Skip to main content
SpringerLink
Log in

Effective properties of materials with random micro-cavities using special boundary elements

  • Published:
Journal of Materials Science Aims and scope Submit manuscript

Abstract

In this work, a general boundary element procedure is proposed to obtain the effective elastic tensor of solids containing randomly distributed micro-cavities in terms of its primary elastic properties. The average-field theory and a special boundary element formulation are combined to carry out a statistical analysis on the numerical results obtained for a Representative Volume Element (RVE). The two-dimensional isotropic material is simulated as a homogeneous matrix containing cylindrical holes. In the proposed implementation each hole boundary is modeled with a single boundary element. The average variables of the micro-field are evaluated using boundary-only data, which leads to a formulation particularly suitable for Boundary Element Methods. Expressions for effective elastic properties as a function of the micro-fields for both isotropic and transversally isotropic hypothesis are derived. Finally, the methodology is illustrated with some application examples and the results are compared with analytical and experimental results.

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

Similar content being viewed by others

References

  1. Hu N, Wang B, Tan GW, Yao ZH, Yuan WF (2000) Compos Sci Technol 60:1811

    Article  Google Scholar 

  2. Yang Q-S, Qin Q-H (2004) Eng Anal Bound Elem 28:919

    Article  Google Scholar 

  3. Dong CY, Lee KY (2006) Eng Anal Bound Elem 30:662

    Article  Google Scholar 

  4. Okada H, Fukui Y, Kumazawa N (2001) Comput Struct 79:1987

    Article  Google Scholar 

  5. Dong CY, Lo SH, Cheung YK (2002) Eng Anal Bound Elem 26:471

    Article  Google Scholar 

  6. Knight MG, Wrobel LC, Henshall JL (2003) Compos Struct 62:341

    Article  Google Scholar 

  7. Wang J, Crouch SL, Mogilevskaya SG (2003) Eng Anal Bound Elem 27:789

    Article  Google Scholar 

  8. Mogilevskaya SG, Crouch SL (2001) Int J Numer Meth Eng 52:1069

    Article  Google Scholar 

  9. Zohdi TI (2002) In: Computational modeling and design of new random microheterogeneous materials. CISM Course notes, Udine Italy

  10. Nemat-Nasser S, Hori M (1999) In: Micromechanics: overall properties of heterogeneous solids. Elsevier

  11. Hori M, Nemat-Nasser S (1999) Mech Mater 31:667

    Article  Google Scholar 

  12. Henry DP, Banerjee PK (1991) Int J Numer Meth Eng 31:369

    Article  Google Scholar 

  13. Buroni FC, Marczak RJ (2006) In: Cardona A, Nigro N, Songzogni V, Storti M (eds) Mecánica Computacional, vol XXV. Santa Fé, Argentina, p 2747

  14. Buroni FC, Marczak RJ (2007) A family of hole boundary elements for modeling materials with cylindrical voids. Accepted in: Eng Anal Bound Elem. doi:https://doi.org/10.1016/j.enganabound.2007.11.003

    Article  Google Scholar 

  15. Buroni FC, Marczak RJ (2006) In: Cardona A, Nigro N, Songzogni V, Storti M (eds) Mecánica computacional, vol XXV. Santa Fé, Argentina, p 2835

  16. Banerjee PK (1994) In: The boundary element methods in engineering. Mc Graw-Hill

  17. Brebbia CA, Dominguez J (1992) Boundary elements. An introductory course. Computational Mechanics Publications. McGraw-Hill

    Google Scholar 

  18. Kanit T, Forest S, Galliet I, Mounoury V, Jeulin D (2003) Int J Solids Struct 40:3647

    Article  Google Scholar 

  19. Dong CY (2006) Int J Solids Struct 43:7919

    Article  Google Scholar 

  20. Öchsner A (2004) Numerical simulation of the mechanical behavior of cellular materials. Research report, Mechanical Engineering Department, Universidade de Aveiro, Portugal

