Skip to main content
SpringerLink
Log in

Explicit solutions of quasi-static problems in the coupled theory of poroelasticity

  • Original Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract

This paper is concerned to study quasi-static boundary value problems of coupled linear theory of elasticity for porous circle and for plane with a circular hole. The Dirichlet type boundary value problem for a circle and the Neumann boundary value problem for a plane with a circular hole are solved explicitly. All the formulas are presented in explicit ready-to-use form. The solutions are represented by means of absolutely and uniformly convergent series.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Biot, M.A.: General theory of three dimensional consolidation. J. Appl. Phys. 12, 155–164 (1941)

    Article  ADS  Google Scholar 

  2. Nunziato, G.W., Cowin, S.C.: A nonlinear theory of elastic materials with voids. Arch. Ration. Mech. Anal. 72, 175–201 (1979)

    Article  MathSciNet  Google Scholar 

  3. Cowin, S.C., Nunziato, G.W.: Linear theory of elastic materials with voids. J. Elast. 13, 125–147 (1983)

    Article  Google Scholar 

  4. de Boer, R.: Theory of Porous Media. Highlights in the Historical Development and Current State. Springer, Berlin (2000)

    Book  Google Scholar 

  5. Straughan, B.: Mathematical aspects of multi-porosity continua. In: Advances in Mechanics and Mathematics, vol. 38. Springer, Switzerland (2017)

  6. Straughan, B.: Stability and Wave Motion in Porous Media. Springer, New York (2008)

    MATH  Google Scholar 

  7. Svanadze, M.: Potential Method in Mathematical Theories of Multi-porosity Media, Interdisciplinary Applied Mathematics, vol. 51. Springer, Switzerland (2019)

    MATH  Google Scholar 

  8. Ieşan, D., Quintanilla, R.: On a theory of thermoelastic materials with a double porosity structure. J. Therm. Stresses 37, 1017–1036 (2014)

    Article  Google Scholar 

  9. Svanadze, M.: Boundary integral equations method in the coupled theory of thermoelasticity for porous materials. In: Proceedings of the ASME 2019 International Mechanical Engineering Congress and Exposition. v. 9: Mechanics of Solids, Structures, and Fluids. Salt Lake City, Utah, USA. November 11–14, 2019. V009T11A033. ASME. https://doi.org/10.1115/IMECE2019-10367

  10. Mikelashvili, M.: Quasi-static problems in the coupled linear theory of thermoelasticity. J. Therm. Stresses 44(2), 236–259 (2021)

    Article  Google Scholar 

  11. Ieşan, D.: A theory of thermoelastic materials with voids. Acta Mech. 60, 67–89 (1986)

    Article  Google Scholar 

  12. Cheng, A.H.D.: Poroelasticity. Springer, Basel (2018)

    MATH  Google Scholar 

  13. Bitsadze, L., Tsagareli, I.: The solution of the Dirichlet BVP in the fully coupled theory for spherical layer with double porosity. Meccanica 51, 1457–1463 (2016)

    Article  MathSciNet  Google Scholar 

  14. Bitsadze, L.: Explicit solution of the Dirichlet boundary value problem of elasticity for porous infinite strip. Z. Aangew. Math. Phys. 71(5), 145 (2020)

    Article  MathSciNet  Google Scholar 

  15. Bitsadze, L.: Explicit solutions of the BVPs of the theory of thermoelasticity for an elastic circle with voids and microtemperatures. J. Appl. Math. Mech. (ZAMM) 100(10) (2020)

  16. Bitsadze, L.: On Some Solutions in the plane equilibrium theory for solids with triple porosity. Bull. TICMI 21(1), 9–20 (2017)

    MathSciNet  MATH  Google Scholar 

  17. Bitsadze, L.: Explicit solutions of boundary value problems of elasticity for circle with a double voids. J. Brazil. Soc. Mech. Sci. Eng. 41, 383 (2019)

    Article  Google Scholar 

  18. Bitsadze, L.: On one BVP for a thermo-microstretch elastic space with spherical cavity. Turk. J. Math. 42(5), 2101–2111 (2018)

    Article  MathSciNet  Google Scholar 

  19. Svanadze, M.: On the linear equilibrium theory of elasticity for materials with triple voids. Math. Mech. Solids 24(4), 919–938 (2019)

    Article  MathSciNet  Google Scholar 

  20. Svanadze, M.: Fundamental solutions in the theory of elasticity for triple porosity materials. Meccanica 51(8), 1825–1837 (2016)

