The focus of this investigation is to study the mechanics of the knee joint using new ligament/bone insertion site constraint models that require the integration of multibody system and large displacement finite element algorithms. Two different sets of clamped end conditions at the ligament/bone insertion site are examined using nonlinear large displacement absolute nodal coordinate formulation (ANCF) finite elements. The first set of end conditions, called the partially clamped joint, eliminates only the translations and rotations at a point, allowing for the cross section stretch and shear at the ligament/bone connection. The second joint, called the fully clamped joint, eliminates all the translation, rotation, and deformation degrees of freedom of the cross section at the ligament/bone insertion site. In the case of the fully clamped joint, the gradient vectors do not change their length and orientation, allowing for the use of the constant strain assumptions. The partially clamped joint, on the other hand, allows for the change in length and relative orientation of the gradient vectors at the bone/ligament insertion site, leading to the cross section deformation induced by knee movements. Nanson’s formula is applied as a measure of the deformation of the cross section in the case of the partially clamped joint. In this study, the major bones in the knee joint consisting of the femur, tibia, and fibula are modeled as rigid bodies while the ligaments structures are modeled using the large displacement ANCF finite elements. Two different ANCF finite element models are developed in this investigation: the first model employs the fully parameterized three-dimensional beam element while the second model employs the three-dimensional cable element. The three-dimensional fully parameterized beam element allows for a straight forward implementation of a neo-Hookean constitutive model that can be used to accurately predict the large displacement as experienced in knee flexation and rotation. At the ligament bone insertion site, the ANCF fully parameterized beam element is used to define a fully or partially constrained joint while the ANCF cable element can only be used to define one joint type. The fully and partially clamped joint constraints are satisfied at the position, velocity, and acceleration levels using a dynamic formulation that is based on an optimum sparse matrix structure. The approach described in this investigation can be used to develop more realistic models of the knee and is applicable to future research studies on ligaments, muscles, and soft tissues. In particular, the partially clamped joint representation of the ligament/bone insertion site constraints can be used to develop improved structural mechanics models of the knee.

1.
Bartel
,
D. L.
,
Davy
,
D. T.
, and
Keaveny
,
T. M.
, 2006,
Orthopaedic Biomechanics: Mechanics and Design in Musculoskeletal Systems
,
Pearson
,
Upper Saddle River, NJ
.
2.
Silva
,
J. F.
, and
Flores
,
P.
, 2007, “
Dynamic Modeling and Analysis of the Knee Joint
,”
Proceedings of the National Conference on the Dynamics of Multibody Systems
,
P.
Flores
and
M.
Silva
, eds., Actas da DSM2007: National Conference on Multibody System Dynamics, Guimaraes, Portugal, Dec. 6–7, p.
88
.
3.
Yamaguchi
,
G.
, 2001,
Dynamic Modeling of Musculoskeletal Motion
,
Kluwer Academic
,
Dordrecht, The Netherlands
.
4.
Weed
,
D.
,
Maqueda
,
L. G.
,
Brown
,
M.
, and
Shabana
,
A. A.
, 2008, “
A Multibody/Finite Element Nonlinear Formulation of a Two-Ligament Knee Joint
,”
Proceedings of the 2008 ASME International Mechanical Engineering Congress and Exhibition
, Boston, MA, Oct. 31–Nov. 6.
5.
Sugiyama
,
H.
,
Escalona
,
J. L.
, and
Shabana
,
A. A.
, 2003, “
Formulation of Three-Dimensional Joint Constraint Using the Absolute Nodal Coordinates
,”
Nonlinear Dyn.
0924-090X,
31
, pp.
167
195
.
6.
Shabana
,
A. A.
, 2005,
Dynamics of Multibody Systems
,
3rd ed.
,
Cambridge University Press
,
Cambridge
.
7.
Nordin
,
M.
, and
Frankel
,
V. H.
, 2001,
Basic Biomechanics of the Musculoskeletal System
,
3rd ed.
,
Lippincott Williams & Willkins
,
Baltimore, MD
.
8.
Drake
,
R. L.
,
Vogl
,
W.
, and
Mitchell
,
A. W. M.
, 2005,
Gray’s Anatomy for Students
,
Elsevier
,
London
.
9.
Shabana
,
A. A.
, 2008,
Computational Continuum Mechanics
,
Cambridge University Press
,
Cambridge
.
