Research Papers

Multiphasic Finite Element Framework for Modeling Hydrated Mixtures With Multiple Neutral and Charged Solutes

[+] Author and Article Information
Gerard A. Ateshian

Department of Mechanical Engineering,
Columbia University,
New York, NY 10027

Jeffrey A. Weiss

Department of Bioengineering,
University of Utah,
Salt Lake City, UT 84112

A steady-state analysis is performed by setting time derivatives to zero in Eq. (2.15). This option is available in febio by setting a switch in the input file.

Contributed by the Bioengineering Division of ASME for publication in the Journal of Biomechanical Engineering. Manuscript received October 17, 2012; final manuscript received June 2, 2013; accepted manuscript posted June 17, 2013; published online September 23, 2013. Assoc. Editor: Mohammad Mofrad.

J Biomech Eng 135(11), 111001 (Sep 23, 2013) (11 pages) Paper No: BIO-12-1493; doi: 10.1115/1.4024823 History: Received October 17, 2012; Revised June 02, 2013; Accepted June 17, 2013

Computational tools are often needed to model the complex behavior of biological tissues and cells when they are represented as mixtures of multiple neutral or charged constituents. This study presents the formulation of a finite element modeling framework for describing multiphasic materials in the open-source finite element software febio.1 Multiphasic materials may consist of a charged porous solid matrix, a solvent, and any number of neutral or charged solutes. This formulation proposes novel approaches for addressing several challenges posed by the finite element analysis of such complex materials: The exclusion of solutes from a fraction of the pore space due to steric volume and short-range electrostatic effects is modeled by a solubility factor, whose dependence on solid matrix deformation and solute concentrations may be described by user-defined constitutive relations. These solute exclusion mechanisms combine with long-range electrostatic interactions into a partition coefficient for each solute whose value is dependent upon the evaluation of the electric potential from the electroneutrality condition. It is shown that this electroneutrality condition reduces to a polynomial equation with only one valid root for the electric potential, regardless of the number and valence of charged solutes in the mixture. The equation of charge conservation is enforced as a constraint within the equation of mass balance for each solute, producing a natural boundary condition for solute fluxes that facilitates the prescription of electric current density on a boundary. It is also shown that electrical grounding is necessary to produce numerical stability in analyses where all the boundaries of a multiphasic material are impermeable to ions. Several verification problems are presented that demonstrate the ability of the code to reproduce known or newly derived solutions: (1) the Kedem–Katchalsky model for osmotic loading of a cell; (2) Donnan osmotic swelling of a charged hydrated tissue; and (3) current flow in an electrolyte. Furthermore, the code is used to generate novel theoretical predictions of known experimental findings in biological tissues: (1) current-generated stress in articular cartilage and (2) the influence of salt cation charge number on the cartilage creep response. This generalized finite element framework for multiphasic materials makes it possible to model the mechanoelectrochemical behavior of biological tissues and cells and sets the stage for the future analysis of reactive mixtures to account for growth and remodeling.

