STRESS IN THE CARTILAGE OF THE HUMAN HIP JOINT
by
THOMAS MACIROWSKI
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREES OF
BACHELOR OF SCIENCE
and
MASTER OF SCIENCE
and
DOCTOR OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 1983
Signature of Author.,
.
Department of M,
anical Engineerin•,
February 1983
Certified by ............................
Thesis
Accepted by
.......
' OF T•• IS
TECHNOLOGY
JUN 23 1•<!I ID1
PIFSp
..........................
De p a rtm e n t al
Archives
Supervisor
CTr
Chairman
- 2-
STRESS IN THE CARTILAGE OF THE HUMAN HIP JOINT
by
Thomas Macirowski
Submitted to th
February 15, 1983 in
the Degrees of Bachel
of Science.
Department of Mechanical
Engineering on
rtial fulfil Iment of the requirements
for
of Science, Master of Science, and Doctor
ABSTRACT
factors
of the mechanical
To understand the possible ro
which determine tissue deformat n and fluid flow in normal
of ost eoarthritis, the state
synovial joints in the pathogenesi
of stress in the cartilage in situ i n the human hip joint is
In particular, the nature of the interarticular
described.
boundary condition and its impli cations for the function of the
synovial joint are discussed.
The geometry and the poro elastic constitutive properties of
normal
adult articular c artil age in the human hip joint have been
The time response of both the surface
measured experimenta lly.
stress distribution and the s urface displacement of the cartilage
by
in the acetabulum of the human hip joint when loaded
instrumented endopr ostheses has been measured. Modelling of the
human hip joint, inc orporatin g the measurements described has been
the sur f ace boundary conditions governing
predict
to
used
interarticular flui d flow, i.e.
the the local and
global
resistance to fluid flow in t he interarticular space.
Fluid flow
The result is the
toward and parallel to the spa ce are included.
first experimentall y determi ned estimate of the time dependent
resistance to fluid flow in
the interarticular space and its
effects on the solid stress and fluid pressure in the cartilage.
The ratio of the resistance to
ow
in
the
fluid
interarticular space to the resistance to flow i the cartil age
layer increases under static load from about one to twen ty.
The
flow in the
interarticular space relative to tha t in the layer
decreases in about the same proportion.
It appears that for good
i nterarticul ar
sealing to occur the interarticular space needs to
be much smaller than the unloaded shape of the carti lage surface.
The fluid pressure typically supports ninety pe rcent of the
load, even after twenty minutes; corre spondingly the stress in
the solid matrix remains small.
This has
important
implications
for the load bearing and lubrication functions of articular
cartilage.
Thesis Supervisor:
Titl e:
Robert W. Mann
Whitaker Professor of Biomedical Engineering
Department of Mechanical Engineering
- 3-
To Chris,
My sine qua non
-4-
ACKNOWLEDGMENTS
First, I would like to thank Professor Robert W. Mann,
my advisor, for the opportunities he gave me. I have
benefited greatly from his experience, insight, guidance,
and support.
Most of all, he provided us with a wonderful
environment to learn and grow in.
Slobodan Tepic, my friend and c ollaborator,
his
magnificant skills, insight, and patience
Our projects have always been done j ointlv, and
this thesis could have been so also.
Doctor William H. Harris has provided
with all his clinical and medical expertise.
for his time, effort, and enthusiasm.
contri buted
nselfi shly.
onlyv wish
this project
I am grateful
Professors Klaus-Jurgen Bathe and Michael P.
Cleary
were responsible not only for teaching me the fundamentals
of mechanics but also the phy sical
in sight necessary to
apply them in a sensible way.
Thei r readiness to offer
advice and criticism is most ap peciated.
Paul Rushfeldt, who began the in vitro
program, left an enormous amount of working
software, without which none of the experiments
been possible.
John Dent, who introduced
project, did an excellent job of tackling
subject of cartilage mechanics. in a clear and
The roots of this work trully c ome from Paul'
investigations and discoveries.
experimental
hardware and
would have
me to this
e difficult
actical way.
and John's
I thank Dave Palmer for his technical and other
contributions
to this project.
The hip simulator he
designed and built has performed admirably.
Ralph Burgess has always been able and willi ng to lend
hand
a
helping
with
the
electronics
and general
trouble-shooting.
Dr.
Charles McCutchen, who pioneered the concept of
interarticular fluid flow, has provoked many interesting
discussions.
Erik Antonsson was always willing to share his talents,
times of distress.
in
particula rly
The laboratory's
computers haver run as well.
Brad
computing
friend.
Hunter provided much needed
electronic
and
help in the early days and has continued as a
- 5Andy Hodge were always
Dr.
Joe Halcomb and Dr.
helpful on cl inical and other matters. Their willingness to
contribute in an engineering environment is app reciated.
Maureen Hayes kept our financial and other
in
order and was always willing
affairs
sympathetic e ar.
bueracratic
to lend a
Lucille Blake typed too many abstracts and proposals on
too short deadlines without complaint.
and
Fijan,
Steve Madreperla, Bob
contributed their respective tal ents to
general.
all.
Cary Abul-Haj was a friend
and
Harrigan
Tim
the project in
confidant
through
it
The Department of Mechanical Engineeri ng Staff made MIT
a socialble place to work.
I thank my parents-in-law Richard and Mary Torbinski,
who treated me the best way they knew -- 1 ike a son, and my
parent s who only wanted me to do my best.
Her love and
Finally, I am indebted to Christina.
always been
has
advice
her
and
devotion are unequaled
her.
without
it
done
have
insightful. I couldn't
This work was conducted in the Eric P. and Evelyn
Human
and
Biomechanics
for
Laboratory
E. Newman
Engineering
Mechanical
of
Department
Rehabilitation in the
at the Massachusetts Institute of Technology, and was
The Kroc Foundation, The Office of the
supported by:
Provost at MIT, and the Whitaker Professorship of Biomedical
Engineering. The National Institutes of Health Grant Number
5-RO1-AM16116 supported earlier phases of the study.
- 6-
TABLE OF CONTENTS
Page
ABSTRACT
ACKNOWLEDGEMENTS
TABLE OF CONTENTS
LIST OF FIGURES
CHAPTER 1 -
INTRODUCTION
CHAPTER 2 -
BACKGROUND
2.1
The Synovial Joint
2.1 .1 Cartilage
2.1.2 Performance
2.2
Osteoarth ritis
2.2.1 Description
2.2.2 Cl assificiation
2.2.3 Pathogenesis
2.3
Previous
2.3.1
2.3.2
2.3.3
CHAPTER 3 -
Work
Lubrication
Cartilage Properties
Failure
THEORETICAL ANALYSIS
3.1
Porelastic Materials
3.1.1 Compressible Constituents
3.1.2 Incompressibl e Constituents
3.2
One-Dimensional Consolidation
3.2.1 Vertical Flow
3.2.2 Lateral Flow
3.2.3 Frequency Response
3.3
Interarticular Resistance
- 7 -
CHAPTER 4
EXPERIMENTAL TECHNIQUE
4.1
Equipment
4.2
Geometry
4.3
Swelling
4.4
Pressure Distribution
4.5
Consolidation
CHAPTER 5 - RESPONSE OF THE CARTILAGE LAYER
5.1
Simulation
5.1.1 Description of the Model
5.1.2 Finite Element Formulation
5.1.3 Interarticular Boundary Condition
5.2
Results
5.3
5.3
5.3
99
107
Solid and Fl uid Stress
Interarticul ar Flow
Interarticul ar Resistance
CHAPTER 6 - CONCLUSIONS
120
APPENDIX A - COMPUTER PROGRAMS
131
APPENDIX
REFERENCES
- FREQUENCY RESPONSE
175
181
-8LIST OF FIGURES
2-1.
Schematic Cross-Section of an
Layer
2-2.
Frontal View of the Hip Joint
2-3.
Side View of the Hip Joint
Articular
Cartilage
2-4.
Axes of Motion at the Hip Joint
2-5.
Friction and Deformation versus Time for
on Glass
Cartilage
2-6.
Weeping Lubrication
3-1.
Vertical Flow Model Geometry
3-2.
Consolidation versus Time for Vertical
3-3.
Lateral Flow Model Geometry
3-4.
Phase Shift for Compressible Constitiu ents
3-5.
Surface Flow Model
4-1.
In Vitro Facility
4-2.
Instrumented Prosthesis
4-3.
Prosthesis for Consolidation Measurement
4-4.
Ultrasonic Apparatus
4-5.
Femoral
4-6.
Acetabulum Mounting
4-7.
Cartilage Surface Deviations from Spheri city
4-8.
Cartilage
to
Calcified-Cartilage
Deviations from Sphericity
4-9.
Cartilage Thickness
4-10.
Swelling Surface Displacement
4-11.
Log Surface Displacement
Fl ow
Head Mounting
Interface
- 94-12
Best Linear Fit
4-13
Swelling Surface Displacement
4-14
Best Linear Fit
4-15
Surface Stress Measurement
4-16
Surface Stress:
0 Minutes
4-17
Surface Stress:
5 Minutes
4-18
Surface Stress:
20 Minutes
4-19
Consolidation Measurement
4-20
Prosthesis Position
4-21
Prosthesis Displacement along Load Axis
4-22
Log Prosthesis Displacement
5-1.
Surface Flow Model
5-2.
Coupled Solid-Fluid Analysis Flowchart
5-3.
Calculated Surface Flow:
0 Minutes
5-4.
Calculated Surface Flow:
5 Minutes
5-5.
Calculated Surface Flow:
20 Minutes
5-6.
Calculated Solid Stress:
0 Minutes
5-7.
Calculated Solid Stress:
5 Minutes
5-8.
Calculated Solid Stress:
20 Minutes
r.
0
Average Gap Conductance:
450 N
5-1
Average Gap Conductance:
900 N
5-1
Biot Number
5-1
Average Stress
B-I.
Cartilage Plug Confinement for the Frequency
Response Measurements
B-2.
Phase Shift Between Surface Displacement and Load
- 10 -
CHAPTER 1
INTRODUCTION
- 11 -
Synovial or diarthroidal joints p rovide mobility to the
Healthy
skeleton.
human
carrying loads that can
weight
joints
reach
perform
three
extremely well,
five
to
times
body
lower extremities during normal walking,
in the
at
of
velocities ranging down to near zero , with a coefficient
friction as low as 0.01, over a life -time of several mill ion
cycles per
in
material
Unfortunately,
year.
synovial
the
biological
art icular cartilage, is not
joints,
Its incidence
degenerative disease known as osteoa rthrit is.
and severity generally increase with age;
are
treatment [1].
a
is
The most widespread joint affliction
indestructible.
Americans
bear ing
over
16
mill ion
affected seriously enough to require medi cal
In fact, the almost ubitqu itous occurence of
this type of joint failu re has been the major motivation for
the study of
articular
the
mecha nics
cartilage.
of
Despite
the
synovial
extensive
epidemiol ogi cal research the etiology of
joints
biological
osteoarthritis
and
and
is
unknown [106
Both the inherent load-bearing function of diarthroidal
joints
and
the morphology of the lesions that characterize
osteoarthritis suggest that mechanical factors are important
in
The
the
etiology and subsequent development of the disease.
changes
fraying
and
in
osteoarthritic
splitting
at
areas and vary in severity at
affected joint [45].
cartilage,
most
notably
the surface, occur in localized
different
locations
in
the
- 12 -
The research hypothesis we advance
local
here
assumes
that
mechanical factors which determine tissue deformation
and fluid flow
important
in
role
normal
synovial
joints
play
in the pathogenesis of osteoarthritis.
goal of this work is to explicate the
mechanical
also
importance
of
an
The
these
factors for the behavior of the cartilage in the
synovial joint and their possible role
in
its
failure
in
osteoarthritis.
The determination of the state of stress
in
synovial
joints
in cartilage
and its relation to mechanical factors
such as joint load magnitude, direction , and
the
geometry
layers,
along
tissue,
is
certain
and
propert ies
mechanical
information
with
essential
on
the
frequency
of the cartilage
strength
physiological
conditions
and
the
circumstances
failure
inevitable.
The goal of this thesis is to develop
estimate
of
to support the hypothesis that under
mechanical
to
and
of
the
tissue
is
likely
or
even
a
model
stress in the cartilage 1 ayer of the human hip
joint.
Chapter 2 contains background information
on
synovial
joint mechanics.
Chapter 3 develops some simple theoretical
models
compression
for
conditions.
the
cartilage
under
various
Chapter 4 describes the experimental techniques
and the results of
cartilage
of
layer
measurements
and
the
stress
of
the
on
the
geometry
surface
of
the
of the
- 13 -
cartilage in the intact acetabulum.
the experimental
Chapter 5
inco rporates
results and theoretical analysis in a model
of the cartilage layer.
Finally
Chapter
6
conta ins
conclusions and recommendations for future research.
the
- 14 -
CHAPTER 2
BACKGROUND
- 15 -
2.1
THE SYNOVIAL JOINT
The
bones
the
skeleton
joints.
In
of
or
articulations
of
parts
diarthroses the
the
are
the
bones
at
connected
joints
movable
the
(usually
or
ends)
the joint are covered by articular cartilage, which
forming
forms the joint surface, and enclosed in a capsule formed of
hence
capsule is lined with a synovial membrane,
The
ligaments.
The
joint.
membrane
transparent fluid -- synovial
viscous,
thick,
a
secretes
synovial
synonym,
common
a
fluid.
2.1.1
Cartilage
The articular cartilage is of the hyaline
distingui shed
from fibro-cartilage.
variety,
as
The tissue consists of
a sparse distribution of cells (chondrocytes) immersed in an
The matrix is mostly water, typical ly
extracell ular matrix.
The remainder is a meshwork
70 to 80 percent by weight.
collagen
proteogly cans
chains
(PG).
The
as
composed
of
of charged disaccharides known as glycosaminoglyca ns
affinity
solution.
The swell ing of
of
the
PG
the
PG's
is
resi
In cartilage the intrinsi
gel
is
greater
than
a
have
The PG molecules
for water, tending to expand their
collagen fiber mesh.
volume
are
proteoglycans
(GAG) att ached to a protein core.
high
known
nonfibrous - filler
a
and
fibers
of
volume in
ed
by
the
eaui 1 ibrium
the
actual
- 16 -
physiologi cal
restrained
i.e.
equil ibrium;
collagen
the
by
tensioned by the PG' s [90].
a molecula r
scal e
or
influ en ce
effects
network,
swelli ng
which
in
level
phenomenon
propagation
in
PG's are
turn
is
appears to be
This "tensioning"
fiber
wave
the
the
its
since
ultrasonic
(5-20 MHz) range (se e section 4.2).
Morphologically,
neither
homogeneous
the
composition
nor isotropic.
of
cartilage
is
The cartilage layer is
classified into four descriptive zones, distinguished by the
of
morphology
(Figure 2-1).
formed
from
the
collagen
fibers
and
the
cells
The most superficial few microns appear to be
a
layer
of
fine
fibers known as the lamina
splendons (LS) [701 that covers the articular cartilage.
order
In
of increasing distance from the cartilage surface the
layers are:
1.
The superficial
fibers
are
(tangential) zone in which the
oriented primarily parallel to the surface.
Locally there is a preferred
with
joint
collagen
orientation
type and location.
which
varies
The cells are generally
oval shaped with long axes also parallel to the surface.
This
[101].
layer
has the greatest collagen and water content
- 17 -
StVpERI AL ZIoK E
TRi'srio
r
- ZONME
CACt.c.tC.
i
CAlTiL.AGE
Su.Zc.wkoms.bAo,.
Figure
2-1. Cross-Section of an Articular Cartilage Layer
Bota
- 18 -
2.
The transitional zone in which the fibers
are
arranged
obliquely in a more random network.
3.
The deep zone in which
radial to the surface.
4.
the
fibers
are
predominately
The cells are large and round.
The calcified zone in which there are few cells and
matrix
It
contains many salt crystals.
the deep zone
visible
by
the
"tidemark"
the
i s divided from
(TM),
a
blue
line
In older
in hematoxyline stained preparati ons.
tissue multiple tidemarks are often visibl e [115].
The cells or chondrocytes seem to
and
meiotic activity.
has been observed
near
lack
DNA
However, cell division and synthesis
clefts
and
defects
cartilage and the cells do divide in culture.
in
are
as injury [82],
currently
cartilage
the
arthritic
The cells are
metabolically active and the effects of external
(such
synthesis
influences
pressure [107], and temperature [81])
subject
of
intense
research.
The
is devoid of nerves, lymphs, a'nd vascularization.
This implies that information, nutrients, and wastes must be
exchanged through the matrix via the synovial fluid.
2.1.2
Function
Synovial joints and articular cartilage
lifetime
often
under a wide range of demanding conditions.
are two basic functional requirements for
synovial
last
a
There
joints;
- 19 -
carriage and articulation.
load
The loads
load across the bones of the joint.
transferring
while
articulation
during
constraint
kinematic
smooth
The joint surfaces provide
that act across the joints have two components;
those
that
from the gravitational and inertial forces acting on
result
acting
the limbs and the forces of the muscles
cases, including the weight-bearing lower
In many
joint.
the
across
major
the
provide
forces
muscle
extremities,
the
contribution.
Since this thesis is concerned with the human
hip joint it
is
consider
to
instructive
functional
its
requirements in more detail.
The bal 1 and socket geometry of the hip joint
2-2 & 2-3)
the
reflects
the
movemen ts
flexion/extension
n
adduction/abducti on
n
rotation
along
the
standing on one l eq
the
center
of
axis
The mus c-le
the femur.
of
motion
provide both stabilty and
small1 (relative to the
ma ss)
of
w alking
The moment arm of the abductor
orces
v
phase
so
the
muscle
from
distance
force
comparatively large to balance the torque generated
weight of the body.
and
plane,
medial-lateral
duri ng the stance
require large mus cl e
muscles is relati vel
plane,
anterior-posterior
the
of
the s implest cases of equilibriu m when
Ev
joint.
consis t
2-4)
(Figure
the
must
forces across the h
to
hi p
long
th
demands for mobility;
functional
the
(Figures
must
by
be
the
Typically the net load across the joint
is estimated to be two to three
times
body
weight
during
- 20 -
Figure
2-2. Frontal View of the Hip Joint
- 21 -
Su
Figure 2-3.
Side View of the
Hip Joint
- 22 -
supgmeridian"
I Oxt
%equ
Figure 2-4. Axes of Motion at the Hip Joint
- 23 -
to ten times body weight during running
seven
and
walking
and jumping [32],[109].
to
It is important
that
note
in
cases the muscle forces represent an estimate based on
most
motion;
the force required to generate a measured kinematic
possible
are
joint
the
about
produce
which
forces
muscle
higher
the same net torques
the
if
are
muscles
co-contracting.
cartilage
These loads must be supported by the
of
the
joi nt
and the femoral head.
acetabulum
tot al
the
typ ically
cartil age
area
in
Du ring normal
1000 mm .
the
walking
bod y weight or 750 N dict ates an averaqe str
on
the adult hip
cetabulum
load of twice
of
s
1.5 MPa
The hip
join t
will
routin e ly
experi ence
load duri ng one mil lion walking cycles e ach year.
thi s
is
howev er we have found the maximum stress
the cartila ge;
is much higher.
layers
The
per •formance of healthy synovia 1 joints under t hese condi tons
good.
The friction coeff i cient of whole
is
astonishing ly
hip
joints has been measu red in the range of 0 .005 to
[24 1.
Many
This
joints
seem
unfortunately
synovial joints
people.
better
is
many
can
than Teflon against Teflon (0.04).
perform
to
do
be
0 .024
not.
very
well
The
high,
for
a
lifetime;
frequency of failure of
especially
in
older
-
2.2
24 -
OSTEOARTHRITIS
The most common disease of the joints
for
cause
their
osteoarthritis.
disease
failure)
a
Despite its suffix it
characterized
cartilage (hence a less
disease).
is
Rheumatoid
by
disease
name
arthritis,
major
known
as
is a non-inflammatory
of
degenera tion
used
the
(and
--
the
articular
degenerative
joint
in contrast, attacks the
synovial membranes.
Osteoarthritis is widespread, increasing
age
with
[1].
