A Linear Viscoelastic Biphasic Model for Soft Tissues Based on the Theory of Porous Media

[+] Author and Article Information
Wolfgang Ehlers, Bernd Markert

Institute of Applied Mechanics (Civil Engineering), University of Stuttgart, Stuttgart, Germany

J Biomech Eng 123(5), 418-424 (Apr 25, 2001) (7 pages) doi:10.1115/1.1388292 History: Received November 30, 1999; Revised April 25, 2001
Copyright © 2001 by ASME
Your Session has timed out. Please sign back in to continue.


Bowen, R. M., 1976, “Theory of Mixtures,” in: Continuum Physics, Vol. III, Eringen, A. C., ed., Academic Press, New York, pp. 1–127.
Bowen,  R. M., 1980, “Incompressible Porous Media Models by Use of the Theory of Mixtures,” Int. J. Eng. Sci., 18, pp. 1129–1148.
de Boer, R., and Ehlers, W., 1986, “Theorie der Mehrkomponentenkontinua mit Anwendung auf bodenmechanische Probleme,” Forschungsberichte aus dem Fachbereich Bauwesen, Vol. 40, Universität Essen, Essen.
Ehlers, W., 1989, “Poröse Medien—ein kontinuumsmechanisches Modell auf der Basis der Mischungstheorie,” Forschungsberichte aus dem Fachbereich Bauwesen, Vol. 47, Universität Essen, Essen.
Ehlers, W., 1993, “Constitutive Equations for Granular Materials in Geomechanical Context,” in: Continuum Mechanics in Environmental Sciences, CISM Courses and Lectures, Vol. 337, Hutter, K., ed., Springer-Verlag, Wien, pp. 313–402.
Ehlers,  W., 1996, “Grundlegende Konzepte in der Theorie Poröser Medien,” Technische Mechanik, 16, pp. 63–76.
Prévost,  P., 1982, “Nonlinear Transient Phenomena in Saturated Porous Media,” Comput. Methods Appl. Mech. Eng., 30, pp. 3–18.
Diebels,  S., and Ehlers,  W., 1996, “Dynamic Analysis of a Fully Saturated Porous Medium Accounting for Geometrical and Material Non-linearities,” Int. J. Numer. Methods Eng., 39, pp. 81–97.
Ehlers,  W., and Markert,  B., 2000, “On the Viscoelastic Behavior of Fluid-Saturated Porous Materials,” Granular Matter, 2, No. 3.
Ehlers, W., Diebels, S., Ellsiepen, P., and Volk, W., 1997, “Localization Phenomena in Liquid-Saturated Soils,” in: Proc. NAFEMS World Congress 1997, Stuttgart, pp. 287–298.
Ehlers,  W., and Volk,  W., 1998, “On Theoretical and Numerical Methods in the Theory of Porous Media Based on Polar and Non-polar Elasto-plastic Solid Materials,” Int. J. Solids Struct., 35, pp. 4597–4617.
Diebels, S., Ellsiepen, P., and Ehlers, W., 1998, “A Two-Phase Model for Viscoplastic Geomaterials,” in: Dynamics of Continua, Besdo, D., and Bogacz, R., eds., Shaker-Verlag, Aachen, pp. 103–112.
Bachrach,  N. M., Mow,  V. C., and Guilak,  F., 1998, “Incompressibility of the Solid Matrix of Articular Cartilage Under High Hydrostatic Pressures,” J. Biomech., 31, pp. 445–451.
Hayes,  W. C., and Bodine,  A. J., 1978, “Flow-Independent Viscoelastic Properties of Articular Cartilage Matrix,” J. Biomech., 11, pp. 407–419.
Mow,  V. C., Kuei,  S. C., Lai,  W. M., and Armstrong,  C. G., 1980, “Biphasic Creep and Stress Relaxation of Articular Cartilage in Compression: Theory and Experiments,” ASME J. Biomech. Eng., 102, pp. 73–84.
Woo,  S. L.-Y., Simon,  B. R., Kuei,  S. C., and Akeson,  W. H., 1980, “Quasi-linear Viscoelastic Properties of Normal Articular Cartilage,” ASME J. Biomech. Eng., 102, pp. 85–90.
Mow, V. C., Lai, W. M., and Holmes, M. H., 1982, “Advanced Theoretical and Experimental Techniques in Cartilage Research,” in: Biomechanics: Principles and Applications, Huiskes, R., Van Campen, D., and De Wijn, J., eds., Martinus Nijhoff Publishers, The Hague, pp. 47–74.
Mow, V. C., and Ratcliff, A., 1997, “Structure and Function of Articular Cartilage and Meniscus,” in: Basic Orthopaedic Biomechanics, 2nd ed., Mow, V. C., and Hayes, W. C., eds., Lipincott-Raven Publishers, Philadelphia, pp. 113–177.
Mak,  A. F., 1986, “The Apparent Viscoelastic Behavior of Articular Cartilage—The Contributions From the Intrinsic Matrix Viscoelasticity and Interstitial Fluid Flows,” ASME J. Biomech. Eng., 108, pp. 123–130.