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


Truesdell, C., and Toupin, R., 1960, The Classical Field Theories, (Handbuch der Physik), Vol. III/1, Springer, Heidelberg.
Bowen, R., 1976, Theory of Mixtures, (Continuum Physics), Vol. 3, Academic, New York.
Kenyon, D. E., 1976, “Transient Filtration in a Porous Elastic Cylinder,” ASME J. Appl. Mech., 43(4), pp. 594–598. [CrossRef]
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(1), pp. 73–84. [CrossRef]
Armstrong, C. G., Lai, W. M., and Mow, V. C., 1984, “An Analysis of the Unconfined Compression of Articular Cartilage,” ASME J. Biomech. Eng., 106(2), pp. 165–173. [CrossRef]
Holmes, M. H., and Mow, V. C., 1990, “The Nonlinear Characteristics of Soft Gels and Hydrated Connective Tissues in Ultrafiltration,” J. Biomech., 23(11), pp. 1145–1156. [CrossRef] [PubMed]
Lai, W. M., Hou, J. S., and Mow, V. C., 1991, “A Triphasic Theory for the Swelling and Deformation Behaviors of Articular Cartilage,” ASME J. Biomech. Eng., 113(3), pp. 245–258. [CrossRef]
Huyghe, J. M., and Janssen, J. D., 1997, “Quadriphasic Mechanics of Swelling Incompressible Porous Media,” Int. J. Eng. Sci., 35(8), pp. 793–802. [CrossRef]
Gu, W. Y., Lai, W. M., and Mow, V. C., 1993, “Transport of Fluid and Ions Through a Porous-Permeable Charged-Hydrated Tissue, and Streaming Potential Data on Normal Bovine Articular Cartilage,” J. Biomech., 26(6), pp. 709–723. [CrossRef] [PubMed]
Gu, W. Y., Lai, W. M., and Mow, V. C., 1997, “A Triphasic Analysis of Negative Osmotic Flows Through Charged Hydrated Soft Tissues,” J. Biomech., 30(1), pp. 71–78. [CrossRef] [PubMed]
Lai, W. M., Mow, V. C., Sun, D. D., and Ateshian, G. A., 2000, “On the Electric Potentials Inside a Charged Soft Hydrated Biological Tissue: Streaming Potential Versus Diffusion Potential,” ASME J. Biomech. Eng., 122(4), pp. 336–346. [CrossRef]
Sun, D. D., Guo, X. E., Likhitpanichkul, M., Lai, W. M., and Mow, V. C., 2004, “The Influence of the Fixed Negative Charges on Mechanical and Electrical Behaviors of Articular Cartilage Under Unconfined Compression,” ASME J. Biomech. Eng., 126(1), pp. 6–16. [CrossRef]
Lu, X. L., Sun, D. D. N., Guo, X. E., Chen, F. H., Lai, W. M., and Mow, V. C., 2004, “Indentation Determined Mechanoelectrochemical Properties and Fixed Charge Density of Articular Cartilage,” Ann. Biomed. Eng., 32(3), pp. 370–379. [CrossRef] [PubMed]
Wan, L. Q., Guo, X. E., and Mow, V. C., 2010, “A Triphasic Orthotropic Laminate Model for Cartilage Curling Behavior: Fixed Charge Density Versus Mechanical Properties Inhomogeneity,” ASME J. Biomech. Eng., 132(2), p. 024504. [CrossRef]
Likhitpanichkul, M., Guo, X. E., and Mow, V. C., 2005, “The Effect of Matrix Tension-Compression Nonlinearity and Fixed Negative Charges on Chondrocyte Responses in Cartilage,” Mol. Cell Biomech., 2(4), pp. 191–204. [CrossRef] [PubMed]
Haider, M. A., Schugart, R. C., Setton, L. A., and Guilak, F., 2006, “A Mechano-Chemical Model for the Passive Swelling Response of an Isolated Chondron Under Osmotic Loading,” Biomech. Model. Mechanobiol., 5(2-3), pp. 160–171. [CrossRef] [PubMed]
Frijns, A. J. H., Huyghe, J. M., and Janssen, J. D., 1997, “Validation of the Quadriphasic Mixture Theory for Intervertebral Disc Tissue,” Int. J. Eng. Sci., 35(15), pp. 1419–1429. [CrossRef]
Huyghe, J. M., Houben, G. B., Drost, M. R., and van Donkelaar, C. C., 2004, “An Ionised/Nonionised Dual Porosity Model of Intervertebral Disc Tissue: Experimental Quantification of Parameters,” Biomech. Model. Mechanobiol., 2(4), pp. 3–19. [CrossRef]
Azeloglu, E. U., Albro, M. B., Thimmappa, V. A., Ateshian, G. A., and Costa, K. D., 2008, “Heterogeneous Transmural Proteoglycan Distribution Provides a Mechanism for Regulating Residual Stresses in the Aorta,” Am. J. Physiol. Heart Circ. Physiol., 294(3), pp. H1197–H1205. [CrossRef] [PubMed]
