Research Articles

Tortuosity and the Averaging of Microvelocity Fields in Poroelasticity

[+] Author and Article Information
S. C. Cowin

e-mail: sccowin@gmail.com
Department of Biomedical Engineering,
City College of New York,
City University of New York,
New York, NY 10031;
New York Center for Biomedical Engineering,
City College of New York,
City University of New York,
New York, NY 10031;
Grove School of Engineering,
City College of New York,
City University of New York,
New York, NY 10031

Manuscript received August 23, 2011; final manuscript received March 17, 2012; accepted manuscript posted October 25, 2012; published online February 4, 2013. Assoc. Editor: Younane Abousleiman.

J. Appl. Mech 80(2), 020906 (Feb 04, 2013) (5 pages) Paper No: JAM-11-1302; doi: 10.1115/1.4007923 History: Received August 23, 2011; Revised March 17, 2012

The relationship between the macro- and microvelocity fields in a poroelastic representative volume element (RVE) has not being fully investigated. This relationship is considered to be a function of the tortuosity: a quantitative measure of the effect of the deviation of the pore fluid streamlines from straight (not tortuous) paths in fluid-saturated porous media. There are different expressions for tortuosity based on the deviation from straight pores, harmonic wave excitation, or from a kinetic energy loss analysis. The objective of the work presented is to determine the best expression for tortuosity of a multiply interconnected open pore architecture in an anisotropic porous media. The procedures for averaging the pore microvelocity over the RVE of poroelastic media by Coussy and by Biot were reviewed as part of this study, and the significant connection between these two procedures was established. Success was achieved in identifying the Coussy kinetic energy loss in the pore fluid approach as the most attractive expression for the tortuosity of porous media based on pore fluid viscosity, porosity, and the pore architecture. The fabric tensor, a 3D measure of the architecture of pore structure, was introduced in the expression of the tortuosity tensor for anisotropic porous media. Practical considerations for the measurement of the key parameters in the models of Coussy and Biot are discussed. In this study, we used cancellous bone as an example of interconnected pores and as a motivator for this study, but the results achieved are much more general and have a far broader application than just to cancellous bone.

Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.


