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On the permeability of fibre-reinforced porous materials. (English) Zbl 1151.74016
Summary: Biological tissues can be considered as composite materials comprised of a porous matrix filled with interstitial fluid and reinforced by impermeable collagen fibres. Motivated by studies on fluid flow in articular cartilage, we would like to quantify the undeformed configuration permeability of fibre-reinforced composite materials. If there is a sufficient scale separation between the internal structure of porous matrix and the arrangement of fibres, the matrix can be taken as a porous continuum at the fibre scale. In this case, the fibres can be treated as inclusions in porous continuum, and the overall permeability of the composite can be evaluated using homogenisation procedures. For an isotropic homogeneous matrix, the symmetry of the system is governed by the orientation of fibres. Here, we propose to retrieve the overall permeability through geometrical considerations and directional averaging methods. The special case of transverse isotropy is discussed in detail, with particular attention to the sub-cases of aligned fibres and fibres lying on a plane.

74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74E30 Composite and mixture properties
74Q15 Effective constitutive equations in solid mechanics
74L15 Biomechanical solid mechanics
92C10 Biomechanics
Full Text: DOI
[1] Dormieux, L.; Kondo, D.; Ulm, F. -J.: Microporomechanics, (2006) · Zbl 1112.76002
[2] Farquhar, T.; Dawson, P. R.; Torzilli, P. A.: A microstructural model for the anisotropic drained stiffness of articular cartilage, Journal of biomechanical engineering 112, 414-425 (1990)
[3] Federico, S.; Grillo, A.; Herzog, W.: A transversely isotropic composite with a statistical distribution of spheroidal inclusions: a geometrical approach to overall properties, Journal of the mechanics and physics of solids 52, No. 10, 2309-2327 (2004) · Zbl 1115.74358 · doi:10.1016/j.jmps.2004.03.010
[4] Federico, S.; Grillo, A.; La Rosa, G.; Giaquinta, G.; Herzog, W.: A transversely isotropic, transversely homogeneous microstructural-statistical model of articular cartilage, Journal of biomechanics 38, No. 10, 2008-2018 (2005)
[5] Federico, S., Herzog, W., in press. On the anisotropy and inhomogeneity of permeability in articular cartilage. Biomechanics and Modeling in Mechanobiology, http://dx.doi.org/10.1007/s10237-007-0091-0.
[6] Fung, Y. C.: Biomechanics — mechanical properties of living tissues, (1993)
[7] Gasser, T. C.; Holzapfel, G. A.; Ogden, R. W.: Hyperelastic modelling of arterial layers with distributed collagen fibre orientations, Journal of the royal society interface 3, 15-35 (2006)
[8] Han, S.; Gemmell, S. J.; Helmer, K. G.; Grigg, P.; Wellen, J. W.; Hoffman, A. H.; Sotak, C. H.: Changes in ADC caused by tensile loading of rabbit achilles tendon: evidence for water transport, Journal of magnetic resonance 144, No. 2, 217-227 (2000)
[9] Hedlund, H.; Mengarelli-Widholm, S.; Reinholt, F.; Svensson, O.: Stereological studies on collagen in bovine articular cartilage, Acta pathologica, microbiologica et immunologica scandinavica 101, 133-140 (1993)
[10] Holmes, M. H.; Mow, V. C.: The non-linear characteristics of soft gels and hydrated connective tissues in ultrafiltration, Journal of biomechanics 23, 1145-1156 (1990)
[11] Holzapfel, G. A.; Gasser, T. C.; Ogden, R. W.: A new constitutive framework for arterial wall mechanics and a comparative study of material models, Journal of elasticity 61, 1-48 (2000) · Zbl 1023.74033 · doi:10.1023/A:1010835316564
[12] Kolmogorov, A. N.; Fomin, S. V.: Elements of the theory of functions and functional analysis, (1999) · Zbl 0235.46001
[13] Landau, L. D.; Lifshitz, E. M.: Electrodynamics of continuous media, (1960) · Zbl 0122.45002
[14] Lanir, Y.; Lichtenstein, O.; Imanuel, O.: Optimal design of biaxial tests for structural material characterization of flat tissues, Journal of biomechanical engineering 118, 41-47 (1996)
[15] Långsjö, T. K.; Hyttinen, M.; Pelttari, A.; Kiraly, K.; Arokoski, J.; Helminen, H. J.: Electron microscopic stereological study of collagen fibrils in bovine articular cartilage: volume and surface densities are best obtained indirectly (from length densities and diameters) using isotropic uniform random sampling, Journal of anatomy 195, 281-293 (1999)
[16] Maroudas, A.; Bullough, P.: Permeability of articular cartilage, Nature 219, 1260-1261 (1968)
[17] Mclaughlin, R.: A study of the differential scheme for composite materials, International journal of engineering science 15, 237-244 (1977) · Zbl 0349.73050 · doi:10.1016/0020-7225(77)90058-1
[18] Nicholson, C.: Diffusion and related transport mechanisms in brain tissue, Reports on progress in physics 64, 812-884 (2001)
[19] Norris, A. N.: A differential scheme for the effective moduli of composites, Mechanics of materials 4, 1-16 (1985)
[20] Ogden, R. W.: Nonlinear elasticity, anisotropy, material stability and residual stresses in soft tissue, Biomechanics of soft tissue in cardiovascular systems, CISM courses and lectures series no. 441, 65-108 (2003) · Zbl 1151.74386
[21] Podzniakov, S.; Tsang, C. -F.: A self-consistent approach for calculating the effective hydraulic conductivity of a binary, heterogeneous medium, Water resources research 40, 1-13 (2004)
[22] Quinn, T. M.; Dierickx, P.; Grodzinsky, A. J.: Glycosaminoglycan network geometry May contribute to anisotropic hydraulic permeability in cartilage under compression, Journal of biomechanics 34, 1483-1490 (2001)
[23] Pollack, G. H.: Cells, gels and the engines of life, (2001)
[24] Rajagopal, K. R.: Diffusion through polymeric solids undergoing large deformations, Materials science and technology 19, 1175-1180 (2003)
[25] Shvidler, M. I.: Stochastic hydrodynamics of porous medium (in russian), (1985) · Zbl 0574.76098
[26] Soulhat, J.; Buschmann, M. D.; Shirazi-Adl, A.: A fibril-network-reinforced biphasic model of cartilage in unconfined compression, Journal of biomechanical engineering 121, 150-159 (2000)
[27] Walpole, L. J.: Elastic behavior of composite materials: theoretical foundations, Advances in applied mechanics 21, 169-242 (1981) · Zbl 0512.73056 · doi:10.1016/S0065-2156(08)70332-6
[28] Wellen, J.; Helmer, K. G.; Grigg, P.; Sotak, C. H.: Application of porous-media theory to the investigation of water ADC changes in rabbit achilles tendon caused by tensile loading, Journal of magnetic resonance 170, 49-55 (2004)
[29] Zimmerman, R. W.: Elastic moduli of a solid containing spherical inclusions, Mechanics of materials 12, 17-24 (1991)
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