Although
in
incidence
not an inevitable consequence of
it is the most common cause of work dis ability [129].
aging
If for no other reason than its omnipresence, it has been a
major
subject for medical research.
2.2.1
Description
Disintegration of the articular cartilage is
common
feature
of
osteoarthritis.
(fibrillation) appears.
most
Early in the course of
the disease the cartilage becomes soft and
surface
the
fraying
of
the
Meanwhile the chondrocytes
increase their metabolic rate, both with respect to the rate
of
synthesis of proteoglycans and collagen [87],
along with
cell division and proliferation near splits and mechanically
damaged
cartilage.
As the disease progresses the cartilage
often becomes thinner, leading in some cases to
of
the
cartilage,
total
exposing the bone (eburnation).
loss
As the
- 25 -
pain
incipient
after
joint
disease
of
the
by the
impaired
further
be
may
function
symptom of
course
the
in
later
osteoarthritis;
classical
the
is
activity
trabeculae.
the
of
thickening
new bone (osteophytes) and
Joint
via
including formation of bone cysts and
remodelling;
active
changes
dramatic
undergoes
it
exposed
bone becomes
remodelling of the bone and joint surfaces.
Classification
2.2.2
Osteoarthriti s is clas sif ied by the (lack of)
of
in
factors
etiological
"Secondary"
disease.
the
evidence
arthritis results from changes in the joint previ ous to
the
onset of the disea;se, such as gross anatomical abnormalities
or
other
majority
(such
disease
arthri tis).
rheumatoid
The
of cases (58 percent by one estimate [134]) do not
C ases.
idiopathic
these are the so-called
have a kn 0 wn cause
primary
as
A pervasive
problem in identifying first
causes is
that the progressive changes associated
disease
have
any
o bliterated
deformati e s (such as
seen
or
after
changes.
primary
a
with
epiphysis
slipped
the
Mild
or
Legg-Pert hes disea se) have been identified in a majority (79
percent) of cases in
osteoarth ritis
[134].
directly to mechan ical
a
particular
Evidence
damage
of
study
that
the
of
such
"idiopathic"
changes
cartilage
through excessive stress) is still lacking.
lead
(perhaps
The association
of mechanical factors with the initiation and development of
- 26 -
osteoarthritis is the primary hypothesis of this research.
Pathogenesis
2.2.3
Although the mechanical
is
joints
inescapable,
factors su
as
establ ishi
is
and
in
on
stress
obsc ure.
failure
its
the
loads
and
activity.
synovial
the
boundary
nce the way
i
the
mechanics
r
pathogenesis
Further
work
on
of
the
of
the
to
during daily
in
a
normal
the role of mechanical factors in
osteoarthritis
abnormalities
can
be
addressed.
of ca rtilage, for example,
strength
might be suggested by predictions
effects
imposed
the
thesis
cartilage
fun ctions
cartilage
of
This
osteoarth ritis.
conditions
understood,
functioning
This is the first step toward
attempts to relate the state of stress in the
the
synovial
in
exact influence of mechanical
a relationsh ip between
and
joint
the
geometry
int
synovial
role of cartilage
of
the
details
of
the
that p resumably can lead to
osteoarthritis.
2.3
PREVIOUS WORK
Bi omec hanical studies of synovial joints have
primarily
on
two topics:
articular c artilage.
its
role
in
the
lubrication and the mechanics of
Cartilage is a
of
function
understood and the subject
focussed
of
complex
synovial
much
material
and
joints is poorly
research
and
intense
- 27 -
The most common cause of the confusion has been the
debate.
synovial joint as a
the
and/or
material
a
as
cartilage
in this
research
A brief history of the relevant
system.
to
models
engineering
traditional
of
misapplication
field is given here to orient the reader with respect to the
work described in this thesis.
2.3.1
Lubrication
(]L) In vitro whole
categories.
an average coef fice nt
joint studies have measured
fricti on
of
joints under va riou s conditions.
different
two
of
Joint lubrication studies have been
ani mal
and
human
for
human hip joint ) ha ve ranged from 0.003 [78] to 0.05
Even
values
high est
the
are
types of man-ma4de bearings
hydrodynamic
b eari ngs.
and
more
pr ecis e
and
geometry
and
experiments of this
sliding
all
to
comparable
fluid-f ilm
(2) Exp eriments on small cartil age
been
used
to
achieve
McCutchen
steady-state
than
rather
performed the first
[94]
type with sma 1ll bovine
cartilage
pl ugs
The most sign iificant result was that
on gla SS.
to
calculation of the loaded contact a rea
slidinig.
oscillatory
[14 9].
much better than almost
specimens again st an artificial s urface have
allow
the
The quoted values (for
the
friction began low (0.002 to 0.02 ) and rose with time (up to
0.2)
as
separated
replaced,
the
from
cart i lage consol idated.
the
glass
fo r
a
If the ca rtilage was
few
seconds
and
t hen
the friction was reduced but quickly rose when the
- 28 -
load was reapplied
(Figure 2-5).
expe riments
cartilage samples in a rotating geometry
using
A
more
set
recent
of
demo nstrated that the friction and deformation under dynamic
load ing is much less than under static loading.
Various
engineering
lubrication
from
theories
have been "adap ted" to explain the phenomenally
low friction coeeficient of cartilage.
every
traditional
lubrication
regime
In
fact,
virtually
has been proposed for cartilage.
The list includes [135]:
1.
Hydrodynamic
mechanism
Lubrication
depends
upon
the
maintenance
wedge-shaped film of synovial fluid
surfaces.
This
is
[79]).
(MacConaill
unlikely
given
of
between
This
a thick,
the
joint
the reciprocating
motion of a synovial joint.
2.
Boundary Lubrication (Charnley [24]).
molecules,
A thin
presumably from the synovial
fluid,
layer
of
attaches
to the cartilage surfaces and provides the low friction.
Charnley
suggested
this
could
account
for
the near
independence of the friction with sliding speed
3.
Elastohydrodynamic (Dintenfass [36]).
1.,
An
extension
of
compliance of the bearing surfaces and the change in
lubricant viscosity are included in order to enhance the
maintenance of a layer of fluid separating the surfaces.
- 29 -
C
EU
2
0
Ii
0
C
S
U
S
0
ha
10
20
Duration of loading (minutes)
0
Figure 2-5. Friction and Deformation versus Time for Cartilage on Glass
't,
.. oo,.,^
-
i-gure..
-
-
::.
0..
.
2-6.WepingLu.bricat
Figure 2-6.
Weeping Lubrication
.
:-
.. :::. ::
.:: -
- 30 -
4.
al.
Boosted Lubrication (Walker, et
molecules
large
of
the
synovial
enter the small (6 nm) pores of
If
[150]).
the
fluid are u nable to
the
then
cartilage
a
concentrated hyaluronate would be left on the surface to
lubricate.
[96]
The most obvious problem
is that it
with
this
theory
requires fluid to flow from the squeeze
film into the cartilage layers.
5.
Weeping Lubrication (McCu tchen [94]).
This
proposes
type of self-pressurized hydrostatic bearing,
of the load
is
manner)
the
by
supporte d
slow
compression of the
(in
a
l ayers,
between the layers (Figur e 2-6).
frictionless
into
the
by
the
synovi al
space
Solid-solid contact of
the asperities is the sou rce of the friction
well-lubricated
where most
of fluid, supplied from
seepage
carti lage
nearly
a
fluid
(which
is
if the solid
stress is less than 0.45 MPa [98 1) and of the local seal
to
prevent
the
pressur ized fl uid from quickly flowing
out of the interarticular space.
rate
of
seepage
He noted
that
actual
would depend on the microstructure of
the cartilage surface [93].
The question of whether a squeeze film can prevent
opposing
surface
of cartilage from making contact has been
analyzed in much detail by Piotrowski
the
effects
the
[111].
He
included
of shear-thinning of the synovial fluid.
Dent
[34] applied these equations to a hypothetical bearing 13 mm
- 31 -
in
with
radius,
applied
an
The
700 N.
of
load
This is the
thickness was predicted to reach 2 um in 0.2 s.
of
order
the
tertiary roughness of the cartil age surface,
hence the she ar-thinning of synovial fluid can
contact
unde r
film
not
prevent
conditions simulating heel-strik e during the
stance phase of walking.
Cartilage Properties
2.3.2
experimental
previous
Most
of
investigations
the
properties of cartilage have been limited to the measurement
of
behavior
time-dependent
the
cartilage
isolated
of
specimens, usually small plugs for compression tests or thin
The major problem
strips for tensile tests.
these
tests
has
been
stress in the sample;
simply
the results
studies of cartilage is ultimately
state
stress
of
in
cartilage
the
prediction
the
under
constitutive
model should be capable
cartilage
if
best
model
in mind.
predicting
of
the
physiological
tests
conditions it is important to perform the
specific
at
therefore
are
the goal of most biomechanical
Since
descriptive.
of
the exact state of
to
inattention
most
with
with
some
In addition such a
the
stress
in
the
the loading and boundary conditions are known.
- 32 -
Compression -
2.3.2.1
For
when
to
exampl e,
the time-dependent response
of
cartilage
it is loaded by an indentor while it is still
the
underl ying
investiga tors
bone
(e.g.
has
been
measured
many
The incomplete recovery of the
[65]).
cartilage after the load was removed led
that
attached
to
the
discov e ry
the ."creep" behavior observed was due primarily to the
expressio n of t he interstitial water.
frequentl y
use d to quantify the
These tests have been
"topographi cal variation of
rel ate
the resistance of cartilage to indentation" [65] and
this
"creep
modul us"
cartilage
(the
so-called
"2
GAG
to
and
the
collagen
d epth
to
a ds
appl ied
compressiv e sti
correl ate
lower
Finally
[ 43];
in
The
of indentation on rubber sheets
and
indentor
geom etries.
w as
The
found
to
A similar result wa s obtaine d by
in addiition the GAG content wa s found t o be
to
tissue
modulus
creep
permeability [67],
proportional
[64].
of
extent with the GAG content but not
conten t.
fibrillated
the
contents)
nesses measured in this way
large
to
with the collag
Freeman
con stituents
second creep modulus" was determ ined from the
equations rel ating th
the
chemical
(and
the
correlated
stra ins
hiqh er).
v
inversely with the
which also has been shown to be inversely
the
GAG
content [91].
None of the above
experiments found any correlation of the stiffness with age.
- 33 -
The equi librium compressive stiffness
plug
a
cartilag e
by a porous loader in compression was measure d
loaded
by McCutchen [94].
strains
of
The
about
than
less
was
stiffness
40
percent.
linear
quite
for
The stiffness for
cartilage soa ked in tap water was nearly twice that measure d
in
NaCl
This was the first illustration of the
sol ution.
Donnan contri bution of the charged GAG's to the stiffness of
the cartilage .
2.3.2.2
The salt neutralizes many of these charges.
Tension -
In an analagous manner the
response
of
cartilage
to
tensi le stres s has also been measured and compared with age,
patho logical
The
most
appearance, and various
dumbbell shaped ca
condyles
of
the
"Din
have
been
contents.
performed on 200 um
i 1 age specimens excised from the femoral
k nee
huma
aligned with the 1
to
tests
extensive
bio chemical
joints
[66].
The specimens were
g axis either parallel or
or
Dri
perpendicular
"split line" patterns.
These are
elongate d cleavage li nes that are visible when the point
a
round
pin
is
in s erted into the cartilage surface.
lines te 'nd to foll ow t he dominant orientation of the
fib ers
surface
collagen
joint to
joint but is
[101].
consistent
of
The
nearby
This pattern varies from
for any one type of joint.
- 34 -
The specimens were loaded
5 mm/min.
in
tension
at
a
rate
of
The tensile stiffness and fracture strength were
strongly rel a ted
collagen
co
to
ent.
the
collagen
fiber
and
Those loaded parallel to the predomi nant
the collagen fibers had
oreintation
orientation
a
stiff ness
higher
and fracture
trength than those loaded perpendicular to the
fiber direct
n. The stiffness decreases with age
depth from t
and strenath
on
slightly
surface.
the
cartilage wit h
lowers
the
especially
proteoglycan
proteolytic
tensile
in
the
the
coll agen
enzymes
zone
to
at
not
aff ected.
collagenase
1 owers
Treatment
of
both
of
the
degrade
loiw stress
the
perpendicular to the fiber direction.
was
content
and
Treatment of the
content.
stiffness
deep
the
The tensile prope rties of stiff ness
stronalv on
eDend
and
PG's
val ues,
cartilage
and
The fracture stre ngth
the
the
tensile
of
cartilage
with
cartilage
stiffness
and
the
fracture stre ngth.
2.3.2.3
Fatigue -
The fatigue strength
interest
since
one
of
is
of
the hypotheses for failure is that
surface fibrillation is the result of fatigue.
fatigue
The
tensile
strength of similar specimens from 30 femoral heads
was measured by Weightman [153].
cartilage
particular
exhibits
typical
The results indicate
fatigue
behavior
stress and number of cycles to cause failure
are
(i.e.
that
the
inversely
- 35 -
one
fatigue resistance varied considerably from
The
related).
femor al
age.
with
decreases
and
another
to
head
co mpressive loads by an indentor on the surface of
Periodic
a femoral head also produced damage that appeared similar to
fibrillati on .
Again the major problem with these results is
cartilage
how to rel ate the tensile stresses and strains to
in vivo.
Permeability -
2.3.2.4
of
Since the time response
on
dependent
flow
the
permeability of cartilage
parameter.
The
cartilage
interstitial
the
of
also
is
to
an
loading
water,
important
is
permeability
the
same
premeability in the middle zone [91].
exhibit
a
the
materi al
permeability for flow perpendicular to the
surfaces decreases with the depth of the cartilage
tangential
is
as
the
and
the
typic al
Cartilage appears
to
non-linear dependence on the compressive strai n;
become smaller.
presumably the "pores"
Synovial Joint Modelling
2.3.3
McCutchen [98] has reviewed the
analyze
the
fluid
flow
pointed out many errors
in
in
previous
attempts
the cartilage layers.
describing
problem and the mathematical analysis.
the
real
to
He has
physic al
- 36 -
Two
published
[97]
McCutchen
models
has
are
to
relevant
the
described
fluid
opposing skeletons of cartilage touch.
this
work.
when
flow
He assumed there
the
is
a conductive path for fluid flow due to the channels between
contact, which determines
cartilage.
He
the
estimated
flow
and
skeletal
pressure
stress
is
tangential fluid flow in the cartilage layers is
in
the
least when
about
the
between
the
same as in the gap.
Kenyon [68] calculated the amount
of flow
surfaces and the pressure in the fluid film as a function of
the resistance to flow in the gap.
if
the
surface
resistance
necessarily preve nted,
is
He points out that
high,
even
the gap flow is not
although the sol id stress is minimal.
- 37 -
CHAPTER 3
THEORETICAL ANALYSIS
- 38 -
3.1
POROELASTICITY
The physical structure of cartilage ( Section 2.1.1 ) and
its
response
materials
to
(Section 2.3.2) are similar to many
load
known
as
porous
media.
Any
material
so
(including cartilage) is comprised of a solid
characterized
matrix (in this case the collagen fibers, proteoglycans, and
cells) which contains pores saturated with some interstitial
fluid (the water).
been
motivated
Most of the research on porous media has
by
and
applied
geo logical
to
following the classical work of Biot
the
of
application
porous
medi a
spite of the
popular
and
[116]
notoriety
More
recently
theory of mixtu res of interacting
the
continua has become a
[11] .
materials
to
approach
of
a ffo
study
of
articular c atilage [103].
In
the
ed
theory" (I suppose because a
the
s 0 -called
"biphasic
nary mixtu r e is analyzed) the
results thus far have provided no
additional
insight
over
the Biot formulation.
A more serious problem with any porous medium model
cartilage
may
be
the
characterization
network and the proteoglycan gel as
network.
While
the
networks
one
of
the
equivalent
of
collagen
single
are structurely coupled and
appear to move together, this characterization
ignores
the
complex
interaction of the elastic and osmotic forces which
make up
the
externally
single
supplied
equivalent
force
load.
fact, we (and others [92])
In
which
balances
an
- 39 -
have exploited this unique coupling of the components of the
matrix
of
stress
analysis
discussed
to osmotically load the cartilage.
In terms
to
loading
of
herein
response
the
of
cartilage
concept of a single equivalent solid
the
matrix proves useful and adequate.
Compressible Constituents
3.1.1
The simplest approach [116] to the formulation
governing
equations
variables.
and
volume
It
is
of
the pore fluid.
these
the
assumed these are related to the
per
mass
unit
Rice and Cleary have shown that
for linear isotropic materials (with
compressibility)
as
pressure
pore
in the solid matrix and the
strains
the
for the response of a porous medium is
to define total stresses
state
of
arbitrary
constituent
can be given in terms of
relations
four elastic constants as:
(1)
(2)
ZG I+)+V')
Where:
B
-0jZr-d)
0+V1,C
Fluid mass flow J; is assumed to
be
governed
by
D'Arcy's
law:
r£;
=i
(3)
- 40 -
It is conventional to define an 'effective stress' <U;>
the
portion
strain
in
as
of the stress which is directly related to the
the
matrix
via
the
conventional
elasticity
equations.
2G~u = <@> - 3<K(4a)
<
j> = OE + Twpdid
(4b)
Equilibrium of the forces and mass conservation lead to:
O
(5)
+-
O
(6)
Finally, they observe that the usual
equation
can
be
obtained
form
of
a
diffusion
in terms of the stress and pore
pressure:
CVZ.
where
K+
(7)
the diffussivity is given by:
2-G
-1)
Comparison with Equation
'( TIvAý
2
-tzv))
shows
satisfies a diffusion equation.
the
(8)
fluid
mass
always
Note that the first term in
brackets is the drained uniaxial modulus.
- 41 -
Incompressible Constituents
3.1.2
The change in
mass
incompressible
for
formulation
easily obtained from these equations.
be
can
constituents
content
is
strain
volumetric
simply
pore pressure does not affect the solid matrix)
the
(since
Biot
classical
The
and the effective or matrix stress is the total stress minus
the
pore
In terms of the elastic parameters the
pressure.
undrained Poisson's ratio V_
6 =
= 1/2 and
Equations
1.
1, 2, and 4b become:
(9)
2II3+
ZG(lGY)
<~Y)>3
(10)
>Q5'
(11)
The constituents of cartil age are usually assumed to be
incompressible.
Cartilage is 70 to 80 percent water, which
has a bulk modulus (2.2 Gpa)
maximum
1 00
times
pressures measured in the hip joint.
number
molecules typically
pore
pressures
of
cells)
considered
induced
in
are
also
incompressible
t:ypical
plug
the
The remaining
and
solids (collagen fibers, proteo glycan gel,
insignificant
than
greater
a
probably
composed
[91].
experiments
of
The
on
cartilage are usually less thar S0.2 MPa in order to keep the
solid
matrix
strains
in the elastic range.
The frequency
- 42 -
response of cartilage was measured herein in order to
check
this assumption (Section 3.2.3).
3.2
ONE DIMENSIONAL CONSOLIDATION
There
are
equati ons
very
few
3.1.
Section
of
analytical
solutions
Simplifying
to
assumptions
usuall y made abou t the state of strain
(such
uniaxi al
of fluid flow.
strain)
and
the
direction
as
simple cases are considered here for two reasons.
is
their
plugs.
Th
stress
in
are
or
Some
The first
pplicability to experiments on isolated cartilage
secon d is to illustrate the
ondit i ons
boundary
plane
the
the
consolidat i on
importance
of
the
for fluid flow and their effect on the
In
cartilage.
( d isplacement
all
cases
the
cartilage
of the solid matrix) is assumed
to be limi t ed to the direction of load application.
For the
plug exper i ments this implies the plugs are tightly confined
so
latera
appl icatio n
bul
ing
of
the
to
the
cartilage layer in the intact synovial
plug
joint thi s trans l ates into the radial
cartilage
is
not
possible.
displacement
of
In
the
layer is uniform or at least that shear is small.
This is discussed later in
Section 5.1.1.
- 43 -
3.2.1
Vertical Flow
Cartilage plugs are often tested
compression.
