Finite element (FE) models of articular joint structures do not typically implement the fully nonlinear viscoelastic behavior of the soft connective tissue components. Instead, contemporary whole joint FE models usually represent the transient soft tissue behavior with significantly simplified formulations that are computationally tractable. The resultant fidelity of these models is greatly compromised with respect to predictions under temporally varying static and dynamic loading regimes. In addition, models based upon experimentally derived nonlinear viscoelastic coefficients that do not account for the transient behavior during the loading event(s) may further reduce the model’s predictive accuracy. The current study provides the derivation and validation of a novel, phenomenological nonlinear viscoelastic formulation (based on the single integral nonlinear superposition formulation) that can be directly inputted into FE algorithms. This formulation and an accompanying experimental characterization technique, which incorporates relaxation manifested during the loading period of stress relaxation experiments, is compared to a previously published characterization method and validated against an independent analytical model. The results demonstrated that the static and dynamic FE approximations are in good agreement with the analytical solution. Additionally, the predictive accuracy of these approximations was observed to be highly dependent upon the experimental characterization technique. It is expected that implementation of the novel, computationally tractable nonlinear viscoelastic formulation and associated experimental characterization technique presented in the current study will greatly improve the predictive accuracy of the individual connective tissue components for whole joint FE simulations subjected to static and dynamic loading regimes.

References

References
1.
Fung
,
Y. C.
, 1972, “
Stress-Strain-History Relations of Soft Tissues in Simple Elongation
,”
Biomechanics: Its Foundations and Objectives
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
2.
Abramowitch
,
S. D.
,
Woo
,
S. L. Y.
,
Clineff
,
T. D.
, and
Debski
,
R. E.
, 2004, “
An Evaluation of the Quasi-Linear Viscoelastic Properties of the Healing Medial Collateral Ligament in a Goat Model
,”
Ann. Biomed. Eng.
,
32
(
3
), pp.
329
335
.
3.
Elliott
,
D. M.
,
Robinson
,
P. S.
,
Gimbel
,
J. A.
,
Sarver
,
J. J.
,
Abboud
,
J. A.
,
Iozzo
,
R. V.
, and
Soslowsky
,
L. J.
, 2003, “
Effect of Altered Matrix Proteins on Quasilinear Viscoelastic Properties in Transgenic Mouse Tail Tendons
,”
Ann. Biomed. Eng.
,
31
(
5
), pp.
599
605
.
4.
Funk
,
J. R.
,
Hall
,
G. W.
,
Crandall
,
J. R.
, and
Pilkey
,
W. D.
, 2000, “
Linear and Quasi-Linear Viscoelastic Characterization of Ankle Ligaments
,”
ASME J. Biomech. Eng.
,
122
(
1
), pp.
15
22
.
5.
Iatridis
,
J. C.
,
Setton
,
L. A.
,
Weidenbaum
,
M.
, and
Mow
,
V. C.
, 1997, “
The Viscoelastic Behavior of the Non-Degenerate Human Lumbar Nucleus Pulposus in Shear
,”
J. Biomech.
,
30
(
10
), pp.
1005
1013
.
6.
Johnson
,
G. A.
,
Tramaglini
,
D. M.
,
Levine
,
R. E.
,
Ohno
,
K.
,
Choi
,
N. Y.
, and
Woo
,
S. L. Y.
, 1994, “
Tensile and Viscoelastic Properties of Human Patellar Tendon
,”
J. Orthop. Res.
,
12
(
6
), pp.
796
803
.
7.
Kwan
,
M. K.
,
Lin
,
T. H. C.
, and
Woo
,
S. L. Y.
, 1993, “
On the Viscoelastic Properties of the Anteromedial Bundle of the Anterior Cruciate Ligament
,”
J. Biomech.
,
26
(
4–5
), pp.
447
452
.
8.
Lucas
,
S. R.
,
Bass
,
C. R.
,
Salzar
,
R. S.
,
Oyen
,
M. L.
,
Planchak
,
C.
,
Ziemba
,
A.
,
Shender
,
B. S.
, and
Paskoff
,
G.
, 2008, “
Viscoelastic Properties of the Cervical Spinal Ligaments Under Fast Strain-Rate Deformations
,”
Acta Biomater.
,
4
(
1
), pp.
117
125
.
9.
