The Effect of Cyclic Loading on the Articular Cartilage of the FemoroAcetabular Joint
by
Taylor J. Castagna
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
Master of Engineering
Major Subject: Mechanical Engineering
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, CT
December, 2013
i
CONTENTS
The Effect of Cyclic Loading on the Articular Cartilage of the Femoro-Acetabular Joint i
LIST OF TABLES ............................................................................................................ iv
LIST OF FIGURES ........................................................................................................... v
LIST OF SYMBOLS ....................................................................................................... vii
ACRONYMS AND DEFINITIONS .............................................................................. viii
KEYWORDS ..................................................................................................................... x
ACKNOWLEDGMENTS ................................................................................................ xi
ABSTRACT .................................................................................................................... xii
1. Introduction.................................................................................................................. 1
1.1
Background ........................................................................................................ 1
1.2
Problem Description........................................................................................... 2
2. Theory and Methodology ............................................................................................ 4
2.1
Theoretical Background ..................................................................................... 4
2.2
Numerical Analysis - Modeling ......................................................................... 5
2.2.1
Parts, Part Geometry, and Assembly...................................................... 6
2.2.2
Property Definition................................................................................. 7
2.2.3
Step Definition, Boundary Conditions, and Applied Loads ................ 10
2.2.4
Mesh ..................................................................................................... 15
3. Results and Discussion .............................................................................................. 17
3.1
Baseline Model Results .................................................................................... 17
3.2
Cyclic Load Results ......................................................................................... 21
3.2.1
Load Case Comparison ........................................................................ 21
3.2.2
Weight Comparison ............................................................................. 23
4. Conclusions................................................................................................................ 26
5. References.................................................................................................................. 28
6. Appendices ................................................................................................................ 29
ii
6.1
Excel Data for Strain Dependent Cartilage and Labrum ................................. 29
6.2
Excel Data for Loads as a Function of %BW .................................................. 30
6.3
Excel Data for Varying Load Cases as Inputs ................................................. 31
6.4
Excel Data for Varying Weights as Input Loads ............................................. 33
iii
LIST OF TABLES
Table 3.1 Tabular Results for Pore Pressure, Strain and Normal Contact Force for
Differing Load Cases ....................................................................................................... 22
Table 3.2 Tabular Results for Pore Pressure, Strain and Normal Contact Force for
Differing Weights ............................................................................................................ 24
Table 6.1 Input Data for Material Property of Strain Dependent Cartilage and Labrum 29
Table 6.2 Data Used for Development of Polynomial Functions for Walking, Jogging
and Sprinting Loads ......................................................................................................... 30
Table 6.3 ABAQUS Input Data Developed from Polynomial Functions for Walking,
Jogging and Sprinting (150 Lb Bodyweight) .................................................................. 32
Table 6.4 Varying Weights Represented as Contact Forces for ABAQUS Input
Amplitudes....................................................................................................................... 33
iv
LIST OF FIGURES
Figure 1.1 (A) 3D View of the hip joint (B) View of the labral and cartilage attachment
points [2] ............................................................................................................................ 1
Figure 2.1 Structure of articular cartilage with representation of proteoglycans, collagen
and water concentration varying with depth [8] ................................................................ 5
Figure 2.2 Axisymmetric finite element geometry representation of the cartilage on the
femoral head (dark green) and the cartilage with intact labrum (light green orange)
which is attached to the subchondral bone of the acetabulum. .......................................... 7
Figure 2.3 Material orientation assignment for the in-plane and out-of-plane material
properties in the labrum ................................................................................................... 10
Figure 2.4 ABAQUS screenshot of the interaction module (top-row) detailing the rigid
surfaces and the contact interaction surfaces. The load module is also shown detailing
the areas where pore pressure = 0 boundary condition is employed and necessary
boundary conditions for axisymmetry are shown (bottom-row) ..................................... 11
Figure 2.5 Initial contact step with applied displacement ............................................... 12
Figure 2.6 Walking, Jogging, and Sprinting Polynomial Functions as Percentage of Body
Weight.............................................................................................................................. 13
Figure 2.7 Cyclic Load Representation for 150 Lb Conditions of Walking, Jogging, and
Sprinting .......................................................................................................................... 14
Figure 2.8 Graphical Representation of Varying Weights for ABAQUS Amplitude Input
......................................................................................................................................... 15
Figure 2.9 Mesh of CAX4P elements for both the intact labrum (top) and resected
labrum (bottom) ............................................................................................................... 16
Figure 3.1 Fluid velocity vectors for an intact labrum (top) and resected labrum (bottom)
for a 200 lb force being held for 100 seconds ................................................................. 18
Figure 3.2 Normal contact force represented as vectors at each element for an intact
labrum (top) and resected labrum (bottom) at 1 second of full load application for a 200
lb load. ............................................................................................................................. 19
Figure 3.3 In-Plane strain for an intact labrum (left) and resected labrum (right) at 1
second of full load application for a 200 lb load. ............................................................ 20
v
Figure 3.4 Graphical Results for Normal Contact Force for Differing Load Cases ........ 23
Figure 3.5 Graphical Results for Normal Contact Force for Differing Weights ............. 25
vi
LIST OF SYMBOLS
Symbol/Variable
n
Description
Volume fraction of the voids to total volume
Units
-
Vvoids
Total volume of the voids
in3
Vtotal
Total volume of the solid matrix
in3
e0
Void ratio as defined by ABAQUS at t=0
-
λ
Lame’s first constant for the solid matrix
psi
μ
Lame’s second constant for the solid matrix
psi
𝜎𝑖𝑒
Principal elastic stressed for hyperelastic model
psi
𝜆𝑖
Principal stretch ratios for hyperelastic model
-
U
Strain energy density function
-
𝛼𝑖
Material parameter for hyperelastic model
𝛽
Material parameter for hyperelastic model
k0
Permeability at t=0
in/s
Strain dependent permeability
in/s
k(e)
M
Material constant for strain dependence
-
κ
Material constant for strain dependence
-
Ep
In-plane modulus of labrum
psi
Et
Transverse modulus of labrum
psi
νp
In-plane Poisson’s ratio of labrum
νpt
νtp
Poisson’s ratio (labrum) for strain transverse resulting
from stretch normal to it
Poisson’s ratio (labrum) for strain in plane resulting from
stretch normal to it
-
-
Gp
Shear modulus, in-plane of labrum
psi
Gt
Shear modulus, transverse of labrum
psi
%BW
Load as a percentage of body weight
-
vii
ACRONYMS AND DEFINITIONS
1. Femoro-acetabular Joint –The joint in the human hip consisting of the femoral
head, acetabulum, joint capsule, associated ligaments and articular cartilage.
2. Labrum – Located in the joint capsule attached to the acetabulum and made of
primarily collagen. The primary focus of this paper, the labrum has been studied
for its unique function which has been proposed as creating a seal for the joint
fluid between the femur and acetabulum.