The
specimens
are
in
lateral
confined
put in a closely fitting
(usually cylindrical chamber) and loading in compression via
a
very
applied
permeable loader Figure 3-1).
to
compressive
assure
load
the
is
plugs
fit
applied
A preload is usually
tightly.
A
constant
and the time response of the
thickness of the specimen is measured.
Since the problem is one dimensional the only
variables
w(z,t).
the
are
pressure
non-zero
p(z,t) and the z displacement
The effective stress in the load direction <(Tla)
directly
related
to
the
strain
LZ_:
constant E the uniaxial strain modulus.
via
is
single elastic
If the material is
isotropic this is simply:
(12)
(1-2v)
If not E can be derived simply from the
(see
[34]
material).
for
an
example
Equation 10
gives
of
a
the
elastic
constants
tran'sversely
isotropic
change
in
fluid
content which is directly proportional to the strai n.
since the total stress is constant
the
diffusion
mass
Also,
equation
(7) simply becomes, in terms of the fluid pressure:
(13)
- 44 -
RIC>b PORO U5
Figure 3-1.
Vertical Flow Model Geometry
- 45
The boundary and initial condtions are:
(14a)
(14b)
c):Z-
(14c)
The solution is given by:
where
where
'b:
the
fundamental
the
fundamental
(15)
time
constant
and
the
eigenvalues are:
I
I
(16a)
2--t
• 2_--
(16b)
The surface displacement is related to pressure by:
W
(0-)
=
-P( --))
(17)
a synovi al joint under
It is unlikely that cartilage
load
has
zero flux or a free drainin g surfac e.
to flow at the surface is analyzed
here
boundary condition Equation 14b to be:
by
Resistance
modifying
the
[34]
(14b')
- 46 -
where Rd is an effective resistance at the
cartilage
layer.
This
is
called
a
surface
of
the
convective boundary
condition from the analogy to heat flow.
The
solution
is
then[34]
(18)~
--
where:
r't-
-+cA(
(19a)
-
,
The results (Figure 3-2) depend on the
If Rd
(19b)
surface
resistance.
is large the layer consolidation is dominated by the
surface resistance and the consolidation is nearly constant.
3.2.2
Lateral Flow
This model assumes the cartilage is loaded by
rigid, impermeable loader [34].
very
porous
walls
(Figure 3-3).
confinement
flow toward the
p(r,t)
and
written as:
is
loader
w(r,t).
not
the
necessary.
only
lateral
edges
If the cartilage is
attached to bone and h << a then bulging is
lateral
flat,
The specimen is also a disk
of thickness h and radius a, confined at the
by
a
negligible
and
Since there is no
non-zero
variables
are
Mass conservation (Equation 6) can be
SURFACE RESISTANCE = 0. 0. 0. 1. 0. 5. 1. 0,. 5. 0.
10. 0
1. 00
0. 90
0. 80
0. 70
0. 60
0. 50
0. 40
30
0.20
0.10
0. 00
0.0
.1.0
2.0
.TIME
3.0
Figure 3-2. Consolidation versus Time for Vertical Flow
4.0
5
- 48 -
F
RI
g
b
1AGsRbt\f£
L4
N%
*. *ID)
x xo
?0OA5
R~G~
tS\\
p~cERME
\bl
Figure 3-3. Lateral Flow Model Geometry
6o
.
- 49 -
(20)
r~
Integrating using the boundary condtions:
brs ~:
C
(21a)
(21b)
9A~
(r,
Vertical
(22)
'
v
equilibrium requires:
ý-x
SI
+
(23)
This can be integrated as:
/
where '
7G
+
(24)
the radial time constant is:
/?-,
=
(25)
1-X2
,;a8$
The solution to Equation 21 assuming zero initial
f ýý
e(L
-F4
1 y
~Qrr&) ~:
-
stra-in is:
(26a)
~
C'ý t /
(26b)
- 50 -
3.3
FREQUENCY RESPONSE
the
Recently
sinusoidal
a
cartilage
plug
to
load was measured by Lee, et al.
varying
The response of
of
response
of
plug
a
cartilage
with
the
[72].
boundary
conditions of Section 3.2.1 was derived by Lee and confirmed
by Tepic [139]
using
a
lumped
model.
The
results
for
compressible constituents are derived here.
For uniaxial
is
E
related
to
strain the effective vertical
the
total
effective
stress
stress <
< 6KK >
>i-
since
= G~E=O.
(27)
<t-e-
equl ibrium requires:
Vertical
N'i
C
2GI-7
(28)
The equation for mass conservartion can be rewritten as:
I<
P
E (C-2v)
ZGE +V
u
3 c)4-
+
(29)
Finally, in terms of surface displacement and fluid pressure
the equations of motion are:
a
r
3 Is
K b 2P
-+
5c
Lw
3_3~ - 7-y)) /2)?-- 32,G-tVL) ýB-5e)'P
(30)
(31)
This is the same form as Equations 2.6 and 2.7 in Wijesinghe
- 51 -
and Kingsbury [1561.
The phase shift (Figure 3-4) is:
(32)
where the time constant is:
/L
K
Where E
(33)
is drained and •
is undrained, the
phase
shift
will be 45 degrees only if:
I<
3.4
<
&
)
(34)
INTERARTICULAR RESISTANCE
The fluid pressure and solid stress
in
the
cartilage
clearly depend on the resista nce to fluid flow at the loaded
surface
resistance
cartilage.
the
of
to
This
concept
of
surface
fluid flow was first introduce d by Dent [34]
and Kenyon [68] in reference to the
boundary
condition
at
the loaded surface of compres sed cartilage spe cimens, but it
is equally applicable to the
flow
in
in
whole
situ
i nterarticul ar
sy novial
joints.
resistance will depend on the thickness of the
the
cartilage
roughness,
fluid flow paths.
cartilage
to
They noted the
film,
fluid
the effective length of the
It is like ly to vary with
time,
as
the
surface compresses under load and the average gap
height decreases.
(and
a nd
resistance
length)
This can happen at many different
height
scales ranging from the physical roughness of
Comressible Constituents
Compressible Constituents
1. 50
1.00
0.50
0.00
-0. 50
0.00
Lo9 ('TLZw)
0. 50
1.00
Figure 3-4. Phase Shift for Compressible Constituents
1.50
- 53 -
the cartilage surface (0.5 to 2.0 pm [1261)
deviations
to
global
the
sphericity measured by Rushfeldt [1211 and
from
Tepic [1381 (RMS d 75 um).
The resistance to interarticular flow is important
both
of
functions
These are in
casual
observation
than
fact
more
rel ated
intimately
As
suggest s.
consolidates fluid must flow out of
the
the
cartilage
cartilage ,
either
and then through the interartic ular space or laterally
into
The stress supported by the solid
through the layer itself.
matrix
is
directly
related to the strain in the matrix by
the drained uniaxial strain modulus,
In
load support and
articular cartilage --
1ow friction.
for
vitro
experiments
by
Rushfeldt [121] and myself [80],
performed by lo ading human
with
a cetabu
demonstra te d
have
minutes,
1 MPa.
typically about
th
the
2250 N
for
30
total
maximum
Since cartilage should
compression is less than 30 p ercent.
be nearly in equilibrium th is corres onds to a stress in the
stress
on
surface
the
The
1 MPa)
solid matrix of 0.3 MPa (
i
total
measured
typical y 7 to 10 MPa over this
s
time period ind icating the fluid pre s sure is supporting
difference,
i. e.
the
more tha n 95 perc e nt of the total stress.
The resistance to fluid flo w between the layers must be high
I
enough to su ppo rt this pres su re acro s s the radius of contact
(about
15
to
20 mm).
The
lubric ation
quality
of
the
cartilage depends only upon supportin g this relatively small
v
solid stress component at the
locations
where
solid-solid
- 54 -
contact occurs.
A simple model
between
(Figure 3-5)
illustrates
res istance
interarticular
and
magnitudes of lateral flow and film flow and
fluid
stress.
A
of
layer
cartilage
the
the
of
relation
relative
solid
of thickness h and
radius L is consolidating with fluid flow both laterally
the
and
layer
flow in the gap is
K/y
t.
The
Z
average
(near the bone say)
flow Gý,,
the f'ilm.
toward
and
and
the
fluid
Pý in
and
in
The conductance to fluid
cartilage
pressures
the gap.
permeability
is
are ?_ in the layer
The
total
vertical
lateral flow in the 1ayerpL, and gap flow are:
V=V1 %TL
WL.)~ILP
(35)
(36)
QS·jni) Z P2,L
(37)
The gap flow must come from the vertical flow so a:=,
and
the rati os of flows and pressures are:
(38)
£)Q12+I
(39)
where R is defined as
(40)
- k6
- 55 -
S.
Figure 3-5. Surface Flow Model
Figure 3-5. Surface Flow Model
- 56 -
Note this
flow
in
is just the ratio of the macroscopic resistance to
the gap (over length L) to flow in the layer (over
length h).
2-/.
L--/ý
~/A-r-
Experimental estimates for the resistance to
described in Chapter 5.
(41)
flow
are
- 57 -
CHAPTER 4
EXPERIMENTAL TECHNIQUE AND RESULTS
- 58 -
4.1
EQUIPMENT
The equipment used to perform the experiments described
in this
facility,
thesis
is part of the in vitro hip joint testing
located
in the
Eric
P. and
Evelyn
E. Newman
Laboratory for Biomechanics and Human Rehabilitation at MIT.
This unique facility, Figure 4-1, was designed
vitro
measurement
of
Rushfeldt
the
in
the pressure distribution on and the
geometry of the cartilage layer in the human
Paul
for
[121]
and
David
Palmer
acetabulum
[122].
by
The
instrumentation for the measurement of geometry and acoustic
impedance
of
the
cartilage
covering the femoral head was
added by Slobodan Tepic [138].
measuring
technique
Tepic and
I
automated
the
by adding the capability for recording
snd off-line processing of the ultrasonic signals.
The major components of the facility are:
1.
A
three
degree
servo-control 1 ed
of
freedom,
hip
hydraulical ly
simulator
which
replicate physiol ogical motions and loads
is
powered,
used
across
to
human
hip joints.
2.
Instrumented femoral head prostheses, which are used
load
to
the cartilage in the acetabulum and measure either
the stress on
the
surface
thickness under load.
of
the
cartilage
or
its
- 59 -
4-J
i9,·
-
~Cu
0
.I>rJ
sl
0
4d
- 60 -
3.
A control console
which
provides
the
interfaces
for
manual and/or computer control of the hip simulator.
4.
A two degree of freedom, stepper motor
system
which
is
used
to
ultrasonically
geometry of the components of
ultrasonic
equipment
is
driven
human
hip
computer
scanning
measure the
joints.
interfaced
The
via
a
waveform recorder.
5.
A DEC PDP 11/60 computer and interfaces which
for
are
used
control and data acquisition during the experiments
and subsequent analysis and plotting.
The laborat ory resources include
an
addi tional
multi-user
DEC PDP 11/ 60 computer for data reduction and color graphics
display wit h DECnet connections to the in vitro
the
Joint
Computer
Mechanical, Civil
,
Facility
th e
(of
Aero and Astro,
and
11/60
Departments
0 cean
and
of
Engineering)
DEC VAX 11/ 782.
4.1.1
Hip Simulator
The
hip
actuators
that
hip joint.
4500 N
simulator
has
three
independent
hydraulic
simulate load, flexion, and rotation of the
The simulator can apply compressive loads up
over
a
to
dynamic bandwidth of 15 Hz while a combined
load and
torque
signals.
The load actuator position is measured with a DCDT
cell
provides
measurement
and
feedback
- 61 -
stroke
is
down
and
up
range
of
can
be
electric motor drive to
an
with
frame
loading
The
deformation of the cartilage.
positioned
2.5 mm
a second DCDT to measure the
with
instrumented
center
the
over the fu11 50 mm stroke;
adjust for specimen size and mounting.
A rotary actuator attached to the
the
for
capability
range of 280 degrees.
precision
around
rotation
provides
cell
load
the load axis over a
Rotation position is measured with
which
potentiometer
geared
is
to
the
a
rotary
actuator.
The loading frame can be rotated about a fixed axis
extension while the load is applied.
and
flexion
simulate
Although the loading frame has a large moment of inertia,
large
at
a
actuator can achieve 140 degrees of flexion
rotation
10 Hz,
to
adequate
physiological movements.
walking
simulate
to
and
other
The flexion angle is also measured
with a precision potentiometer.
The specime ns, immersed
are
saline
bath,
device
provides
mounted
adjustment
in
on
of
relative to the simulator axes.
temperature
controlled
rotary base.
A mounting
a
a
the
specimen
orientation
- 62 -
4.1.2
Instrumented Prostheses
A specially instrumented
used
to
apply
load
to
femoral
the
pressure distribution ove r the
The
design
was
develop e d
by
head
acetabul um
surface
prosthesis
and
of
is
measure the
the
cartilage.
Carlson [18 ] to measure the
pressure distribution ove r the cartilage in the hip joint in
a
consenting
human
nee d ing a femoral head prosthesis.
It
utilizes fourteen pressur e tranducers which are an
integral
part
Fourteen
of
the wall of the prosthesis (Fi au re 4-2).
diaphragms (3 mm diameter
into the hemisphere.
and
0.25 mm
thic k
are
A s mall pin rests on the center of the
diaphram and transmits the small deflect ion of the
under
external
machined
diaphram
pressure (sensitivity i s 4.um / 7 MPa)
to a
single-crystal silicon be am with four st rai n gauges diffused
onto
its
surface.
A
brid ge
Wheatstone
electrical output proport ional to the
p ress ure
produces
applied
an
to
the external surface of the prosthesis.
The pre ssure transducers are located in pairs at
ten
degrees
of latitude measured from the location through
which the re sultant load vector
passes.
prost hesis
the
180
degrees
about
By
rotating
of
the
A special apparatus rotates one transducer from
its 1 ocation at 10 degrees latitude into the pole under
1 oad vector.
the
load axis in 10 degree
steps the pr essure transducers sweep the full range
longi tude.
every
the
- 63 -
re
1
2
Diophragm
Wall of
Hemisphere
Pin
Beam
Strain Gages
Figure 4-2. Instrumented Prosthesis
- 64 -
A
second
mounted
with
flush
an
has
prosthesis
ultrasonic
(Figure 4 -3).
surface
the
transducer
This
transducer is used to measured the distance from the ball to
the
cartilage
to calcified-cartilage interface when a load
The transducer can
is applied to the joint.
be
positioned
at the pole or 30 degrees latitude.
4.1.3
Control Console
The console contains the feedback control circuits
the
hydraulic
displays
actuators,
for
of the values of load,
stroke, torque, flexion, and rotation, and limit meters that
activate an inte rlock system to shut off the hydraulic power
if any parameter exceeds a preset limit.
signals
to
control
the simulator can be set internally by a set of
potenti ometers
on
connection
the
bridge
The input
to
outputs
the
console
computer
from
time-multiplexed
onto
or
the
a
,
or
a
externally
signal generator.
pressure
single
from
signal.
state signals are amplified for output to
a
The
transducers
are
and the simulator
the
computer
or
display on an oscilloscope.
A "menu
digital
box"
input
is
connected
via
the
console
and output registers of the computer
used to send digital
instructions to the computer or
the experimenter for input during an experiment.
to
the
It
is
prompt
- 65 -
Figure 4-3. Prosthesis for Consolidation Measurement
- 66 -
4.1.4
Ultrasonic Apparatus
The scanning motions for the ultra son ic measurement are
by
provided
two
The first (longitude) is
stepper motors.
rotation of the base on which the speci men
second
(la titude)
is
rotati on
the
acetabulum .
scanner has an addition al
moving
the
femoral
head
or
carrier fo r the femoral head
The
degree
of
freeedom,
capable
of
the transducer about its focal poirnt anywhere within
a solid ang le of eight
motor.
The
the a rm which carries the
ultrasonic transducer, either around
inside
is mounted.
A
degrees,
driven opei n
loop
by
digital
sent to eac
position
DC
controller converts an exte rna 1 pulse train from
the compute r into the s ignals required to drive the
Additional
a
ci rcuitry
motors.
th e number of pulses
counts
motor and calculates and displays on LED' s
each
o
axis
in degrees.
Manual control
the
of the
scanners is also pos sible.
The ultrasonic signals are generated and amplified by a
Panametrics PR 52 pulser/receiver (Figure 4-4).
The signals
are sampled for 20.48 ps at 100 MHz and digitized with 8 bit
resolution
by
a
Biomation
sampling is initiated by a
synchronized
8100
signal
wavefo rm
from
the
recorder.
The
computer
and
with the pulser/receiver by a Tektronix FG 501
function generator;
in this way averaging of the signals to
improve the SNR is facilitated.
An analog reconstruction of
the signal stored in the waveform recorder is displayed on a
0'
Figure 4-4. Ultrasonic Apparatus
- 68 -
465B
Tektronix
the
oscilloscope;
transferred to the computer where it
data
can
also
is averaged and
be
stored
on disk for proces sing.
4.1.5
Computer Interfaces
There are tV
vitro
equipment.
interfaces from the PDP 11/60 to
incl udes
outputs,
anal og
16
input
digital
and
inputs
which
channels
16
igital
are mul
pl exed
through a 12 bit A/D converter, 2 programmable relay
D/A outputs.
channels
the
generate
control
11-D
by
the
waveform recorder.
11/60
The
a
high
speed
Direct Memory Access (DMA) transfer s between
the PDP- 11 memory and an exte rnal device;
Biomatio n
signals
and stepper controller.
The second interface is a DECkit
for
he hip
provide various control functions f or the
simulator, wavefo rm recorder,
interfac e
and 2
The a nalog signals are used to monitor
simulator states and
digital
in
The first is a DEC Lab Peripherals System
(LPS) which
8
the
and
in this case
Once data ta king is
complete d
by
Biomati on the
the
initiated
data
is
transfer red into the computer memory withou t interv ention by
the proc essor, which is free to
with
the
previous samples,
begin
ave ra.ging
for example.
the data transfer (""D.5 million 8-bit
the
data
The high speed of
samples
per
second)
enables sampling averaging of about 250 records/seco nd.
- 69 -
4.2
GEOMETRY
The specimens used i n these studies
the
at
autopsy
General
into
and
pelvis
the
reference
as
used
axes
at
by
Dr.
are
rschner wires
rpendic ular t o the
femur
These
sagittal and frontal planes of the body.
later
obtained
Hospital
Harris and his associates.
William H.
inserted
Massach usetts
were
mounting
when
are
wires
joint
the
components for the geometry and pressure measurements.
The intact joint with capsule is removed,
section.
hemipelvic
size
the
of
u sually aS
a
The joint is x-rayed to cal cul ate the
[121] ,[124]
Rushfeldt
acetabulum.
has
demonstrated the fit of the prosthesis in the acetabulum has
distributions.
a strong affect on the pressure
which
acetabulae
selected
for
fit
components
joint
natural
fit
the
closely
very
prosthesis
the pressure studies.
size
in
plastic
bags
at
-20 C.
[138]
the
only
are
( 49 mm)
Nearly any size can be
accomodated for the geometry measurements.
stored
Since
The
joints
are
We have compared the
pressure distributions for one acetabulum which was
studied
immediately after autopsy with data taken after freezing for
both 12 hours and 18 months and
then
thawing.
little change in the pressure distribution.
There
was
- 70 -
room
The specimens are thawed in
The
capsule
femoral
is
is
head
Clinical
opened
measured
evalu at ion
performed by Dr
drawings
and
of
removed.
with
the
saline.
temperature
The diameter of the
calipers
and
recorded.
condition of the cartilage is
Harris or one of
his
associates.
After
ph otographs are made, India ink is swabbed is
and
onto the cartil ag e and rinsed off using saline solution,
described
by
Meachim
[100].
Slight
as
fibrillation of the
cartilage surfa ce , one of the early signs of osteoarthritis,
is made more eas ily visible by this technique.
taken if
of photographs is
the
cartilage
Another set
there is any discernible fault in
c ondition.