  21. Batista RG, Cisilino AP, Iturrioz I (2003) In: Rosales MB, Cortínez VH, Bambill DV (eds) Mecánica computacional, vol XXII. p 1878

  22. Pundale SH, Rogers RJ, Nadkarni GR (1998) Am Foundrymen Soc Trans 106:99

    CAS  Google Scholar 

  23. Berdin C, Dong MJ, Prioul C (2001) Eng Fract Mech 68:1107

    Article  Google Scholar 

  24. Zhang KS, Bai JB, François D (1999) Int J Solids Struct 36:3407

    Article  Google Scholar 

  25. Ortiz JE, Cisilino AP, Otegui JL (2001) Fatigue Fract Engng Mater Struct 24:591

    Article  Google Scholar 

  26. Gross D, Seelig T (2006) In: Fracture mechanics with an introduction to micromechanics. Springer

  27. Rokhlin V (1985) J Comput Phys 60:187

    Article  Google Scholar 

  28. Fu YH, Klimkowski KJ, Rodin GJ, Berger E, Browne JC, Singer JK, Geijn RA (1998) Int J Numer Meth Eng 42:1215

    Article  Google Scholar 

Download references

Acknowledgements

The first author wishes to express his gratitude to CNPq (Brazil) for the financial support and to Prof. Adrián P. Cisilino (National University of Mar del Plata, Argentina) for useful discussions. The authors are also grateful to Prof. V. Mantič for his comments on the original manuscript. This work was partially financed by PROSUL 490185/2005-3 and CAPES/SECYP 048/03 projects.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Federal University of Rio Grande do Sul, Rua Sarmento Leite 425, 90050-170, Porto Alegre, RS, Brazil

    Federico C. Buroni & Rogério J. Marczak

Authors
  1. Federico C. Buroni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rogério J. Marczak
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rogério J. Marczak.

Appendix

Appendix

The shape functions used by the hole elements are presented below.

  • 3-noded element (as proposed by Henry and Banerjee [4]):

    $$ \begin{aligned} M_1 \left(\theta \right)=&\frac{1}{3}+\frac{2}{3}\hbox{cos}\, \theta \\ M_2 \left(\theta \right)=&\frac{1}{3}+\frac{\sqrt{3}}{3}\hbox{sin}\, \theta -\frac{1}{3}\hbox{cos}\, \theta \\ M_3 \left(\theta \right)=&\frac{1}{3}-\frac{\sqrt{3}}{3}\hbox{sin}\, \theta -\frac{1}{3}\hbox{cos}\, \theta \end{aligned} $$

  • 4-noded element:

    $$ \begin{aligned} M_1 \left(\theta \right)=&\frac{\left(1+\hbox{cos}\, \theta \right)}{2}\hbox{cos}\, \theta \\ M_2 \left(\theta \right)=&\frac{1}{2}+\frac{1}{2}\hbox{sin}\, \theta -\frac{1}{2}\hbox{cos}^{2}\theta \\ M_3 \left(\theta \right)=&\frac{\left(-1+\hbox{cos}\, \theta \right)}{2}\hbox{cos}\, \theta \\ M_4 \left(\theta \right)=&\frac{1}{2}-\frac{1}{2}\hbox{sin}\, \theta -\frac{1}{2}\hbox{cos}^{2}\theta \end{aligned} $$

  • 5-noded element:

    $$ \begin{aligned} M_1 \left(\theta \right)=&\frac{\hbox{cos}^{2}\theta +\hbox{cos} \left(\frac{\pi}{5} \right)\hbox{cos}\, \theta -\hbox{cos} \left(\frac{2\pi}{5} \right)\hbox{cos}\, \theta -\hbox{cos} \left(\frac{\pi}{5} \right)\hbox{cos} \left(\frac{2\pi}{5} \right)}{\hbox{cos} \left(\frac{\pi}{5} \right)\hbox{cos} \left(\frac{2\pi}{5} \right)-\hbox{cos} \left(\frac{\pi}{5} \right)+\hbox{cos} \left(\frac{2\pi}{5} \right)-1}\\ M_2 \left(\theta \right)=&\frac{\hbox{cos} \left(\frac{2\pi}{5} \right)\hbox{cos}\, \theta \hbox{sin}\, \theta -\hbox{cos} \left(\frac{\pi}{5} \right)\hbox{sin}\, \theta +\hbox{cos} \left(\frac{\pi}{5} \right)\hbox{cos} \left(\frac{2\pi}{5} \right)\hbox{sin}\, \theta -\hbox{cos}\, \theta \hbox{sin}\, \theta}{\Uppsi}\\ &+\frac{\hbox{sin} \left(\frac{2\pi}{5} \right)\hbox{cos}^{2}\theta -\hbox{sin} \left(\frac{2\pi}{5} \right)\hbox{cos}\, \theta +\hbox{sin} \left(\frac{2\pi}{5} \right)\hbox{cos} \left(\frac{\pi}{5} \right)\hbox{cos}\, \theta -\hbox{sin} \left(\frac{2\pi}{5} \right)\hbox{cos} \left(\frac{\pi}{5} \right)}{\Uppsi}\\ M_3 \left(\theta \right)=&\frac{\hbox{cos} \left(\frac{2\pi}{5} \right)\hbox{cos} \left(\frac{\pi}{5} \right)\hbox{sin}\, \theta +\hbox{cos} \left(\frac{2\pi}{5} \right)\hbox{sin}\, \theta -\hbox{cos} \left(\frac{\pi}{5} \right)\hbox{cos}\, \theta \hbox{sin}\, \theta -\hbox{cos}\, \theta \hbox{sin}\, \theta}{\Upsigma}\\ &+\frac{\hbox{sin} \left(\frac{\pi}{5} \right)\hbox{cos}^{2}\theta -\hbox{sin} \left(\frac{\pi}{5} \right)\hbox{cos} \left(\frac{2\pi}{5} \right)\hbox{cos}\, \theta -\hbox{sin} \left(\frac{\pi}{5} \right)\hbox{cos}\, \theta +\hbox{sin} \left(\frac{\pi}{5} \right)\hbox{cos} \left(\frac{2\pi}{5} \right)}{\Upsigma}\\ M_4 \left(\theta \right)=&\frac{-\hbox{cos} \left(\frac{2\pi}{5} \right)\hbox{cos} \left(\frac{\pi}{5} \right)\hbox{sin}\, \theta -\hbox{cos} \left(\frac{2\pi}{5} \right)\hbox{sin}\, \theta +\hbox{cos} \left(\frac{\pi}{5} \right)\hbox{cos}\, \theta \hbox{sin}\, \theta +\hbox{cos}\, \theta \hbox{sin}\, \theta}{\Upsigma}\\ &+\frac{\hbox{sin} \left(\frac{\pi}{5} \right)\hbox{cos}^{2}\theta -\hbox{sin} \left(\frac{\pi}{5} \right)\hbox{cos} \left(\frac{2\pi}{5} \right)\hbox{cos}\, \theta -\hbox{sin} \left(\frac{\pi}{5} \right)\hbox{cos}\, \theta +\hbox{sin} \left(\frac{\pi}{5} \right)\hbox{cos} \left(\frac{2\pi}{5} \right)}{\Upsigma}\\ M_5 \left(\theta \right)=&\frac{-\hbox{cos} \left(\frac{2\pi}{5} \right)\hbox{cos} \left(\frac{\pi}{5} \right)\hbox{sin}\, \theta -\hbox{cos} \left(\frac{2\pi}{5} \right)\hbox{sin}\, \theta \hbox{cos}\, \theta +\hbox{cos} \left(\frac{\pi}{5} \right)\hbox{sin}\, \theta +\hbox{cos}\, \theta \hbox{sin}\, \theta}{\Uppsi}\\ &+\frac{\hbox{sin} \left(\frac{2\pi}{5} \right)\hbox{cos}^{2}\theta +\hbox{sin} \left(\frac{2\pi}{5} \right)\hbox{cos} \left(\frac{\pi}{5} \right)\hbox{cos}\, \theta -\hbox{sin} \left(\frac{2\pi}{5} \right)\hbox{cos}\, \theta -\hbox{sin} \left(\frac{2\pi}{5} \right)\hbox{cos} \left(\frac{\pi}{5} \right)}{\Uppsi} \end{aligned} $$

