Exact and “Exact” Formulae in the Theory of Composites

  • Igor AndrianovEmail author
  • Vladimir Mityushev
Conference paper
Part of the Trends in Mathematics book series (TM)


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.


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

Mathematics Subject Classification (2010)

74A40 35C05 35C20 33E05 30E25 74Q15 



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


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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

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