Toms
,
S. R.
,
Dakin
,
G. J.
,
Lemons
,
J. E.
, and
Eberhardt
,
A. W.
, 2002, “
Quasi-Linear Viscoelastic Behavior of the Human Periodontal Ligament
,”
J. Biomech.
,
35
(
10
), pp.
1411
1415
.
10.
Weiss
,
J. A.
, and
Gardiner
,
J. C.
, 2001, “
Computational Modeling of Ligament Mechanics
,”
Crit. Rev. Biomed. Eng.
,
29
(
3
), pp.
303
371
.
11.
Woo
,
S. L.
,
Gomez
,
M. A.
, and
Akeson
,
W. H.
, 1981, “
The Time and History-Dependent Viscoelastic Properties of the Canine Medical Collateral Ligament
,”
ASME J. Biomech. Eng.
,
103
(
4
), pp.
293
298
.
12.
Woo
,
S. L.
,
Simon
,
B. R.
,
Kuei
,
S. C.
, and
Akeson
,
W. H.
, 1980, “
Quasi-Linear Viscoelastic Properties of Normal Articular Cartilage
,”
ASME J. Biomech. Eng.
,
102
(
2
), pp.
85
90
.
13.
ABAQUS
, 2009,
ABAQUS Analysis User’s Manual (Version 6.9)
,
Dassault Systèmes Simulia Corp.
,
Providence, RI
.
14.
Hingorani
,
R. V.
,
Provenzano
,
P. P.
,
Lakes
,
R. S.
,
Escarcega
,
A.
, and
Vanderby
,
R.
, Jr
., 2004, “
Nonlinear Viscoelasticity in Rabbit Medial Collateral Ligament
,”
Ann. Biomed. Eng.
,
32
(
2
), pp.
306
312
.
15.
Provenzano
,
P.
,
Lakes
,
R.
,
Keenan
,
T.
, and
Vanderby
,
R.
, Jr
., 2001, “
Nonlinear Ligament Viscoelasticity
,”
Ann. Biomed. Eng.
,
29
(
10
), pp.
908
914
.
16.
Troyer
,
K. L.
, and
Puttlitz
,
C. M.
, 2011, “
Human Cervical Spine Ligaments Exhibit Fully Nonlinear Viscoelastic Behavior
,”
Acta Biomater.
,
7
(
2
), pp.
700
709
.
17.
Ambrosetti-Giudici
,
S.
,
Gedet
,
P.
,
Ferguson
,
S. J.
,
Chegini
,
S.
, and
Burger
,
J.
, 2010, “
Viscoelastic Properties of the Ovine Posterior Spinal Ligaments are Strain Dependent
,”
Clinical Biomech.
,
25
(
2
), pp.
97
102
.
18.
Thornton
,
G. M.
,
Oliynyk
,
A.
,
Frank
,
C. B.
, and
Shrive
,
N. G.
, 1997, “
Ligament Creep Cannot Be Predicted From Stress Relaxation at Low Stress: A Biomechanical Study of the Rabbit Medial Collateral Ligament
,”
J. Orthop. Res.
,
15
(
5
), pp.
652
656
.
19.
Duenwald
,
S. E.
,
Vanderby
,
R.
, Jr.
, and
Lakes
,
R. S.
, 2009, “
Viscoelastic Relaxation and Recovery of Tendon
,”
Ann. Biomed. Eng.
,
37
(
6
), pp.
1131
1140
.
20.
Bonifasi-Lista
,
C.
,
Lake
,
S. P.
,
Small
,
M. S.
, and
Weiss
,
J. A.
, 2005, “
Viscoelastic Properties of the Human Medial Collateral Ligament Under Longitudinal, Transverse and Shear Loading
,”
J. Orthop. Res.
,
23
(
1
), pp.
67
76
.
21.
Troyer
,
K. L.
,
Estep
,
D. J.
, and
Puttlitz
,
C. M.
, 2012, “
Viscoelastic Effects During Loading Play an Integral Role in Soft Tissue Mechanics
,”
Acta Biomater.
,
8
(
1
), pp.
234
243
.
22.
Moon
,
D. K.
,
Woo
,
S. L.
,
Takakura
,
Y.
,
Gabriel
,
M. T.
, and
Abramowitch
,
S. D.
, 2006, “
The Effects of Refreezing on the Viscoelastic and Tensile Properties of Ligaments
,”
J. Biomech.
,
39
(
6
), pp.
1153
1157
.