3. Excision – The term for removal of an organ, tissue, bone or tumor from a body.
Significant to this research paper due to the surgical techniques employed for
labrum pain consisting of removal of the labrum in its entirety or in a small
section. Used interchangeably with the term “resection”.
4. Articular Cartilage – Soft, porous, composite material made up mostly of water
and collagen found mainly in the joints of the human body as a bearing surface.
Modeled in this paper as a biphasic material consisting of a solid matrix and a
fluid that can permeate through the voids of the solid.
5. Cartilage Consolidation – The process by which a sustained load will cause
increasing loads in the cartilage solid matrix due to exudation of fluid from the
surface in contact.
6. Biphasic – A system that has two phases. In this case, the articular cartilage has a
solid phase in the presence of collagen fibers and a fluid which can permeate
between the voids made by the solid matrix.
7. Poroelasticity – The mechanical model detailing a porous elastic material
permeated by an incompressible fluid. In this paper, used to model the articular
cartilage and labrum.
8. ABAQUS – Commercial finite element code used for this research paper.
9. Gait cycle – The cycle by which a human walks; consists of the time span from
when one foot contacts the ground to when that foot contacts the ground again.
10. Hyperelastic – A material model in which the stress-strain relationship derives
from a strain energy density function. The cartilage in the finite element models
found in this research paper are modeled by a hyperelastic (HYPERFOAM)
viii
model due to the large change in cartilage volume that can occur when
undergoing compressive loading.
11. Transversely Isotropic – The description of a material in which the material
properties are equivalent in the pain but differ perpendicular to the plane. The
labrum is modeled as this type of material in the finite element code due to its
specific orientation of collagen in the circumferential direction, or around the rim
of the acetabulum.
12. CAX4P – A continuum axisymmetric 4-noded bilinear pore-pressure and bilinear
displacement quadrilateral element. The element type chosen for the analyses
performed in this paper due to its ability to transmit fluid pressure through
compressive loading and to converge in a contact governed model.
13. POR – The abbreviation for the field variable in ABAQUS which calculates the
pore pressure of the fluid at each integration point.
14. LE MAX PRINCIPAL – The abbreviation for the field variable in ABAQUS that
represents the maximum principal logarithmic strain calculated at each
integration point.
15. CNORM – The abbreviation for the field variable in ABAQUS that represents
the normal contact force calculated at a surface specified in contact for each
integration point on that surface.
16. Reference Point – The point in the model that all degrees of freedom are tied to
when considering a rigid surface or part. In this paper abbreviated as “RP”.
ix
KEYWORDS
1. Labrum
2. Femoro-acetabular joint
3. Articular Cartilage
4. Finite Element Modeling
5. Poroelasticity
x
ACKNOWLEDGMENTS
I would first like to thank my Project Adviser, Professor Ernesto, for hours spent
offering helpful hints, guidance and pointed reviews on various topics.
Secondly, I would like to thank any and all of the contributors to my reference
page who have published similar papers on this subject. Without the thorough
knowledge base put together by these members, this project would have never been
possible.
I would also like to thank my friends and family for the tons of support I was
provided, including the very heartfelt understanding that they would not be seeing me
(as much) until the completion of this degree.
Lastly, I would like to thank my lovely girlfriend Allison. She was a constant
source of inspiration and understanding when the times got tough, the hours got short,
and the nights got late. I would have never made it through this without her.
xi
ABSTRACT
The labrum of the femoro-acetabular joint provides an effective seal for the
articulating cartilage surfaces on the head of the femur and in the acetabulum. Due to
advancement in surgical techniques, removal of the labrum or excision is now
commonly used to alleviate pain in patients subjected to labrum tears. The lack of a
labrum providing an adequate seal for the cartilage in the joint may lead to increased
cartilage consolidation during loading and therefore accelerated wear, creating more pain
for the patient. The commercial finite element program, ABAQUS, was used to quantify
the increase in the contact force between the articulating surfaces in the joint subjected
to increasingly strenuous activities as well as changes in bodyweight. The results showed
the normal contact force may increase up to 200% due to the lack of a functional labrum,
implying a substantial risk for an increase in wear rates of the cartilage. Furthermore, the
results showed increases in contact forces due to greater body weight and more
strenuous activity, such as jogging when compared with walking in the absence of a
labrum.
xii
1. Introduction
1.1 Background
In recent years, the management of hip and groin injuries has broadened
significantly due to advancements in arthroscopic procedures. Minimally invasive
surgical techniques allow for a relatively fast recovery for athletes in highly competitive
environments and often a return to normal activity without pain. The advancements in
magnetic resonance imaging (MRI) help explain the source of pain stemming from
damage or deformities interior to the femoro-acetabular joint. A major result of
advanced imaging techniques was the evaluation of acetabular labral tears, which are
often left untreated. Given improvements in detecting tears, the acetabular labrum
became a main focus for research due to a lack of understanding with regard to its
function.
The labrum in the human femoro-acetabular joint, located in the capsule of the
hip, is attached to the circumference of the acetabular perimeter. As shown in Figure 1,
the transverse acetabular ligament is connected to the labrum both anteriorly and
posteriorly. The labrum is thinner in the anterior inferior section and thicker, with a
slight roundness in appearance, in the posterior section. Free nerve endings have been
identified within labral tissue, which may explain the pain pathway in a patient with a
labral tear [1].
Figure 1.1 (A) 3D View of the hip joint (B) View of the labral and cartilage attachment points [2]
1
In an attempt to fully understand the pathology of, and study the range of surgical
techniques to remove pain associated with, acetabular labrum tears, experiments and
finite element modeling have been used to explain the labrum’s function as a part of the
femoro-acetabular joint. Using a finite element model, Ferguson et al. demonstrate one
of the most important functions of the joint, namely the labrum’s function as a seal for
escaping fluid under normal loading between the articular cartilage on the femoral head
and acetabulum [3], [4]. The labrum effectively prevents fluid from escaping the joint in
order to retain a thin fluid film between the articulating surfaces allowing lubrication and
transfer of the load via fluid pressure, which prevents premature wear of the cartilage
surfaces by reducing cartilage consolidation. The authors attempted to replicate the
findings of the model using an in vitro experiment, with similar results [5]. Song et al.
used experimental results on cadaveric hips to show the friction increase from partial
removal or complete removal of the acetabular labrum, further validating the
hypothesized sealing function [6].
Using MRI techniques, a patient can be diagnosed with an acetabular labral tear
and may choose to undergo surgery. In the event that the labrum cannot be fully
repaired, excision, otherwise known as debridement, which is a complete or partial
removal of the torn area, is implemented to relieve the pain. The surgery may also
uncover a significant amount of work or damaged articular cartilage and may require
micro-fracturing to elicit growth of new cartilage. As previously proven by Ferguson et
al., if the labrum is no longer functioning as a seal to the joint fluid, cartilage
consolidation will greatly increase [3]. Continued rotation of the femoral head within the
joint during walking or exercising will wear away cartilage due to the increased friction
and lack of fluid to develop hydrostatic pressure to carry the load. Therefore, a surgical
technique used to re-grow cartilage may provide short-term pain relief but the long term
effects of a debrided or damaged labrum will be problematic.
