Skip to main content
Log in

Boundary element analysis for effective stiffness tensors: effect of fabric tensor determination method

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

Second-rank fabric tensors have been extensively used to describe structural anisotropy and to predict orthotropic elastic constants. However, there are many different definitions of, and approaches to, determining the fabric tensor. Most commonly used is a fabric tensor based on mean intercept length measurements, but star volume distribution and star length distribution are commonly used, particularly in studies of trabecular bone. Here, we investigate the effect of the fabric tensor definition on elastic constant predictions using both synthetic, idealized microstructures as well as a micrograph of a porous ceramic. We use an efficient implantation of a symmetric Galerkin boundary element method to model the mechanical response of the various microstructures, and also use a boundary element approach to calculate the necessary volume averages of stress and strain to obtain the effective properties of the media.

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

Access this article

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. Benn DI (1994) Fabric shape and the interpretation of sedimentary fabric data. J Sediment Res A64: 910–915

    Google Scholar 

  2. Berger JR (2011) Fabric tensor based boundary element analysis of porous solids. Eng Anal Boundary Elem 35: 430–435

    Article  Google Scholar 

  3. Cowin SC (1985) The relationship between the elasticity tensor and the fabric tensor. Mech Mater 4: 137–147

    Article  Google Scholar 

  4. Cruz-Orive LM, Karlsson LM, Larsen SE (1992) Characterizing anisotropy: a new concept. Micron Microscop Acta 23: 75–76

    Article  Google Scholar 

  5. Elmabrouk B, Berger JR, Phan A-V, Gray LJ (2011) Apparent stiffness tensors for porous solids using symmetric Galerkin boundary elements. Comput Mech. doi:10.1007/s00466-011-0650-1

  6. Fisher NI, Lewis TL, Embleton BJ (1987) Statistical analysis of spherical data. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  7. Gibson LJ, Ashby MF (1999) Cellular solids: structure and properties, Cambridge solid state science series. 2 edn Cambridge University Press, Cambridge

    Google Scholar 

  8. Harrigan TP, Mann RW (1984) Characterization of microstructural anisotropy in orthotropic materials using a second rank tensor. J Mater Sci 19: 761–767

    Article  Google Scholar 

  9. Hashin Z (1965) On elastic behaviour of fibre reinforced materials of arbitrary transverse phase geometry. J Mech Phys Solids 13: 119–134

    Article  Google Scholar 

  10. Hashin Z, Shtrikman S (1963) A variational approach to the theory of the elastic behaviour of multiphase materials. J Mech Phys Solids 11: 127–140

    Article  MathSciNet  MATH  Google Scholar 

  11. Hazanov S, Amieur M (1995) On overall properties of elastic heterogeneous bodies smaller than the representative volume. Int J Eng Sci 33: 1289–1301

    Article  MATH  Google Scholar 

  12. Hazanov S, Huet C (1994) Order relationships for boundary conditions effect in heterogeneous bodies smaller than the representative volume. J Mech Phys Solids 42: 1995–2011

    Article  MathSciNet  MATH  Google Scholar 

  13. Hill R (1963) Elastic properties of reinforced solids: some theoretical principles. J Mech Phys Solids 11: 357–372

    Article  MATH  Google Scholar 

  14. Hollister SJ, Kikuchi N (1992) A comparison of homogenization and standard mechanics analyses for periodic porous composites. Comput Mech 10: 73–95

    Article  MATH  Google Scholar 

  15. Huet C (1990) Application of variational concepts to size effects in elastic heterogeneous bodies. J Mech Phys Solids 38: 813–841

    Article  MathSciNet  Google Scholar 

  16. Kanatani K (1984) Distribution of directional data and fabric tensors. Int J Eng Sci 22: 149–164

    Article  MathSciNet  MATH  Google Scholar 

  17. Ketcham RT, Ryan TM (2004) Quantification and visualization of anisotropy in trabecular bone. J Microscop 213: 158–171

