Skip to main content
SpringerLink
Log in

Stress Concentration Due to the Presence of a Rigid Elliptical Inclusion in Porous Elastic Solids Described by a New Class of Constitutive Relations

  • Published:
Journal of Elasticity Aims and scope Submit manuscript

Abstract

In a large class of porous elastic solids such as cement concrete, rocks, ceramics, porous metals, biological materials such as bone, etc., the material moduli depend on density. When such materials undergo sufficiently small deformations, the usual approach of appealing to a linearized elastic constitutive relation to describe their response will not allow us to capture the dependence of the material moduli on the density, as this would imply a nonlinear relationship between the stress and the linearized strain in virtue of the balance of mass as dependence on density implies dependence on the trace of the linearized strain. It is possible to capture the dependence of the material moduli on the density, when the body undergoes small deformations, within the context of implicit constitutive relations. We study the stress concentration due to a rigid elliptic inclusion within a new class of implicit constitutive relations in which the stress and the linearized strain appear linearly, that allows us to capture the dependence of the material moduli on the density. We find that the stress concentration that one obtains employing the constitutive relation wherein the material moduli depend on the density can be significantly different from that obtained by adopting the classical linearized elastic constitutive relation to which it reduces to when the density dependence of the material moduli are ignored.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  1. Cauchy, A.L.B.: Recherches sur l’équilibre et le mouvement intérieur des corps solides ou fluides. Élastiques ou non élastiques (1822)

  2. Cauchy, A.L.B.: Sur les equations qui experiments des conditions de equilibre ou le lois du mouvement interieur, d’un corps solide. Elastique un non elastique (1828)

  3. Cristescu, N.: Rock Rheology, vol. 7. Springer, Berlin (2012)

    MATH  Google Scholar 

  4. Helgason, B., Perilli, E., Schileo, E., Taddei, F., Brynjólfsson, S., Viceconti, M.: Mathematical relationships between bone density and mechanical properties: a literature review. Clin. Biomech. 23(2), 135–146 (2008)

    Article  Google Scholar 

  5. Hirose, N., Tanaka, S.I., Tanaki, T., Asami, J.: The relationship between elastic modulus and porosity of sintered Fe–Cu system alloys. J. Jpn Soc. Powder Powder Metall. 51(5), 315–322 (2004)

    Article  Google Scholar 

  6. Inglis, C.E.: Stresses in a plate due to the presence of cracks and sharp corners. Trans. R. Inst. Nav. Archit. 55, 219–241 (1913)

    Google Scholar 

  7. Kirsch, C.: Die theorie der elastizitat und die bedurfnisse der festigkeitslehre. Z. Ver. Dtsch. Ing. 42, 797–807 (1898)

    Google Scholar 

  8. Kováčik, J.: Correlation between Young’s modulus and porosity in porous materials. J. Mater. Sci. Lett. 18(13), 1007–1010 (1999)

    Article  Google Scholar 

  9. Li, G., Zhao, Y., Pang, S.S., Li, Y.: Effective Young’s modulus estimation of concrete. Cem. Concr. Res. 29(9), 1455–1462 (1999)

    Article  Google Scholar 

  10. Luo, J., Stevens, R.: Porosity-dependence of elastic moduli and hardness of 3Y-TZP ceramics. Ceram. Int. 25(3), 281–286 (1999)

    Article  Google Scholar 

  11. Lydon, F., Balendran, R.: Some observations on elastic properties of plain concrete. Cem. Concr. Res. 16(3), 314–324 (1986)

    Article  Google Scholar 

  12. Manoylov, A., Borodich, F.M., Evans, H.P.: Modelling of elastic properties of sintered porous materials. Proc. R. Soc. A, Math. Phys. Eng. Sci. 469(2154), 20120689 (2013)

    MATH  Google Scholar 

  13. Munro, R.G.: Analytical representations of elastic moduli data with simultaneous dependence on temperature and porosity. J. Res. Natl. Inst. Stand. Technol. 109(5), 497 (2004)

    Article  Google Scholar 

  14. Murru, P., Rajagopal, K.R.: Stress concentration due to the bi-axial deformation of a plate of a porous elastic body with a hole. Z. Angew. Math. Mech. 101, e202100103 (2021)

    Article  MathSciNet  Google Scholar 

  15. Murru, P.T., Rajagopal, K.R.: Stress concentration due to the presence of a hole within the context of elastic bodies. Mater. Des. Process. Commun. 3, e219 (2021)

