Exact and “Exact” Formulae in the Theory of Composites

Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

The effective properties of composites and review literature on the methods of Rayleigh, Natanzon–Filshtinsky, functional equations and asymptotic approaches are outlined. In connection with the above methods and new recent publications devoted to composites, we discuss the terms analytical formula, approximate solution, closed form solution, asymptotic formula, etc…frequently used in applied mathematics and engineering in various contexts. Though mathematicians give rigorous definitions of exact form solution the term “exact solution” continues to be used too loosely and its attributes are lost. In the present paper, we give examples of misleading usage of such a term.

Keywords

Composite material Effective property Closed form solution Weierstrass function Eisenstein summation Asymptotic solution 

Mathematics Subject Classification (2010)

74A40 35C05 35C20 33E05 30E25 74Q15 

Notes

Acknowledgements

Authors thanks Dr Galina Starushenko for fruitful discussions and providing the working notes with the calculation data and figures presented in Supplement.

References

  1. 1.
    I.V. Andrianov, J. Awrejcewicz, New trends in asymptotic approaches: summation and interpolation methods. Appl. Mech. Rev. 54, 69–92 (2001)CrossRefGoogle Scholar
  2. 2.
    I.V. Andrianov, H. Topol, Asymptotic analysis and synthesis in mechanics of solids and nonlinear dynamics (2011). arxiv.org/abs/1106.1783.Google Scholar
  3. 3.
    I.V. Andrianov, J. Awrejcewicz, B. Markert, G.A. Starushenko, Analytical homogenization for dynamic analysis of composite membranes with circular inclusions in hexagonal lattice structures. Int. J. Struct. Stab. Dyn. 17, 1740015 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    B.Ya. Balagurov, Effective electrical characteristics of a two-dimensional three-component doubly-periodic system with circular inclusions. J. Exp. Theor. Phys. 92, 123–134 (2001)Google Scholar
  5. 5.
    B.Ya. Balagurov, Electrophysical Properties of Composite: Macroscopic Theory (URSS, Moscow, 2015) (in Russian)Google Scholar
  6. 6.
    B.Ya. Balagurov, V.A. Kashin, Conductivity of a two-dimensional system with a periodic distribution of circular inclusions. J. Exp. Theor. Phys. 90, 850–860 (2000)Google Scholar
  7. 7.
    B.Ya. Balagurov, V.A. Kashin, Analytic properties of the effective dielectric constant of a two-dimensional Rayleigh model. J. Exp. Theor. Phys. 100, 731–741 (2005)Google Scholar
  8. 8.
    D.I. Bardzokas, M.L. Filshtinsky, L.A. Filshtinsky, Mathematical Methods in Electro-Magneto-Elasticity (Springer, Berlin, 2007)MATHGoogle Scholar
  9. 9.
    L. Berlyand, V. Mityushev, Increase and decrease of the effective conductivity of a two phase composites due to polydispersity. J. Stat. Phys. 118, 481–509 (2005)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    L. Berlyand, A. Novikov, Error of the network approximation for densely packed composites with irregular geometry. SIAM J. Math. Anal. 34, 385–408 (2002)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    L.C. Botten, N.A. Nicorovici, R.C. McPhedran, C.M. de Sterke, A.A. Asatryan, Photonic band structure calculations using scattering matrices. Phys. Rev. E 64, 046603 (1971)CrossRefGoogle Scholar
  12. 12.
    J. Bravo-Castillero, R. Guinovart-Diaz, F.J. Sabina, R. Rodriguez-Ramos, Closed-form expressions for the effective coefficients of a fiber-reinforced composite with transversely isotropic constituents - II. Piezoelectric and square symmetry. Mech. Mater. 33, 237–248 (2001)MATHGoogle Scholar
  13. 13.
    A.M. Dykhne, Conductivity of a two-dimensional two-phase system. Sov. Phys. JETP 32, 63–65 (1971)Google Scholar
  14. 14.