The femoral head is mounted on a base plate as shown in
Figure 4-5
using
polymethyl
methacrylate
lateral-medial axis is positioned vertical
the
superior-inferior
The base plate
circulating
and
upward
The
and
axis is horizontal and to the right.
is mounted
saline
cement.
bath
on
the
rotary
table
in the
controlled at body temperature of
37 C.
The corresponding acetabulum i s mounted
orientation
(Figure 4-6).
Since
the
in
a
similar
joint will later be
loaded by the instrumented prosthes is care is taken to mount
the
acetabulum
in
stable and secure.
an
orientatio n and manner that will be
The following procedure has
proven
the
- 71 -
----
C__
_____
__
---
__·_
(base
rotation
positive s
for
scan)
Figure 4-5. Femoral Head Mounting
~-'
~-~~~--
-`
-
~-~--~~-' ----~-
---
-
- 72 -
(base rotation tor
positive P scan )
Figure 4-6. Acetabulum Mounting
- 73 -
The joi nt is prepared for mounting by making
most reliable.
A
guide.
of
as
wires
Kirschner
a
cut is made 50 mm above the superior
transverse
edge of the acetabulum and
medial
the
using
two cuts in the pelvis,
the acetabu lum.
just
is made
cut
sagittal
a
The acetabulum is mounted on an
L-shaped aluminum base using methylmethacrylate cement.
The
in
the
and
the
is positi oned
acetabulum
horizontal plane
superior
cut
for
in
the
the
the
with
measurements
geometry
plane
horizontal
cut
medial
for
the pressure
measurements.
The joint is first centered relative
axes
scanning
the
to
approximately by eye and then accurately by monitoring
the position and amplitude of the ultrasonic reflections
as
and scanning arm are rotated back and forth.
It
the
joint
is important to get the joint properly
since
centered
the
amplitude of the ultrasonic reflections is decreased greatly
if the incidence of the ultrasonic wave axis is
normal
not
to the cartilage.
The
computer
rotation.
ultrasonic
boundaries
cartilage
storage
at
are
equidistant
20
inputted
points
The scanning arm is manually advanced
reflections
manually
of
to
the base
until
the
from the surface become indistinct.
This position of the arm is input to the computer and stored
for
use
later during plotting and to define the limits for
the automatic
scanning
of
the
cartilage
geometry.
The
- 74 -
automatic
scanning locations are at even 9 degree intervals
of base rotation (longitude) and the roots of
polynomial
of
degree
the
Legendre
40 (about every 4.5 degrees) for the
scanning arm (longitude).
The scanner is positioned automatically under
at
control
computer
every location within the cartilage boundaries.
The ultrasonic reflections are sampled
at
by
100 MHz
the
waveform recorder and transferred to the computer, where 128
records 1024 samples long (10.24 ps) at
each
location
are
averaged and stored on disk.
Complete scanning of a femoral
head produces about 700 data
locations
hour.
in
less
than
one
There are about 350 data locations on the acetabulum
cartilage which is much smaller in area.
The recorded ultrasonic
usi ng
a
correlated
signal
receiver
is
off-line
analyzed
techni que.
cross
The
correlation of a st anda rd reflection from a steel ball
the
reflections
using
location
correlation
cartilage
interface.
from
the cartilage is calculated for each
Fouri er
are
The
transforms.
ident ified
and
surface
as
cartilage
the
to
(1533 m/s
The
travel
maxima
of
times
calcified
the
to the
cartilage
trave 1 times are reduced to a set of radii
which describe each sur face, using the
saline
with
@
sp eed
of
sound
in
37 C) and in cartila ge (1760 * 90 m/s)
and the calibration measurements from the steel ball.
- 75 -
The cartilage surfaces are described in two ways.
The
first is by fitting a least squa red error sphere to each and
calculating the
at
deviations
each
location.
Spherical
harmonic series are also fitted to the data [138], providing
a smooth function closely approx imating the
interface.
plots
Contour
of
sphericity (Figures 4-7 & 4-8)
(Figure 4-9)
cartilage
constant
and
are
shape
the
of
each
deviation
from
thickness
generated
from
the
of
the
series
representations.
4.3
SWELLING
The swelling experiments described
strain equilibrium modulus.
osmotic-mechanical
noted
by
coupling
[ 9 2],
Maroudas
proteoglyca ,ns
(based
osmotic
when
cartilage
the
osmotic
pressure
charge
the
supports
compressed
the
(since
Conversely,
by
of the solution bathing the
cartilage, compression can be effected.
note
is
position
equilibrium
unique
pressure
stresses in the collagen fibers are zero).
varying
to
As has been
of the car tilage.
on the extrafibrillar water)
physiologic al
the
used
The method exploits th e
the
the entire applied pre ssure
from
are
diffusivity of the cartil age and the uniaxial
the
estimate
here
It is important
to
that this is not exactly the same as varying the local
on
concentrati on
the
of
pro teoglycans
the
bath
since
diffusion oif salt ions through the
by
varying
in
the
salt
the latter case the
cartilage
introduces
a
SUPERIOR
INFERIOR
Figure 4-7. Cartilage Surface Deviations from Sphericity [um]
SUPERIOR
INFERIOR
Figure 4-8. Cartilage to Calcified-Cartilage Interface Deviations from Sphericity [um]
0
SUPERIOR
INFERIOR
Figure 4-9. Cartilage Thickness [um]
i
- 79 -
In fact our initial
complicating coupled diffusion problem.
in the
changes
when
cartilage
of
response
time
attempts to measure the
equilibrium potential of the proteoglycans
bath
were induced by varying the salt concentrations of the
complicated
were
by the nearly equal diffusivitites of the
water and salt [53],[91].
Our next set
of
rel ied
experiments
on
varying
the
osmotic potential of the PG gel by exposing the cartilage to
humid air [1391.
conv ection
free
For
in
air
time
associated with the dryi ng was about two orders of
constant
magnitude longer than the layer ti me consta nt.
in
therefore
The layer is
quasi-equilibrium throughout the dehydration.
displacement
techniques
and
for
as
equipment
the
the
same ultrasonic
the
using
measured
and
bath
The layer was then resubmerged in the salin e
surface
the
geometry
static
measurements.
First, since
There were two problems with this method.
dehydration would continue well past' the ten to fifteen
the
give
percent compression desired to
while
displacement
adequate
keeping nonlinear (permeability) effects to a minimum
some control over the dehydration was
required.
This
was
hard to judge accurately since the dehydration rate depended
on
the
orientation
influences
and
the
of
the
amount
surface
of
and
dehydration
other
external
could
measured without re-introducing the bathing solution.
not be
- 80 -
been
An alternative method has
uses
which
developed
solutions of polyethylene glycol (PEG 20,000) to osmotically
The concentration of the PEG determines
load the cartilage.
thermodynamic considerations.
The method is similar to
and
the experiments of Maroudas [921 on cartilage
by
motivated
from
determined
be
can
and
pressure
osmotic
the
slices.
There are two major
First,
the
equilibrium
the
attractions
displacement
applied
method.
this
to
be
can
and
precisely
the
controlled
via
equlibrium
modulus can be measured by varying the generated
pressure
second,
osmotic pressures.
~rA
olutions are prepared by dissolving 100
Ftl
n
PEG
of
pressures
of 0.15 M NaCl.
1 1
are
0.6 MPa.
The cor responding osmotic
The
PEG
solutions
in sacs of Spectrapor 2 dialysis tubing (MW
inserted
12-14000)
taking ca r e the sacs are not fully
b ath level i
saline
to
u.i
s
hours
the
After .2 to
the sac is removed (initial expe riments for longer
saline bat h is restored.
surface displ acement and the
from
The
lowered and a PEG sac is positioned in
times pro duced no add itional compression of
and
are
cutoff
filled.
the aceta bulum, tota 1ly covering the carti lage.
3
250 g
to
the
cartilage)
The ti me response of the
amplitude
the cartilage surface is measured.
of
the
reflection
A set of locations
at one longitude are scanned during one swelling cycle.
- 81 -
The surface displacement response at one location
osmotic pressure was 0.5 MPa)
equil i brium
physiological
different
from
i.e.
equation;
that
it
is shown in Figure 4-1 0.
the
response
by
predicte d
is
a
Near
signi ficantly
simple
i s not well d escribed by a
constant (Fig ure 4-11 ).
(the
diffusion
sing le
time
The coll agen fibers appear
to limit
'
the expansion of the proteoglycan gel near equ il ibrium,
in the osmotic pressu re of the pro teoglycans
difference
the tissue swelling p ressure bein g the tensile
stress in the
tissue
of
(Mar oudas
found
typi cal
values
0.2 MPa).
Consequently the unre strained equ ilibrium voluime of the
the
(which
physiological
of
volume
is
1 ayer
moving
toward
)is
greater
equil ib rium
cartil age .
prior
than
to
linear
relative
to
fit
to
the
ge 1
reaching
the
in
situ
The equil ibrium volume was estimated
by finding th e equili brium volume of the gel t h at
best
gel
the
log
gave
the
of the surfac e displacement
equilibr ium
(Figure 4-12).
The
estimated equ ilibrium volume was 32 percent gre ater than the
physi ologi cal equilibrium volume and the time
2400
was
S.
A second set of experiments on
same
constant
the
location is shown in Figure 4-13.
pressure
was
(Figure 4-14)
0.4 MPa
and
was
percent
29
the
cartilage
gel
volume
than physiological
equilibrium and the time constant was 2300 s.
Equation
the
The applied osmotic
calculated
greater
at
Refering
to
16 in Chapter 3, the permeability at this location
z*
m
0)
C)
o
0
0
en
I-a
Ln
m
M *L
((1
(12
r·
IU)
U
- ze -
SURFACE DISPLACEMENT [Microns]
U-.
--
C
-,
0
0
o
DRI: SWO762. DAT
0. 0
-
-1. 0
--
-2. 0 -
-3.0 -
-4. 0
-5.0
1
500.
TIME
I
1000.
(Seconds]
Figure 4-11. Log Surface Displacement
1500.
DR1: SW0762. DAT
0. 00
-0. 50
-1. 00
-1.50
500.
TIME
1000.
[Seconde]
Figure 4-12. Best Linear Fit
1500.
DRi: SW0562. DAT
90.
aI
80.
a
a
70.
SI
60.
m_
0
L
0
50.
m,
.rG
-.
z
I
00.
40.
mr
ma~
U
30
maI
wu
20.
m,
S0.
maI
0.
500.
TIME
1000.
[Seoondls]
Figure 4-13. Swelling Surface Displacement
1500.
DR1s SW0562. DAT
0. 00
-0. 10
-0. 20
-0. 30
-0. 40
-0. 50
-0. 60
-0. 70
-0. 80
-0. 90
0.
500.
TIME
1000.
[Seconds]
Figure 4-14. Best Linear Fit
1500.
- 87 -
was therefore estimated to be 5 X 10
4.4
m /Ns.
PRESSURE DISTRIBUTION
The measurement of the time-history of the total stress
on
the surface of the cartilage is performed as depicted in
Figure 4-15.
A static load is
using
instrumented
the
intervals (usually
the
load
is
applied
'prosthesis.
the
At
acetabulum
selected
0,1,2,3,5,10,15,20,25,&30 minutes
applied)
time
after
the prosthesis is rotated about the
load axis in 10 degree increments
mapping
to
of the surface stress.
to
generate
a
complete
The prosthesis can be moved
through this sequence by the hip
simulator
under
computer
control in about 30 seconds, during which time the pressures
vary very little (cartilage layer time constants are on
the
order of 30 minutes).
Contou r plots of the surface
4-18),
axes,
in
a
coordinate
(Figures 4-16
to
system fixed relati ve to the body
are generated by fitting
dimensional
stress
Fourier series.
the
data
pressure
to
two
The average error is 1es s than
0.05 MPa.
4.5
CARTILAGE CONSOLIDATION
The time-history of the consolidation of the
surface
is measured
as shown in Figure 4-19.
the prosthesis about an axis 15 degrees off
the
cartilage
By rotating
load
axis
- 88 -
PRESSURE
,TRA5DVC•ER DTAL
DIAPKHRAG
Figure 4-15. Surface Stress Measurement
rar~c
~oc~
+
MEASURED
PRESSURE IKPaJ
0 Minutes
Figure 4-16. Surface Stress: 0 Minutes
2a-w- --
+
MEASURED
PRESSURE [KPal
5 Minutes
Figure 4-17. Surface Stress: 5 Minutes
+
MEASURED
PRESSURE CKPal
20 Minutes
Figure 4-18. Surface Stress: 20 Minutes
- 92 -
4EDUI
LATER
ULTRAS6Nt8C
~cER
TRANSb
bETAIL
Figure 4-19. Consolidation Measurement
- 93 -
the
di stance
from
the
prosthesis
to
the
calcifi ed-calcified interface is measured at
several
locations
of 30 degrees latitude.
the bal 1 relative to
therfor e
the
the
surface
center
displacment
joint) is calculated (Figu re 4-20).
the
load
axis
is
scale.
the
pole
and
The positio
acetabulum
o0
and
of the cartilage in the
The displ acement
along
with loads of 2 25,
45 0,
and
Fi gure 4 -22 shows the data from one test on lo g-log
In all c a ses the r ate
prosthe sis
into
the
of
the
of pene trat ion i
ace tabulum decreases ve ry quic kly;
in
to load.
reference
to
The
a
The initi al rate
act ually less for higher loa ds;
rate is proportional
the
of
penetr ati on
resul t whic h was note d by Rushfeldt [121].
results
the
to
shown in Figure 4-21 for three s eparate
experiments on on e acetabu lum,
900 N.
of
cartilage
implications
th e final
of
these
the resistance to interarticular
flow are discussed in Chapter 5.
1 0
DR : CD6601. DAT
0. 200
E
E
(0
i
0. 150
i
c0
0
a
0
a
n.
0
(0
0.100
(D
-,I
4r
c0
(0
-cr
4,
(I
0. 050
xa-
0.000
.e0
C
>-
-0.050
0.
500.
1000.
TIME [Secondsl
Figure 4-20. Prosthesis Position
I
1500.
2000.
DR1: CD6501. DAT
0. 200
r-i
E
E
L,
4)
0
E
0
0
C,
O
-r4
0. 150
0.100
0
0
-c
0
0
L
a.
0
0
0. 050
03
-J
0.000
500.
1000.
TIME [Seconde3
1500.
Figure 4-21. Prosthesis Displacement along Load Axis
2000.
DR CD0
DRI: CD6601. OAT
2.50
-
0
3E
2. 00 -
4-)
*
C
0
E
0
0
g
0
1.50*-C
0)
,1
._
1.00
-
_
1.50I
1.00
1.50
__
I
I
2.50
2,00
LOG TIME [Second&]
Figure 4-22. Log Prosthesis Displacement
3.00
3.50
- 97 -
CHAPTER 5
RESPONSE OF THE CARTILAGE LAYER
- 98 -
The, development of a model
layers
of
the
features of the
properties,
chapter.
and
The
synovi al
joint
cartil age
for the articular
which reflects the salient
geometry,
bound ary
dynamic
conditions
incorporates
model
cartilage
constitutive
is discussed in this
the
results
of
the
experiments described i n Chapter 4, namely:
1.
Geometry of
cartilage
the
layer
in
the
acetabulum,
measured with our ultrasonic technique;
2.
Permeability and uniaxial strain modulus, estimated from
the swelling experiments;
3.
Surface stress distribution time-history, measured
with
the instrumented endoprosthesis;
4.
Displacement
of
the
cartilage
surface
time-history,
measured with the ultrasonic prosthesis.
The
conditions
model
will
for
fluid
be
flow
the interarticul ar gap,
solid
stress
cartilage layer.
and
used
to
the
boundary
at the cartilage surface,
and describe
fl uid
estimate
pressure
the
response
distributions
of
i.e.
the
in the
- 99 -
5.1
SIMULATION
The goal of the analysis described in t his
the
formulation
an
of
model
idealized
layers which is s uitable for numerical
section
is
of the cartilage
solu tion.
A
major
problem with mode ls used by other investigat ors has been the
assumption that 1 oad sharing at the surface is dependent
porosity
on
The approach taken here is to consider the
[102].
boundary conditonIs at the interarticular sur face unknown and
to
results to estimate the actual boundary
mod el
the
use
conditions for fluid flow.
Description
5.1.1
Our capacity
of
variation
to
data
acquire
cartil age
the
of
the
cartilage
geometry
the
is
well
topographical
constitutive
and
displacemen t
properties and the stress and
surface
on
sui ted
at
the
for
construction of valid models of the cartil age layer
acetabulum.
In
The
simplest
cartilage, consistent with
Chapter
4,
is
the
in
the
the
order to fully exploit this capability the
finite element method was us ed to
equations.
loaded
that
formula te
constituitiv e
the
measureme nts
the
model
continuum
for
described
the
in
of a porous medium with intrinsically
incompressible constituents.
- 100 -
The model
shown schematically in
strain
the
in
the
1aye r
porous
displacements of the solid matri x
vertical
(radial)
was used due to
Figure 5-1,
Shear and lateral
thick ness
ratio
cartilage area and the fixation to bon e.
concerning
pressures
can
The
layer.
based
on
the
vary
are
the
both
of
measurements
loaded
No assumptions are
fluid
flow;
hence
bone
assumed
is
sti ffness
relative
the
the
and laterally in the
to
surfac e
model
obtained
rigid
so
subchondral
are
The
the
and
boundary
measured
total
the surface displacements
will
be
compared
ult rasonic
the
from
of
cart ilage.
stress and an unknown fluid flow
by
of
vert ically
high
conditions on the loaded
predicted
geom etry).
zero at the cartil age to bone interface,
known
bone
cancellous
direction
supporting
displacements
joint
strains in the carti lage are assumed small
due to the large width to
made
the
coordinate system
the
ref lect
i.e.
to
con strained
direction (a spher ical
accurately
uniaxial,
iS
are
assumes
with
the
prosthesis to
predict the flow at the surface.
A
finite
incorporating
element
these
model
for
the
assumptions
cartilage
was
layer
developed.
The
cartilage layer in the acetabulum was discretized into three
equal
layers
orthogonal
interpolation
radi al ly,
and
circumferential
was
used
every
9
degrees
directions.
in
both
Quadratic
for the displacements of the solid
matrix and linear interpolation
for
the
fluid
pressures.
- 101 -
TOTAL SujRFfC-E
SURIFAc.--
-
NPL
.
Ae - MIkr
w
KNO)WN SORFACE
FLutlD
PCe &)r)
FLoW) qs(0))
50LIb
DISPLACErNT
t
bRAIKO
MDbULOS
PERMrA2BUlY
Figure 5-1. Cartilage Layer Model
rS'k (9)
- 102 -
The model had 1,300 degrees of freedom.
Finite Element Formulation
5.1.2
The
derived
finite
from
element
the
velocities [9].
virtual
equations
principles
for
equilibrium
are
of virtual displacements and
These state that for any compatible, small,
displacements
and
velocities imposed on the body,
the total internal and external virtual work and energy
are
equal :
V +S
V
V/d~r
± oflp~v j~dvv
S; ldVrc
where A
vir tual
are the
dGI
displacements,
a re
actual stresses, and C
variables
in
any
51fPPc
stra ins,
the
(1)
el ement
actual
are
are
cUL
virtual
are the
(2)
the
pressures,
flows.
'1
a
are the
The
approximated
interpolation matrices which rel ate them as
virtual
field
by
the
function
of
the nodal values.
1(k
P
ý A,
(3)
N=
k
(4)
The
strains
and
pressure
gradients
are
obtained
appropriately differentiating the interpolation matrices.
- 103 -
)=B(A)G
(5)
(6)
The stresses and flows are related to the displacements
and
pressures by the material matrices.
Ai('
cM)
ovt
('P
9P)17
(7)
P
(8)
Also conservation of mass is expressed as:
Ut.
(9)
Equations 1 and 2 are rewritten as sums of integrations over
the volume and areas of the elements:
SJvfd~
(M 4)1 (A
TLJcb~)
-/)in
J
Substituting
pressures,
,~ilfs~i~d~Pk)T_ /
÷z
rvý~'ri~
ý(,q)
5t
(M)
(MLi~
(11)
Bsi9,j] dVi M¾1
cpio.V
the
expressions
for
the
(10)
displacements
and
strains and pressure gradients, and stresses and
flows we obtain:
Jý C-F-dv) 0
r
-
I
Ads]
A.dV)p
=
T
Ad
O dvP (§pI~h3 dv --·
cry(V
S
Imposing unit virtual
A~It icL3
(12)
(13)
displacements and pressures at all1 the
- 104 -
the equilibri um eqiations can be written as
nodes
LP-
KLAL4
R.