Bryant, M. R., and McDonnell, P. J., 1998, “A Triphasic Analysis of Corneal Swelling and Hydration Control,” ASME J. Biomech. Eng., 120(3), pp. 370–381. [CrossRef]
Elkin, B. S., Shaik, M. A., and Morrison, B., 3rd, 2010, “Fixed Negative Charge and the Donnan Effect: A Description of the Driving Forces Associated With Brain Tissue Swelling and Oedema,” Philos. Trans. R. Soc. London, 368(1912), pp. 585–603. [CrossRef]
Ateshian, G. A., Likhitpanichkul, M., and Hung, C. T., 2006, “A Mixture Theory Analysis for Passive Transport in Osmotic Loading of Cells,” J. Biomech., 39(3), pp. 464–475. [CrossRef] [PubMed]
Albro, M. B., Chahine, N. O., Caligaris, M., Wei, V. I., Likhitpanichkul, M., Ng, K. W., Hung, C. T., and Ateshian, G. A., 2007, “Osmotic Loading of Spherical Gels: A Biomimetic Study of Hindered Transport in the Cell Protoplasm,” ASME J. Biomech. Eng., 129(4), pp. 503–510. [CrossRef]
Albro, M. B., Petersen, L. E., Li, R., Hung, C. T., and Ateshian, G. A., 2009, “Influence of the Partitioning of Osmolytes by the Cytoplasm on the Passive Response of Cells to Osmotic Loading,” Biophys. J., 97(11), pp. 2886–2893. [CrossRef] [PubMed]
Mauck, R. L., Hung, C. T., and Ateshian, G. A., 2003, “Modeling of Neutral Solute Transport in a Dynamically Loaded Porous Permeable Gel: Implications for Articular Cartilage Biosynthesis and Tissue Engineering,” ASME J. Biomech. Eng., 125(5), pp. 602–614. [CrossRef]
Albro, M. B., Li, R., Banerjee, R. E., Hung, C. T., and Ateshian, G. A., 2010, “Validation of Theoretical Framework Explaining Active Solute Uptake in Dynamically Loaded Porous Media,” J. Biomech., 43(12), pp. 2267–2273. [CrossRef] [PubMed]
Gu, W. Y., Lai, W. M., and Mow, V. C., 1998, “A Mixture Theory for Charged-Hydrated Soft Tissues Containing Multi-electrolytes: Passive Transport and Swelling Behaviors,” ASME J. Biomech. Eng., 120(2), pp. 169–180. [CrossRef]
Spilker, R. L., Suh, J. K., and Mow, V. C., 1990, “Effects of Friction on the Unconfined Compressive Response of Articular Cartilage: A Finite Element Analysis,” ASME J. Biomech. Eng., 112(2), pp. 138–146. [CrossRef]
Spilker, R. L., Suh, J. K., and Mow, V. C., 1992, “A Finite Element Analysis of the Indentation Stress-Relaxation Response of Linear Biphasic Articular Cartilage,” ASME J. Biomech. Eng., 114(2), pp. 191–201. [CrossRef]
Suh, J. K., and Spilker, R. L., 1994, “Indentation Analysis of Biphasic Articular Cartilage: Nonlinear Phenomena Under Finite Deformation,” ASME J. Biomech. Eng., 116(1), pp. 1–9. [CrossRef]
Almeida, E. S., and Spilker, R. L., 1997, “Mixed and Penalty Finite Element Models for the Nonlinear Behavior of Biphasic Soft Tissues in Finite Deformation: Part I—Alternate Formulations,” Comput. Methods Biomech. Biomed. Eng., 1(1), pp. 25–46. [CrossRef]
Biot, M., 1941, “General Theory of 3-Dimensional Consolidation,” J. Appl. Phys., 12, pp. 155–164. [CrossRef]
Bowen, R., 1980, “Incompressible Porous Media Models by Use of the Theory of Mixtures,” Int. J. Eng. Sci., 18(9), pp. 1129–1148. [CrossRef]
Mow, V. C., and Lai, W. M., 1980, “Recent Developments in Synovial Joint Biomechanics,” SIAM Rev., 22(3), pp. 275–317. [CrossRef]
Simon, B. R., Liable, J. P., Pflaster, D., Yuan, Y., and Krag, M. H., 1996, “A Poroelastic Finite Element Formulation Including Transport and Swelling in Soft Tissue Structures,” ASME J. Biomech. Eng., 118(1), pp. 1–9. [CrossRef]
Sun, D. N., Gu, W. Y., Guo, X. E., Lai, W. M., and Mow, V. C., 1999, “A Mixed Finite Element Formulation of Triphasic Mechano-Electrochemical Theory for Charged, Hydrated Biological Soft Tissues,” Int. J. Numer. Methods Eng., 45(10), pp. 1375–1402. [CrossRef]
Kaasschieter, E. F., Frijns, A. J. H., and Huyghe, J. M., 2003, “Mixed Finite Element Modelling of Cartilaginous Tissues,” Math. Comput. Simul., 61(3-6), pp. 549–560. [CrossRef]