Mak,  A. F., 1986, “Unconfined Compression of Hydrated Soft Viscoelastic Tissues: A Biphasic Poroviscoelastic Analysis,” Biorheology, 23, pp. 371–383.
Fung, Y. C., 1972, “Stress-Strain-History Relations of Soft Tissues in Simple Elongation,” in: Biomechanics: Its Foundations and Objectives, Fung, Y. C., Perrone, N., and Anliker, M., eds., Prentice-Hall, Englewood Cliffs, NJ, pp. 181–208.
Suh,  J.-K., and Bai,  S., 1998, “Finite Element Formulation of Biphasic Poroviscoelastic Model of Articular Cartilage,” ASME J. Biomech. Eng., 120, pp. 195–201.
Coleman,  B. D., and Gurtin,  M. E., 1967, “Thermodynamics With Internal State Variables,” J. Chem. Phys., 47, pp. 597–613.
Le Tallec,  P., Rahier,  C., and Kaiss,  A., 1993, “Three-Dimensional Incompressible Viscoelasticity in Large Strains: Formulation and Numerical Approximation,” Comput. Methods Appl. Mech. Eng., 109, pp. 133–258.
Reese,  S., and Govindjee,  S., 1998, “A Theory of Finite Viscoelasticity and Numerical Aspects,” Int. J. Solids Struct., 35, pp. 3455–3482.
Spilker,  R. L., and Suh,  J.-K., 1990, “Formulation and Evaluation of a Finite Element Model for the Biphasic Model of Hydrated Soft Tissues,” Comput. Struct., 35, pp. 425–439.
Diebels,  S., Ellsiepen,  P., and Ehlers,  W., 1999, “Error-Controlled Runge–Kutta Time Integration of a Viscoplastic Hybrid Two-Phase Model,” Technische Mechanik, 19, pp. 19–27.
Brenan, K. E., Campbell, S. L., and Petzold, L. R., 1989, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, North-Holland, New-York.
Hairer, E., and Wanner, G., 1991, Solving Ordinary Differential Equations II—Stiff and Differential-Algebraic Problems, Springer-Verlag, Berlin.
Hassanizadeh,  S. M., and Gray,  W. G., 1979, “General Conservation Equations for Multi-Phase-Systems: 2. Mass, Momenta, Energy and Entropy Equations,” Adv. Water Resour., 2, pp. 191–203.
de Boer, R., Ehlers, W., Kowalski, S., and Plischka, J., 1991, “Porous Media—A Survey of Different Approaches,” Forschungsberichte aus dem Fachbereich Bauwesen, Vol. 54, Universität Essen, Essen.
Eipper, G., 1998, “Theorie und Numerik finiter elastischer Deformationen in fluidgesättigten porösen Festkörpern,” Dissertation, Bericht No. II-1 des Instituts für Mechanik (Bauwesen), Universität Stuttgart, Stuttgart.
Tschoegl, N. W., 1989, The Phenomenological Theory of Linear Viscoelastic Behavior: An Introduction, Springer-Verlag, New York.
Ehlers,  W., and Ellsiepen,  P., 1997, “Zeitschrittgesteuerte Verfahren bei stark gekoppelten Festkörper-Fluid-Problemen,” Z. Angew. Math. Mech., 77, pp. S81–S82.
Ellsiepen, P., 1999, “Zeit- und ortsadaptive Verfahren angewandt auf Mehrphasenprobleme poröser Medien,” Dissertation, Bericht Nr. II-3 des Instituts für Mechanik (Bauwesen), Universität Stuttgart, Stuttgart.
Hambly,  E. C., 1969, “A New True Triaxial Apparatus,” Geotechnique, 19, pp. 307–309.
Lade,  P. V., and Duncan,  J. M., 1973, “Cubical Triaxial Tests on Cohesionless Soils,” J. Geotech. Eng., 99, pp. 793–812.
Zhu,  W. B., Lai,  W. M., and Mow,  V. C., 1986, “Intrinsic Quasi-linear Viscoelastic Behavior of the Extracellular Matrix of Cartilage,” Trans. Annu. Meet. — Orthop. Res. Soc., 11, p. 407.
Mow,  V. C., Gibbs,  M. C., Lai,  W. M., Zhu,  W. B., and Athanasiou,  K. A., 1989, “Biphasic Indentation of Articular Cartilage—II. A Numerical Algorithm and an Experimental Study,” J. Biomech., 22, pp. 853–861.
Setton,  L. A., Zhu,  W., and Mow,  V. C., 1993, “The Biphasic Poroviscoelastic Behavior of Articular Cartilage: Role of the Surface Zone in Governing the Compressive Behavior,” J. Biomech., 26, pp. 581–592.
Suh,  J.-K., and DiSilvestro,  M. R., 1999, “Biphasic Poroviscoelastic Behavior of Hydrated Biological Soft Tissue,” ASME J. Appl. Mech., 66, pp. 528–535.