1.2 Problem Description
The function of the acetabular labrum as a seal for the femoro-acetabular joint
has been widely established through finite element modeling and in vitro
experimentation. This knowledge allows refinement of surgical techniques and physical
2
therapy programs for patients suffering hip pain. If a patient requires excision of the
labrum to relieve immediate pain, the long term effects on cartilage consolidation must
be considered.
A well known principle in tribology is an increase in the normal force on a
material will lead to increased frictional forces and this often may expedite wear on the
surface of the material. This project will seek to utilize the ability of finite element
software to model a material as poroelastic, in this case the biphasic (liquid and solid)
configuration of cartilage, and to explore how exposure to different loading conditions
affect the strains and stresses in the solid matrix. The loading conditions in this case
would be normal forces into the femoro-acetabular joint from daily activities such as
walking and even more strenuous conditions including jogging. The intent is to
determine whether there is a regimen a patient can follow after being diagnosed with a
torn labrum that will limit the wear in articular cartilage and subsequent pain, ultimately
leading to improved health and freedom from pain later in life.
3
2. Theory and Methodology
2.1 Theoretical Background
A porous medium can be modeled in a commercial finite element code in which
the medium is considered a biphasic material and adheres to the effective stress
principle, which considers the total stress acting at a point to be made up of an average
pressure stress and the solid matrix stress, in order to describe its behavior. The porous
medium is considered to consist of a solid matrix and voids that can contain liquid. The
constitutive behavior of the material is governed by the response of the liquid and solid
matter to local pressure, or fluid flow, and the response of the solid matrix to effective
stress [7].
The importance of a commercial code being able to model a porous medium is
critical in analyzing geological systems such as soil containing ground water and the
effect of forces on that system. For this project, the ability to model a porous media is
also critical when considering biological systems, such as articular cartilage.
Articular cartilage can be described as a soft, porous, composite material made up of
collagen, proteoglycans, and water. Visually, articular cartilage is white with a smooth,
shiny surface. The collagen and proteoglycans in the cartilage are intertwined to create a
solid matrix of material. Typically, the volume of cartilage is made up of 80% water [8].
Cartilage is found between bony contact surfaces in the human body, otherwise
known as joints. The percent compositions of the materials that make up cartilage vary
with the depth of the cartilage, as shown in Figure 2.1. At the surface of the cartilage, the
collagen fibers are oriented parallel to the surface, in the middle zone the orientation
becomes more angled, and in the deep zone the collagen fibers begin to orient
themselves perpendicular to the bone interface in order to properly anchor into the bone
[8].
4
Figure 2.1 Structure of articular cartilage with representation of proteoglycans, collagen and water
concentration varying with depth [8]
Due to the biphasic make up of articular cartilage, the intrinsic mechanical
properties of each phase, the liquid and solid matrix, as well as the interaction between
the phases, correspond to the interesting mechanical properties of the cartilage as a
whole. Mow et al. was able to apply a linear nonhomogenous theory to accurately
represent test data for an aggregate elastic modulus and permeability of the tissue [9].
Adaptations of these findings are used in conjunction with a commercial finite element
code in order to run complex analyses to represent joints in the human body.
2.2 Numerical Analysis - Modeling
The biphasic cartilage model detailed by Mow et al. demonstrated the
mechanical properties of articular cartilage through an analytical solution [9]. In order to
adequately analyze the joint contact mechanics within the irregular geometry of a human
joint, an appropriate finite element code is required. J.Z. Wu, W. Herzog, and M. Epstein
demonstrated the biphasic cartilage model can be implemented in the finite element code
ABAQUS [10]. The results achieved in ABAQUS were comparable to analytical
solutions as well as other finite element codes for three numerical tests: an unconfined
indentation test, a test with the contact of a spherical cartilage surface with a rigid plate,
and an axi-symmetric joint contact test [11].
Since ABAQUS has previously demonstrated a capacity to analyze the biphasic
cartilage model proposed by Mow et al. and contact mechanics, it will be used herein.
An axisymmetric model is used to reduce the number of elements in comparison to a
three dimensional model. Although it has been shown the joint can be modelled as a two
5
dimensional plane-strain finite element mesh [4], the axisymmetric model alleviates the
use of in-plane truss elements to represent the out-of-plane stiffness of the labrum.
2.2.1
Parts, Part Geometry, and Assembly
In the ABAQUS part module, two 2D axisymmetric parts were created to
represent the joint. The parts represent the articulating cartilage surfaces in the hip with
an intact labrum and on the head of the femur, as shown in Figure 2.2. The light green
represents the acetabular articular cartilage with the attachment to the rigid acetabulum,
the dark green represents the femoral articular cartilage with the attachment to the rigid
femur, and the orange represents the labrum with the attachment to the acetabulum. All
corners, including the labrum to acetabulum interface, were modelled as sharp corners
without radii for simplicity and ease in meshing. This assumption is reasonable as
demonstrated by results in Haemer, Carter and Giori [12].
In the assembly, the parts were oriented with a common part reference point,
which is used to tie the bone interface surface in each the femur and acetabulum. In this
way, the loads and boundary conditions can be applied directly to the distinct reference
points for each part. The axis of symmetry details the axis in which the each part is
revolved around to represent the joint in three dimensional space.
A second model was built in which the labrum was removed to represent a
resected, or removed, labrum. The radius of the femur was set as 1.02 in. and the bone
was modeled as rigid and impermeable. The articulating cartilage surfaces were modeled
to have a thickness of 0.11 in. The joint was modeled as being fully congruent and the
labrum was modeled as being in continuity with the articular cartilage [3], [12].
6
Figure 2.2 Axisymmetric finite element geometry representation of the cartilage on the femoral head
(dark green) and the cartilage with intact labrum (light green orange) which is attached to the
subchondral bone of the acetabulum.
2.2.2
Property Definition
The property module in ABAQUS allows for the definition of material properties
and orientations. Since the material properties are equivalent for the acetabular and
femoral articular cartilage, only two material definitions were required to represent both
the cartilage and the labrum.
For the cartilage, a hyperporoelastic material model was used due to the porosity
of the material allowing large volumetric changes. The hyperelastic constitutive relation
was based on the following:
𝜎𝑖𝑒 =
𝜕𝑈
𝜕𝜆𝑖
𝑖 = 1,2,3
2𝜇
(1)
1
𝛼𝑖
𝛼𝑖
𝛼𝑖
𝑖
−𝛼𝑖 𝛽
𝑈 = ∑𝑁
− 1)]
𝑖=1 𝛼2 [𝜆1 + 𝜆2 + 𝜆3 − 3 + 𝛽 ((𝐽)
𝑖
7
(2)
where 𝜎𝑖𝑒 and 𝜆𝑖 (i=1,2,3) are the principal elastic stresses and the principal stretch ratios
respectively; U is the strain energy function; J = 𝜆1 𝜆2 𝜆3 is the volume ratio; 𝛼𝑖 , 𝜇𝑖
(i=1…N), and β are material parameters [10]. The material parameters 𝛼𝑖 are determined
from equations 𝜆 = 4𝛼0 𝛼2 , 𝜇 = 2(𝛼1 + 𝛼2 )𝛼0, and 𝛽 = 𝛼1 + 2𝛼2 in which 𝜆=1.89 psi,
𝜇=49.2 psi, and 𝛽=0.761 [12].
