Skip to main content
SpringerLink
Log in

Functional Equations for Analytic Functions and Their Application to Elastic Composites

  • Conference paper
  • First Online:
Book cover New Trends in Analysis and Interdisciplinary Applications

Part of the book series: Trends in Mathematics ((RESPERSP))

  • 1030 Accesses

Abstract

Two-dimensional elastic composites with non-overlapping inclusions is studied by means of the boundary value problems for analytic functions following Muskhelishvili’s approach. We develop a method of functional equations to reduce this problem for a circular multiply connected domain to functional-differential equations. Analytical formulae for the effective constants are deduced.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. 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)

    Article  Google Scholar 

  2. P. Drygaś, Generalized Eisenstein functions. J. Math. Anal. Appl. 444 (2), 1321–1331 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. P. Drygaś, Functional-differential equation in a class of analytic functions and its application to elastic composites. Complex Variablesand Elliptic Eqn. 61 (8), 1145–1156 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. P. Drygaś, V. Mityushev, Effective conductivity of unidirectional cylinders with interfacial resistance. Q. J. Mech. Appl. Math. 62, 235–262 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. 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. 217–262

    Google Scholar 

  6. E.I. Grigolyuk, L.A. Filshtinsky, Perforated Plates and Shells (Nauka, Moscow, 1970) (in Russian)

    Google Scholar 

  7. E.I. Grigolyuk, L.A. Filshtinsky, Periodic Piecewise-Homogeneous Elastic Structures (Nauka, Moscow, 1992) (in Russian)

    Google Scholar 

  8. P. Kurtyka, N. Ryko, Structure analysis of the modified cast metal matrix composites by use of the RVE theory. Arch. Metall. Mater. 58, 357–360 (2013)

    Article  Google Scholar 

  9. P. Kurtyka, N. Rylko, T. Tokarski, A. Wojcicka, and A. Pietras, Cast aluminium matrix composites modified with using FSP process-changing of the structure and mechanical properties. Compos. Struct. 133, 959–967 (2015)

    Article  Google Scholar 

  10. V. Mityushev, Transport properties of two-dimensional composite materials with circular inclusions. Proc. R. Soc. Lond. A455, 2513–2528 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. V. Mityushev, Thermoelastic plane problem for material with circular inclusions. Arch. Mech. 52 (6), 915–932 (2000)

    MATH  Google Scholar 

  12. 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, 115–127 (2006)

    MATH  Google Scholar 

  13. V. Mityushev, S.V. Rogosin, Constructive Methods for Linear and Nonlinear Boundary Value Problems for Analytic Functions. Theory and Applications (Chapman & Hall CRC, London, 1999)

    Google Scholar 

  14. V. Mityushev, N. Rylko, Maxwell’s approach to effective conductivity and its limitations. Q. J. Mech. Appl. Math. 66 (2), 241–251 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. V.V. Mityushev, E. Pesetskaya, S.V. Rogosin, Analytical Methods 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 

  16. N.I. Muskhelishvili, Some Mathematical Problems of the Plane Theory of Elasticity (Nauka, Moscow, 1966)

    MATH  Google Scholar 

  17. V.Y. 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 

  18. J.W. Rayleigh, On the influence of obstacles arranged in rectangular order upon the properties of a medium. Philos. Mag. 32, 481–491 (1892)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

Piotr Drygaś was partially supported by the Centre for Innovation and Transfer of Natural Science and Engineering Knowledge of University of Rzeszow (grant No. WMP/GD-04/2015).

Author information

Authors and Affiliations

  1. Department of Differential Equations and Statistics, Faculty of Mathematics and Natural Sciences, University of Rzeszow, ul. Pigonia 1, 35-959, Rzeszow, Poland

    Piotr Drygaś

  2. Pedagogical University, ul. Podchorazych 2, 30-084, Krakow, Poland

    Vladimir Mityushev

Authors
  1. Piotr Drygaś
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vladimir Mityushev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Piotr Drygaś .

Editor information

Editors and Affiliations

  1. Faculty of Information Technology, Macau University of Science and Technology, Taipa, Macao

    Pei Dang

  2. Department of Mathematics, University of Aveiro, Aveiro, Portugal

    Min Ku

  3. Faculty of Science and Technology, Department of Mathematics, University of Macau, Taipa, Macao

    Tao Qian

  4. Department of Mathematics, University of Torino, Torino, Italy

    Luigi G. Rodino

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Drygaś, P., Mityushev, V. (2017). Functional Equations for Analytic Functions and Their Application to Elastic Composites. In: Dang, P., Ku, M., Qian, T., Rodino, L. (eds) New Trends in Analysis and Interdisciplinary Applications. Trends in Mathematics(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-48812-7_4

Download citation

Publish with us

Policies and ethics

Search

Navigation