    E.S. Ferguson, How engineers lose touch. Invent. Technol. 8, 16–24 (1993)Google Scholar
  15. 15.
    L.A. Fil’shtinskii, Heat-conduction and thermoelasticity problems for a plane weakened by a doubly periodic system of identical circular holes. Tepl. Napryazh. Elem. Konstr. 4, 103–112 (1964) (in Russian)Google Scholar
  16. 16.
    L.A. Fil’shtinskii, Stresses and displacements in an elastic sheet weakened by a doubly periodic set of equal circular holes. J. Appl. Math. Mech. 28, 530–543 (1964)MathSciNetCrossRefGoogle Scholar
  17. 17.
    L.A. Fil’shtinskii, Toward a solution of two-dimensional doubly periodic problems of the theory of elasticity. PhD thesis (Novosibirsk University, Novosibirsk, 1964) (in Russian)Google Scholar
  18. 18.
    L.A. Fil’shtinskii, Physical fields modelling in piece-wise homogeneous deformable solids. Publ. SSU, Sumy (2001) (in Russian)Google Scholar
  19. 19.
    L. Filshtinsky, V. Mityushev, Mathematical models of elastic and piezoelectric fields in two-dimensional composites, in Mathematics Without Boundaries, ed. by P.M. Pardalos, Th.M. Rassias. Surveys in Interdisciplinary Research (Springer, New York, 2014), pp. 217–262Google Scholar
  20. 20.
    A.Ya. Findlin, Peculiarities of the use of computational methods in applied mathematics (on global computerization and common sense). Applied Mathematics: Subject, Logic, Peculiarities of Approaches. With Examples from Mechanics, ed. by I.I. Blekhman, A.D. Myshkis, Ya.G. Panovko (URSS, Moscow, 2007), pp. 350–358 (in Russian)Google Scholar
  21. 21.
    F.D. Gakhov, Boundary Value Problems, 3rd edn. (Nauka, Moscow, 1977) (in Russian); Engl. transl. of 1st edn. (Pergamon Press, Oxford, 1966)MATHGoogle Scholar
  22. 22.
    Y.A. Godin, The effective conductivity of a periodic lattice of circular inclusions. J. Math. Phys. 53, 063703 (2012)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    G.M. Golusin, Solution to basic plane problems of mathematical physics for the case of Laplace’s equation and multiply connected domains bounded by circles (method of functional equations). Matematicheskij zbornik 41, 246–276 (1934) (in Russian)Google Scholar
  24. 24.
    E.I. Grigolyuk, L.A. Filshtinsky, Perforated Plates and Shells (Nauka, Moscow, 1970) (in Russian)Google Scholar
  25. 25.
    E.I. Grigolyuk, L.A. Filshtinsky, Periodical Piece-Homogeneous Elastic Structures (Nauka, Moscow, 1991) (in Russian)Google Scholar
  26. 26.
    E.I. Grigolyuk, L.A. Filshtinsky, Regular Piece-Homogeneous Structures with Defects (Fiziko-Matematicheskaja Literatura, Moscow, 1994) (in Russian)Google Scholar
  27. 27.
    E.I. Grigolyuk, L.M. Kurshin, L.A. Fil’shtinskii, On a method to solve doubly periodic elastic problems. Prikladnaya Mekhanika (Appl. Mech.) 1, 22–31 (1965) (in Russian)Google Scholar
  28. 28.
    R. Guinovart-Diaz, J. Bravo-Castillero, R. Rodriguez-Ramos, F.J. Sabina, Closed-form expressions for the effective coefficients of a fibre-reinforced composite with transversely isotropic constituents – I. Elastic and hexagonal symmetry. J. Mech. Phys. Solids 49, 1445–1462 (2001)CrossRefMATHGoogle Scholar
  29. 29.
    R. Guinovart-Diaz, R. Rodriguez-Ramos, J. Bravo-Castillero, F.J. Sabina, G.A. Maugin, Closed-form thermoelastic moduli of a periodic three-phase fiber-reinforced composite. J. Therm. Stresses 28, 1067–1093 (2005)MathSciNetCrossRefGoogle Scholar
  30. 30.
    J. Gleik, Chaos: Making a New Science (Viking Penguin, New York, 1987)Google Scholar
  31. 31.
    S. Gluzman, V. Mityushev, W. Nawalaniec, Computational Analysis of Structured Media (Elsevier, Amsterdam, 2017)MATHGoogle Scholar
  32. 32.
    A.N. Guz, V.D. Kubenko, M.A. Cherevko, Diffraction of Elastic Waves (Naukova Dumka, Kiev, 1978) (in Russian).Google Scholar
  33. 33.