ABAQUS represents the solid volume fraction as a void ratio (e0) based on the
following:
𝒏=
𝑽𝒗𝒐𝒊𝒅𝒔
(3)
𝑽𝒕𝒐𝒕𝒂𝒍
𝒏
𝒆𝟎 = (𝟏−𝒏)
(4)
where e0 for cartilage is 4. The specific weight of the pore fluid is γ=0.0361008 lb/in3 [3].
The permeability is dependent on the strain, which can be related directly to the
void ratio based on the following equation:
𝑒 𝜅
𝑀
1+𝑒 2
𝑘(𝑒) = 𝑘0 (𝑒 ) exp { 2 [(1+𝑒 ) − 1]}
0
0
(5)
where k0=2.89355E-009 in/s, e is the void ratio, and e0 is the void ratio of the
undeformed state, as defined above. M and κ are material constants which have been
determined for cartilage to be 4.638 and 0.0848, respectively. A tabular form imported
into the ABAQUS material property definition was used to define the strain dependent
permeability over a void ratio range of 1.7 to 5 and is shown below in Appendix 6.1
[12].
The permeability of the labrum was set at one-sixth of articular cartilage, which is
comparable to experimental results found by Ferguson et al. The strain-dependence of
labrum permeability is not well defined and since the transverse properties of the
cartilage compared with the labrum are of the same order, the strain-dependent
permeability was adapted from the material constants of cartilage. However, an initial
void ratio for the labrum was defined as 3, in lieu of 4 for cartilage as detailed in
Appendix 6.1 [12].
The labrum is modeled as a transversely isotropic permeable elastic material in
which the circumferential direction is the out-of-plane direction with differing material
8
properties considering the labrum’s unique pathology in which fibrils run in the
circumferential direction, resulting in a greater stiffness. ABAQUS relates the stresses to
the strains in each direction by the following tensor:
where p and t stand for “in-plane” and “transverse” or out-of-plane, respectively [7].
In the case of Poisson’s ratio νtp characterizes the strain in the plane of isotropy
resulting from stress normal to it, while νpt characterizes the transverse strain in the
direction normal to the plane resulting from stress in the plane. These quantities are
related by the following:
𝜈𝑡𝑝
𝜈
⁄𝐸 = 𝑝𝑡⁄𝐸
𝑡
𝑝
(6)
For the labrum, the specific engineering constants were defined as Ep=80 psi,
Et=29000 psi, νp=0.05, νtp=0.05, νpt=0.0001, and Gt=3.77 psi [12].The shear modulus inplane, Gp, is defined by:
𝐸𝑝
𝐺𝑝 = 2(1+𝜈
𝑝)
(7)
where Gp=38 psi.
When engineering constants are used, a specific material orientation must be
defined. The transverse isotropy defined for the labrum material is assigned with the
orientation shown below, noting the 1 and 2 directions are in-plane while the 3rd
direction, representing the circumferential oriented fibers in the labrum, is out-of-plane
(Figure 2.3).
9
Figure 2.3 Material orientation assignment for the in-plane and out-of-plane material properties in
the labrum
2.2.3
Step Definition, Boundary Conditions, and Applied Loads
Since the articular cartilage and labrum are modeled as poroelastic materials, a
coupled pore fluid diffusion and stress analysis was employed and a *SOILS step is
required [7]. The opposing faces of cartilage are defined as contact surfaces, including
the labrum surface in the applicable model, see Figure 2.4. Defining contact will allow
pore fluid to flow between the surfaces that come into contact. The degree of freedom
for pore fluid flow is no fluid flow across defined surfaces as the default in ABAQUS.
Since this is the case, any free surface will employ a boundary condition that sets the
pressure at this surface to zero. Through the analysis, fluid will enter or leave this
surface to maintain this boundary condition. As previously stated, the surfaces in which
the cartilage and labrum attach to bone are made rigid and impermeable. This
assumption is reasonable since the material properties of bone are many magnitudes
different compared to cartilage. The rigid surfaces are tied to reference points on the axis
of symmetry. All loads and boundary conditions are applied to these reference points
since ABAQUS will define all degrees of freedom for nodes on the rigid body by the
10
reference node. The acetabulum reference point is restrained in the radial and vertical
directions. The loads are applied at the femur reference point in the vertical direction and
are defined in section 2.2.3.2. In addition, axisymmetric boundary conditions are applied
along the axis of symmetry to prevent movement and rotation (Figure 2.4).
Figure 2.4 ABAQUS screenshot of the interaction module (top-row) detailing the rigid surfaces and
the contact interaction surfaces. The load module is also shown detailing the areas where pore
pressure = 0 boundary condition is employed and necessary boundary conditions for axisymmetry
are shown (bottom-row)
11
2.2.3.1 Initial Contact Step
The first step of the analysis is used to ensure good contact between the faces due to
any differences in geometry. A small displacement of 0.002 in. is used on the reference
point (RP) of the femur, shown in Figure 2.5. This boundary condition will be active in
this step and then removed in all subsequent steps for load application. A separate
boundary condition is applied and propagated to all subsequent steps to prevent left-right
movement of the RP. A simply supported boundary condition (U1=U2=0) is applied to
the RP of the acetabulum to prevent rigid body motion. For this step, a time period of 1
second is used and the steady-state consolidation assumption is used since the transient
affects of fluid flow are not required at this point.
Figure 2.5 Initial contact step with applied displacement
2.2.3.2 Load Application Step
Once initial contact has been established, the necessary loads can be applied at the
Femur RP. All loads were derived from the Fz direction detailed by Fabry, Hermann,
Kaehler, Klinkenberg, Woernle and Bader for the walking load case [13]. Data points
were developed from the hip contact force, as a percentage of bodyweight (%BW), over
a normalized time scale (Appendix 6.2). A conservative estimate of cycle time for the
12
gait cycle at a moderate walking pace was taken to be 1 second. The data points were fit
to a 6th order polynomial which will be used to develop the tabular input to be used in
ABAQUS as an amplitude function (Figure 2.6). To develop load conditions for jogging
and sprinting, the walking loads were amplified by 150% bodyweight and 200%
bodyweight, respectively. The time scale was also modified from 1 second to 0.8
seconds for a jogging cycle and 1 second to 0.55 seconds for a sprinting cycle. Each
cycle is taken to be from initial heel contact to the next increment of heel contact on the
same leg.
