scholar
  34. 34.
    V. Mityushev, N. Rylko, Maxwell’s approach to effective conductivity and its limitations. Q. J. Mech. Appl. Math. 66(2), 241–251 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Mathematics and Natural SciencesUniversity of RzeszówRzeszówPoland

Personalised recommendations