Yao, H., and Gu, W. Y., 2007, “Three-Dimensional Inhomogeneous Triphasic Finite-Element Analysis of Physical Signals and Solute Transport in Human Intervertebral Disc Under Axial Compression,” J. Biomech., 40(9), pp. 2071–2077. [CrossRef] [PubMed]
Magnier, C., Boiron, O., Wendling-Mansuy, S., Chabrand, P., and Deplano, V., 2009, “Nutrient Distribution and Metabolism in the Intervertebral Disc in the Unloaded State: A Parametric Study,” J. Biomech., 42(2), pp. 100–108. [CrossRef] [PubMed]
van Loon, R., Huyghe, J. M., Wijlaars, M. W., and Baaijens, F. P. T., 2003, “3D FE Implementation of an Incompressible Quadriphasic Mixture Model,” Int. J. Numer. Methods Eng., 57(9), pp. 1243–1258. [CrossRef]
Wu, J. Z., and Herzog, W., 2002, “Simulating the Swelling and Deformation Behaviour in Soft Tissues Using a Convective Thermal Analogy,” Biomed. Eng. Online, 1, p. 8. [CrossRef] [PubMed]
Ateshian, G. A., Rajan, V., Chahine, N. O., Canal, C. E., and Hung, C. T., 2009, “Modeling the Matrix of Articular Cartilage Using a Continuous Fiber Angular Distribution Predicts Many Observed Phenomena,” ASME J. Biomech. Eng., 131(6), p. 061003. [CrossRef]
Sengers, B. G., Oomens, C. W., and Baaijens, F. P., 2004, “An Integrated Finite-Element Approach to Mechanics, Transport and Biosynthesis in Tissue Engineering,” ASME J. Biomech. Eng., 126(1), pp. 82–91. [CrossRef]
Steck, R., Niederer, P., and Knothe Tate, M. L., 2003, “A Finite Element Analysis for the Prediction of Load-Induced Fluid Flow and Mechanochemical Transduction in Bone,” J. Theor. Biol., 220(2), pp. 249–259. [CrossRef] [PubMed]
Zhang, L., and Szeri, A., 2005, “Transport of Neutral Solute in Articular Cartilage: Effects of Loading and Particle Size,” Proc. R. Soc. London, Ser. A, 461(2059), pp. 2021–2042. [CrossRef]
Deen, W. M., 1987, “Hindered Transport of Large Molecules in Liquid-Filled Pores,” AIChE J., 33(9), pp. 1409–1425. [CrossRef]
Ateshian, G. A., Albro, M. B., Maas, S., and Weiss, J. A., 2011, “Finite Element Implementation of Mechanochemical Phenomena in Neutral Deformable Porous Media Under Finite Deformation,” ASME J. Biomech. Eng., 133(8), p. 081005. [CrossRef]
Maas, S. A., Ellis, B. J., Ateshian, G. A., and Weiss, J. A., 2012, “Febio: Finite Elements for Biomechanics,” ASME J. Biomech. Eng., 134(1), p. 011005. [CrossRef]
Albro, M. B., Banerjee, R. E., Li, R., Oungoulian, S. R., Chen, B., Del Palomar, A. P., Hung, C. T., and Ateshian, G. A., 2011, “Dynamic Loading of Immature Epiphyseal Cartilage Pumps Nutrients Out of Vascular Canals,” J. Biomech., 44, pp. 1654–1659. [CrossRef] [PubMed]
Ateshian, G. A., and Weiss, J. A., 2013, “Finite Element Modeling of Solutes in Hydrated Deformable Biological Tissues,” Computer Models in Biomechanics: From Nano to Macro, G. A.Holzapfel and E.Kuhl, eds., Springer, New York.
Katzir-Katchalsky, A., and Curran, P. F., 1965, Nonequilibrium Thermodynamics in Biophysics, Harvard University, Cambridge, England.
Ateshian, G. A., 2007, “On the Theory of Reactive Mixtures for Modeling Biological Growth,” Biomech. Model. Mechanobiol., 6(6), pp. 423–445. [CrossRef] [PubMed]
McNaught, A. D., and Wilkinson, A., 1997, Compendium of Chemical Terminology: IUPAC Recommendations, 2nd ed., Blackwell Science, Oxford.
Ogston, A. G., and Phelps, C. F., 1961, “The Partition of Solutes Between Buffer Solutions and Solutions Containing Hyaluronic Acid,” Biochem. J., 78, pp. 827–833. [PubMed]
Laurent, T. C., and Killander, J., 1963, “A Theory of Gel Filtration and Its Experimental Verification,” J. Chromatogr., 14, pp. 317–330. [CrossRef]
Tinoco, I., Sauer, K., and Wang, J. C., 1995, Physical Chemistry: Principles and Applications in Biological Sciences, 3rd ed., Prentice Hall, Englewood Cliffs, NJ.
Bonet, J., and Wood, R. D., 1997, Nonlinear Continuum Mechanics for Finite Element Analysis, Cambridge University, Cambridge, England.
