Porous-permeable tissues have often been modeled using porous media theories such as the biphasic theory. This study examines the equivalence of the short-time biphasic and incompressible elastic responses for arbitrary deformations and constitutive relations from first principles. This equivalence is illustrated in problems of unconfined compression of a disk, and of articular contact under finite deformation, using two different constitutive relations for the solid matrix of cartilage, one of which accounts for the large disparity observed between the tensile and compressive moduli in this tissue. Demonstrating this equivalence under general conditions provides a rationale for using available finite element codes for incompressible elastic materials as a practical substitute for biphasic analyses, so long as only the short-time biphasic response is sought. In practice, an incompressible elastic analysis is representative of a biphasic analysis over the short-term response δtΔ2C4K, where Δ is a characteristic dimension, C4 is the elasticity tensor, and K is the hydraulic permeability tensor of the solid matrix. Certain notes of caution are provided with regard to implementation issues, particularly when finite element formulations of incompressible elasticity employ an uncoupled strain energy function consisting of additive deviatoric and volumetric components.

1.
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.
0148-0731,
102
, pp.
73
84
.
2.
Cohen
,
B.
,
Lai
,
W. M.
, and
Mow
,
V. C.
, 1998, “
A Transversely Isotropic Biphasic Model for Unconfined Compression of Growth Plate and Chondroepiphysis
,”
ASME J. Biomech. Eng.
0148-0731,
120
, pp.
491
496
.
3.
Soulhat
,
J.
,
Buschmann
,
M. D.
, and
Shirazi-Adl
,
A.
, 1999, “
A Fibril-Network-Reinforced Biphasic Model of Cartilage in Unconfined Compression
,”
ASME J. Biomech. Eng.
0148-0731,
121
, pp.
340
347
.
4.
Soltz
,
M. A.
, and
Ateshian
,
G. A.
, 2000, “
A Conewise Linear Elasticity Mixture Model for the Analysis of Tension-Compression Nonlinearity in Articular Cartilage
,”
ASME J. Biomech. Eng.
0148-0731,
122
, pp.
576
586
.
5.
Bachrach
,
N. M.
,
Mow
,
V. C.
, and
Guilak
,
F.
, 1998, “
Incompressibility of the Solid Matrix of Articular Cartilage Under High Hydrostatic Pressures
,”
J. Biomech.
0021-9290,
31
, pp.
445
451
.
6.
Armstrong
,
C. G.
,
Lai
,
W. M.
, and
Mow
,
V. C.
, 1984, “
An Analysis of the Unconfined Compression of Articular Cartilage
,”
ASME J. Biomech. Eng.
0148-0731,
106
, pp.
165
173
.
7.
Brown
,
T. D.
, and
Singerman
,
R. J.
, 1986, “
Experimental Determination of the Linear Biphasic Constitutive Coefficients of Human Fetal Proximal Femoral Chondroepiphysis
,”
J. Biomech.
0021-9290,
19
, pp.
597
605
.
8.
Mak
,
A. F.
,
Lai
,
W. M.
, and
Mow
,
V. C.
, 1987, “
Biphasic Indentation of Articular Cartilage. I. Theoretical Analysis
,”
J. Biomech.
0021-9290,
20
, pp.
703
714
.
9.
Ateshian
,
G. A.
,
Lai
,
W. M.
,
Zhu
,
W. B.
, and
Mow
,
V. C.
, 1994, “
An Asymptotic Solution for the Contact of Two Biphasic Cartilage Layers
,”
J. Biomech.
0021-9290,
27
, pp.
1347
1360
.
10.
Armstrong
,
C. G.
, and
Mow
,
V. C.
, 1982, “
Variations in the Intrinsic Mechanical Properties of Human Articular Cartilage With Age, Degeneration, and Water Content
,”
J. Bone Jt. Surg., Am. Vol.
0021-9355,
64
, pp.
88
94
.
11.
Chahine
,
N. O.
,
Wang
,
C. C.
,
Hung
,
C. T.
, and
Ateshian
,
G. A.
, 2004, “
Anisotropic Strain-Dependent Material Properties of Bovine Articular Cartilage in the Transitional Range From Tension to Compression
,”
J. Biomech.
0021-9290,
37
, pp.