    Google Scholar 

  16. Murru, P.T., Grasley, Z., Torrence, C., Rajagopal, K.R., Garboczi, E.: Density driven damage mechanics (D3-M) model for concrete II: fully coupled chemo-mechanical damage. Int. J. Pavement Eng. 23, 1175–1185 (2022)

    Article  Google Scholar 

  17. Murru, P.T., Torrence, C., Grasley, Z., Rajagopal, K.R., Alagappan, P., Garboczi, E.: Density-driven damage mechanics (D3-M) model for concrete I: mechanical damage. Int. J. Pavement Eng. 23, 1161–1174 (2022)

    Article  Google Scholar 

  18. Nguyen, L., Beaucour, A., Ortola, S., Noumowé, A.: Influence of the volume fraction and the nature of fine lightweight aggregates on the thermal and mechanical properties of structural concrete. Constr. Build. Mater. 51, 121–132 (2014)

    Article  Google Scholar 

  19. Pauw, A.: Static modulus of elasticity of concrete as affected by density. University of Missouri (1960)

  20. Pilkey, W.D., Pilkey, D.F., Bi, Z.: Peterson’s Stress Concentration Factors. Wiley, New York (2020)

    Book  Google Scholar 

  21. Rajagopal, K.R.: On implicit constitutive theories. Appl. Math. 48(4), 279–319 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Rajagopal, K.R.: The elasticity of elasticity. Z. Angew. Math. Phys. 58(2), 309–317 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Rajagopal, K.R.: An implicit constitutive relation for describing the small strain response of porous elastic solids whose material moduli are dependent on the density. Math. Mech. Solids 26(8), 1138–1146 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  24. Spencer, A.J.: Part III. Theory of invariants. Contin. Phys. 1, 239–353 (1971)

    Google Scholar 

  25. Truesdell, C., Noll, W.: The non-linear field theories of mechanics. In: The Non-linear Field Theories of Mechanics, pp. 1–579. Springer, Berlin (2004)

    Chapter  MATH  Google Scholar 

  26. Vajipeyajula, B., Murru, P., Rajagopal, K.R.: Stress concentration due to an elliptic hole in a porous elastic plate. Math. Mech. Solids 28, 854–869 (2023)

    Article  MathSciNet  Google Scholar 

  27. Vanleene, M., Rey, C., Tho, M.H.B.: Relationships between density and Young’s modulus with microporosity and physico-chemical properties of Wistar rat cortical bone from growth to senescence. Med. Eng. Phys. 30(8), 1049–1056 (2008)

    Article  Google Scholar 

  28. Zhang, L., Gao, K., Elias, A., Dong, Z., Chen, W.: Porosity dependence of elastic modulus of porous Cr3C2 ceramics. Ceram. Int. 40(1), 191–198 (2014)

    Article  Google Scholar 

Download references

Acknowledgements

K. R. Rajagopal thanks the Office of Naval Research for support of this work.

Author information

Authors and Affiliations

  1. The Department of Engineering Technology and Industrial Distribution, Texas A&M University, 106 Ross Street, College Station, TX, 77843, USA

    Bhaskar Vajipeyajula

  2. Texas A&M University, College Station, TX, 77843, USA

    Pavitra Murru

  3. J. Mike Walker ’66 Department of Mechanical Engineering, Texas A&M University, College Station, TX, 77843, USA

    K. R. Rajagopal

Authors
  1. Bhaskar Vajipeyajula
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pavitra Murru
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. K. R. Rajagopal
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Bhaskar Vajipeyajula and Pavitra Murru carried out the computation, produced all the figures and documented the results. K. R. Rajagopal developed the theory and formulated the problem. All authors reviewed the manuscript.

Corresponding author

Correspondence to K. R. Rajagopal.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Dedicated to Roger Fosdick on his Eighty-Fifth Birthday

Publisher’s Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vajipeyajula, B., Murru, P. & Rajagopal, K.R. Stress Concentration Due to the Presence of a Rigid Elliptical Inclusion in Porous Elastic Solids Described by a New Class of Constitutive Relations. J Elast 154, 255–273 (2023). https://doi.org/10.1007/s10659-023-10027-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10659-023-10027-8

Keywords

Mathematics Subject Classification

Search

Navigation