(14)
dLTK
K~iTA,
(15)
where the "stiffness" matrices and "load" vectors are:
(16)
Lc·
L5,~dV
B Bcdv
(17)
GJKGidv
(18)
T.
T3ýS ds
I
(19)
Al;d
(20)
The finite element equations fI and /1 are solved via a
fully
i mplicit
time
integration
s.cheme.
This
is
unconditi onally stable and has the advantage tha t the system
matrix
can
be
assembled once, triangularized, and stored:
the solut ion at each time step simply
and
back
substitution
of
the
requires
appropriate
calculation
1oad
vector,
Equations 21 and 22.
JK
L3
PI
PJ
(21)
T
tK,
A,
- 105 -
L
· :~1·
These are the same
as
(22)
m~+n
Equation
in
12-38
Desai
and
into
the
Christian [35].
Interarticular Boundary Condition
5.1.3
The
general
boundary
condition
for
flow
interarticular space is of the form:
(23)
where
R5
is
the
"surface
resistance".
This
can
be
incorporated as (using the equation for flow):
' K
If R_
of
-
(24)
is known then this can be moved to the left hand side
equation;
the
unknowns.
They can
otherwise we regard the surface flows as
be
found
by
the
following
sequence
(Figure 5-2).
1.
Find the sol ution vectors Um (surface displacements) for
a unit flow at the m'th surface node.
2.
Assembl e and triangularize the matrix Kq where the
column in Kq is the vector Um.
At each time step:
m'th
- 106 -
Figure 5-2. Coupled Solid-Fluid Analysis Flowchart
- 107 -
1.
and
22
Calculate the error in the surface displacements
Ue
Find the solution Ur(t+dt) to Equations
21
using R(t) for Q(t)=O.
2.
from
the
surface displacement
(measured)
desired
Us(t+dt) as Ue = Us - Ur.
3.
Backsubstitute Ue into Kq to
required
for
the
find
ow
Qs(t)
resulting
the total displacement at
from the load vector R(t) + Qs(t).
4.
Uq(t+dt)
Apply the flow Qs(t) and add the resulting
The
Ur(t+dt).
to
total
will equal
displacement
Us(t+dt).
5.2
RESULTS
The model was used to calculate the
two
static
load
cases:
surface
450 N and 900 N.
30 s was used for the simulation (about 0.01
time
constant).
steps ;
The
solution
flow
for
A time step of
of
the
layer
was calculated for 40 time
the consol idation rate changes very little after
20
minut es (Figure 4-21).
The calculated flow (Figu res 5-3 to
stress
(Figures 5-6
time
to
steps
5-5)
and
surface
5-8) over the cartilage surface is
of
the
shown for
3
decreases
dramatically with time.
900 N
case.
The
flow
The calculated flows are
0
0 Minutee
FLOW [cu. mm/10Ms]
VERTICAL
+
C.,-4
Figure 5-3. Calculated Surface Flow: 0 Minutes
a
O
I
Kj;-ý
+
VERTICAL
FLOW ocu. mm/10Ms3
5 Minutes
Figure 5-4. Calculated Surface Flow: 5 Minutes
S
0)
+
VERTICAL
FLOW Ecu. mm/10Ms3
20 Minutes
Figure 5-5. Calculated Surface Flow: 20 Minutes
I
I
+
SOLID
STRESS [KPal
0 Minutes
Figure 5-6. Calculated Solid Stress: 0 Minutes
0
S
0
w
U
+
SOLID
STRESS [KPa3
5 Minutee
Figure 5-7. Calculated Solid Stress: 5 Minutes
w
H
+
SOLID
STRESS (KPa3
20 Minutes
Figure 5-8. Calculated Solid Stress: 20 Minutes
- 114 -
generally highest at the locations
where
the
gradient
in
surface stress is largest.
An average conductance to flow
gap,
motivated
by
the
simple
model
calculated from the ratio of total
pressure.
the
in
of
flo w
interarticul ar
Section 3.4, is
to
average
flu id
Figures 5-9 and 5-10 show the time-history of the
ratio of this average conductance to the conductance of
cartilage layer (permeability times thi ckness).
the
The initi al
conductance is higher for the smaller 1 oad, consistent
wi th
the observation that the consolidation rate is greater.
The
conductance decreases with time to a mi nimum value
that
is
nearly the same for both cases.
A
measure
conductance
vertical
of
in
By
(Figure 5-11 ) is
the
for
pressure
vertical
direction
is dominated
to
heat
ve ry small
cart ilage
in
of
the
layer,
the
ca
car tilage 1
fluid pressu res and flows induced
calculated
by
solving
cartilage geometries,
by
the
layer
that
illustrates
the
high
surface
ransfer the Biot number
This s ggests that
epic [139] has used su
s
to
conductance
p redict tne pressures
cartilage.
dynamic
surface
anal ogy
g adie nt
realisticall
the
the
equilib ri um
resistance.
model
the
the
igno'ring
ilage
a
simpler
vertical
layer,
could
nd displacements in the
a
model
to
simulate
er during walking.
the
contact
are very similar to
cartilage
problem
the
The
1 ayer,
using
the
experimental
1. 50
w
z
-
1.00-
a.3
0
0
I
7)
I
0.50
w
Fz
w
.I-
I.
0-.
00 SI
0.
500.
TIME [SECONDS]
Figure 5-9. Average Gap Conductance: 450 N
I
1000.
1500.
0
1.
0·
0,
0
kbi
w
z
I0
3
1.00
z
0
F-,
I
n-0.00
'IH
~I-I-
n,,
0. 00
0.
500.
TIME ISECONDS]
1000.
Figure 5-10. Average Gap Conductance: 900 N
1500.
0. 0250
900 N
0. 0200
0.0150
0. 0100 -
0. 0050 -
L,
0. 0000
500.
TIME ESECONDS]
Figure 5-11. Biot Number
1000.
1500.
- 118 -
and theoretical results decscribe herein.
An overall
layer
view of the average stress
is shown in Figure 5-12.
most of the load ,
between
the
fluid
even
pressure
which produces flow in
with time.
for
the
The fluid
long
at
times.
n the
cartilage
ressure supports
The
difference
the bone and the surface,
interarticul ar
gap,
decreases
900 NN
LOAD == 900~
LOAD
2.00
-
DEEP PRESSURE
1.50 FILM PRESSURE
1.00 -
0.50 -
SOLID STRESS
•,
0.00
;. -----
Q.T-•
_-
--.
---
'-----
•'
-
•.
...
.T- -•
_----~ --
~
I
500.
TIME [SECONDS]
Figure 5-12. Average Stress
-- "-
..
..
•--'g
I
1000.
1500.
-
120 -
CHAPTER 6
CONCLUSIONS
- 121 -
The importance of both global in
the
of
response
and
properties
situ
cartilage
physically based models is demonstrated.
nature
measurement
of
combined with
In particular, the
of the interarticular condition and its implications
(and
for the normal
abnormal)
possibly
of
function
the
synovial joint is illuminated.
The geometry and the stat ic
adult
normal
of
properties
and
dynamic
constitutive
articular cartilage in situ in
the human hip joint have been measured experimentally.
the
includes
strai n
uniaxial
in
the
of
acetabulum
includes
both
the
surface
surface displacement.
particular
the
endoporostheses
instrumented
the
equilibrium modulus and the
The t ime response of the
hydraulic permeability.
cartilage
human hip joint when loaded by
has
stress
been
measured.
hip
joint,
the
joint,
in
incorporating
measurements described has bee n used to predict the
boundary
conditions
governin g
This
and
distribution
Modelli ng of the synovial
human
This
interarticular
the
surface
fluid.flow.
The model is used to estimate the solid stress in the matrix
and fluid pressure in the cart ilage.
Much of this work depends on refinement and application
of
experimental
techniques
devel oped in this laboratory.
and
theoretical
concepts
- 122 -
6.1
GEOMETRY
Ultrasonic measurment of the geometry of the
cartilage
layer in the acetabulum of the human hip joint was initiated
by Rushfeldt [121].
Tepic [138] extended the
the
head, providing conclusive quantitative
human
femoral
evidence of the congruency of the
More
recently,
natural
bone.
and
ultrasonic
joint.
signal
and
reflections from the cartilage layer and the underlying
Automated computer control of
off-line
processing
of
the
the
in
turn
made
the
data
acquisition
signal have improved the
resolution of the distance measurement to
This
synovial
to
using a waveform recorder we have added the
capability to sample and record the
the
technique
osmotic
less
swelling
than
and
2 um.
cartilage
consolidation experiments feasible.
6.2
CONSTITUTIVE PROPERTIES
Osmotic
exploiting
loading
the
of
the
cartilage
electromechanical
layer
properties
in
situ,
of articular
cartilage, has provided the means to obtain precise,
easily
interpretable measurements of the constitutive properties of
cartilage under simple loading conditions which mimic the in
vivo
environment.
An
osmotic
loading technique has been
developed
to
solutions
of high molecular weight polyethylglycol (PEG) to
achieve
this.
Selective
cartilage through a layer of dialysis
application
membrane
lowers
of
the
- 123 -
osmotic pressure of the proteoglycan gel and uniformly loads
the solid matrix.
ultrasonic techniques used for the
same
the
via
measured
equilibrium,
The return to physiological
estimate
an
only
geometry, provided not
the
of
dynamic
of the cartilage but also the nonlinear behavior
properties
near equilibrium.
This corroborates the static measurements
of Maroudas [921.
6.2.1
Ultrastructure
We
to
cartilage
swelling
also
have
its properties
through
of
impedance
the
time
and
cartilage.
during
response
of
measurement
simultaneous
the
the
of
ultrastructure
the
related
the
We be lieve these experiments
demonstrate that th e nonlinearity of the bulk properties and
the time response is due to the confi nement and restraint of
the collagen fiber
pressure
of
the
network
the
and
proteoglycan
or
osmotic
swelling
which keeps the fibers
gel
This is only possible when the overall volume
"turgid".
near physiological
is
the fibers are tightly
equilibrium (i.e.
stretched).
We have
loading
used
estimate
to
transient
the
the
response
after
fundamental time constant of the
layer by fitting the response to a linear model (as
proteoglycan
volume).
gel
Combined
was
with
osmotic
approaching
the
static
its
own
if
the
equilibrium
measurement
of
the
- 124 -
equilibrium
displacement of the layer under various osmotic
pressures and the thickness of the layer,
the
permeability
of the cartilage has been estimated.
The application of Biot's model [11] for
to
porous
media
cartilage is predicated on the assumption that the fluid
and solid constituents are intrinsically incompressible.
have
attempted
to
verify
that
consistent with this model;
We
the frequency response is
in particular we
attempted
to
the frequency range over which the phase shift between
find
the surface stress and displacement
(Appendix
B).
Although
our
investigators,
the
at
expe rimental
er ror
eliminated the likely sources of
other
remains
degrees
technique
that
s hift
phas e
45
have
has
plagued
(Fi gure B-1) falls
below 45 degrees for frequencies abo ve 0.1 Hz or two decades
beyond
the
time
constant
of
the
layer.
At
such high
frequencies the total displaceme nt ( for this system) is less
than
of
1 um.
The roughness of the 1 oader and the assumption
homogeneity
continuum
over
(i.e.
the
that
the
material
behaves
as
a
relevant time and distance scales) may
not be appropriate.
6.3
SURFACE STRESS
The measurements of
over
the
cartilage
in
the
the
surface
human
stress
distribution
acetabulum
instrumented endoprosthesi s loaded in vitro
using
using
our
an
hip
125 -
-
simulator
wer e
first
Additional
tes ts
since
conducted
and
then
by
Rushfeldt
[121].
applicati on
their
as
described in this thesis have demonst rated the useful nes s of
this
other
Recently,
approach.
u sing
investigators
dissimilar tech niques have obtained c orroborating res ult
pressure
0.375 mm
24
Brown [14] mounted
S.
piezoresistive
thick
transducers in recesses machined in the surface of
the cartilage layer of the femoral head and 1oaded the joint
in
are
distributions
resul ting
The
orientations.
various
qualitatively
and
quantitatively
pressure
remarkably similar to our results.
6.4
CONSOLIDATION
Armstrong [7] measured the deformation of the c artil age
in
loaded
hip
resolution of I to 2
cartilage
roentgenograms.
via
joints
percent
in
He c laimed a
measu rement
the
thic kness using a magnification of 30 X.
deformations were 10 percent after 30 min'utes
at
of
the
Typical
1oads
of
of
the
five times body weight.
The influence of the
cartilage
illustrated
orientation
geometry
in
on
the
the
on
idiosyncratic
the
comparisons
pressure
pressure
of
the
character
distribution
is
effect
of
1oad
and
of
1oad
distribution
magnitude on the time response of the cartilage deformati on.
- 126
Small,
physiologically relevent,
direction
of
the
variations in the
the
load vector relative to the cartilage in
the acetabulum have a dramatic effect on local peaks in the
pressure
distribution.
Interaction
areas is mediated by the flow of
with
the surrounding
interarticular
fluid
and
hence the surface resistance.
Furthermore,
the
short-time
independent of the applied load.
while causing
rapid
more
interarticular
response
is
nearly
The higher surface stress,
also
consolidation,
space more rapidly.
seals
the
In all cases studied an
abrupt change in the rate of consolidation occurs at a total
consolidation
of
about
100-150 um,
about
average of the deviations from sphericity
twice
the rms
the
surface.
at
It appears that for adequate interarticular sealing to occur
the interarticular spacing needs to be small (on
of
the
the
order
the roughness of the cartilage surface) compared to
the unloaded shape.
The
measurement
consolidation
automated
has
of
the
response
anal ysis
and
of
the
reflections
using
transducer.
For the first time measurement of
of
the
dynamics
These were used,
surface
the
of
of
the
greatl y improved by the high speed
been
sampling
time
ultrasonic
prosthe sis with integral ultrasonic
the
together
the
details
cons olidation has been possible.
with
the
measurements
of
the
stress on the cartilage , in a model to estimate the
- 127 -
fluid
boundary condition for
in
flow
the
interarticular
space.
6.5
SURFACE RESISTANCE
The model
us ed to
l ayer
cartilage
characterize
project.
cartilage
pressure in the
geometry
of
and
its
The
of
surface
on
of
dep end
of
surface
the
the
fluid flow and
complicated
time-varying
cartilage
layer in the
the
as
zero
The general boundary condition of a
(along
finite resistance to fluid flow
will
the
joint precludes such simple cases
or zero pre ssure.
space)
at
effect
cartilage.
the
natural synovial
flow
previously
both
In particular, they addressed the nature of
the fluid flow boundary condition
loaded
the
been motivated by the experiments and
has
analysis of Dent [34] and Kenyon [68],
this
of
behavior
the
the
interarticular
on the fluid film thickness, cartilage
surface roughess and geometry, and the length and tortuosity
of the flow paths [34].
Analytical treatment of the problem of
surface
resistance
and
Kenyon has shown
the
can be high, greatly affecting the proportioning
of stress between the fluid and solid phases.
the
the
its effect on load carriage by the
cartilage is fraught with difficulty.
resistance
estimating
pressure
in
the
compressed cartilage
fluid
plugs
Dent measured
film at the loaded surface of
5 mm
in
diameter.
He
found
- 128 -
significant
load
carriage
(25 to 75 percent) by the fluid
pressure even for long times (20 minutes).
The
final
part
of
this
thesis
incorporates
the
measurements
of the geometry of the cartilage layers, their
constitutive
properties,
displacement
in
joint.
The goal
and
the
surface
stress
and
a model of the cartilage layers in the hip
was
to
estimate
the
local
and
global
resistance to fluid flow in the interarticular space.
flow toward the space and parallel to it are included.
result
the
is
the
time
The
first experimentally determined estimate of
dependent
space
interarticular
Fluid
resistance
and
to
fluid
flow
in
the
its effects on the stress in the
cartilage.
The relative conductance of the interarticular space to
the
conductance
of the cartilage layer decreases with time
from about one to less than 0.05.
The relative flow in
the
interarticular space to that in the layer decreases in about
the
proportion.
same
conductance
of
the
than for lateral flow;
This
is
becau'se
the
relative
path for vertical flow is much greater
it
is
likely
the
fluid
pressure,
which supports ninety percent of the load, even after twenty
minutes,
is
approximately
location in the joint.
constant
with
depth
at
any
- 129 -
6.6
STRESS IN THE CARTILAGE
Further, the model,
condition,
is used
incorporating this
The
stress
in the
solid
remains low, never above 0.3 MPa and typically about
0.1 MPa.
at
boundary
to predict the stress in the fluid and
solid phases of the cartilage.
matrix
final
The severe nonlinearity of the equilibrium modulus
approximately
30
percent
compression
(0.3 MPa
solid
stress) suggests that significant irreversible damage may be
occurring
to
the matrix.
Cyclic loading could fatigue the
fibers via buckling, such as the apparent compaction of
the
fibers in the STZ, as suggested by Tepic [139].
6.7
IMPLICATIONS
It
is c lea r the cartilage in
supporting
the
load
the
joint
functions
mainly by fluid press ure.
both through the cartilage layer and into
a nd
interarticul ar space is therefore important.
by
Fluid flow
through
the
Even under the
severe condi tio ns of a static load, the soli d stress in the
cartilage is mi nimal.
When the joint is unl oaded (as during
walking) flu id can freely flow
interarticul ar
resistance
is
the
into
likely
very small.
simulation, suc h as performed by Tepic [139] ,
the
since
c artilage,
Dynamic
incorporating
estimat es of interarticular resistance provided by this
thesis, are 1ik ely to produce very
good
estimates
of
the
solid stress in the cartilage throughout the walking cycle.
- 130 -
The underlying hypothesis for this work has
mechanical
which
factors
function in a normal
failure
will clearly result
The
range
of
important
example,
in
the
Los s of interarti cular sealing
n increased stress
in
the
cartilage.
seal ng coefficie nts and their dependence on
cartilage properties and conditio n need to
For
that
the cartilage
synovial joi nt may play a role
osteoar hritis.
by
are
been
norma 1
roughness --
does
experiments
on
it
the
adult
seal
be
established.
car tilage has a 1arger surface
les s
strength
physiological condit i ons such as
of
effectively ?
cartilage,
those
Meanwhile
under
more
in
this
desc ribed
thesis, would provid e ev dence that increased stress through
loss of sealing can lead to destruction of the cartilage.
- 131 -
APPENDIX
Programs
- 132 -
C
C
C
CSFA
C
C
COUPLED SOLID-FLUID ANALYSIS PROGRAM
C
C
TOM MACIRCWSKI
C
C
*
Adapted from: STAP in K.J. Bathe, Finite Element Procedures in Engineering
Analysis [9].