Ateshian, G. A., Maas, S., and Weiss, J. A., 2010, “Finite Element Algorithm for Frictionless Contact of Porous Permeable Media Under Finite Deformation and Sliding,” ASME J. Biomech. Eng., 132(6), p. 061006. [CrossRef]
Kedem, O., and Katchalsky, A., 1958, “Thermodynamic Analysis of the Permeability of Biological Membranes to Non-electrolytes,” Biochim. Biophys. Acta, 27(2), pp. 229–246. [CrossRef] [PubMed]
Overbeek, J. T., 1956, “The Donnan Equilibrium,” Prog. Biophys. Biophys. Chem., 6, pp. 57–84. [PubMed]
Frank, E. H., and Grodzinsky, A. J., 1987, “Cartilage Electromechanics–I. Electrokinetic Transduction and the Effects of Electrolyte pH and Ionic Strength,” J. Biomech., 20(6), pp. 615–627. [CrossRef] [PubMed]
Frank, E. H., and Grodzinsky, A. J., 1987, “Cartilage Electromechanics–II. A Continuum Model of Cartilage Electrokinetics and Correlation With Experiments,” J. Biomech., 20(6), pp. 629–639. [CrossRef] [PubMed]
Sokoloff, L., 1963, “Elasticity of Articular Cartilage: Effect of Ions and Viscous Solutions,” Science, 141, pp. 1055–1057. [CrossRef] [PubMed]
Canal Guterl, C., Hung, C. T., and Ateshian, G. A., 2010, “Electrostatic and Non-Electrostatic Contributions of Proteoglycans to the Compressive Equilibrium Modulus of Bovine Articular Cartilage,” J. Biomech., 43(7), pp. 1343–1350. [CrossRef] [PubMed]
Flory, P. J., 1961, “Thermodynamic Relations for High Elastic Materials,” Trans Faraday Soc., 57, pp. 829–838. [CrossRef]
Simo, J. C., and Taylor, R. L., 1991, “Quasi-incompressible Finite Elasticity in Principal Stretches. Continuum Basis and Numerical Algorithms,” Comput. Methods Appl. Mech. Eng., 85(3), pp. 273–310. [CrossRef]
Maroudas, A., 1979, Physicochemical Properties of Articular Cartilage, 2nd ed., Pitman Medical, Kent, pp. 215–290.
Maroudas, A., and Bannon, C., 1981, “Measurement of Swelling Pressure in Cartilage and Comparison With the Osmotic Pressure of Constituent Proteoglycans,” Biorheology, 18(3-6), pp. 619–632. [PubMed]
Ehrlich, S., Wolff, N., Schneiderman, R., Maroudas, A., Parker, K. H., and Winlove, C. P., 1998, “The Osmotic Pressure of Chondroitin Sulphate Solutions: Experimental Measurements and Theoretical Analysis,” Biorheology, 35(6), pp. 383–397. [CrossRef] [PubMed]
Chahine, N. O., Chen, F. H., Hung, C. T., and Ateshian, G. A., 2005, “Direct Measurement of Osmotic Pressure of Glycosaminoglycan Solutions by Membrane Osmometry at Room Temperature,” Biophys. J., 89(3), pp. 1543–1550. [CrossRef] [PubMed]
Buschmann, M. D., and Grodzinsky, A. J., 1995, “A Molecular Model of Proteoglycan-Associated Electrostatic Forces in Cartilage Mechanics,” ASME J. Biomech. Eng., 117(2), pp. 179–192. [CrossRef]
Maroudas, A., Wachtel, E., Grushko, G., Katz, E. P., and Weinberg, P., 1991, “The Effect of Osmotic and Mechanical Pressures on Water Partitioning in Articular Cartilage,” Biochim. Biophys. Acta, 1073(2), pp. 285–94. [CrossRef] [PubMed]
Wilson, W., Huyghe, J. M., and van Donkelaar, C. C., 2007, “Depth-Dependent Compressive Equilibrium Properties of Articular Cartilage Explained by Its Composition,” Biomech. Model. Mechanobiol., 6(1-2), pp. 43–53. [CrossRef] [PubMed]
Shklyar, T. F., Dinislamova, O. A., Safronov, A. P., and Blyakhman, F. A., 2012, “Effect of Cytoskeletal Elastic Properties on the Mechanoelectrical Transduction in Excitable Cells,” J. Biomech., 45(8), pp. 1444–1449. [CrossRef] [PubMed]
Bowen, R. M., 1968, “Thermochemistry of Reacting Materials,” J. Chem. Phys., 49(4), pp. 1625–1637. [CrossRef]
Ateshian, G. A., Costa, K. D., Azeloglu, E. U., Morrison, B., R., and Hung, C. T., 2009, “Continuum Modeling of Biological Tissue Growth by Cell Division, and Alteration of Intracellular Osmolytes and Extracellular Fixed Charge Density,” ASME J. Biomech. Eng., 131(10), p. 101001. [CrossRef]
Ateshian, G. A., 2011, “The Role of Mass Balance Equations in Growth Mechanics Illustrated in Surface and Volume Dissolutions,” ASME J. Biomech. Eng., 133(1), p. 011010. [CrossRef]