1251
1261
.
12.
Huang
,
C. Y.
,
Stankiewicz
,
A.
,
Ateshian
,
G. A.
, and
Mow
,
V. C.
, 2005, “
Anisotropy, Inhomogeneity, and Tension-Compression Nonlinearity of Human Glenohumeral Cartilage in Finite Deformation
,”
J. Biomech.
0021-9290,
38
, pp.
799
809
.
13.
Kempson
,
G. E.
,
Freeman
,
M. A.
, and
Swanson
,
S. A.
, 1968, “
Tensile Properties of Articular Cartilage
,”
Nature (London)
0028-0836,
220
, pp.
1127
1128
.
14.
Hayes
,
W. C.
,
Keer
,
L. M.
,
Herrmann
,
G.
, and
Mockros
,
L. F.
, 1972, “
A Mathematical Analysis for Indentation Tests of Articular Cartilage
,”
J. Biomech.
0021-9290,
5
, pp.
541
551
.
15.
Eberhardt
,
A. W.
,
Keer
,
L. M.
,
Lewis
,
J. L.
, and
Vithoontien
,
V.
, 1990, “
An Analytical Model of Joint Contact
,”
ASME J. Biomech. Eng.
0148-0731,
112
, pp.
407
413
.
16.
Carter
,
D. R.
, and
Beaupre
,
G. S.
, 1999, “
Linear Elastic and Poroelastic Models of Cartilage Can Produce Comparable Stress Results: A Comment on Tanck Et Al. (J Biomech 32:153–161, 1999)
,”
J. Biomech.
0021-9290,
32
, pp.
1255
1257
.
17.
Wong
,
M.
, and
Carter
,
D. R.
, 1990, “
Theoretical Stress Analysis of Organ Culture Osteogenesis
,”
Bone (N.Y.)
8756-3282,
11
, pp.
127
131
.
18.
Bowen
,
R. M.
, 1980, “
Incompressible Porous Media Models by Use of the Theory of Mixtures
,”
Int. J. Eng. Sci.
0020-7225,
18
, pp.
1129
1148
.
19.
Huyghe
,
J. M.
, and
Janssen
,
J. D.
, 1997, “
Quadriphasic Mechanics of Swelling Incompressible Porous Media
,”
Int. J. Eng. Sci.
0020-7225,
35
, pp.
793
802
.
20.
Holmes
,
M. H.
, and
Mow
,
V. C.
, 1990, “
The Nonlinear Characteristics of Soft Gels and Hydrated Connective Tissues in Ultrafiltration
,”
J. Biomech.
0021-9290,
23
, pp.
1145
1156
.
21.
Lai
,
W. M.
, and
Mow
,
V. C.
, 1980, “
Drag-Induced Compression of Articular Cartilage During a Permeation Experiment
,”
Biorheology
0006-355X,
17
, pp.
111
123
.
22.
Gu
,
W. Y.
,
Yao
,
H.
,
Huang
,
C. Y.
, and
Cheung
,
H. S.
, 2003, “
New Insight Into Deformation-Dependent Hydraulic Permeability of Gels and Cartilage, and Dynamic Behavior of Agarose Gels in Confined Compression
,”
J. Biomech.
0021-9290,
36
, pp.
593
598
.
23.
Bonet
,
J.
, and
Wood
,
R. D.
, 1997,
Nonlinear Continuum Mechanics for Finite Element Analysis
,
Cambridge University Press
,
Cambridge
.
24.
Simo
,
J. C.
,
Taylor
,
R. L.
, and
Pister
,
K. S.
, 1985, “
Variational and Projection Methods for the Volume Constraint in Finite Deformation Elastoplasticity
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
51
, pp.
177
208
.
25.
Weiss
,
J. A.
,
Maker
,
B. N.
, and
Govindjee
,
S.
, 1996, “
Finite Element Implementation of Incompressible, Transversely Isotropic Hyperelasticity
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
135
, pp.
107
128
.
26.