10.
Peña
,
E.
,
Calvo
,
B.
,
Martínez
,
M. A.
, and
Doblaré
,
M.
, 2006, “
A Three-Dimensional Finite Element Analysis of the Combined Behavior of Ligaments and Menisci in the Healthy Human Knee Joint
,”
J. Biomech.
0021-9290,
39
(
9
), pp.
1686
1701
.
11.
Weiss
,
J.
,
Maker
,
B.
, and
Govindjee
,
S.
, 1996, “
Finite Element Implementation of Incompressible, Transversely Isotropic Hyperelasticity
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
135
, pp.
107
128
.
12.
Gerstmayr
,
J.
, and
Shabana
,
A. A.
, 2006, “
Analysis of Thin Beams and Cables Using the Absolute Nodal Coordinate Formulation
,”
Nonlinear Dyn.
0924-090X,
45
, pp.
109
130
.
13.
Hussein
,
B. A.
,
Weed
,
D.
, and
Shabana
,
A. A.
, 2009, “
Clamped End Conditions and Cross Section Deformation in the Finite Element Absolute Nodal Coordinate Formulation
,”
Multibody Syst. Dyn.
1384-5640,
21
(
4
), pp.
375
393
.
14.
Ogden
,
R. W.
, 1984,
Non-Linear Elastic Deformations
,
Dover
,
New York
.
15.
Garcia-Vallejo
,
D.
,
Mayo
,
J.
,
Escalona
,
J. L.
, and
Dominguez
,
J.
, 2007, “
Modeling Three-Dimensional Rigid-Flexible Multibody Systems by Using Absolute Coordinates
,”
12th IFToMM World Congress
, Besancon, Jun. 18–21.
16.
García-Vallejo
,
D.
,
Mayo
,
J.
, and
Escalona
,
J. L.
, 2008, “
Three-Dimensional Formulation of Rigid-Flexible Multibody Systems With Flexible Beam Elements
,”
Multibody Syst. Dyn.
1384-5640,
20
(
1
), pp.
1
28
.
17.
Benjamin
,
M.
,
Toumi
,
H.
,
Ralphs
,
J. R.
,
Bydder
,
G.
,
Best
,
T. M.
, and
Milz
,
S.
, 2006, “
Where Tendons and Ligaments Meet Bone: Attachment Sites (‘Entheses’) in Relation to Exercise and/or Mechanical Load
,”
J. Anat.
0021-8782,
208
, pp.
471
490
.
18.
Weiss
,
J. A.
, and
Gardiner
,
J. C.
, 2001, “
Computational Modeling of Ligament Mechanics
,”
Crit. Rev. Biomed. Eng.
0278-940X,
29
(
4
), pp.
1
70
.
19.
Moffat
,
K. L.
,
Sun
,
W. S.
,
Pena
,
P. E.
,
Chahine
,
N. O.
,
Doty
,
S. B.
,
Ateshian
,
G. A.
,
Hung
,
C. T.
, and
Lu
,
H. H.
, 2008, “
Characterization of the Structure-Function Relationship at the Ligament-to-Bone Interface
,”
Proc. Natl. Acad. Sci. U.S.A.
0027-8424,
105
, pp.
7947
7952
.
20.
LaPrade
,
R. F.
,
Engebretsen
,
A. H.
,
Thuan
,
V. L.
,
Johansen
,
S.
,
Wentorf
,
F. A.
, and
Engebretsen
,
L.
, 2007, “
The Anatomy of the Medial Part of the Knee
,”
J. Bone Jt. Surg.
0021-9355,
89
, pp.
2000
2010
.
21.
Meister
,
B. R.
,
Michael
,
S. P.
,
Moyer
,
R. A.
,
Kelly
,
J. D.
, and
Schneck
,
C. D.
, 2000, “
Anatomy and Kinematics of the Lateral Collateral Ligament of the Knee
,”
Am. J. Sports Med.
0363-5465,
28
, pp.
869
878
.
22.
Pedowitz
,
R. A.
,
O'Connor
,
J. J.
, and
Akeson
,
W. H.
, 2003,
Daniel's Knee Injuries: Ligament and Cartilage, Structure, Function, and Repair
,
2nd ed.
,
Lippincott Williams & Willkins
,
Baltimore, MD
.
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