PROGRAM CS7A
REAL*4
COMMON
COMMON
COMMON
COMMON
COMMON
COMMON
EI.TA,REST
A(17500O)
/SOL/ NUMNP,NEO,NWK,NUMEST,MIDEST,MAXEST,MK
/DIM/ N1,N2,N3,N4,N
e,N7,N8,N,N9,NO,N11,N12,N13,N14,N15
/EL/ IND,NPAR(10),NUMEG,MTOT,NFIRST,NLAST,ITWO
/VAR/ NG,MODEX,RBEST,DELTA
/TAPES/ IELMNT,ILOAD,IIN,ICUT,IDISP,ISTATE,IRCRG,IKTRI
DIMENSION TIM(5),EEDf(20•
DIMENSION IA(1)
C
C
C
*
EOUIVALENCE tA(1),IA(1))
BYTE FF,IFILE(15),OFILE(15)
DATA FF/ 14/
DATA IFILE/D',',1 '',,
'E',
,
'M' 'F','6',
,
'2',
* . 'D"A"
S "'T'/0/
DATA CFILE/" ', K , 1, : , 0,
0
, ,'L''S , 'T 0/
MTOT IS TEi MAXIMUM CORE STORAGE AVAILABLE
MTOT=175000
ITWO=1
C
C
THE FOLLCWING DEVICE NUMBERS ARE USED
C
C
IELMNT = ELEMENT DATA
ILCAD = LOAD VECTORS
C
IIN
= INPUT DATA
C
IOUT
= OUTPUT LISTING
C
IDISP
C
C
= SURFACE DISPLACEMENTS
ISTATE = STATE VECTOR
IKORG = ORIGINAL STIFFNESS (K) MATRIX
C
IKTPI
= TRIANGULARIZED K MATRIX
I'IMNT = 11
ILOAD = 12
= 1IIN
IOUT
= 14
IDISP
ISTATE
IKORG
IKTRI
=
=
=
=
15
16
17
18
C
CALL ASSIGN lIELMNT,'DK1:IELMNT.DAT',14)
CALL ASSIGN (ILOAD,'DK1:ILOAD.DAT',13)
CALL ASSIGN (IIN,IFILE,14)
DO 2070 I=5,10
- 133 -
207e
C
200
OFILE(I )=IFILE(I)
CALL ASSIGN(ICUT,OFILE,14)
CALL ASSIGN (IDISP,'DK1:ILISP.DAT',13)
CALL ASSIGN (ISTATE,'DK1:ISTATE.DAT',14)
CALL ASSIGN (IKORG, DK1:IKORG.DAT',13)
CALL ASSIGN (IKTRI,'DK1:IKTRI.DAT',13)
NUTEST=O
MAXEST=O
C
C
C
C
C
C
C
C
C
C
C
C
PHA SE
~
I N P U T
C
C
C
C
*
CALL SECCNl(TIM(1))
READ
CCNTR OL
0
IN F
RMA
I
N
READ (IIN,1~00 E!ID,NUMNP,NUMIG,NLCASE,MOCDEX,REEST,ELTA
IF(NUMNP.EQ.O)GO TO 999
WRITE(IOUT,2e00) FF,HED,NUMNP,NUMEG,NLCASE,MODEX,REEST,DELTA
READ
NODAL
P 0 INT
DATA
N1=1
N2=NI+2*NUMNT
N3=N2+NUMN?*ITWO
N4=N3+NUMNP*ITWO
N5=N4+NUýNP*ITWO
IF(N5.GT.MTOT) CALL ERROR(N.-MTOT,1)
IF(N5.GT.MAX)MAX=N5
6000
C
TYPE 6000,4*MAX
FORMAT(' Maximum memory used is:',I8,' Bytes ')
CALL INPUT(A(N1),A(N2),A(N3),A(N4),NUMNP,NEQ)
C
C
C
C
C
NEQ1=NEQ+1
AND
CALCULATE
STO0
NE=N5+NEO*ITWO
IF(N6.GT.MTOT) CALL ERROR(N6-MTOT,2)
IF(Ne.GT.MAX) MAX=NE
TYPE 6000,4*'AX
WRITE(IOUT,2e05)
F?
C
REWIND ILOAD
C
DC 300 L=1,NICASE
IF(L.GT.1)WRITE(IOUT,2004)
FF
E
LOAD
VECT
RS
- 134 -
2004
FORMAT(1X,A1)
READ( IIN, 1010)LII,NLOAD,NPRES
WRITE IOUT,2010I
LL,NLOAD,NPRES
IF(]L.EO.I)GOTO 310
WRITE(IOUT,2020)
STOP
C
310
300
C
C
CONTINUI
CALL LOADS (A (N2
,A(N3,A(N5),A(N1),NLCAD,NPRES ,N
CONTINUE
READ
/
GENERATE /
ELEMENT
AND
STORE
DATA
CLEAR STORAGE
10
N6=N5+NEQ*ITWO
DO 10 I=N5,N6
A(I)=e
IND=1
C
C
CALL EL CAL
CALL SECOND(TIM(2))
TYPE 3000,(TIM(2)-TIM(1))
3000Q
FORMAT(/'
S*
DATA INPUT
*c S O L U T I O N
*****************************
ASSEMELE
:', 12.2 " Sec')
PHASE
* *
************
STIFFNESS
CALL ADDRES(A(N2),A(N5))
MM=NWK/NEQ
NZ=N2+NEQ+1
N4=N3+NWK*I TWO
NS=N4+NEQ*I TWO
NE=N5+MAXEST
N7=NE+NEQ*ITWO
IF(N7.GT.VTOT) CALL ERROR (N7-MTOT,4)
IF(N7.GT .MAX)MAX=N7
TYPE 60ee,4*MAX
WRITE TOTAL SYSTEM DATA
WR ITE(IOUT ,225 ) FF,NEQ ,NWK,MK,MM
C
*
IF DATA CHECK SKIP FURTHER CALCULATIONS
MATRIX
,NUMNP)
- 135 -
IF(MODEX.GT.0e
GO TO 100
CALL SECOND(TIM(3))
CALL SECCND(TIM(4))
CALL SECOND(TIM(5))
GO TO 120
C
C
C
100
CLEAR STCRAGE
NNL=NWK+NEQ
CALL CLEAR(A(N3),NNL)
C
C
INE=2
CALL ASSEM(A(N5))
C
READ(IIN,1005) NSUP
oe1005 FCRMAT(IS)
NSIZE=NSUR*(NSUR+1)/2
NE=N7+NSUR
Nc=N8+NSI ZE*ITWO
N10=N9+NSUR+1
N11=N1O+NSUR*ITWO
IF(N11.GT.ETCT) CALL .RROR(N11-MTOT,5)
IF(N11.GT.MAX)MAX=N11
TYPE 6000,4*MAX
CALL DISPL(A(N7),A(N10),NSUR,NLCASE')
C
3010
CALL SECONr(TIM(3))
TYPE 3010,(TIM(3)-TIM(1))
FORMAT(' ASSEMPLE MATRIX
', F12.2)
C
TR I AN GULARI Z E
ST I FFNESS
MATRIX
KTR=1
WRITE (IKORG) NWK, (A(IO), IOQ=N3, N3-1+NWK*ITWO )
CALL COLSOL (A(N3 ),A(N4),A(N2) ,NEQ,NWK,NEQ1 ,KTR)
WRITE(IKTRI) NWI , (A(IQ),IQ=N3, N3-1+NWK ITWO )
35
30 20
C
CALL SECOND(TIM(4))
TYPE Z020,(TIM(4)-TIM(1))
FORMAT(' TRIANGULARIZATION
:' ,F12.2)
KTR=2
IND=3
C
CALL rASSEM (A(N1),A(N7'",A(NE),A(N9',A(N4).,NSURNLCA-E)
C
CALL CIEAR(A(N6),NEQ)
REWIND ILOAD
REWIND IDISP
LO 400 L=1 ,NICASE
CALL SAVE(A(N4),A(N6),NQFQ
CALL LOALV(A(N4),NEQ)
SAVE U(t) in
R(t+dt)
A(N6)
in A(N41
- 136 -
CALL DISPV(A(N10),NSUR)
! Us(t+dt) in A(N10)
REWINr IKCRG
READ(IKORG) NWK,(A(IQ) ,IC=NZ,
N3-1+NWK*ITWO)
CALL UPDATE(A(N4),A(N6),A(N1) ,A(N2),A(N3),NUMNP)! ADD LTxU(t)
CALL SAVF(A(N4),A(NE),NEQ)
I SAVE R(t) in A(NE)
CAL CUL
ATION
CF
D I S PL A CE MENT S
REWIND IKTRI
READ(IKTR)l NWK, (A(IQ), IO=N3, N3%-1NWK*ITWO)
CALL COLSCL(A(N3 ),A(N4) ,A(N2) ,NEQ,NWK,NEQ1,KTR)
CALL FLCW(A(N4),A(N10),A(N1),. A(N8),A(N9),A(N7),NSUR,A(Ne))
CALL COLSOL(A(N3),A(N6),A(N2) ,NEQ,NWK,NEQ1,KTR)
WRITE (ICUT,2015) FF,L
CALL WRITE(A(Ne),A(NI),NEQ,NUIMNP,A(N1•),A(N7),NSUR)
CALCULATI
N
CF
STRESSES
CALL STRESS(A(N5S)
CONTINUE
C
CALL SECOND(TIM(5))
TYPE 3030,(TIM(5)-TIM(1))
FORMAT(' LOAD SOLUTIONS
:
3030
C
C
C
120
,F12.2)
PRINT SOLUTICN TIMES
TT=0.
DO 500 I=1,4
TIM(I)=TIM(I+1)-TIM(l)
TT=TT+TIM(I)
WRITE(ICUT,2e3e) fF,HlD,(TIM(I),I=1,4),TT,FF
500
C
READ NEXT ANALYSIS CASE
GO TO 200
1010e
10,10
FORMAT(20A4/415,2F10.0)
FORMAT(3I5)
2000
FCMAT(1X,A1 ,20A4///
CONTROl
INFORM
ATI 0 N '//5X
2' NUMBER CF NODAL POINTS
15//5X
(NUMNP)
3, NUMBER OF ELE M ENT GROUPS
(NUMEG)
I5//5X
4' NUMBER OF LOAD CASES .......
(NLCASE)
5' SOLUTICN MODE ..............
(MODEX)
I5/5X,
6'
EQ.0, DATA CFECK'/5XX,
7'
EQ.1, EXECUTION '//5X, · · ·
8' RADIUS .......................
......
(RPEST)
,F9.3//5X,
9' TIME STEP ..........................
(DEITA) = ,Fg9./)
FORMAT(1X,A1,
20 05
L 0 A D
C A S F
D A T A ')
1' ,
- 137 -
2010
2015
2020
2025
203 e
S=',//i4X,
FORMAT(////4X,' LOAD CASE NUMBER
1'
NUMBER OF CONCENTRATED LOADS .....
2'
NUMBER CF PPESSURE LOADS .........
15//4X,
=', I5/)
=',
FOPMAT(1X,A1,' LOAD CASE ',I
LOAD CASES ARE NOT IN ORDER
FORMAT(' *** ERROR
FOPMAT(IX,A1,
TOTAL SYSTE M DATA '///5X,
NUMBER OF EQUATIONS
(NEQ)
NUMBER CF VATRIX EMNTS....(NWK)
(MK)
MAXIMUM HALF ANDWIDTH ......
MEAN HALF BANDWIDTH
(MM)
FOPMAT(IX,A1,' S 0 L U T I CN T I M E L
*
SECONDS'//
112X,'FOR PROBLEM'//1X,20A4 ////SX,
2' TIME FCR INPUT PHASE
3' TIME FOR CALCULATION OF STIFFNESS MATRIX
4' TRIANGULARIZATION OF STIFFNESS MATRIX ......
5' TIME FOR LOAD CASE SOLUTIONS ...............
6' TOTAL
SOLUTION
TI VE ....
C
999
CALL
CALL
CALL
CALL
CALL
CALL
CALL
CALL
END
CLCS~ I(
IMNT)
CLOSE (ILOAD)
CLCSE(IIN)
CLOSE(IOUT)
CLOSE(IDISP)
CLOSE(ISTATE)
CLOSE (IORG)
CLOSE(IKTRI)
,I8//5X,
,18//NX,
0 G
,I
1)
I N
',F12
',F12
,12
,F12
.2//5X,
.2//5X,
.2//5X,
.2//5X,
.2/1X,A1)
- 138 C *******************************************************
*C
INPUT -- REAr AN• PRINT NODAL POINT INPUT DATA
-- CALCULATE AND STORE EQUATION NUMBERS
SUEROUTINE INPUT(ID,P,T,R,NUMNP,NEQ)
I'PLICIT RIAL*8 (A-H,O-Z)
COMMON /TAPES/ IEIMNT,ILOAD,IIN,IOUT,IDISP,I STA TE, IKORG, IKTRI
COMMON /ONED/NUMET,DELTA
DIMENSION P(1),T(1),R(1),IE(2,NUMNP)
BYTE F?
DATA FF/"14/
C
C
1000
10
C
C
C
C
C200
C
READ NODAL PCINT DATA
DO 10 IQ=1,NUMNP
READ(IIN,1000 NvID(1,N) ,ID(2,N),P(N),T(N),R(N)
FORMAT(315,3E15.5)
CONTINUE
WRITE(IOUT ,200 0) FF
WRITE(I CUT ,2015)
WRITE(IOUT ,202020)
DO 200 N=1 ,NTMNP
WRITE(IOUT ,20'30 N,(ID(I,N),I=1,2),P(N),T(N),R(N)
NUMBER UNKNOWNS
120
110
100
C
C
C
NEQ=O
DO 100 N=1,NUMNP
DO 100 I=1,2
IF(ID(I,N)) 110,120,110
NEO=NEQ+1
ID(I ,N)=NEQ
GO TO 100
ID(I,N)=0
CONTINUE
WRITE EQUATICN NUMBERS
WRITE(IOUT,2040) FF,(N,(ID(I, N),I=1,2),N=1,NUMNP)
RETURN
2000
2015
2020
2030
2040
FOPMAT(1X,A1,'N 0 D A L
P 0 I NT
D • T A '//
FORMAT(' GENERATED NODAL DATA'//)
FORMAT(' NODE ',c-,'EOUNDARY',24X,'NODAI POINT',17X,
/' NUMBER',5X,'CONDITION CODES',21X,'COORDINATES'
//15X, X
F',30X,' PHI',PX,'THETA',8X, 'RADIUS')
FORMAT(I5,EX,2I5,22X,3F13 .3)
FORMAT(1X,A1' EQUATION NUMBERS'//,4X,'NODE',9X,
'DEGREE CF FFEEDOM'/3X,'NUMBER'//,
'
N',13X,'X
P'/(1X,I5,9X,215))
ENE
- 139 -
LOADS ---
READ NODAL POINT DATA
CALCULATE LOAD VFCTOR R FOR EACH LOAD CASE AND
WRITE TO DISC FILE ILOAD
C
SUROUTINY LCADS (PH,TH,R,ID,NLCAD,NPRES,N
EQ,NUMNP)
COMMON /VAR/ NG,MODEX,RBEST,DELTA
COMMON /TAPES/
I!EMNT,ILOAD,IIN,IOUT,IDISP,ISTATE,IKORG,IKTRI
DIMENSION PHNUMNP(NUMNP'vTF(NUMNP),R(NEQ)
DIMENSION ID(2,NUMNP),NODPR(4),PRESS(4),PP(4),TP(4).RP(4)
DC 210 I=1,NEQ
210
R(I)=0.
IF(NLCAD.EQ.e)GO TO 100
WRITE(IOUT ,2000
C
C
C
IF(MODFX.EQ.0)RFTURN
IF(NLOAD.EQ.0)GO TO 100
C
DC 220 L=1,NIOAD
READ(IIN,1000) NOD,IDIRN,FIOAD
WRITE(IOUT,2e10) NOD, IDIRN,FLOAr
LN=NOD
LI=IDIRN
II=ID(LI,LN)
IF(II) 220,220,240
240
C
220
C
C
C
100
C
R(II)=R(II)+FLOAD
C
WRITE(IOUT,2020)
CONTINUE
PRESSURE LOADS
IF(NPRES.EQ.-)GO TO 300
RADIAN=57.2c577951
DO 25e L=1,NPPES
READ(IIN,1010)
IEI, (NODPR(I),I=1,4),(PRESS(I),I=1,4)
DO 260 K=1,4
260
PP(K)=PH(NODPP(K))/ RADIAN
TP(K)=TH(NODPR(K' )/ RADIAN
CALL PLOAD(PP,TP,PRESS,RP)
WRITE(IOUT,2040)
DO 270 K=1,4
LI=1
LN=NOEPR(K)
WRITE(IOUT,203Qe)
NODPR(K),LI,RP(K),PRESS(K)
II=ID(LI,LN)
IF(II.GT.0) R(II )=R(II)+RP(K)
- 140 -
270
250
CONTINUE
CCNTINUE
300
C
WRITE(ILOAD)R
200
CONTINUE
C
FORMAT(2I5 ,Fie.0)
FOPMAT( 515,4?10 .1)
FORMAT(////'
NODE
rIRECTION
LOAD '/
1'
NUMBER',I9X,' T'AGNITUE' )
2010
FOPMAT(1X ,!,9X,I4,7X,E12.
2020
FORMAT(////'
NODE
DIRECTICN
LOAD
1
PRESSURE
"/
2'
NUMBER',19X,' MAGNITUDE
MAGNITUDE
2030
FORMAT( X,I ,9X,I4,2E19.5)
2040
FORMAT(/)
RETUPN
1000
1010
2000
END
- 141 -
C
C
ELEMNT --
CAIL THE APPROPRIATE ELEMENT SUBROUTINE
C
*
C
SUEROUTINE ELEMNT
COMMON A(1)
COMMON /EL/ IND,NPAR(10),NUMEG,MTOT,NFIRST,NLAST,ITWO
C
NPAR1=NPAR (1)
C
C
1
C
2
3
4
5
C
GO TO (1,2,3,4,5),NPAR1
RETURN
RETURN
CALL THREED
RETURN
RETURN
RETURN
END
- 142 C
•
•
•~~~s~~~~~~~~~*****
PLCAD --
CALCULATE CCNSISTENT PRESSURE LOADING
C
C*****************************************clC~Frr~:~~~~C
SUEROUTINE PLCAD (PHI,THETA,PR!SS,RP)
COMMON /SOL/ NUMNP,NEO,NWK,NUMEST,MIDEST,MAXEST,MK
CCMMON /VAR/ NG,MODEX,R]EST,DELTA
COMMON /DIM/ N1,N2,N3,N4,N5,N6,N7,N&,N9,N10,N11,N12,N13,N14,N15
COMMON /EL/ IND,NPAR(10),NUMEG,MTOT,NFIRST,NLAST,ITWO
COMMON /TAPES/ IELMNT,ILOAD,IIN,ICUT,IDISP,ISTATY,IKCRG,IKTRI
DIMENSIOCN PHI(1),THETA(1),PRESS(1),RP(1),H(4)
DIMENSION XX(2,4),XG(4,4),WGT(4,4)
DO 600 J=1,4
600
XX(2,J)=PHI(J)
XX(1,J)=THETA(J)
CONTINUE
NINT=2
XG STORES G-L SAMPLING POINTS
DATA XG /
0.,
-. 577350269189E,
0.,
.5773502691895,
-. 77459692415,
•
-. 8611363115941 ,-.3399810435849,
0e,
0.,
0.,
0.861363
.7745966692415,
.3399810435849, .S611363115941/
WGT STCPES G-L WEIGBTING FACTCRS
DATA WGT / 2.00,
1 .0000000000000
. 5555555555556, .P868888888889
.6521451548E25
.3478548451375,
1 .00000009000,
0.,
0.,
e0.,
0.,
.5555555555556,
0.,
.6521451548625, .3478548451375/
DC 30 J=1,4
RP(J)=0.0
DO 80 LX=1,NINT
RI=XG(LX,NINT)
DO 80 LY=1,NINT
SI=XG(LY,NINT)
EVALUATE DERIVATIVE H AND JACOBIAN DET
CALL DM(XX,B,DET,RI,SI)
WT=WGT(LX,NINT)*WGT(LY,NINT)*DET
DC 40 K=1,4
DO 40 L=1,4
RP(K)=RP(K)+H(K)*B(L)*PRESS(L)*WT
CONTINUE
RETURN
END
- 143 -
DM --
EVALUATE THE MATRIX H
AT PCINT (R,S) FOR A QUAD ELEMENT
c* *
******
** * ** *4*44*4*4*
*****
****
****
* ** ***
***
***
SU-ROUTINE DM(XX,E,DET,R,S)
CCMMON /VAR/ NG,MCDEX,RBEST,DELTA
COMMON /TAPES/ IELMNT,ILOAD,IIN,IOUT,IDI
IMENSION XX(2,1) ,H(4),P(J(,4),XJ2,2),XJ
RP
SP
RM
SM
=
=
=
=
1. +
1. +
1. 1. -
****
*****
':
* ** *4 * *4* ** * * *
SP,ISTATE, IKORG,IKTRI
I(2,2)
R
S
R
S
INTERPOLATION FUNCTIONS
)=e.25*RP*SP
)=0.25*PM*SM
)=k .25*RP*SM
NATURAL COORDINATE DERIVATIVES W.R.T.