    A.L. Kalamkarov, I.V. Andrianov, G.A. Starushenko, Three-phase model for a composite material with cylindrical circular inclusions. Part I: application of the boundary shape perturbation method. Int. J. Eng. Sci. 78, 154–177 (2014)Google Scholar
  34. 34.
    A.L. Kalamkarov, I.V. Andrianov, G.A. Starushenko, Three-phase model for a composite material with cylindrical circular inclusions. Part II: application of Padé approximants. Int. J. Eng. Sci. 78, 178–219 (2014)Google Scholar
  35. 35.
    A.L. Kalamkarov, I.V. Andrianov, P.M.C.L. Pacheco, M.A. Savi, G.A. Starushenko, Asymptotic analysis of fiber-reinforced composites of hexagonal structure. J. Multiscale Model. 7, 1650006 (2016)MathSciNetCrossRefGoogle Scholar
  36. 36.
    L.V. Kantorovich, V.I. Krylov, Approximate Methods of Higher Analysis (Noordhoff, Groningen, 1958)MATHGoogle Scholar
  37. 37.
    I. Kushch, Stress concentration and effective stiffness of aligned fiber reinforced composite with anisotropic constituents. Int. J. Solids Struct. 45, 5103–5117 (2008)CrossRefMATHGoogle Scholar
  38. 38.
    I. Kushch, Transverse conductivity of unidirectional fibrous composite with interface arc cracks. Int. J. Eng. Sci. 48, 343–356 (2010)CrossRefMATHGoogle Scholar
  39. 39.
    I. Kushch, Micromechanics of Composites (Elsevier/Butterworth-Heinemann, Amsterdam/Oxford, 2013)Google Scholar
  40. 40.
    I. Kushch, S.V. Shmegera, V.A. Buryachenko, Elastic equilibrium of a half plane containing a finite array of elliptic inclusions. Int. J. Solids Struct. 43, 3459–3483 (2006)CrossRefMATHGoogle Scholar
  41. 41.
    R.C. McPhedran, D.R. McKenzie, The conductivity of lattices of spheres. I. The simple cubic lattice. Proc. Roy. Soc. London A359, 45–63 (1978)CrossRefGoogle Scholar
  42. 42.
    R.C. McPhedran, L. Poladian, G.W. Milton, Asymptotic studies of closely spaced, highly conducting cylinders. Proc. Roy. Soc. London A415, 185–196 (1988)CrossRefGoogle Scholar
  43. 43.
    S.G. Mikhlin, Integral Equations and their Applications to Certain Problems in Mechanics, Mathematical Physics, and Technology, 2nd edn. (Pergamon Press, Oxford, 1964)MATHGoogle Scholar
  44. 44.
    V. Mityushev, Boundary value problems and functional equations with shifts in domains, PhD Thesis, Minsk (1984) (in Russian)Google Scholar
  45. 45.
    V. Mityushev, Plane problem for the steady heat conduction of material with circular inclusions. Arch. Mech. 45, 211–215 (1993)MathSciNetMATHGoogle Scholar
  46. 46.
    V. Mityushev, Generalized method of Schwarz and addition theorems in mechanics of materials containing cavities. Arch. Mech. 45, 1169–1181 (1995)MathSciNetMATHGoogle Scholar
  47. 47.
    V. Mityushev, Functional equations and its applications in mechanics of composites. Demonstratio Math. 30, 64–70 (1997)MathSciNetGoogle Scholar
  48. 48.
    V. Mityushev, Transport properties of regular array of cylinders. ZAMM 77, 115–120 (1997)CrossRefMATHGoogle Scholar
  49. 49.
    V. Mityushev, Steady heat conduction of the material with an array of cylindrical holes in the non-linear case. IMA J. Appl. Math. 61, 91–102 (1998)MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    V. Mityushev, Exact solution of the \(\mathbb R\)-linear problem for a disk in a class of doubly periodic functions. J. Appl. Funct. Anal. 2, 115–127 (2007)Google Scholar
  51. 51.
    V.V. Mityushev, S.V. Rogosin, Constructive Methods for Linear and Nonlinear Boundary Value Problems for Analytic Functions. Theory and Applications. Monographs and Surveys in Pure and Applied Mathematics (Chapman & Hall/CRC, Boca Raton, 2000)Google Scholar
  52. 52.
    V. Mityushev, N. Rylko, Maxwell’s approach to effective conductivity and its limitations. Q. J. Mech. Appl. Math. 66, 241–251 (2013)MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    A.B. Movchan, S. Guenneau, Split-ring resonators and localized modes. Phys. Rev. B 70, 125116 (2004)CrossRefGoogle Scholar