    where

    $$ \begin{aligned} \Uppsi =&2\hbox{sin} \left(\frac{2\pi}{5} \right)\left[ \hbox{cos}^{2}\left(\frac{2\pi}{5} \right)-\hbox{cos} \left(\frac{2\pi}{5} \right)+\hbox{cos} \left(\frac{2\pi}{5} \right)\hbox{cos} \left(\frac{\pi}{5} \right)-\hbox{cos} \left(\frac{\pi}{5} \right) \right]\\ \Upsigma =&4\hbox{sin} \left(\frac{\pi}{5} \right)\left[ \hbox{cos}^{2}\left(\frac{\pi}{5} \right)+\hbox{cos} \left(\frac{2\pi}{5} \right)+\hbox{cos} \left(\frac{2\pi}{5} \right)\hbox{cos} \left(\frac{\pi}{5} \right)+\hbox{cos} \left(\frac{\pi}{5} \right) \right] \end{aligned} $$

  • 6-noded element:

    $$ \begin{aligned} M_1 \left(\theta \right)=&-\frac{1}{6}-\frac{1}{6}\hbox{cos}\, \theta +\frac{2}{3}\hbox{cos}^{2}\theta +\frac{2}{3}\hbox{cos}^{3}\theta\\ M_2 \left(\theta \right)=&\frac{\sqrt{3}}{6}\hbox{sin}\, \theta +\frac{\sqrt{3}}{3}\hbox{sin}\, \theta \hbox{cos}\, \theta -\frac{1}{3}\hbox{cos}^{2}\theta -\frac{2}{3}\hbox{cos}^{3}\theta +\frac{2}{3}\hbox{cos}\, \theta +\frac{1}{3}\\ M_3 \left(\theta \right)=&\frac{\sqrt{3}}{6}\hbox{sin}\, \theta -\frac{\sqrt{3}}{3}\hbox{sin}\, \theta \hbox{cos}\, \theta -\frac{2}{3}\hbox{cos}\, \theta -\frac{1}{3}\hbox{cos}^{2}\theta +\frac{2}{3}\hbox{cos}^{3}\theta +\frac{1}{3}\\ M_4 \left(\theta \right)=&-\frac{1}{6}+\frac{1}{6}\hbox{cos}\, \theta +\frac{2}{3}\hbox{cos}^{2}\theta -\frac{2}{3}\hbox{cos}^{3}\theta\\ M_5 \left(\theta \right)=&-\frac{\sqrt{3}}{6}\hbox{sin}\, \theta +\frac{\sqrt{3}}{3}\hbox{sin}\, \theta \hbox{cos}\, \theta -\frac{2}{3}\hbox{cos}\, \theta -\frac{1}{3}\hbox{cos}^{2}\theta +\frac{2}{3}\hbox{cos}^{3}\theta +\frac{1}{3} \\ M_6 \left(\theta \right)=&-\frac{\sqrt{3}}{6}\hbox{sin}\, \theta -\frac{\sqrt{3}}{3}\hbox{sin}\, \theta \hbox{cos}\, \theta +\frac{2}{3}\hbox{cos}\, \theta -\frac{1}{3}\hbox{cos}^{2}\theta -\frac{2}{3}\hbox{cos}^{3}\theta +\frac{1}{3} \end{aligned} $$

The shape functions for 4, 5, and 6-noded elements were proposed in [13]. It is important to point out that all the functions describe exactly rigid body movements of the holes.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Buroni, F.C., Marczak, R.J. Effective properties of materials with random micro-cavities using special boundary elements. J Mater Sci 43, 3510–3521 (2008). https://doi.org/10.1007/s10853-008-2479-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10853-008-2479-3

Keywords

Search

Navigation