Curnier
,
A.
,
He
,
Q. C.
, and
Zysset
,
P.
, 1995, “
Conewise Linear Elastic Materials
,”
J. Elast.
0374-3535,
37
, pp.
1
38
.
27.
Quapp
,
K. M.
, and
Weiss
,
J. A.
, 1998, “
Material Characterization of Human Medial Collateral Ligament
,”
ASME J. Biomech. Eng.
0148-0731,
120
, pp.
757
763
.
28.
Baer
,
A. E.
,
Laursen
,
T. A.
,
Guilak
,
F.
, and
Setton
,
L. A.
, 2003, “
The Micromechanical Environment of Intervertebral Disc Cells Determined by a Finite Deformation, Anisotropic, and Biphasic Finite Element Model
,”
ASME J. Biomech. Eng.
0148-0731,
125
, pp.
1
11
.
29.
Lanir
,
Y.
, 1983, “
Constitutive Equations for Fibrous Connective Tissues
,”
J. Biomech.
0021-9290,
16
, pp.
1
12
.
30.
Lanir
,
Y.
, 1987, “
Biorheology and Fluid Flux in Swelling Tissues, Ii. Analysis of Unconfined Compressive Response of Transversely Isotropic Cartilage Disc
,”
Biorheology
0006-355X,
24
, pp.
189
205
.
31.
Laasanen
,
M. S.
,
Toyras
,
J.
,
Korhonen
,
R. K.
,
Rieppo
,
J.
,
Saarakkala
,
S.
,
Nieminen
,
M. T.
,
Hirvonen
,
J.
, and
Jurvelin
,
J. S.
, 2003, “
Biomechanical Properties of Knee Articular Cartilage
,”
Biorheology
0006-355X,
40
, pp.
133
140
.
32.
Wayne
,
J. S.
,
Woo
,
S. L.
, and
Kwan
,
M. K.
, 1991, “
Application of the U-P Finite Element Method to the Study of Articular Cartilage
,”
ASME J. Biomech. Eng.
0148-0731,
113
, pp.
397
403
.
33.
Maker
,
B. N.
,
Ferencz
,
R. M.
, and
Hallquist
,
J. O.
, 1990, “
Nike3D: A Nonlinear, Implicit, Three-Dimensional Finite Element Code for Solid and Structural Mechanics
,” LLNL Technical Report No. UCRL-MA 105268.
34.
Kelkar
,
R.
, and
Ateshian
,
G. A.
, 1999, “
Contact Creep of Biphasic Cartilage Layers
,”
ASME J. Appl. Mech.
0021-8936,
66
, pp.
137
145
.
35.
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.
1025-5842,
1
, pp.
25
46
.
36.
Levenston
,
M. E.
,
Frank
,
E. H.
, and
Grodzinsky
,
A. J.
, 1998, “
Variationally Derived 3-Field Finite Element Formulations for Quasistatic Poroelastic Analysis of Hydrated Biological Tissues
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
156
, pp.
231
246
.
37.
Suh
,
J. K.
, and
Spilker
,
R. L.
, 1991, “
Penalty Finite Element Analysis for Non-Linear Mechanics of Biphasic Hydrated Soft Tissue Under Large Deformation
,”
Int. J. Numer. Methods Eng.
0029-5981,
32
, pp.
1411
1439
.
38.
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.
0029-5981,
39
, pp.
81
97
.
39.
Simon
,
B. R.
,
Kaufmann
,
M. V.
,
McAfee
,
M. A.
, and
Baldwin
,
A. L.
, 1993, “
Finite Element Models for Arterial Wall Mechanics
,”
ASME J. Biomech. Eng.
0148-0731,
115
, pp.
489
496
.
40.
Meng
,
X. N.
,
LeRoux
,
M. A.
,
Laursen
,
T. A.
, and
Setton
,
L. A.
, 2002, “
A Nonlinear Finite Element Formulation for Axisymmetric Torsion of Biphasic Materials
,”
Int. J. Solids Struct.
0020-7683,
39
, pp.
879
895
.
This content is only available via PDF.
You do not currently have access to this content.