  54. 54.
    A.B. Movchan, N.V. Movchan, Ch.G. Poulton, Asymptotic Models of Fields in Dilute and Densely Packed Composites (World Scientific, London, 2002)CrossRefMATHGoogle Scholar
  55. 55.
    A.B. Movchan, N.V. Movchan, R.C. McPhedran, Bloch–Floquet bending waves in perforated thin plates. Proc. Roy. Soc. London A463, 2505–2518 (2007)MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    V. Mityushev, Poincaré α-series for classical Schottky groups and its applications, in Analytic Number Theory, Approximation Theory, and Special Functions, ed. by G.V. Milovanović, M.Th. Rassias (Springer, Berlin, 2014), pp. 827–852CrossRefGoogle Scholar
  57. 57.
    E.S. Nascimento, M.E. Cruz, J. Bravo-Castillero, Calculation of the effective thermal conductivity of multiscale ordered arrays based on reiterated homogenization theory and analytical formulae. Int. J. Eng. Sci. 119, 205–216 (2017)CrossRefGoogle Scholar
  58. 58.
    V.Ya. Natanzon, On stresses in a tensioned plate with holes located in the chess order. Matematicheskii sbornik 42, 617–636 (1935) (in Russian)Google Scholar
  59. 59.
    N.A. Nicorovici, G.W. Milton, R.C. McPhedran, L.C. Botten, Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance. Opt. Express 15, 6314–6323 (2007)CrossRefGoogle Scholar
  60. 60.
    W. Parnell, I.D. Abrahams, Dynamic homogenization in periodic fibre reinforced media. Quasi-static limit for SH waves. Wave Motion 43, 474–498 (2006)CrossRefMATHGoogle Scholar
  61. 61.
    W.T. Perrins, D.R. McKenzie, R.C. McPhedran, Transport properties of regular arrays of cylinders. Proc. Roy. Soc. London A369, 207–225 (1979)MathSciNetCrossRefGoogle Scholar
  62. 62.
    Rayleigh, On the influence of obstacles arranged in rectangular order upon the properties of medium. Phil. Mag. 34, 481–502 (1892)Google Scholar
  63. 63.
    R. Rodriguez-Ramos, F.J. Sabina, R. Guinovart-Diaz, J. Bravo-Castillero, Closed-form expressions for the effective coefficients of a fibre-reinforced composite with transversely isotropic constituents – I. Elastic and square symmetry. Mech. Mater. 33, 223–235 (2001)MATHGoogle Scholar
  64. 64.
    N. Rylko, Transport properties of a rectangular array of highly conducting cylinders. Proc Roy. Soc. London A472, 1–12 (2000)MathSciNetMATHGoogle Scholar
  65. 65.
    N. Rylko, Structure of the scalar field around unidirectional circular cylinders. J. Eng. Math. 464, 391–407 (2008)MathSciNetMATHGoogle Scholar
  66. 66.
    F.J. Sabina, J. Bravo-Castillero, R. Guinovart-Diaz, R. Rodriguez-Ramos, O.C. Valdiviezo-Mijangos, Overall behaviour of two-dimensional periodic composites. Int. J. Solids Struct. 39, 483–497 (2002)CrossRefMATHGoogle Scholar
  67. 67.
    G.P. Sendeckyj, Multiple circular inclusion problems in longitudinal shear deformation. J. Elast. 1 83–86 (1971)CrossRefGoogle Scholar
  68. 68.
    A.B. Tayler, Mathematical Models in Applied Mechanics (Clarendon Press, Oxford, 2001)MATHGoogle Scholar
  69. 69.
    Y. Wang, Y. Wang, Schwarz-type problem of nonhomogeneous Cauchy-Riemann equation on a triangle. J. Math. Anal. Appl. 377, 557–570 (2011)MathSciNetCrossRefMATHGoogle Scholar
  70. 70.
    A. Weil, Elliptic Functions According to Eisenstein and Kronecker (Springer, Berlin, 1976)CrossRefMATHGoogle Scholar
  71. 71.
    S. Yakubovich, P. Drygas, V. Mityushev, Closed-form evaluation of 2D static lattice sums. Proc. Roy. Soc. London A472, 20160510 (2016)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für Allgemeine MechanikRWTH Aachen UniversityAachenGermany
  2. 2.Institute of Computer SciencesPedagogical UniversityKrakowPoland

Personalised recommendations