Combined % BW Loading With Trendlines
Walking % BW
500
450
Jogging % BW
400
350
% Load (Fz)
Sprinting % BW
300
250
Poly. (Walking % BW)
y = -27962x6 + 87465x5 - 103145x4 + 58370x3
- 17298x2 + 2580x + 66.19
R² = 0.9898
Poly. (Jogging % BW)
200
150
100
y = -204262x6 + 505163x5 - 473217x4 +
213052x3 - 49737x2 + 5749.6x + 68.754
R² = 0.9943
Poly.
(Sprinting
% BW) 4
6
5
50
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Time (seconds)
1
y = -2,858,684.52x + 4,840,189.88x - 3,109,233.68x +
960,451.20x3 - 153,272.38x2 + 12,035.17x + 71.32
R² = 1.00
Figure 2.6 Walking, Jogging, and Sprinting Polynomial Functions as Percentage of Body Weight
The polynomial functions developed for the three load cases are based on %BW
and they are appropriately scaled to represent the bodyweight of a 150 lb human. %BW
was used in order to easily scale between differing body weights and magnitudes of
loads. A model was created for all three cases, one set of each for the resected and intact
labrum, totaling six models in all. The tabular data developed from the scaled
13
polynomial functions (Appendix 6.3) was input into ABAQUS as an amplitude function
and applied at the Femur RP. Each load case was applied over five cycles (Figure 2.7).
These inputs represent a consolidated system of various load conditions that can be
imparted on the hip during daily activity or physical exertion.
Combined Load Conditions (150 LB)
700
600
Load (Lbs)
500
Sprinting
150 Lb
400
300
Walking
150 Lb
200
Jogging
150 LB
100
0
0
1
2
3
Time (seconds)
4
5
Figure 2.7 Cyclic Load Representation for 150 Lb Conditions of Walking, Jogging, and Sprinting
As a comparison, walking loading conditions were developed for a 200 lb and
250 lb human (Appendix 2.4) and are shown graphically in Figure 2.8. This requires
creating two models each for an intact and resected labrum and providing the necessary
inputs as an amplitude function. Similar to the varying load conditions for walking,
jogging and running, these loads are applied over 5 cycles and in this case are a 5 second
total step time.
These conditions will serve to represent the effect on weight for intact and
resected labrums. Inherently, an increase in weight is an increase in contact force since
the formulation used is based on %BW.
14
Combined Weights for Walking
600
500
Load (Lbs)
400
200 Lb Human
150 Lb Human
300
250 Lb Human
200
100
0
0
0.1
0.2
0.3
0.4
0.5 0.6
Time (s)
0.7
0.8
0.9
1
Figure 2.8 Graphical Representation of Varying Weights for ABAQUS Amplitude Input
2.2.4
Mesh
Figure 2.9 details the mesh refinement required for an accurate result. A mesh
convergence study was performed resulting in the mesh density used. CAX4P, 4-noded
axisymmetric quadrilateral, bilinear displacement, bilinear pore pressure elements were
used to handle the effective stress of coupled pore pressure diffusion and stress analysis.
4-noded elements were used in lieu of 8-noded since the 4-noded elements behave better
in contact. This resulted in a much denser mesh due to half the number of nodes used for
each element.
15
Figure 2.9 Mesh of CAX4P elements for both the intact labrum (top) and resected labrum (bottom)
16
3. Results and Discussion
3.1 Baseline Model Results
As a baseline, to ensure the properties and boundary conditions were reasonable, a
200 lb load was applied over 1 second and then held for 1000 seconds. Specifically, the
fluid velocities represented as vector quantities at each node, the normal contact force
between the articulating cartilage surfaces, and the in-plane strains were evaluated for an
intact and resected labrum.
The baseline model showed expected results for fluid velocities and normal contact
forces; however it did present an unexpected result for strain. Numerical results are not
tabulated for these results since the intent was to show representative strain, force, and
fluid velocity results for an arbitrary load. The results are only for a comparison of visual
representations of requested field variables. The results provided herein were of the same
order of magnitude when compared to results found by Haemer, Carter and Giori [12].
For the fluid velocity, the field variable was taken at 100 seconds into the 1000
seconds step. The fluid velocity vectors for a model with an intact labrum, as displayed
in Figure 3.1, show the labrum effectively “choking” the fluid as it is forced from the
surface of the cartilage. The velocity significantly decreases at the interface and
dissipates as the fluid continues into the cross section. There is a significant increase in
the fluid velocity vectors on the periphery of the labrum over a very short length.
Compared with the resected labrum, the fluid is escaping over a much larger area, which
is congruent with the boundary conditions set for the free surface. The behavior of the
fluid flow in the model shows the inputs are yielding expected results.
17
Figure 3.1 Fluid velocity vectors for an intact labrum (top) and resected labrum (bottom) for a 200
lb force being held for 100 seconds
The normal contact force field variable was evaluated after the 1 second ramp load
was applied, as shown in Figure 3.2, to obtain the maximum contact force before the
cartilage began to consolidate and the fluid pressure was redistributed. As expected, the
labrum adds a significant amount of contact area to the articulating cartilage surface of
the femur. This allows the force to be more evenly distributed over the cartilage of the
18
femoral head, resulting in a relatively smaller contact force. Since the model with the
resected labrum has a diminished contact area, the contact force showed a higher peak
force and a distribution that shifted the force towards the centerline of the joint.
Figure 3.2 Normal contact force represented as vectors at each element for an intact labrum (top)
and resected labrum (bottom) at 1 second of full load application for a 200 lb load.
Similar to the normal contact force field variable, the in-plain strain was evaluated
after 1 second, which is the maximum value prior to cartilage consolidation. The results
19
showed a higher strain in the femoral and acetabular cartilage in the resected labrum
model compared with the intact labrum as expected, however the area of highest strain
contour was of particular interest. Figure 3.3 shows the areas of highest strain (light
green) were at the cartilage to bone interface, not at the articulating surface (light blue).
The smaller magnitude of strain at the contact surface does not explain how patients will
show degraded and worn cartilage at the articular surface during hip arthroscopy. The
significance is that even with the baseline model the results show the normal contact
force will be the driving component when evaluating cartilage wear.
Figure 3.3 In-Plane strain for an intact labrum (left) and resected labrum (right) at 1 second of full
load application for a 200 lb load.
20
3.2 Cyclic Load Results
For comparison of the results, the field variables of pore pressure (POR), maximum
strain (LE Max Principal), and normal contact force (CNORM) are extracted from the
steps at the peak of each load cycle. The results are broken into two separate sections for
comparison in order to correlate between the differing load cases of walking, running
and sprinting as well as the effect of the increase in weight on the extracted field
variables from the analyses.
3.2.1
Load Case Comparison
As shown in Table 3.1, the pore pressure, strain, and normal contact force are
reported at the peak load application in each model.