23.
Santoni
,
B. G.
,
Mcgilvray
,
K. C.
,
Lyons
,
A. S.
,
Bansal
,
M.
,
Turner
,
A. S.
,
Macgillivray
,
J. D.
,
Coleman
,
S. H.
, and
Puttlitz
,
C. M.
, 2010, “
Biomechanical Analysis of an Ovine Rotator Cuff Repair Via Porous Patch Augmentation in a Chronic Rupture Model
,”
Am. J. Sports Med.
,
38
(
4
), pp.
679
686
.
24.
Hee
,
C. K.
,
Dines
,
J. S.
,
Dines
,
D. M.
,
Roden
,
C. M.
,
Wisner-Lynch
,
L. A.
,
Turner
,
A. S.
,
Mcgilvray
,
K. C.
,
Lyons
,
A. S.
Puttlitz
,
C. M.
, and
Santoni
,
B. G.
, 2011, “
Augmentation of a Rotator Cuff Suture Repair Using Rhpdgf-Bb and a Type I Bovine Collagen Matrix in an Ovine Model
,”
Am. J. Sports Med.
,
39
(
8
), pp.
1630
1639
.
25.
Einat
,
R.
, and
Yoram
,
L.
, 2009, “
Recruitment Viscoelasticity of the Tendon
,”
ASME J. Biomech. Eng.
,
131
(
11
), p.
111008
.
26.
Sverdlik
,
A.
, and
Lanir
,
Y.
, 2002, “
Time-Dependent Mechanical Behavior of Sheep Digital Tendons, Including the Effects of Preconditioning
,”
ASME J. Biomech. Eng.
,
124
(
1
), pp.
78
84
.
27.
Butler
,
D. L.
,
Grood
,
E. S.
,
Noyes
,
F. R.
,
Zernicke
,
R. F.
, and
Brackett
,
K.
, 1984, “
Effects of Structure and Strain-Measurement Technique on the Material Properties of Young Human Tendons and Fascia
,”
J. Biomech.
,
17
(
8
), pp.
579
596
.
28.
Shetye
,
S. S.
,
Malhotra
,
K.
,
Ryan
,
S. D.
, and
Puttlitz
,
C. M.
, 2009, “
Determination of Mechanical Properties of Canine Carpal Ligaments
,”
Am. J. Vet. Res.
,
70
(
8
), pp.
1026
1030
.
29.
Gibbons
,
M. J.
,
Butler
,
D. L.
,
Grood
,
E. S.
,
Bylskiaustrow
,
D. I.
,
Levy
,
M. S.
, and
Noyes
,
F. R.
, 1991, “
Effects of Gamma-Irradiation on the Initial Mechanical and Material Properties of Goat Bone-Patellar Tendon-Bone Allografts
,”
J. Orthop. Res.
,
9
(
2
), pp.
209
218
.
30.
Mcgilvray
,
K. C.
,
Santoni
,
B. G.
,
Turner
,
A. S.
,
Bogdansky
,
S.
,
Wheeler
,
D. L.
, and
Puttlitz
,
C. M.
, 2011, “
Effects of (60)Co Gamma Radiation Dose on Initial Structural Biomechanical Properties of Ovine Bone–Patellar Tendon–Bone Allografts
,”
Cell Tissue Bank
,
12
(
2
), pp.
89
98
.
31.
Provenzano
,
P. P.
,
Lakes
,
R. S.
,
Corr
,
D. T.
, and
Vanderby
,
R.
, Jr
., 2002, “
Application of Nonlinear Viscoelastic Models to Describe Ligament Behavior
,”
Biomech. Model. Mechanobiol.
,
1
(
1
), pp.
45
57
.
32.
Lakes
,
R. S.
, 1999,
Viscoelastic Solids
,
CRC Press
,
Boca Raton
.
33.
Findley
,
W. N.
,
Lai
,
J. S.
, and
Onaran
,
K.
, 1976,
Creep and Relaxation of Nonlinear Viscoelastic Materials, With an Introduction to Linear Viscoelasticity
,
Elsevier/North Holland
,
Amsterdam/New York
.
34.
Troyer
,
K. L.
, and
Puttlitz
,
C. M.
, 2012, “
Nonlinear Viscoelasticty Plays an Essential Role in the Functional Behavior of Spinal Ligaments
,”
J. Biomech.
,
45
(
4
), pp.