Grahic Jump Location
Fig. 2

Cell volume response to osmotic loading: V is the current cell volume and Vr is its volume in the reference configuration. Black curves are the responses from the K–K model and gray curves are the finite element solutions. For the finite element model, the membrane thickness is 10 nm and its properties are ϕrs=0.3, μ=2.5kPa, κ=2.5MPa, k=5×10-7μm4/nN·s, d0p=1.43×10-3μm2/s, and κ∧p=1; the protoplasm has a radius of 10μm, and its properties are ϕrs=0.3, EY=2.5kPa, ν=0, k=5×10-3μm4/nN·s, d0p=d0n=14.3μm2/s, and κ∧p values are indicated on the graph. For the K–K model, the equivalent membrane properties are reported in Ref. [22]. For these analyses, c0=0.3M and c1=1M (see text).

Grahic Jump Location
Fig. 5

One-dimensional current flow in NaCl. The concentration of NaCl along the entire length of the domain is provided at selected time points. Black curves represent the analytical solution and gray curves represent the finite element solution.

Grahic Jump Location
Fig. 1

Basic algorithmic scheme for finite element implementation of multiphasic materials

Grahic Jump Location
Fig. 4

Schematic for current flow in an electrolyte

Grahic Jump Location
Fig. 6

Schematic for configuration of current-generated stress analysis in cartilage

Grahic Jump Location
Fig. 3

Donnan osmotic swelling of a triphasic material (ϕrs=0.2) with a neo-Hookean solid (λs=0, μs=0.25 MPa). The relative volume J=λ3 is plotted for four different values of crF (ranging from =100 to =400 mEq/L by increments of =100 mEq/L) over a range of values for the bath concentration c* (θ = 293 K). The numerical solution to Eq. (3.1) is given by the thin black curves, and the finite element solution is represented by the thicker gray curves.

Grahic Jump Location
Fig. 7

Current-generated stress. (a) Stress response σzz as a function of time, inclusive of the tare stress resulting from the initial Donnan osmotic pressure. (b) Concentrations c+ of Na+ and c of Cl along the axis z of the tissue at selected time points t (t = 0 s corresponds to the state immediately prior to current application; remaining time points increase from 100 s to 104 s by decade).

Grahic Jump Location
Fig. 8

Effect of cation charge on the creep response of cartilage: The compressive engineering strain is reported for three solutions corresponding to a monovalent (NaCl), divalent (CaCl2), and trivalent (AlCl3) cation



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