INTERPOLATION FUNCTIONS
1. W.R.T
1,1)=e.25*SP
1,2)=-PF( 1,11)
1,3)=-0.25*SM
1,4 )=-F \1i,.
2.
W.R.T
P(2,1)=0.25*RP
P(2,2)=0.25*RM
P(2,3)=-P(2,2)
P(2,4)=-P(2,1)
EVALUATE JACCBIAN MATRIX AT (R,S)
DO 30 1=1,2
DO 30 J=1,2
DUM=0.0
20
30
C
C
DO 20 K=1,4
DUM=DUM+P(I,K)XX (J,K)
XJ(I,J)=DUM
COMPUTE DETERMINATE OF JACOBIAN MATRIX AT (R,S)
DET=XJ(1,1)*XJ(2,2)-XJ(2,1)*XJ(1,2)
01) GO TO 4e
IF(DET.GT..0.0000e
WRITE(IOUT,2000)
STOP
COMPUTE INVERSE OF JACOBIAN MATRIX
-
C
40
50
2000
144 -
DUM=1./DIT
XJI (1,1 )=XJ(2,2)*DUM
XJI(1,2)=-XJ(1,2)*DUM
XJI(2,1)=-XJ(2,1)*DUM
XJI (2,2)=XJ (1,1 )*DUM
THETA=e .e
DO 50 I =1,4
THETA=TRETA+H(IQ)*XX(1,IQ)
]ET=DET*RBEST*RBEST*SIN(THETA)
FOPMAT(' NEGATIVE OR ZERO JACOBIAN IN PRESSURE LOADING ')
RETURN
END
- 145 -
C
C
C
C
ELCAL -- LOOP OVER ELEMENT GPCUPS FOR READING,
GENERATING, AND STORING THE ELEMENT DATA
C
SUBROUTINE ELCAL
COMMON /SOL/ NUMNP,NEQ,NWK,NUMEST,MIDEST,MAXEST,MK
COMMON /EL/ IND,NPAR(10),NUMEG,MTOT,NFIRST,NLAST,ITWO
COMMON /TAPES/ IEIMNT,ILOAL,IIN,IOUT,IDISP,ISTATE,IKORG,IKTRI
COMMON A(1)
BYTE FF
EATA FF/ 14/
C
C
REWIND IELMNT
WRITE(ICUT,200'0) FF
C
C
C
LOOP OVER ALL ELEMENTS
DO 100 N=1,NUMEG
C
READ(IIN,1000WNPAR
C
CALL ELEMNT
C
IF(MIDEST.GT.MAXEST)MAXEST=MIDEST
C
WRITE (IELMNT)MIDEST,NPAR,(A(I),I=NFIRST,NIAST-1)
C
C
100
C
CONTINUE
RETURN
C
1000
2000
2010
C
FORMAT(1015)
FORMAT(1X,A1,' E L F M E N T
FORMAT(1X,A1)
ENr
GR 0 U P
D A T A '///)
- 146 C
C
THREED --
SET UP STORAGE FOR PCRCUS SUERCUTIN*
C
SUBROUTINE THREED
CCMMON /SOL/ NUMNP,NEQ,NWK,NUMEST,MIDEST,MAXEST,MK
COMMON /DIM/ Ni,N2,N3,N4,N5,Ne,N7,NS,N9,N10,N11,N12,N13,N14,N15
COMMON /EL/ IND,NPAR(10),NUMEG,MTOT,NFIRSTNLAS,
T,ITWO
COMMON /TAPFS/
COMMON A(1)
IEIMNT,ILOAD,IIN,ICUT,IDISP,ISTATEIKCRG,IKTRI
DIMENSION IA(ll
c
EQUIVALENCE (NPAR(2),NUMY),(NPAR(3),NUMMAT)
EQUIVALENCE (A(1),IA(1))
C
NFIRST=NE
IF(IND.GT.1)NFIRST=N5
N101=NFIRST
N102=N101+NUMMAT*ITWO
N103=N102+NUtMATT*ITWO
N104=N102+2 *NUM E
N105=N104+36*NUME*ITWO
N106=N105+NUME
NLAST=N1o~
C
IF(IND.GT.1)GO TO 100
IF(NLAST.GT.MTOT) CALL ERROR(NLAST-MTOT,3)
GO TO 200
100
C
200
C
IF(NLAST.GT.rTOT) CALL ERROR(NLAST-MTOT,4)
MIDEST=NIAST-NFIRST
*
CALL POROUS(A(N1),A(N2),A(N%),A(N4),A(N4),A(N5),
A(N1•1),A(N102),A(N103),A(N104),A(N105))
C
RETURN
C
ENr
- 147 -
POROUS -- POROUS ELEM(ENT SUER CUTI NE
C~~~~8~~~~~~~~~8~8~~~~~4~~~~~~~8~~~~~~~~
SUBROUTINE POROUS (ID,P,T,R,U,MHT,E,PERM,LM,XYZ,MATP)
IMPLICIT REAL*8 (A-H.0-Z)
COMMON
COMMON
COMMON
COMMON
COMMON
/SOL/ NUMNP,NEQ,NWK,NUPEST,MIDEST,MAXEST,MK
/rIM/ N1,N2,N3,N4,,N5,N67,NS,N8,N9,N10,N11,N12,N14,N15
/fL/ IND,NPAR(10),NUMEG,MTOT,NFIRST,NLAST,ITWC
/VAR/ NG,MODEX,RBEST,DELTA
/TAPES/ IELYNT,ILOAD,IIN,ICUT,IDISP,ISTATF,IYORG,IKTRI
COMMON A(1)
REAL*4 A
DIMENSION P (1) ,T(1),R(1) ,ID(2,1),E(1),PERM (1),LM(20,1)
DIMENSION X'YZ(36,1),U(1) ,MET(1),MATP(1)
DIMENSION S1
U(210),NOD(12 S,SUPL(304)
DIMENSION X:X(3,12),E (12) ,HP(8) ,G(3,8) ,XG(4
REAL*4 KU(1I 2,12),KP(e,8) ,KI(12,8)
,4),WGT(4,4)
EQUIVALENCE (NPAR (1),NPAR1),(NPAR (2),NUME),(NPAR(3 ),NUMMAT)
EQUIVALENCE (KUl1 ,1),SUPL(1)),(KP(1,1),SUPL(145) )
EQUIVALENCE (KL(1 ,1),SUPL(209))
BYTE FF
DATA FF/"14/
XG STORES G-1 SAMPLING POINTS
0.
DATA XG /
0.,
0.
-.5773502691896, .5773502691896,
-. 7745966692415,
0., .774596EE92415
-.8611363115941. -. 3399810435849, .3399810435849
0.,
.8611363115941/
WGT STORES G-L WEIGHTING FACTORS
DATA WGT / 2.00
.5555555555556
.3478548451375
0.
.0000000000000,
0.
.8888888888889, .5555555555556
.6521451548625, .6521451548625
0.,
0.,
.,4845137
.3478548451375/
ND=20
RADIAN=57.29577951
NINTX=3
NINTY=3
NINTZ=3
GO TO (300,610,900),IND
G E NERATE
E I E ME N T
IN
FO RM AT I
0 N
- 148 -
MATERIAL INFO
ITE(ICUT,2200)NPAP1,NUME
(NUMMAT.EQ.0) NUMMAT=l
ITE(ICUT,201e
)NUMMAT
WRITE (IOUT,2020)
DO 10 I=1,NUtMAT
READ(IIN,1000)N,E(N) ,PERM(N)
WRITE (IOUT,2030)N,E (N),PERM(N)
READ ELE M ENT INFO
C
C
WRITE (IOUT,24e0) FF
DO 30 NEL=1,NUME
READ(IIN,1030)M,MTYPE,(NOD( IQ) ,IQ=1,12)
1030
FORMAT(14I5)
MATP(NEL)=MTYPE
WRITE (ICUT,2P50)M,(NOD(IQ) ,IQ=1,12),MTYPE
C
150
C
380
390
DO 150 I=13,36,3
XYZ(I-2,NEL)=T(NOD(I
)/RADIAN
XY2(I-1,NEL)=P(NOD(I /3) )/RADIAN
/3) )
XYZ(I-0,NEL)=R(NOD(I /3)
DC 380 L=1,12
LM(L,NEL)=ID(1,NOD(L))
DO 39e L=13,20
LM(L,NEL)=ID(2,NOD(L-1 2))
UPEATE COLUMN HEIGETS AND BANrWIDTH
CALL COLHT(ý M T,ND,LM(1,NEL))
CONTINUE
RETURN
C
C
C
C
610
C
505
c
ASSEMBLE
ST I F
DO 500 NEL=1,NUME
MTYPE=MATP (NFL)
YM=E(MTYPE)
PF=PERM(MTYPE)
DC 505 IQ=1,210
SU(IQ)=0.0
DC 600 J=1,12
600
C
DO 600 I=1,3
INDEX=3*(J-1)+I
XX(I ,J)=XYZ(INDEX,NEL)
NES S
MATRIX
- 149 -
DO 601 10=1,304
601
C
suPp(IO)=0.0
DO 80 LX=1,NINTX
RI=XG(LX,NINTX)
DO 80 LY=1,NINTY
SI=XG(LY,NINTY)
DO 80 LZ=1,NINTZ
TI=XG(LZ,NINTZ)
CALL STDPM (XX,B,EP,G,DET,RI,SI,TI,NEL)
WT=WGT(LX ,NINTX)*WGT(LY,NINTY)*WGT(LZ,
DO 40 IR=1,12
DO 40 JC=1,12
KU(IR,JC)=KU(IR ,JC)+WT*TM*B(IR)*P(JC)
DO 45 IR=1,8
DO 45 JC=1,8
SUM=0 .0
DC 46 IK=1,3
46
45
SUM=SUM-G(IK
,IR
)*G(IK,JC )*PF-FELTA
KP(IP ,JC)=KP(lR ,JC)+WT*SUM
DO 50 IR=1,12
DO 50 JC=1,8
KL(IR,JC)=KL(IR ,JC)+WT*p
CONTINUE
75
C
(IR)*HP(JC)
INDEX=1
DO 60 IR=1,12
DO 65 JC=IR,12
SU(INDEX)=KU(IR,JC)
INDEX=INDEX+1
CCNTINUE
DO 70 JC=1,8
SU(INDEX)=KL(IR,JC)
INEEX=INDEX+1
CONTINUE
CONTINUE
DC 75 IR=1,8
DO 75 JC=IR,e
SU(INDEX)=KP(IR,JC)
INDEX=INDEX+1
CONTINUE
IF (NFL.NE.1)
GOTO 4000
C
3010
3020
3025
3040
WRITE(IOUT,3000) "14
FCRMAT(1X,AI,///' KU MAT RIX '/)
DO 3020 I=1,12
WRITE(IOUT,3010) (KU(I,J ),J=1,12)
FOPMAT(1X,12F10.5)
CONTINUE
WRITE(IOUT,3025)
FCRMAT(//' KI MATRIX /)
DO 303~ I=1,12
(KL(I,J ),J=1 ,8)
WRITE(IOUT,3040)
FORMAT(1X,F1O0.5)
NINTZ)*DET
- 150 -
3030
CONTINUE
WRITE(IOUT ,3050)
FORMAT(//' KP MATRIX '/)
DO 3060 I=1,8
WRITE(ICUT ,370e) (KP(I,J),J=1,8)
FORMAT(1X,SF10.5)
CONTINUE
WRITE(IOUT,308e)
FORMAT(//' SU MATRIX '/)
3050
3070
3060
C
C30E0
C
C
0310
C3090
C
4000
C
C
500
DO 3090 IQ=1,INDEX-1
WRITE (ICUT ,310 )IC,SU (IQ)
FORMAT( Ile ,F10.5)
CONTINUE
CALL ADDBAN(A(N3),A(N2),SU,LM(1,NEL),ND)
CCNTINUE
RETURN
STRESS
C
C
900
C ALCULATI 0 NS
IPRINT=O
DC 830 N=I,NUME
IPRINT=IPRI NT+
IF(IPRINT.GT.50) IP RINT=1
IF(IPRINT.EQ.1)
WRITE(IOUT,2090)NG
MTYPE=MATP(N)
STR=0.
I=LM(L,N)
WRITE(ICUT,2070)N, P,STR
CONTINUE
RETURN
1000
1010
1020
2000
FCPMAT( 15,2F1I.0)
FORMAT(2F10.0)
FORMAT( 51)
FORMAT(' E LE M E N T
D E F I NI
1'
ELEMENT TYPE ',13(' .'), ( NPAR(1 ) )
2'
EC.1, ONE-D FLEMENTS'/,
T I o N '///,
. . =-,15/,
EQ.2, TWO-D ELEMENTS'/,
1/'
GE.3, ELEMENTS CURRENTLY NOT AVAILABLE'/,
4'
FORMAT(' M A T E R IA L
2010
1'
2020
NPAR(2) ) . . =',15//
D E F I N ITI
0 N "///,
.'),'(
NUMBER CF DIFFERENT SETS OF MATER IAL CONSTANTS'/,
3 4 (
.'),
(
NPAR(3)
)
.
. =',IB
//)
FORMAT(///' SET
UNIAXIAL
1' NUMBER
MODULUS',/
2'
2030
2040
NUMBER CF ELEMENTS.' ,10('
Kf
PERMEABILITY'/
')
FORMAT(/I5, 4X,E12.5,3 X,E12.5)
FORMAT(1X,A 1,' E L E MENT
I N F 0 RMA TI 0 N '///,
ELEMENT
NODE
NODE
NODE
NODE
NODE
NODE
NODE
NODE
NOD E
NODE
NODE
NODE
MATERIAL -/,
NUMBER-N
1
2
3
4
5
6
7
- 151 -
*
SET NUMBER'/)
11
12
9
1,
FORMAT(15,8X,12(15,2X),5X,I5)
C A I C U L A T I 0 N S S0 p
FCDMAT(////' S T R E S S
G R 0 U P',I4//,2X,
I. E M E N T
1'
2' ELEMENT
FORCE
STRESS '/,3X,'N UMEER'/)
2070
FCRMAT(1X,I5,11X,113.6,4X,E13 .6)
2050
2060
C
INI
'
- 152 -
C
C
C
C
C
STDPM --
EVALUATE THE STRAIN-DISPLACEMENT-PRESSURE
MATRICES E,N,G AT POINT (R,S,T) FOR A
PCRCUS ELEMENT
C
C
SUEROUTINE STDPM(XX,B,HP,G,DET,R,S,T,NEL)
IMPLICIT REAI*8 (A-E,O-Z)
CCMMON /TAPES/ IELMNT,ILOA,IINO,IOUT,IDPISPISTATE,IKORG,IKTRI
DIMENSION XX(3,1),P(1),HP(1),G(3,1),H(12),P(3,12),PP(3,8)
DIMENSION XJ(3,3),XJI(3,3),AA(3,3)
C
RP
SP
TP
RM
SM
TM
=
=
=
=
=
=
1.
1.
1.
1.
1.
1.
+
+
+
-
R
S
T
R
S
T
C
RR = 1. - R * R
SS = 1. - S
S
TT = 1. - T * T
C
C
C
INTERPOLATION FUNCTIONS
H(1)=0.125*RP*SP*TP
H(2)=e.125*RRV*SP*TP
H(3)=e.125*RP*SM*TP
H(4)=0.125*RP*SM*TP
C
H(5)=0.125*RP*SP*TM
H(6)=0.125*RM*SP*TM
H(7)=e.125*RV*SM*TM
HR()=e.125*RP*SM*TM
C
H(9)=e.25*RP*SP*TT
H(10)=0.25*Rý*SP*TT
H(11)=0.25*RM*SM*TT
H(12)=e.25*RP*SM*TT
C
C
C
C
NATURAL COORDINATE DERIVATIVES W.R.T.
1. W.R.T
C
R
C
P(i,1)=0.125*'P*TP
P(1,2)=-P(1,1)
P (1,3)=-
.125*SM*TP
P(1,4)=-P(1,3)
C
P(1,5)=2.125*SP*TM
P(1,6)=-P (1,5)
P(1,7)=-e.125*SM*TM
P(1,8)=-P(1,7)
INTERPOLATION FUNCTIONS
- 153 -
P (1,9)=0.25*SP*TT
P(1,10)=-P( 1,9)
P(1,11)=-0.25*SM*TT
P(1,12)=-P(1,11)
C
C
C
2. W.R.T
S
P(2,1)=0.125*RP*TP
P(2,2)=0.125*PM*TP
P(2,3)=-P(2,2)
P(2,4)=-P(2,1)
C
P(2,5)=0.125*RP*TM
P(2,e)=0.125*RM*TM
P(2,7)=-P(%2,6)
P(2,8)=-P(2,5)
C
P(2,9)=0.25*RP*TT
P(2,1e)=e.25*RM*TT
P(2,11)=-P(2,10)
P(2,12)=-P(2,9)
C
C
C
3. W.P.T
T
P(3, 1)=e.125*RP*SP
P(3,2)=0.125*RM*SP
P(3,3)=0.125*RM*SM
P(Z,4)=0.125*RP*SM
C
P(3,5)=-2.125*RP*SP
P(3,7)=-0.125*RM*SM
C
P(3,9)=-.25*RP*SP*T
P(3,10) =-0.5*RM*SP*T
P(,I1I)=-e.5*RM*SM*T
P(3,12)=-0.5*RP*St•*T
C
C
C
---------------------------------
DO 100 IoQ=1,
HP(IQ)=H(IO)
DO 10e IP=1,3
100
C
PP(IP,IQ)=P(IP, IQ)
CONTINUE
DO 11
10=1,4
H(IQ)=H(IQ)-H(IQ+E)/2.
)-H (Q+)/2.
H(IQ+4)=H ((IQ+4
DO Iie IP=1,3
P(IP,IQ)=P(IP,IO)-P(IP,IQ+8)/2.
P (IP,IQ+4)=P( IP ,IQ+4)-P(IP,IQ+
110
C
C
C
CONTINUE
C
EVALUATE
)/2.
JACOBIAN MATRIX AT (R,S,T)
- 154 -
20
DO 30 I=1,3
EC 30 J=1,3
DUM=0.0
DO 20 K=1,12
DUM=DUM+?( I ,K)*XX(J
XJ(I ,J)=DUM
30
C
C
,K)
CONTINUE
CCMPUTE DETERMINATE CF JACOBIAN MATRIX AT (R,S,T)
AA(1,1)=+(XJ(2,2)*XJ
AA(1,2)=-(XJ(2,1)*XJ
AA(1,3)=+(XJ(2,1)*XJ
,7)-XJ(
,3)-XJ(
,2) -xJ
)*XJ
(2,3)
)*XJ (2,3)
)*xJ (2,21
DE'=0.
DO 25 I=1,3
DET=DET+XJ(1, I)*AA(1 ,I)
IF(DET.GT .0.0000001) GO TO 40
WRITE(IOUT,2000) NEL
STOP
AA(2,1)=-(XJ(1,2)*XJ (3,3)-XJ (3,2)*XJ(1,3)
AA(2,2)=+(XJ(1,1 )*XJ (3,3)-XJ (3,1 )*XJ(1,3)
AA(2,3)=-(XJ(1,1)*XJ (3,2)-XJ (3,1)*XJ(1,2)
AA(3,1)=+(XJ(1,?)*XJ (2,3)-XJ
(2,2)*XJ(1,3)
(2,1)*XJ(1,3)
(2,3)-XJ
AA(3,2)=-(X J(1,1)*XJ
(2,1)*XJ(1,2)
(2,2)-XJ
AA(3.,3)=+(XJ(191)*XJ
COMPUTE INVERSE OF JACOBIAN MATRIX
DUM=1./DET
rO 35 IROW=1,3
DO 35 JCOI=1,3
XJI( IROW,JCOL)'=AA(JCOL, IROW)*DUM
EVALUATE GICEAL rERIVATIVE OPERATOR
DO 50 K=1,12
B(K)=e.