Coussy, O., 1995, Mechanics of Porous Continua, Wiley, New York, Chap. 2.
Biot, M. A., 1962, “Mechanics of Deformation and Acoustic Propagation in Porous Media,” J. Appl. Phys., 33, pp. 1482–1498. [CrossRef]
Parfitt, A. M., Mathews, C. H. E., Villanueva, A. R., Kleerekoper, M., Frame, B., and Rao, D. S., 1983, “Relationships Between Surface, Volume, and Thickness of Iliac Trabecular Bone in Aging and in Osteoporosis. Implications for the Microanatomic and Cellular Mechanisms of Bone Loss,” J. Clin. Invest, 72, pp. 1396–1409. [CrossRef] [PubMed]
Rehman, M. T., Hoyland, J. A., Denton, J., and Freemont, A. J., 1994, “Age Related Histomorphometric Changes in Bone in Normal British Men and Women,” J. Clin. Pathol., 47(6), pp. 529–534. [CrossRef] [PubMed]
Hildebrand, T., Laib, A., Müller, R., Dequeker, J., and Rüegsegger, P., 1999, “Direct Three-Dimensional Morphometric Analysis of Human Cancellous Bone: Microstructural Data From Spine, Femur, Iliac Crest, and Calcaneus,” J. Bone Miner. Res., 14, pp. 1167–1174. [CrossRef] [PubMed]
Glorieux, F. H., Travers, R., Taylor, A., Bowen, J. R., Rauch, F., Norman, M., and Parfitt, A. M., 2000, “Normative Data for Iliac Bone Histomorphometry in Growing Children,” Bone, 26(2), pp. 103–109. [CrossRef] [PubMed]
Bear, J., 1972, Dynamics of Fluids in Porous Media, Dover, New York, Chap. 4.
Biot, M. A., 1941, “General Theory of Three-Dimensional Consolidation,” J. Appl. Phys., 12, pp. 155–164. [CrossRef]
Biot, M. A., 1956, “Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid. I. Low Frequency Range,” J. Acoust. Soc. Am., 28, pp. 168–178. [CrossRef]
Biot, M. A., 1956, “Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid. II. Higher Frequency Range,” J. Acoust. Soc. Am., 28, pp. 179–191. [CrossRef]
Biot, M. A., 1962, “Generalized Theory of Acoustic Propagation in Porous Dissipative Media,” J. Acoust. Soc. Am., 34, pp. 1254–1264. [CrossRef]
Carman, P. C., 1937, “Fluid Flow Through Granular Beds,” Trans. Inst. Chem. Eng., 15, pp. 150–156. [CrossRef]
Norris, A. N., 1986, “On the Viscoelastic Operator in Biot's Equations of Poroelasticity,” J. Wave-Mater. Interact., 1, pp. 365–380, http://rci.rutgers.edu/∼norris/papers.html
Costa, U. M. S., Andrade, J. S., Makse, H. A., and Stanley, H. E., 1999, “Inertial Effects on Fluid Flow Through Disordered Porous Media,” Phys. Rev. Lett., 82, pp. 5249–5252. [CrossRef]
Johnson, D. L., Koplik, J., and Dashen, R., 1987, “Theory of Dynamic Permeability and Tortuosity in Fluid-Saturated Porous Media,” J. Fluid Mech., 176, pp. 379–402. [CrossRef]
Perrot, C., Chevillotte, F., Panneton, R., Allard, J. F., and Lafarge, D., 2008, “On the Dynamic Viscous Permeability Tensor Symmetry,” J. Acoust. Soc. Am. Express Lett., 124, pp. 210–217. [CrossRef]
Darcy, H., 1856, Les Fontaines Publiques de la Ville de Dijon, Dalmont, Paris.
Hilliard, J. E., 1967, “Determination of Structural Anisotropy,” Stereology—Proc. 2nd Int. Congress for Stereology, Chicago, 1967, Springer, Berlin, p. 219.
Whitehouse, W. J., 1974, “The Quantitative Morphology of Anisotropic Trabecular Bone,” J. Microsc., 101, pp. 153–168. [CrossRef] [PubMed]
Whitehouse, W. J. and Dyson, E. D., 1974, “Scanning Electron Micro scope Studies of Trabecular Bone in The Proximal End of the Human Femur,” J. Anat., 118, pp. 417–444, http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1231543/ [PubMed]
Oda, M., 1976, “Fabrics and Their Effects on the Deformation Behaviors of Sand,” Special Report, Dept. of Foundation Engr., Saitama University, Saitama City, Japan, p. 59.
Oda, M., Konishi, J., and Nemat-NasserS., 1980, “Some Experimentally Based Fundamental Results on the Mechanical Behavior of Granular Materials,” Geotechnique, 30, pp. 479–495. [CrossRef]
Oda, M., Nemat-Nasser, S., and Konishi, J., 1985, “Stress Induced Anisotropy in Granular Masses,” Soils Found. 25, pp. 85–97. [CrossRef]
Cowin, S. C., and Satake, M., eds., 1978, Continuum Mechanical and Statistical Approaches in the Mechanics of Granular Materials, Gakujutsu Bunken Fukyu-Kai, Tokyo.
Satake, M., 1982, “Fabric Tensor in Granular Materials,” Deformation and Failure of Granular Materials, P. A.Vermeer and H. J.Lugar, eds., Balkema, Rotterdam, pp. 63–68.
Kanatani, K., 1983, “Characterization of Structural Anisotropy by Fabric Tensors and Their Statistical Test,” J. Jpn. Soil Mech. Found. Eng., 23, pp. 171–177. [CrossRef]
Kanatani, K., 1984, “Distribution of Directional Data and Fabric Tensors,” Int. J. Eng. Sci., 22, pp. 149–164. [CrossRef]
Kanatani, K., 1984, “Stereological Determination of Structural Anisotropy,” Int. J. Eng. Sci., 22, pp. 531–546. [CrossRef]
Kanatani, K., 1985, “Procedures for Stereological Estimation of Structural Anisotropy,” Int. J. Eng. Sci., 23(5), pp. 587–598. [CrossRef]
Harrigan, T. P., Jasty, M., Mann, R. W., and Harris, W. H., 1988, “Limitations of the Continuum Assumption in Cancellous Bone,” J. Biomech., 21, pp. 269–275. [CrossRef] [PubMed]
Odgaard, A., 1997, “Three-Dimensional Methods for Quantification of Cancellous Bone Architecture,” Bone, 20, pp. 315–328. [CrossRef] [PubMed]
Odgaard, A., Kabel, J., van Rietbergen, B., Dalstra, M., and Huiskes, R., 1997, “Fabric and Elastic Principal Directions of Cancellous Bone are Closely Related,” J. Biomech., 30, pp. 487–495. [CrossRef] [PubMed]
Odgaard, A., 2001, “Quantification of Cancellous Bone Architecture,” Bone Mechanics Handbook, S. C.Cowin, ed., CRC, Boca Raton, FL.
Matsuura, M., Eckstein, F., Lochmüller, E. M., and Zysset, P. K., 2008, “The Role of Fabric in the Quasi-Static Compressive Mechanical Properties of Human Trabecular Bone From Various Anatomical Locations,” Biomech. Model. Mechanobiol., 7, pp. 27–42. [CrossRef] [PubMed]
Cowin, S. C., 1985, “The Relationship Between the Elasticity Tensor and the Fabric Tensor,” Mech. Mater., 4, pp. 137–147. [CrossRef]
Cowin, S. C., 2004, “Anisotropic Poroelasticity: Fabric Tensor Formulation,” Mech. Mater., 36, pp. 666–677. [CrossRef]
Cardoso, L., and Cowin, S. C., 2011, “Fabric Dependence of Quasi-Waves in Anisotropic Porous Media,” J. Acoust. Soc. Am., 129, pp. 3302–3316. [CrossRef] [PubMed]
Cowin, S. C., and Cardoso, L., 2011, “Fabric Dependence of Poroelastic Wave Propagation in Anisotropic Porous Media,” Biomech. Model. Mechanobiol., 10, pp. 39–65. [CrossRef] [PubMed]


Grahic Jump Location
Fig. 1

An image of a cancellous bone section; the gray color areas represent the solid matrix portion and the black regions the pores

Grahic Jump Location
Fig. 2

A cartoon of an enlarged RVE for a continuum model of a porous medium



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In