Explicit solutions of boundary value problems of elasticity for circle with a double-voids structure

  • Lamara BitsadzeEmail author


The present paper is devoted to the explicit solutions of the equilibrium boundary value problems (BVPs) for an elastic circle and for full plane with circular hole with a double-voids structure. The regular solution of the system of equations for an isotropic material with a double-voids structure is constructed by means of the elementary (harmonic, bi-harmonic and meta-harmonic) functions. The Dirichlet-type BVPs for a circle and for a plane with a circular hole are solved explicitly. The obtained solutions are presented as absolutely and uniformly convergent series.


Explicit solutions Circle Plane with a circular hole Material with voids 


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Conflict of interest

The author declares that there is no conflict of interest regarding the publication of this paper.


  1. 1.
    Nunziato GW, Cowin SC (1979) A nonlinear theory of elastic materials with voids. Arch Ration Mech Anal 72:175–201. MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cowin SC, Nunziato GW (1983) Linear theory of elastic materials with voids. J Elast 13:125–147. CrossRefzbMATHGoogle Scholar
  3. 3.
    Ieşan D (1986) A theory of thermoelastic materials with voids. Acta Mech 60:67–89. CrossRefGoogle Scholar
  4. 4.
    Ciarletta M, Scalia A (1993) On uniqueness and reciprocity in linear thermoelasticity of materials with voids. J Elast 32:1–17. MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ciarletta M, Scalia A (1991) Results and applications in thermoelasticity of materials with voids. Le Matematiche XLVI:85–96MathSciNetzbMATHGoogle Scholar
  6. 6.
    Ieşan D, Quintanilla R (2014) On a theory of thermoelastic materials with a double porosity structure. J Therm Stress 37:1017–1036. CrossRefGoogle Scholar
  7. 7.
    Puri P, Cowin SC (1985) Plane waves in linear elastic materials with voids. J Elast 15:167–183. CrossRefzbMATHGoogle Scholar
  8. 8.
    Ieşan D (2004) Thermoelastic models of continua. Kluwer, Boston. CrossRefzbMATHGoogle Scholar
  9. 9.
    Straughan B (2017) Mathematical aspects of multi-porosity continua, vol 38. Advances in mechanics and mathematics. Springer, Berlin. CrossRefzbMATHGoogle Scholar
  10. 10.
    Singh B (2007) Wave propagation in a generalized thermoelastic material with voids. Appl Math Comput 189(1):698–709. MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Singh J, Tomar SK (2007) Plane waves in thermo-elastic materials with voids. Mech Mater 39:932–940. CrossRefGoogle Scholar
  12. 12.
    Ieşan D, Nappa L (2003) Axially symmetric problems for porous elastic solid. Int J Solid Struct 40(20):5271–5286. CrossRefzbMATHGoogle Scholar
  13. 13.
    Eringen AC (1999) Microcontinuum field theories. I. Foundation and Solids. Springer, New-York. CrossRefzbMATHGoogle Scholar
  14. 14.
    Svanadze M (2018) Steady vibration problems in the theory of elasticity for materials with double voids. Acta Mech 229:1517–1536. MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Straughan B (2008) Stability and wave motion in porous media. Springer, New YorkzbMATHGoogle Scholar
  16. 16.
    Svanadze M (2012) Plane waves and boundary value problems in the theory of elasticity for solids with double porosity. Acta Appl Math 122:461–470MathSciNetzbMATHGoogle Scholar
  17. 17.
    Ieşan D (2011) On a theory of thermoviscoelastic materials with voids. J Elast 104:369–384. MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Bitsadze L, Zirakashvili N (2016) Explicit solutions of the boundary value problems for an ellipse with double porosity. Adv Math Phys, Article ID 1810795, 11 pages. MathSciNetCrossRefGoogle Scholar
  19. 19.
    Bitsadze L, Tsagareli I (2016) Solutions of BVPs in the fully coupled theory of elasticity for the space with double porosity and spherical cavity. Math Methods Appl Sci 39:2136–2145. MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Bitsadze L, Tsagareli I (2016) The solution of the Dirichlet BVP in the fully coupled theory for spherical layer with double porosity. Meccanica 51:1457–1463. MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Tsagareli I, Bitsadze L (2015) Explicit solution of one boundary value problem in the full coupled theory of elasticity for solids with double porosity. Acta Mech 226:1409–1418. MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Bitsadze L (2017) On some solutions in the plane equilibrium theory for solids with triple porosity. Bull TICMI 21(1):9–20MathSciNetzbMATHGoogle Scholar
  23. 23.
    Tsagareli I, Bitsadze L (2016) Explicit solutions on some problems in the fully coupled theory of elasticity for a circle with double porosity. Bull TICMI 20(2):11–23MathSciNetzbMATHGoogle Scholar
  24. 24.
    Khalili N, Selvadurai APS (2003) Elastic media with double porosity. Geophys Res Lett 30(24):1–13. CrossRefGoogle Scholar
  25. 25.
    Othman MIA, Tantawi RS, Abd-Elaziz EM (2016) Effect of initial stress on a thermoelastic medium with voids and microtemperatures. J Porous Media 19:155–172. CrossRefGoogle Scholar
  26. 26.
    Eringen AC, Suhubi ES (1964) Nonlinear theory of simple microelastic solids. Int J Eng Sci 2:189–203. CrossRefzbMATHGoogle Scholar
  27. 27.
    Mikhlin SG (1970) Mathematical physic. An advanced course. North-Holland Publishing Company, AmsterdamGoogle Scholar
  28. 28.
    Vekua I (1967) New methods for solving elliptic equations. North-Holland Publishing Company, AmsterdamzbMATHGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  1. 1.I. Vekua Institute of Applied Mathematics of Iv. JavakhishviliTbilisi State UniversityTbilisiGeorgia

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