The pore pressure is shown in order to demonstrate the lack of transient effects from
the loading and unloading in the joint. The pressure is fairly constant at the peak of each
load cycle, dictating the load carried by pore pressure does not change through the
cycles. Also, the pore pressure is much higher in models with a resected labrum but the
pore pressure maximum occurs at the centerline of the joint, dictating that the freely
draining surfaces do not expunge enough fluid to lower the pore pressure at the joint
centerline. Although it provides a reasonable seal in the baseline model the intact labrum
represents an area in which the fluid may redistribute itself into causing pressures on the
order of magnitude of 20% smaller between the model with the resected labrum and
intact labrum.
The maximum strain, as previously shown in the baseline model, occurs at the bone
interface, which does not explain degradation of cartilage on the articulating surfaces.
Since this is the case, the maximum strain is reported in Table 3.1 to show the removal
of a labrum will also increase the maximum strain gradient, which may eventually
propagate to the articulating surface with increasing loads.
Of particular interest, based on results from the baseline model, is the contact force
between opposing faces of cartilage on the acetabulum and femur. The comparison of
the computed contact force, between the intact labrum model and resected labrum
21
model, shows an average percent increase of 160%, 170%, and 155% for walking,
jogging and sprinting.
Table 3.1 Tabular Results for Pore Pressure, Strain and Normal Contact Force for Differing Load
Cases
150 Lb
Walking
Intact Labrum
Step Time
(s)
0.2
1.2
2.2
3.2
Pore Pressure
(psi)
149.7
149.2
148.3
147.0
LE Max.
Principal
0.092
0.091
0.091
0.090
Contact
Force (Lb)
0.276
0.277
0.276
0.274
150 Lbs
Walking
Resected
Labrum
4.2
0.2
1.2
2.2
3.2
146.8
173.6
175.3
172.8
172.0
0.090
0.127
0.128
0.128
0.127
0.274
0.756
0.709
0.709
0.707
4.2
0.2
1.0
1.8
2.6
170.9
222.4
220.9
220.0
220.0
0.126
0.136
0.135
0.135
0.135
0.705
0.359
0.359
0.358
0.358
150 Lb Jogging
Resected
Labrum
3.4
0.2
1.0
1.8
2.6
218.9
257.6
257.6
288.3
254.7
0.134
0.189
0.189
0.188
0.186
0.358
0.974
0.976
0.971
0.967
150 Lb
Sprinting
Intact Labrum
3.4
0.1
0.65
1.2
1.75
253.6
296.0
296.6
293.9
293.9
0.187
0.181
0.181
0.180
0.180
0.966
0.446
0.447
0.445
0.446
150 Lb
Sprinting
Resected
Labrum
2.3
0.1
0.65
1.2
1.75
293.5
344.2
347.3
345.7
346.1
0.180
0.471
0.471
0.471
0.471
0.446
1.135
1.132
1.143
1.145
2.3
345.7
0.470
1.145
Load Case
150 Lb Jogging
Intact Labrum
22
Load Case Comparison Results
1.4
1.2
150 Lbs Walking Intact
Labrum
CNORM (Lbs)
1
150 Lbs Walking Resected
Labrum
0.8
0.6
150 Lbs Jogging Intact
Labrum
0.4
150 Lbs Jogging Resected
Labrum
0.2
150 Lbs Sprinting Intact
Labrum
0
0
1
2
3
Time
4
5
150 Lbs Sprinting Resected
Labrum
(s-1)
Figure 3.4 Graphical Results for Normal Contact Force for Differing Load Cases
3.2.2
Weight Comparison
Similar to the comparison for load cases, the increase in weight has a significant
impact on the normal contact force in the joint. The comparison of the computed contact
force, between the intact labrum model and resected labrum model, shows an average
percent increase of 160%, 220%, and 170% for the differing weights shown in Table 3.2.
Again, the pore pressure and maximum strain are reported but are not of particular
interest. For the weight comparison, the pore pressure and maximum strain show
equivalent traits to the information provided in section 3.2.1.
23
Table 3.2 Tabular Results for Pore Pressure, Strain and Normal Contact Force for Differing
Weights
Load Case
150 Lb
Walking
Intact Labrum
150 Lb
Walking
Resected
Labrum
200 Lb
Walking
Intact Labrum
200 Lb
Walking
Resected
Labrum
250 Lb
Walking
Intact Labrum
250 Lb
Walking
Resected
Labrum
Step Time
(s)
0.2
1.2
2.2
Pore Pressure
(psi)
149.7
149.2
148.3
LE Max.
Principal
0.092
0.091
0.091
CNORM
(Lb)
0.276
0.277
0.276
3.2
147.0
0.090
0.274
4.2
0.2
1.2
2.2
3.2
146.8
173.6
175.3
172.8
172.0
0.090
0.127
0.128
0.128
0.127
0.274
0.756
0.709
0.709
0.707
4.2
0.2
1.2
2.2
170.9
198.8
197.3
197.1
0.126
0.122
0.121
0.121
0.705
0.334
0.333
0.333
3.2
195.7
0.120
0.333
4.2
0.2
1.2
2.2
3.2
195.8
228.7
228.3
227.3
227.6
0.120
0.311
0.311
0.310
0.311
0.333
0.978
1.083
1.063
1.056
4.2
0.2
1.2
2.2
228.4
241.8
248.0
245.1
0.310
0.152
0.152
0.150
1.092
0.387
0.389
0.385
3.2
245.4
0.150
0.380
4.2
0.2
1.2
2.2
3.2
244.8
284.4
283.9
283.6
283.0
0.150
0.390
0.390
0.390
0.389
0.380
1.093
1.031
1.017
1.015
4.2
282.6
0.389
1.011
24
Weight Comparison Results
1.2
150 Lbs Walking Intact
Labrum
CNORM (Lbs)
1
0.8
150 Lbs Walking Resected
Labrum
0.6
200 Lbs Walking Intact
Labrum
0.4
200 Lbs Walking Resected
Labrum
0.2
250 Lbs Walking Intact
Labrum
0
0
1
2
3
4
5
250 Lbs Walking Resected
Labrum
Time (s-1)
Figure 3.5 Graphical Results for Normal Contact Force for Differing Weights
25
4. Conclusions
The following conclusions can be drawn from this research. Specifically, the cyclic
loads, although repetitive in nature, do not influence the cartilage consolidation as much
as the magnitude of the loads. The most significant finding from the results of the
research was the drastic increase of the computed normal contact forces when compared
between models with a labrum and without a labrum. The model results suggest that the
presence of a labrum creates a much larger contact area for distribution of loads,
prevents the exudation of fluid, and allows the fluid an area for redistribution under load.
In the case of the model without the labrum, the solid phase of the cartilage would be
required to carry a greater percentage of the load versus the load being carried by fluid
pressure.
It is well established that the constant of proportionality, the coefficient of friction,
relates the normal load to the friction force. An increase in contact force in the model
with the resected labrum can be correlated to an increase in resulting frictional forces.
Although the model did not provide any rotational motion, one can visualize the effect of
rotation on the joint in the case of increasing contact forces. Therefore, a larger frictional
force would provide a greater force for shearing of the articulating surfaces of the
cartilage. The cartilage on the articulating surfaces would wear away at an accelerated
rate with increasing normal contact loads, eventually resulting in pain through many
cycles and evolutions due to bone-on-bone contact.