684
691
.
35.
Pryse
,
K. M.
,
Nekouzadeh
,
A.
,
Genin
,
G. M.
,
Elson
,
E. L.
, and
Zahalak
,
G. I.
, 2003, “
Incremental Mechanics of Collagen Gels: New Experiments and a New Viscoelastic Model
,”
Ann. Biomed. Eng.
,
31
(
10
), pp.
1287
1296
.
36.
Duenwald
,
S. E.
,
Vanderby
,
R.
, and
Lakes
,
R. S.
, 2009, “
Constitutive Equations for Ligament and Other Soft Tissue: Evaluation by Experiment
,”
Acta Mech.
,
205
(
1–4
), pp.
23
33
.
37.
Lakes
,
R. S.
, and
Vanderby
,
R.
, 1999, “
Interrelation of Creep and Relaxation: A Modeling Approach for Ligaments
,”
ASME J. Biomech. Eng.
,
121
(
6
), pp.
612
615
.
38.
Oza
,
A.
,
Vanderby
,
R.
, and
Lakes
,
R. S.
, 2003, “
Interrelation of Creep and Relaxation for Nonlinearly Viscoelastic Materials: Application to Ligament and Metal
,”
Rheol. Acta
,
42
(
6
), pp.
557
568
.
39.
Ott
,
L.
, and
Longnecker
,
M.
, 2001,
An Introduction to Statistical Methods and Data Analysis
,
Duxbury
,
Pacific Grove
.
40.
Abramowitch
,
S. D.
, and
Woo
,
S. L.
, 2004, “
An Improved Method to Analyze the Stress Relaxation of Ligaments Following a Finite Ramp Time Based on the Quasi-Linear Viscoelastic Theory
,”
ASME J. Biomech. Eng.
,
126
(
1
), pp.
92
97
.
41.
Gimbel
,
J. A.
,
Sarver
,
J. J.
, and
Soslowsky
,
L. J.
, 2004, “
The Effect of Overshooting the Target Strain on Estimating Viscoelastic Properties From Stress Relaxation Experiments
,”
ASME J. Biomech. Eng.
,
126
(
6
), pp.
844
848
.
42.
Yang
,
W.
,
Fung
,
T. C.
,
Chian
,
K. S.
, and
Chong
,
C. K.
, 2006, “
Viscoelasticity of Esophageal Tissue and Application of a Qlv Model
,”
ASME J. Biomech. Eng.
,
128
(
6
), pp.
909
916
.
43.
Puso
,
M. A.
, and
Weiss
,
J. A.
, 1998, “
Finite Element Implementation of Anisotropic Quasi-Linear Viscoelasticity Using a Discrete Spectrum Approximation
,”
ASME J. Biomech. Eng.
,
120
(
1
), pp.
62
70
.
44.
Duenwald
,
S. E.
,
Vanderby
,
R.
, and
Lakes
,
R. S.
, 2010, “
Stress Relaxation and Recovery in Tendon and Ligament: Experiment and Modeling
,”
Biorheology
,
47
(
1
), pp.
1
14
.
45.
Schapery
,
R. A.
, 1969, “
On Characterization of Nonlinear Viscoelastic Materials
,”
Polymer Eng. Sci.
,
9
(
4
), pp.
295
310
.
46.
Brolin
,
K.
, and
Halldin
,
P.
, 2004, “
Development of a Finite Element Model of the Upper Cervical Spine and a Parameter Study of Ligament Characteristics
,”
Spine
,
29
(
4
), pp.
376
385
.
47.
Womack
,
W.
,
Leahy
,
P. D.
,
Patel
,
V. V.
, and
Puttlitz
,
C. M.
, 2011, “
Finite Element Modeling of Kinematic and Load Transmission Alterations Due to Cervical Intervertebral Disc Replacement
,”
Spine
,
36
(
17
), pp.
E1126
E1133
.
48.
Rohlmann
,
A.
,
Burra
,
N. K.
,
Zander
,
T.
, and
Bergmann
,
G.
, 2007, “
Comparison of the Effects of Bilateral Posterior Dynamic and Rigid Fixation Devices on the Loads in the Lumbar Spine: A Finite Element Analysis
,”
Eur. Spine J.
,
16
(
8
), pp.
1223
1231
.
49.