    Article  MathSciNet  Google Scholar 

  18. Ketcham RT (2005) Three-dimensional textural measurements using high-resolution X-ray computed tomography. J Struct Geol 27: 1217–1228

    Article  Google Scholar 

  19. Kröner E (1958) Berechnung der elastischen konstanten des vielkristalls aus den konstanten des einkristalls. Zeitschrift für Physik 151: 504–518

    Article  Google Scholar 

  20. Oda M (1983) A method for evaluating the effect of crack geometry on the mechanical behavior of cracked rock masses. Mech Mater 2: 163–171

    Article  MathSciNet  Google Scholar 

  21. Oda M (1982) Fabric tensor for discontinuous geological materials. Soils Found 22: 96–108

    Article  Google Scholar 

  22. Odgaard A, Kabel J, van Rietbergen B, Dalstra M, Huiskes R (1997) Fabric and elastic principal directions of cancellous bone are closely related. J Biomech 30: 487–495

    Article  Google Scholar 

  23. Odgaard A, Jensen EB, Gundersen HJG (1990) Estimation of structural anisotropy based on volume orientation: a new concept. J Microscop 159: 335–342

    Article  Google Scholar 

  24. Pahr DH, Zysset PK (2008) Influence of boundary conditions on computed apparent elastic properties of cancellous bone. Biomech Model Mechanobiol 7: 463–476

    Article  Google Scholar 

  25. Pindera MJ, Khatam H, Drago AS, Bansal Y (2009) Micromechanics of spatially uniform heterogeneous media: a critical review. Compos Part B 40: 349–378

    Article  Google Scholar 

  26. Saha PK, Wehrli FW (2004) A robust method for measuring trabecular bone orientation anisotropy at in vivo resolution using tensor scale. Pattern Recognit 37: 1935–1944

    Article  Google Scholar 

  27. Smit TH, Schneider E, Odgaard A (1998) Star length distribution: a volume-based concept for the characterization of structural anisotropy. J Microscop 191: 249–257

    Article  Google Scholar 

  28. Stroh AN (1958) Dislocations and cracks in anisotropic elasticity. Philos Mag 3: 625–646

    Article  MathSciNet  MATH  Google Scholar 

  29. Sutradhar A, Paulino v, Gray LJ (2008) The symmetric Galerkin boundary element method. Springer, Berlin

    Google Scholar 

  30. Tabor Z (2011) Equivalence of mean intercept length and gradient fabric tensors: 3d study. Med Eng Phys (in press)

  31. Tabor Z (2009) On the equivalence of two methods of determining fabric tensor. Med Eng Phys 31: 1313–1322

    Article  Google Scholar 

  32. Turner CH, Cowin SC (1987) Dependence of elastic constants of an anisotropic porous material upon porosity and fabric. J Mater Sci 22: 3178–3184

    Article  Google Scholar 

  33. Turner CH, Cowin SC (1988) Errors induced by off-axis measurements of the elastic properties of bone. J Biomech Eng 110: 213–215

    Article  Google Scholar 

  34. Zeng T, Dong X, Mao C, Zhou Z, Yang H (2007) Effect of pore shape and porosity on the properties of porous PZT 95/5 ceramics. J Eur Ceram Soc 27: 2025–2029

    Article  Google Scholar 

  35. Zysset PK, Curnier A (1995) An alternative model for anisotropic elasticity based on fabric tensors. Mech Mater 21: 243–250

    Article  Google Scholar 

  36. Zysset PK, Goulet RW, Hollister SJ (1998) A global relationship between trabecular bone morphology and homogenized elastic properties. J Biomech Eng 120: 640–646

    Article  Google Scholar 

  37. Zysset PK (2003) A review of morphology-elasticity relationships in human trabecular bone: theories and experiments. J Biomech 36: 1469–1485

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to J. R. Berger.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Elmabrouk, B., Berger, J.R. Boundary element analysis for effective stiffness tensors: effect of fabric tensor determination method. Comput Mech 51, 391–398 (2013). https://doi.org/10.1007/s00466-012-0753-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-012-0753-3

Keywords

Navigation