The results of the research, through the different loading conditions and weight
comparisons, suggest that in the absence of a functional labrum, or in the case of a
labrum that has been surgically removed, the normal contact forces increase with an
increase in weight and more strenuous activities. For patients, this would mean focusing
on diet to maintain a reasonable weight and avoiding strenuous activities for long
periods of time, which may limit pain later in life.
26
27
5. References
[1] A. S. Ranawat and B. T. Kelly, “Function of the Labrum and Management of
Labral Pathology,” Hip Arthrosc., vol. 15, no. 3, pp. 239–246, Jul. 2005.
[2] C. R. Henak, B. J. Ellis, M. D. Harris, A. E. Anderson, C. L. Peters, and J. A.
Weiss, “Role of the acetabular labrum in load support across the hip joint,” J.
Biomech., vol. 44, no. 12, pp. 2201–2206, Aug. 2011.
[3] S. J. Ferguson, J. T. Bryant, R. Ganz, and K. Ito, “The acetabular labrum seal: a
poroelastic finite element model,” Clin. Biomech., vol. 15, no. 6, pp. 463–468, Jul.
2000.
[4] S. . Ferguson, J. . Bryant, R. Ganz, and K. Ito, “The influence of the acetabular
labrum on hip joint cartilage consolidation: a poroelastic finite element model,” J.
Biomech., vol. 33, no. 8, pp. 953–960, Aug. 2000.
[5] S. J. Ferguson, J. T. Bryant, R. Ganz, and K. Ito, “An in vitro investigation of the
acetabular labral seal in hip joint mechanics,” J. Biomech., vol. 36, no. 2, pp. 171–
178, Feb. 2003.
[6] Y. Song, H. Ito, L. Kourtis, M. R. Safran, D. R. Carter, and N. J. Giori, “Articular
cartilage friction increases in hip joints after the removal of acetabular labrum,” J.
Biomech., vol. 45, no. 3, pp. 524–530, Feb. 2012.
[7] “ABAQUS 6.11 Documentation.” .
[8] A. Neville, A. Morina, T. Liskiewicz, and Y. Yan, “Synovial joint lubrication —
does nature teach more effective engineering lubrication strategies?,” Proc. Inst.
Mech. Eng. Part C J. Mech. Eng. Sci., vol. 221, no. 10, pp. 1223–1230, Jan. 2007.
[9] V. C. Mow, S. C. Kuei, W. M. Lai, and C. G. Armstrong, “Biphasic Creep and
Stress Relaxation of Articular Cartilage in Compression: Theory and Experiments,”
J. Biomech. Eng., vol. 102, no. 1, pp. 73–84, Feb. 1980.
[10] J. Z. Wu and W. Herzog, “Analysis of the mechanical behavior of chondrocytes in
unconfined compression tests for cyclic loading,” J. Biomech., vol. 39, no. 4, pp.
603–616, 2006.
[11] J. . Wu, W. Herzog, and M. Epstein, “Evaluation of the finite element software
ABAQUS for biomechanical modelling of biphasic tissues,” J. Biomech., vol. 31,
no. 2, pp. 165–169, May 1997.
[12] J. M. Haemer, D. R. Carter, and N. J. Giori, “The low permeability of healthy
meniscus and labrum limit articular cartilage consolidation and maintain fluid load
support in the knee and hip,” J. Biomech., vol. 45, no. 8, pp. 1450–1456, May
2012.
[13] C. Fabry, S. Herrmann, M. Kaehler, E.-D. Klinkenberg, C. Woernle, and R. Bader,
“Generation of physiological parameter sets for hip joint motions and loads during
daily life activities for application in wear simulators of the artificial hip joint,”
Med. Eng. Phys., vol. 35, no. 1, pp. 131–139, Jan. 2013.
28
6. Appendices
6.1 Excel Data for Strain Dependent Cartilage and Labrum
The data provided is established using equation (5) provided below, established in
section 2.2.2:
𝑒 𝜅
𝑀 1+𝑒 2
𝑘(𝑒) = 𝑘0 ( ) exp { [(
) − 1]}
𝑒0
2 1 + 𝑒0
9.81 kN/m^3
7.50E-15 m^4/N*s
8.02E-08 in^4/lb*s
M
k
=
=
=
0.036101 lb/in^3
Specific Weight of Fluid
=
8.02E-08 in^4/lb*s Unit Volume Permeability
=
2.89E-09 in/s
Permeability of Cartilage
=
4.82E-10 in/s
Permeability of Labrum
4.638
0.0848
Table 6.1 Input Data for Material Property of Strain Dependent Cartilage and Labrum
Strain Dependent Cartilage
Permeability (in/s)
5.20556E-10
5.50465E-10
5.8302E-10
6.18501E-10
6.57221E-10
6.99527E-10
7.45808E-10
7.96499E-10
8.5209E-10
9.13129E-10
9.80235E-10
1.0541E-09
1.13552E-09
1.22538E-09
1.32467E-09
1.43454E-09
1.55629E-09
1.69136E-09
1.84144E-09
2.00842E-09
2.19447E-09
Strain Dependent Labrum
Void Ratio
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
3.1
3.2
3.3
3.4
3.5
3.6
3.7
29
Permeability (in/s)
1.30046E-10
1.41521E-10
1.54416E-10
1.68935E-10
1.85316E-10
2.03836E-10
2.24819E-10
2.48642E-10
2.75747E-10
3.06652E-10
3.41968E-10
3.82415E-10
4.2884E-10
4.82248E-10
5.43831E-10
6.15004E-10
6.97453E-10
7.9319E-10
9.0462E-10
1.03463E-09
1.18668E-09
Void ratio
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
3.1
3.2
3.3
3.4
3.5
3.6
3.7
2.40205E-09
2.634E-09
2.89355E-09
3.1844E-09
3.51082E-09
3.87771E-09
4.29069E-09
4.75626E-09
5.28191E-09
5.87632E-09
6.54951E-09
7.3131E-09
8.1806E-09
3.8
3.9
4
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5
1.36494E-09
1.57444E-09
1.82127E-09
2.11281E-09
2.458E-09
2.86774E-09
3.35536E-09
3.93711E-09
4.63293E-09
5.46735E-09
6.47052E-09
7.67971E-09
9.14101E-09
3.8
3.9
4
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5
6.2 Excel Data for Loads as a Function of %BW
The data provided below is adapted from the Fz direction provided in the paper by Fabry,
Hermann, Kaehler, Klinkenberg, Woernle and Bader [13]. This information was plotted
in excel and a polynomial function of the 6th power was fit.