Ayturk
,
U. M.
, and
Puttlitz
,
C. M.
, 2011, “
Parametric Convergence Sensitivity and Validation of a Finite Element Model of the Human Lumbar Spine
,”
Comput. Meth. Biomech. Biomed.
,
14
(
8
), pp.
695
705
.
50.
Bae
,
J. Y.
,
Park
,
K. S.
,
Seon
,
J. K.
,
Kwak
,
D. S.
,
Jeon
,
I.
, and
Song
,
E. K.
, 2012, “
Biomechanical Analysis of the Effects of Medial Meniscectomy on Degenerative Osteoarthritis
,”
Med. Biol. Eng. Comput.
,
50
(
1
), pp.
53
60
.
51.
Baldwin
,
M. A.
,
Clary
,
C. W.
,
Fitzpatrick
,
C. K.
,
Deacy
,
J. S.
,
Maletsky
,
L. P.
, and
Rullkoetter
,
P. J.
, 2012, “
Dynamic Finite Element Knee Simulation for Evaluation of Knee Replacement Mechanics
,”
J. Biomech.
,
45
(
3
), pp.
474
483
.
52.
Eichenseer
,
P. H.
,
Sybert
,
D. R.
, and
Cotton
,
J. R.
, 2011, “
A Finite Element Analysis of Sacroiliac Joint Ligaments in Response to Different Loading Conditions
,”
Spine
,
36
(
22
), pp.
E1446
E1452
.
53.
Phillips
,
A. T. M.
,
Pankaj
,
P.
,
Howie
,
C. R.
,
Usmani
,
A. S.
, and
Simpson
,
A. H. R. W.
, 2007, “
Finite Element Modelling of the Pelvis: Inclusion of Muscular and Ligamentous Boundary Conditions
,”
Med. Eng. Phys.
,
29
(
7
), pp.
739
748
.
54.
Holzapfel
,
G. A.
, and
Gasser
,
T. C.
, 2001, “
A Viscoelastic Model for Fiber-Reinforced Composites at Finite Strains: Continuum Basis, Computational Aspects and Applications
,”
Comput. Meth. Appl. Mech. Eng.
,
190
(
34
), pp.
4379
4403
.
55.
Nguyen
,
T. D.
,
Jones
,
R. E.
, and
Boyce
,
B. L.
, 2008, “
A Nonlinear Anisotropic Viscoelastic Model for the Tensile Behavior of the Corneal Stroma
,”
ASME J. Biomech. Eng.
,
130
(
4
), p.
041020
.
56.
Pena
,
E.
,
Pena
,
J. A.
, and
Doblare
,
M.
, 2008, “
On Modelling Nonlinear Viscoelastic Effects in Ligaments
,”
J. Biomech.
,
41
(
12
), pp.
2659
2666
.
57.
Lanir
,
Y.
, 1983, “
Constitutive Equations for Fibrous Connective Tissues
,”
J. Biomech.
,
16
(
1
), pp.
1
12
.
58.
Bischoff
,
J. E.
, 2006, “
Reduced Parameter Formulation for Incorporating Fiber Level Viscoelasticity into Tissue Level Biomechanical Models
,”
Ann. Biomed. Eng.
,
34
(
7
), pp.
1164
1172
.
59.
Bischoff
,
J. E.
,
Arruda
,
E. M.
, and
Grosh
,
K.
, 2004, “
A Rheological Network Model for the Continuum Anisotropic and Viscoelastic Behavior of Soft Tissue
,”
Biomech. Model. Mechanobiol.
,
3
(
1
), pp.
56
65
.
60.
Vena
,
P.
,
Gastaldi
,
D.
, and
Contro
,
R.
, 2006, “
A Constituent-Based Model for the Nonlinear Viscoelastic Behavior of Ligaments
,”
ASME J. Biomech. Eng.
,
128
(
3
), pp.
449
457
.
61.
Kahn
,
C. J. F.
,
Wang
,
X. O.
, and
Rahouadj
,
R.
, 2010, “
Nonlinear Model for Viscoelastic Behavior of Achilles Tendon
,”
ASME J. Biomech. Eng.
,
132
(
11
), p.
111002
.
62.
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
.
63.
Rajagopal
,
K. R.
, and
Wineman
,
A. S.
, 2009, “
Response of Anisotropic Nonlinearly Viscoelastic Solids
,”
Math. Mech. Solid.
,
14
(
5
), pp.
490
501
.
You do not currently have access to this content.