Grahic Jump Location
REV of cartilage with real microstructure consisting of a collagen–proteoglycan matrix saturated with interstitial fluid 18 and biphasic TPM macro model. The initial volume fractions, i.e., solidity and porosity, for cartilage are approximately n0SS=0.25 and n0SF=0.75.
Grahic Jump Location
Variation of the Darcy permeability parameter kF versus the compressive volumetric strain εV for different values of κ. The initial values n0SS=0.25,n0SF=0.75, and k0F=2.17×10−11 m/s are average values of human patellar cartilage.
Grahic Jump Location
Generalized Maxwell model: σESEQSNEQS denotes the solid extra stress with the equilibrium part σEQS and the nonequilibrium part σNEQS=ΣσnSS=(εSe)n+(εSi)n is the corresponding solid strain.
Grahic Jump Location
Discretization with triangles or tetrahedrals
Grahic Jump Location
Definition of initial boundary value problems: (a) Infinitesimal torsion test. A cylindrical sample (h=1.5 mm,d=6.0 mm) is twisted by a jump to φ0=0.126 rad equal to a shear strain of 2.0 percent. (b) Hydrostatic compression test. A cubic specimen (a=3.0 mm) is charged displacement driven with a compressive volumetric strain εV=−5percent within 5.0 s. (c) Axisymmetric indentation test. An axisymmetric disc (h=3.5 mm,r=7.0 mm) is loaded displacement driven about 5.0 percent of height within 5.0 s with a porous permeable indenter (ri=1.0 mm).
Grahic Jump Location
Tangential displacement (left) and torsional moment of the computed infinitesimal torsion test (right). Squares mark the time steps of the time-adaptive computation.
Grahic Jump Location
Pore-fluid pressure and seepage velocity (arrows) after 5.0×106 seconds of the computed consolidation process of the hydrostatic compression test
Grahic Jump Location
Progression of hydrostatic pressure for short (left) and long time scales (right). Squares mark the time steps of the time-adaptive computation.
Grahic Jump Location
The left figure shows the influence of the deformation dependent permeability on the computed indenter reaction force. The right figure shows the decreasing influence of the skeleton viscoelasticity by increasing the loading time from 5.0 s up to 250 s.




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