Gait Time Jogging=
Gait Time Sprinting=
Load Increase Jog=
Load Increase Sprint=
0.8
s
0.55
s
1.5
Magnitude
2
Magnitude
Table 6.2 Data Used for Development of Polynomial Functions for Walking, Jogging and Sprinting
Loads
Gait Time
(Sprinting)
0
0.022
0.0275
0.0385
0.0495
0.055
0.0605
0.066
Force %BW
Walking
75
140
150
170
195
200
205
211
Force %BW
Jogging
75
210
225
255
292.5
300
307.5
316.5
30
Force %BW
Sprinting
75
280
300
340
390
400
410
422
0.0715
0.07425
0.077
0.0825
0.088
0.0935
0.099
0.1045
0.11
0.1155
0.132
0.143
0.1485
0.154
0.1595
0.165
0.1705
0.198
0.209
0.22
0.231
0.2365
0.242
0.2475
0.253
0.264
0.286
0.308
0.33
0.341
0.4675
0.55
213
215
216
217
219
220
220
221
222
221
219
218
217
215
213
210
208
204
204
204
205
205
203
199
196
189
172
156
120
100
30
75
319.5
322.5
324
325.5
328.5
330
330
331.5
333
331.5
328.5
327
325.5
322.5
319.5
315
312
306
306
306
307.5
307.5
304.5
298.5
294
283.5
258
234
180
150
45
75
426
430
432
434
438
440
440
442
444
442
438
436
434
430
426
420
416
408
408
408
410
410
406
398
392
378
344
312
240
200
60
75
6.3 Excel Data for Varying Load Cases as Inputs
The data provided is developed from the polynomial functions provided in Appendix 6.2
and Figure 2.6. This data was exported to a text file and then read into the ABAQUS
amplitude function to provide the necessary cyclic representation of the gait cycle. The
data provided is applicable to the walking, jogging and sprinting loads during 5 cycles.
31
Table 6.3 ABAQUS Input Data Developed from Polynomial Functions for Walking, Jogging and
Sprinting (150 Lb Bodyweight)
Gait Time
(Walking)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
3.1
3.2
3.3
Force 150
Lb
Bodyweight
Walking
75
300.168282
327.595848
324.061728
310.180872
266.62875
185.881992
93.760068
40.768008
63.240162
114.285
300.168282
327.595848
324.061728
310.180872
266.62875
185.881992
93.760068
40.768008
63.240162
114.285
300.168282
327.595848
324.061728
310.180872
266.62875
185.881992
93.760068
40.768008
63.240162
114.285
300.168282
327.595848
324.061728
Gait
Time
(Jogging)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
3.1
3.2
3.3
Force 150
Lb
Bodyweight
Jogging
75
475.382502
487.563288
472.934088
401.788152
239.23725
82.394712
76.955508
113.173368
475.382502
487.563288
472.934088
401.788152
239.23725
82.394712
76.955508
113.173368
475.382502
487.563288
472.934088
401.788152
239.23725
82.394712
76.955508
113.173368
475.382502
487.563288
472.934088
401.788152
239.23725
82.394712
76.955508
113.173368
475.382502
32
Gait Time
(Sprinting)
Force 150 Lb
Bodyweight
Sprinting
0
0.025
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
1.55
1.6
1.65
75
435.355608
655.7532681
645.6154286
633.2655763
586.4965276
468.603782
293.5648472
132.9782621
74.7623191
133.6134844
113.2245169
435.355608
655.7532681
645.6154286
633.2655763
586.4965276
468.603782
293.5648472
132.9782621
74.7623191
133.6134844
113.2245169
435.355608
655.7532681
645.6154286
633.2655763
586.4965276
468.603782
293.5648472
132.9782621
74.7623191
133.6134844
113.2245169
3.4
3.5
3.6
3.7
3.8
3.9
4
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5
310.180872
266.62875
185.881992
93.760068
40.768008
63.240162
114.285
300.168282
327.595848
324.061728
310.180872
266.62875
185.881992
93.760068
40.768008
63.240162
114.285
3.4
3.5
3.6
3.7
3.8
3.9
4
487.563288
472.934088
401.788152
239.23725
82.394712
76.955508
113.173368
1.7
1.75
1.8
1.85
1.9
1.95
2
2.05
2.1
2.15
2.2
2.25
2.3
2.35
2.4
2.45
2.5
2.55
2.6
2.65
2.7
2.75
435.355608
655.7532681
645.6154286
633.2655763
586.4965276
468.603782
293.5648472
132.9782621
74.7623191
133.6134844
113.2245169
435.355608
655.7532681
645.6154286
633.2655763
586.4965276
468.603782
293.5648472
132.9782621
74.7623191
133.6134844
113.2245169
6.4 Excel Data for Varying Weights as Input Loads
The data provided is developed from the polynomial functions provided in Appendix 6.2
and Figure 2.6. This data was exported to a text file and then read into the ABAQUS
amplitude function to provide the necessary cyclic representation of the gait cycle. The
provided data is applicable to walking with a bodyweight of 150 lbs, 200 lbs and 250 lbs
over 5 cycles.
Table 6.4 Varying Weights Represented as Contact Forces for ABAQUS Input Amplitudes
Gait Time
(Walking)
Force 150
Lb
Bodyweight
Walking
Force 200
Lb
Bodyweight
Walking
Force 250 Lb
Bodyweight
Walking
0
0.1
0.2
99.285
300.168282
327.595848
132.38
400.224376
436.794464
165.475
500.28047
545.99308
33
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4
4.1
4.2
4.3
324.061728
310.180872
266.62875
185.881992
93.760068
40.768008
63.240162
114.285
300.168282
327.595848
324.061728
310.180872
266.62875
185.881992
93.760068
40.768008
63.240162
114.285
300.168282
327.595848
324.061728
310.180872
266.62875
185.881992
93.760068
40.768008
63.240162
114.285
300.168282
327.595848
324.061728
310.180872
266.62875
185.881992
93.760068
40.768008
63.240162
114.285
300.168282
327.595848
324.061728
432.082304
413.574496
355.505
247.842656
125.013424
54.357344
84.320216
152.38
400.224376
436.794464
432.082304
413.574496
355.505
247.842656
125.013424
54.357344
84.320216
152.38
400.224376
436.794464
432.082304
413.574496
355.505
247.842656
125.013424
54.357344
84.320216
152.38
400.224376
436.794464
432.082304
413.574496
355.505
247.842656
125.013424
54.357344
84.320216
152.38
400.224376
436.794464
432.082304
34
540.10288
516.96812
444.38125
309.80332
156.26678
67.94668
105.40027
190.475
500.28047
545.99308
540.10288
516.96812
444.38125
309.80332
156.26678
67.94668
105.40027
190.475
500.28047
545.99308
540.10288
516.96812
444.38125
309.80332
156.26678
67.94668
105.40027
190.475
500.28047
545.99308
540.10288
516.96812
444.38125
309.80332
156.26678
67.94668
105.40027
190.475
500.28047
545.99308
540.10288
4.4
4.5
4.6
4.7
4.8
4.9
5
310.180872
266.62875
185.881992
93.760068
40.768008
63.240162
114.285
413.574496
355.505
247.842656
125.013424
54.357344
84.320216
152.38
35
516.96812
444.38125
309.80332
156.26678
67.94668
105.40027
190.475