Strain-rate Dependent Stiffness of Articular Cartilage in Unconfined Compression

[+] Author and Article Information
L. P. Li

Biosyntech Inc., 475 Armand-Frappier Blvd., Park of Science and High Technology, Laval, Quebec, Canada H7V 4B3

M. D. Buschmann

Department of Chemical Engineering and Institute of Biomedical Engineering, Ecole Polytechnique of Montreal, Canada H3C 3A7

A. Shirazi-Adl

Department of Mechanical Engineering and Institute of Biomedical Engineering, Ecole Polytechnique of Montreal, Canada H3C 3A7

J Biomech Eng 125(2), 161-168 (Apr 09, 2003) (8 pages) doi:10.1115/1.1560142 History: Received March 01, 2001; Revised November 01, 2002; Online April 09, 2003
Copyright © 2003 by ASME
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Grahic Jump Location
Transient compressive stiffness of cartilage at various strain rates as specified in the legends (nominal compressive axial strain per second). For all cases, Ef0=3 MPa;νm=0.38 and Em takes either 0.36 or 2.14 MPa. (a) Stiffness for Efε=1600. (b) Deviations of the stiffness when Efε is reduced from 1600 (same as shown in figure a) to 1200 (shown with symbols). Larger deviations are observed for the cases of higher strain rates. For clarity of the figure, the curves for the strain rate 5%/s are omitted.
Grahic Jump Location
Radial strain at the center of cartilage disk for Efε=1600 vs Efε=1200 while Ef0=3 MPa. For the case of Efε=1200, the strain is shown with unconnected symbols. (a) The radial strain at equilibrium, for which Em and νm are as specified in the legend. (b) The radial strain during the transient for different strain rates, for which Em=1.07 MPa and νm=0.38. The curves for Em=0.78 and νm=0.42 and for Em=1.48 and νm=0.30 would be hard to distinguish from the corresponding curves (while other parameters are the same), were they drawn (very little difference at the low speed, almost no difference at the high speeds).
Grahic Jump Location
Significance of different measures in quantifying cartilage stiffness, for Em=0.36 MPa,νm=0.38,Efε=1600 and Ef0=3 MPa. The tangent modulus is defined as dσz/dεz and the secant modulus is defined as σzz, where σz is the Cauchy stress in the axial direction and εz is the logarithmic strain. The nominal modulus, generally adopted elsewhere in this paper, is the ratio of nominal axial stress over nominal axial strain. For the convenience of comparison, the moduli are shown as functions of the nominal axial strain. (a) Instantaneous moduli as obtained by employing the finite deformation (FD) and small deformation (SD) theories. (b) Transient moduli for the compression rate 0.5%/s (M=10, FD theory only).
Grahic Jump Location
Axial compressive stiffness of cartilage for constant Ef (≡20 MPa) and νm=0.38 when no fluid flow is feasible. The finite deformation and small deformation (SD) theories are employed respectively. In all cases, the stiffness extracted from the SD theory is not axial strain dependent (horizontal lines). (a) Elastic static stiffness for Em=0.36, 1.07 and 2.14 MPa respectively. (b) Instantaneous stiffness for which the fluid is trapped.
Grahic Jump Location
Transient stiffness of cartilage with constant fibrillar modulus (Ef≡20 MPa) for M=10 vs M=0 while Em=0.36 and νm=0.38. No significant qualitative changes in the stiffness are observed for a larger Em and thus the results for the comparison group are not included.
Grahic Jump Location
Transient radial strain at the center of disk for Em=1.07 MPa,νm=0.38 and Ef0=3 MPa. The strain for M=10 (nonlinear permeability) is compared with that for M=0 (constant permeability) while Efε=1600. The strain is shown for M=0 and Efε=1200 with the fine dots.
Grahic Jump Location
Transient compressive stiffness of cartilage when the permeability is taken to be constant (k≡0.003 mm4/Ns,M=0),νm=0.38,Efε=1600 MPa and Ef0=3 MPa. If Efε is reduced to 1200 MPa, the deviations will be invisible (with same 2 digits for the stiffness) at the compression rate 0.05%/s and almost invisible after 10% axial strain at the rate 0.5%/s. The deviations are relatively significant at the compression rate 5%/s.
Grahic Jump Location
Axial compressive stiffness of cartilage when no fluid flow is feasible. For all cases, Ef0=3 MPa. (a) Elastic static stiffness or equilibrium stiffness: νm=0.38 and Efε=1600 MPa except when specified in the legend otherwise. (b) Instantaneous stiffness for Efε=1600, 1200 and 800 for which the fluid is trapped: νm=0.38 and Em is as specified in the legend.
Grahic Jump Location
Axial compressive stiffness of cartilage disks with bone attached, the nonhomogeneous model versus experiments (mean±SD, n=4). The material parameters of cartilage used for simulation are as follows: at the articular surface, Em=0.20 MPa,νm=0.12, the void ratio is 3.6; at the cartilage and bone interface Em=0.80 MPa,νm=0.42, the void ratio is 2.88; k⁁=0.004 mm4/Ns and M=13;Efε=2800 MPa and Ef0=3 MPa at the surface. For the underlying bone, E=18 MPa,ν=0.25,k≡0.0004 mm4/Ns, the void ratio is 0.5.



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