    Article  MathSciNet  Google Scholar 

  21. Maganna, A., Quintanilla, R.: On the spatial behavior of solutions for porous elastic solids quasi-static microvoids. Math. Comput. Model. 44, 710–716 (2006)

    Article  MathSciNet  Google Scholar 

  22. Coussy, O.: Mechanics and Physics of Porous Media. Springer, Dordrecht (2005)

    MATH  Google Scholar 

  23. Svanadze, M., De Cicco, S.: Fundamental solutions in the full coupled linear theory of elasticity for solids with double porosity. Arch. Mech. 65, 367–390 (2013)

    MathSciNet  MATH  Google Scholar 

  24. Straughan, B.: Modelling questions in multi-porosity elasticity. Meccanica 51, 2957–2966 (2016)

    Article  MathSciNet  Google Scholar 

  25. Ciarletta, M., Scalia, A.: On uniqueness and reciprocity in linear thermoelasticity of materials with voids. J. Elast. 32, 1–17 (1993)

    Article  MathSciNet  Google Scholar 

  26. Mikelashvili, M.: Quasi-static problems in the coupled linear theory of elasticity for porous materials. Acta Mech. 231, 877–897 (2020)

    Article  MathSciNet  Google Scholar 

  27. Svanadze, M.: Steady vibration problems in the coupled linear theory of porous elastic solids. Math. Mech. Solids 25(3), 768–790 (2020)

    Article  MathSciNet  Google Scholar 

  28. Vlase, S., Marin, M., Öchsner, A., et al.: Motion equation for a flexible one-dimensional element used in the dynamical analysis of a multibody system. Contin. Mech. Thermodyn. 31(3), 715–724 (2019). https://doi.org/10.1007/s00161-018-0722-y

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Marin, M., Nicaise, S.: Existence and stability results for thermoelastic dipolar bodies with double porosity. Contin. Mech. Thermodyn. 28(6), 1645–1657 (2016). https://doi.org/10.1007/s00161-016-0503-4

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Vekua, I.: New Methods for Solving Elliptic Equations. North-Holland Publ, Oxford (1967)

    MATH  Google Scholar 

  31. Budak, B.M., Fomin. SV.: Multiple Integrals and Series. Science, Moscow (1965)

Download references

Author information

Authors and Affiliations

  1. I.Vekua Institute of Applied Mathematics of Iv., Javakhishvili Tbilisi State University, Tbilisi, Georgia

    Lamara Bitsadze

Authors
  1. Lamara Bitsadze
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lamara Bitsadze.

Ethics declarations

Conflicts of interest

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

Additional information

Communicated by Andreas Öchsner.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

The theory of linear elastic materials with voids is intended for application to solids with small distributed voids. The application of this theory may be found in geological materials like rocks and soils, in biological and manufactured porous materials. Seismology represents one of the many fields where the theory of elasticity of materials with voids is applied. Medicine, various branches of biology, the oil exploration industry and nanotechnology are other important fields of application.

From the point of view of applications, it is interesting to investigate and construct explicit solutions of boundary-value problems of elasticity theory for concrete domains(circle, plane with circular hole, sphere, half-space half-plane, ellipse, etc.). In practice, such BVPs are quite common in many areas of science. Looking for the numerical solution of a problem the engineer needs guarantee that this problem has a solution (to this end serve existence theorems) and how many solutions may have the problem under consideration (to this end serve uniqueness theorems). Otherwise, the engineer will find either the numerical solution (approximated one) for the non-existing solution or, if problem has more then one solutions, it will be not clear which solution (from several ones) of the problem is found numerically.

The theory of linear elastic materials with voids is intended for application to solids with small distributed voids. The application of this theory may be found in geological materials like rocks and soils, in biological and manufactured porous materials. Seismology represents one of the many fields where the theory of elasticity of materials with voids is applied. Medicine, various branches of biology, the oil exploration industry and nanotechnology are other important fields of application. The potential users of the obtained results will be the scientists and engineers working on the problems of solid mechanics, micro and nanomechanics, mechanics of materials, engineering mechanics, engineering medicine, biomechanics, engineering geology, geomechanics, hydro-engineering, applied and computing mechanics.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bitsadze, L. Explicit solutions of quasi-static problems in the coupled theory of poroelasticity. Continuum Mech. Thermodyn. 33, 2481–2492 (2021). https://doi.org/10.1007/s00161-021-01029-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-021-01029-9

Keywords

Search

Navigation