DO 50 I=1,3
B(K)=B(K)+XJI(3, I)*P(I,K)
0o
75
DO 60 K=1,8
DC e0 J=1,3
G(J,K)=0.0
DO 60 I=1,3
G(J,K)=G(J,K)+XJI (JI)*PP(I K)
THITA=0.0
RAD=e .0
DO 75 IQ=1,12
THFTA=THETA+H (IQ)*XX( ,IQ)
RAE=RAD+H(IQ)*XX(3,IQ)
ST=SIN(THETA)
DET=DETRA D*RAD*ST
D1=1 ./RAD
- 155 -
D2=D1/ST
DO 8? K= ',8
G(1,K)=G(1,K)*'D
G(2,K)=G(2,K)*D2
CONTINUE
RETURN
2000
FORMAT(///' *** ERROR , ZERO OR NEGATIVE JACOPIAN FOR
1ELEMENT (',14,') *** '//)
C
ENr
- 156 -
C
CCLHT --
CALCULATE COLUMN HEIGETS
~~''''~'''""
C8~~~Q~~t~~l~~~t~~~~~~~~~f~~~~~~~l~~t~~~
C
SUEROUTINE CCLHT(Et'T,ND, L)
COMMON /SOL/ NUMNP,NEQ,NWK,NUMEST,MkDEST,MAXEST,MK
rIMENSION IM(1),MET(1)
C
LS=10000
DO 100 I=1,ND
IE(LM(I))10,1•0,I10
110
120
100
C
200
C
IF(LM(I)-LS)120,100,100
LS=LM( I)
CONTINUE
DO 200 I=1,ND
II=LM(I)
IF(II.EC.O)GC TO 200
ME=II-LS
IF(ME.GT.MHTVI I ) )MET !(II)=ME
CONTINUE
RETURN
ENE
- 157 -
C
C
*
ADMES -- CALCULATE ADDRESSES OF DIAGONAL ELEMENTS IN
BANDED MATRIX WHOSE CCLUMN BEIGHTS ARE KNOWN
C
C
*
C
MHT = ACTIVE COLUMN HEIGHTS
C
MAXA = ADDRESSES OF DIAGONAL ELEMENTS
*
C
C
SUERCUTINE ADDRES(MAXA,MBT)
COMMON /SOL/ NUMNP,NEQ,NWK,NUMEST,MIDSTEST,
DIMENSION MAXA(1),MBT(1)
C
C
C
20
C
CLEAR ARRAY AREA
NN=NEC+1
DO 20 I=1,NN
MAXA(I)=e.0
MAXA(1)=i
MAXA(2)=2
MK=O
IF(NEQ.EQ.1)GO TO 100
DC 10 I=2,NEC
10
100
IF(MT(I) .GT.MK)MK=MHT(I)
MAXA(I+1)=MAXA(I)+MHT(I)+1
MK=MK+1
NWK=MAXA(NEQ+1 )-MAXA(1)
C
RETURN
END
ST,MK
- 158 -
CLEAR
--
CLEAR ARRAY A
C~~~~~~s~~~~s~+4~~~4~~~~4t~~~~8~~1~~~~~~
SUBROUTINE CLEAR (A,N)
IMPLICIT REAL*8 (A-H,O-Z)
DIMENSION A(1)
DO 10 I=1,N
A(I)=e.
RETURN
END
-
159 -
C
C
ASSEM -- CALL ELEMENT SUBROUTINE FOR ASSEMBLAGE OF THE
STRUCTURE STIFFNES MATRIX
C
C
C
SUBROUTINE ASSEM(AA)
CCMMON /EL/ IND,NPAR(10),NUMEG,MTOT,NFIRST,NLAST,ITWO
COMMON /TAPES/ IELMNT,ILOAD,IIN,IOUT,IDISP,ISTATE,IKORG,IKTRI
DIMENSION AA(1)
C
REWIND IELMNT
C
DO 200 N=1,NUMEG
READ(IELMNT)NUEST,NPAR,(AA(I), I=1,NUMEST)
C
CALL ELEMNT
C
200
CONTINUE
RETURN
C
C
END
- 160 -
C
C
*
ADMBAN --
ASSEMBLE UPPER TRIANGULAR ELEMENT STIFFNESS
*
*
A = GLOBAL STIFFNESS
S = ELEMENT STIFFNESS
C
C
ND = DEGREES OF FREEDCM IN ELEMENT
C
C
S(1
C
S=
C
C
C
C
C
C
C
C
A(1)
A =
S(2)
S(3)
+++
S(ND+2)
+++
S(2*ND)
+++
+++
A(6)
A(5)
A(4)
+++
+++
+++
*
+++
*
A(3)
A(2)
C
C
SUtROUTINE ADDBAN(A,MAXA,S,LM,ND)
IMPLICIT REAI*8 (A-H,O-Z)
DIMENSION A(1),MAXA(1,S(1),LM(1)
C
100
NDI=O
DO 200 I=1,ND
II=LM(I)
IF(II)200,200,100
MI=MAXA(II)
KS=I
D0'220 J=1,ND
JJ=LM(J)
110
IF(JJ)220,220,110
IJ=II-JJ
210
IF(IJ)22 ,210,210
KK=MI+IJ
220
KSS=KS
IF(J.GE.I )KSS=J+NDI
A(KK)=A(KK)+S(KSS)
KS=KS+ND-J
200
NDI=NDI+ND-I
C
RETURN
END
TIFFNESS
S(ND+1)
C
C
*
INTO COMPACTED GLOEAL STIFFNESS
C
C
- 161 C
C
C
*
SECOND --
CBTAIN SYSTEM TIME
C
SUDROUTINE S7CONr(TIM)
TIM=SECNDS (0.)
RETURN
END
- 162 -
C
C
COISOL -- SCIVE FINITE ELEMENT STATIC EQUILIBRIUM EQUATIONS
C
C
IN CORE, USING COMPACTED STORAGE AND COLUMN
REDUCTION SCHEME
C
C
--
C
C
C
C
A(NWK)
= STIFFNESS MATRIX STORED IN COMPACTED FORM
V(NN)
= RIGHT-HAND-SIDE LCAD VECTOR
MAXA(NNM) = VECTOR CONTAINING THE ADDRESSES OF DIAGONAL
ELEMENTS OF STIFFNESS MATRIX IN A
C
NN
= NUMBER OF EQUATIONS
C
NWK
= NUMBER OF ELEMENTS BELOW SKYLINE OF MATRIX
C
NNM
= NN + 1
C
KKK
= INPUT FLAG
INPUT VARIABLES --
C
C
C
IOUT
--
C
A(NWK)
C
C
V(NN)
= LOGICAL UNIT FOR OUTPUT DEVICE
OUTPUT --
= D AND L FACTORS OF STIFFNESS MATRIX
= DISPLACEMENT VECTOR
C
SUBROUTINE COLSOL(A,V,MAXA,NN,NWK,NNM,KKK)
C
C
C
IMPLICIT REAI*8 (A-H,C-Z)
COMMON /TAPES/ IELMNT,IIOAD,IIN,IOUT,IDISPISTATE,IKORG,IKTRI
DIMENSION A(NWK),V(NN),MAXA(NNM)
PERFORM L*D*LT FACTORIZATION OF STIFFNESS MATRIX
IF(KKKK-2)40,150,150
40
50
DO 140 N=1,NN
KN=MAXA(N)
KL=KN+1
KU=MAXA(N+1)-1
KH=KU-KL
IF(KH)110,90,50
K=N-KH
IC=0
KLT=KU
DO 80 J=1,KH
IC=IC+1
KLT=KLT-1
KI=MAXA(K)
ND=MAXA(K+1)-KI-i
IF(ND)80,eO,e?
60
*
EQ.1 TRIANGULARIZATION OF STIFFNESS MATRIX
EQ.2 REDUCTION AND BACK-SUBSTITUTION OF LOAD VECTOR
C
C
C
C
*
KK=MINO(IC,ND)
C=0.
DC 70 L=1,KK
70
C=C+A(KI+L)*A(KIT+L)
e8
90
A(KLT)=A(KIT)-C
K=K+1
K=N
B=e.
*
- 163 -
DO 100 KK=KL,KU
K=K-1
KI=MAXA(K)
C=A(KK)/A(KI)
B=E+C A(KK)
A(KK)=C
110
120
140
C
C
C
150
160
170
180
C
C
C
210
220
230
A(KN)=A(KN)-B
IF(A(KN))140,120,140
WRI T (IOUT ,2000 ) ,A(KN)
STOP
CONTINUE
RETURN
I IGNORE NEGATIVE
REIUCE RIGHT-HAND-SIDE LOAD VECTOR
DO 180 N=1,NN
KL=MAXA (N )+1
KU=MAXA(N+ 1)-1
IF(KU-KL)180,160,160
K=N
c=e.
DO 170 KK=KL,KU
K=K-1
C=C+A(KK)*V(K)
V(N)=V(N)-C
CONTINUE
BACK-SUESTITUTE
DC 200 N=I,NN
K=MAXA(N)
V(N)=V(N)/A(K)
IF(NN .EC.1)RETURN
N=NN
DO 230 L=2,NN
KL=MAXA(N)+1
KU=MAXA(N+1)-1
IF (KU-KL)230,210,210
K=N
DO 220 KK=KL,KU
K=K-1
V(K)=V(K)-A(KK)*V(N)
N=N-1
RETURN
FORMAT(//' STOP - STIFFNESS MATRIX NOT PCSITIVE DEFINITE "//
NONPOSITIVE PIVOT FOR EQUATION ',14,//
PIVOT = ',E2e.12)
END
-
C
C
C
LOADV --
164 -
OBTAIN THE LOAD VECTOR
C
C
SUEROUTINE LOADV(R,NEQ)
IMPLICIT REAL*8 (A-H,C-2)
COMMON /TAPES/ IELMNT,ILOAD,IIN,IOUT,IDISP,ISTATE,IKORG,IKTRI
DIMENSION R(NEQ)
C
READ (ILOAD)R
RETURN
END
- 165 -
C
C
DISPV --
OETAIN THE DISPLACEMENT VECTOR
C
C
C
SUBROUTINE DISPV(U,NSUR)
IMPLICIT REAL*8 (A-H,O-Z)
COMMON /TAPES/ IELMNT,ILOAD,IIN,IOUT,IDISP,ISTATE,IKORG,IKTRI
DIMENSION U(NSUR)
c
READ (IDISP)U
RETURN
END
- 166 ~~~~~~t~44~~~&~~~~1~4t~~~~~~~~~~~~l~~~~~
WRITE -- PRINT DISPLACEMENTS AND FLOWS
C~~~~~~~~~4~~~~~~~~~~~~88~~~~~~~~~~~1~~~
SUBROUTINE WRITE(DISP,ID,NEQ,NUMNP,FLOW,NSNOD,NSUR)
IMPLICIT REAL*8
(A-H,O-Z)
COMMON /TAPES/ IELMNT,ILOAD,IIN,IOUT,IDISP,ISTATE,IKORG,IKTRI
DIfENSION LISP(NIQ),ID(2,NUMNP),D(3),FLOW(NSUR),NSNOD(NSUE)
BYTE FF
DATA FF/" 1 4/
PRINT DISPLACEMENTS
WRITE (IOUT,2000)
IC=4
C
C
DO e10II=1,NUMNP
IC=IC+1
IF(IC.LT.54)GO TO 105
WRITE (iOUT,2020) FF
IC=0
105
110
C
120
C
DO 110 I=1,3
D(I)=e.
DO 120 I=1,2
KK=ID(I,II)
IL=I
IF(KK.NE.0)D(IL)=DISP (KK)
DO 130 IQ=1,NSUR
KK=NSNOD(IQ)
13e
IF(KK.EQ.II )D(3)=FLOW(IQ)
IF(D(3).EQ.0.0)GOTO 100
100
C
C
C
WRITE(IOUT,2010)II,D
CONTINUE
WRITE DISPLACEMENTS AND FLOWS TO FILE
WRITE(ISTATE) (DISP(I) ,I=1 ,NEQ)
WRITE(ISTATE) (F LOW(I) ,I=1 ,NSUR)
RETURN
FCRMAT(////' D I SPLA CEMENT S , P R E S S U R E S,
NODE '
* & SURFACE
F L0 WS '///'
PRESSURE
18X,'
FLOW '/)
DISPLACEMENT
2010
FORMAT( 1X,13,8X, 3E18.E)
2020
FORMAT(1X,A1' D ISP LA CEMEN TS , PRES SURESS
& SURFACE
NODE ',
F L 0 W S '///
PRESSURE
FLOW '/)
18X,' DISPLACEMENT
2000
C
EN]
- 167 -
C
C
STRESS -- CALL THE ELEMENT SUBRCOUTINE TO CALCULATE STRESSES
*
C
C
SUBROUTINE STRESS(AA)
COMMON /VAR/ NG,MODEX,RBEST,DELTA
COMMON /EL/ IND,NFAR(10),NUMEG,MTOT,NFIRST,NLAST,ITWO
COMMON /TAPES/ IEIMNT,ILOAD,IIN,IOUT,IDISP,ISTATE,IKORG,IKTRI
DIMENSION AA(1)
C
C
C
C
LOCP CVER ALU ELEMENT GROUPS
REWIND IELMNT
C
DO 100 N=1,NUMEG
NG=N
C
READ (IELMNT) NUMEST,NPAR,(AA(I),I=1,NUMEST)
C
CALL ELEMNT
C
100
CONTINUE
RETURN
END
- 168 -
0************************************************************************
*e
ERROR -- PRINT ERROR tESSAGES WHEN STORAGE IS EXCEEDED
SUBROUTINE ERRCR(N,I)
COMMON /TAPES/ IEIMNT,IIOAD,IIN,IOUT,IDISP
GO TO (1,2,3,4,5),I
WRITE(IOUT ,2e•0)
GO TO 6
WRITE(IOUT ,210)
GO TO 6
WRITE(IOUT,2020)
GO TO 6
WRITE(ICUT,2'30)
GO TO 6
WRITE(IOUT,2040)
WRITE(IOUT ,205e) N
STOP
FCRMAT(//' NOT ENOUGH STORAGE FOR READ-IN OF ID ARRAY AND'
1/' NODAL POI NT COORDINATES ')
2010
FORMAT(//' NOT ENOUGH STORAGE FOR DEFINITION OF LOAD VECTORS
202
FORMAT(//' NCT ENCUGEH STORAGE FOR ELEMENT DATA INPUT ')
2030
FORMAT(//' NOT ENOUGH STORAGE FOR ASSEMBIAGE OF STRUCTURE'
2/' STIFFNESS ,AND DISPLA CEMENT A ND STRESS SOLUTION PHASE ')
2040
FORMAT(//' NOT ENOUGH STORAGE FOR SURFACE DISPLACEMENT INPUT
2050
FORMAT(//'
ERROR
STORAGE EXCEEDED BY ',I9,' LONGWORDS
END
*** ld
- 169 -
C
C
C
*
SAVE --
SAVE THE DISPLACEMENT VECTOR
C
SUEROUTINE SAVE(U,UOLr,NEQ)
C
IMPLICIT REAt*8 (A-H,C-Z)
DIMENSION U(NEQ),UOLD(NEQ)
C
DO 1~ IQ=1,NIC
10
UOLD(IQ)=U(IQ)
RETURN
END
- 170 -
UPDATE -- UPDATE THE RHS USING PREVIOUS DISPLACEMENTS
SU.ROUTINE DPDATF(R,UOLD,ID,MAXA,A,NUMNP)
IMPLICIT REAI*8 (A-H,O-Z)
COMMON /TAPES/ IELMNT,ILOAD,IIN,IOUT,IDISP,ISTATE, IKORG,IKTRI
EIMENSICN A(1) ,?(1),UOLD(1),MAXA(1),ID(2,1)
DO 200 IP=1,NUMNP
II=ID(2,IP)
IF(II.EQ.0)GOTO 200
IHl=MAXA(II+1 )-MAXA (II)-I
I PRESSURE COLUMNS
SUM=O.
DO 250 IU=1,NUtNP
JJ=ID(1,IU)
IF(JJ.EO.0)GOTO 250
ICOL=II-JJ
! DISPLACEMENT ROW S
IF(ICCL.IE.0)GOTO 250
IF(ICOL.GT.IHT)GOTO 250
INDEX=MAXA(II)+ICOL
SUM=SUM+UOLD(JJ)*A(INDEX)
250
200
CONTINUE
R(II)=R(II)+SUM
CONTINUE
DO 300 IU=1,NUMNP
II=ID(1,IU)
IF(II .EQ.0GOTO '00
IHT=MAXA(II+1)-MAXA(II)-1
DC 350 IP=1,NUMNP
JJ=ID(2,IP)
IF(JJ.EQ.O)GOTO 350
ICOL=II-JJ
IF(ICOL.LE.0)GOTO 350
IF(ICOL.GT.IHT)GOTO 350
IN]EX=MAXA(II)+ICCL
350
300
C
R(JJ)=R(JJ)+UOLD( II)*A(INEX)
CONTINUE
CONTINUE
RETURN
ENL
! DISPLACEMENTS COCLUMNS
! PRESSURE ROWS
-
171 -
C
C
OUT --
CONVERT TO REAL*4 FOR
C
SUIROUTINE OUT(D,R)
REAIL*8 D
REAL*4 R
R=D
RETURN
END
CUTPUT
- 172 CB~~~~~~~~~~~~~~~g~~~~~~~g~~~~~~~~~g~~~~
DISPL -- REAL SURFACE DISPLACEMENT DATA
WRITE TO DISC FILE IDISP
C *******************************************************
C
SUEROUTINE DISPI (NSNOD,SDISP,NSUR,NLCPSE)
COMMON /VAR/ NG,MODEX
COMMON /TAPES/ IEIMNT,ILOAD,IINICUT,ICUTIDISP ,ISTATI,IKORG,IKTRI
DIMENSICN NSNOD(NSURq,SDISP(NSUR)
]YTE FF
DATA FF/"14/
C
1005
200
DO 200 I=1,NSUR
R-AD(IIN,1005) NSNOD(I)
FORMAT(5I)
CONTINUE
C
DO 230 LL=1,NLCASE
WRITE(IOUT,2000) FF,LL
240
DO 240 I=1,NSUR
SDISP(I)=0.
DO 250 N=1,NSUR
C
250
230
C
1000
2000
2010
?EAD(IIN,1000) NOD,DISP
IF(NSNOD(N).NE.NOD)GOTO 250
SDISP(N)=DISP
WRITE(IOUT,2010) N,NOD, SDISP(N)
CONTINUE
WRITE(IDISF)SDISP
CONTINUE
FORMAT(15,FvI.0)
FORMAT(1X,A1 ,1X,
LOAD
CA SE
NODE
DISPLACEMENT '/
NUMBER )2
FORMAT(1X,I5,5X,I 5,F12.3)
RETURN
ENr
-
DASSEM --
173 -
ASSEMBLE SURFACE FLCW MATRIX
SUEROUTINE DASSEM (ID,NSNOD,AKQ,MAXB,DISP,NSUR,NLCASE)
COMMON /VAR/ NG,MODEX,RBEST,DELTA
COMMON /SOL/ NUMNP,NEQ,NWK,NUMEST,MIDEST,MAXEST,MK
COMMON /DIf/ Ni,N2,N3,N4,N5,NE,N7,NB,N9,N9,N,N11,N12,N3,N14,N15
COMMON /TAPES/ IELMNT,ILOAD,IIN,ICUT,IDISP,ISTATE,IKORG,IKTRI
DIMENSION ID(2,1),NSNOD(NSUR),AKQ(1),MAXB(1),DISP(1)
CCFMON A(1)
BYTE FF
DATA FF/"14/
C
INDEX=1
NEQ1=NEO+1
DC 200 I=1,NSUR
MAXE(I)=INDEX
DO 210 IQ=1,NEO
210
DISP(IQ)=o.0
IF=ID(?,NSNOD(I))
DISP( IF)=DELTA
CALL COLSCL(A(NZ),DI SP,A(N2),NEQ,NWK,NEQ1,2)
DO 230 N=I,1,-l
NUM=ID(I,NSNOD(N))
230
200
C
AKQ(INDEX)=DISP(NUM)
INDEX=INDEX+1
CONTINUE
CCNTINUE
NSUR1=NSUR+1
MAXB(NSUR1)=INDEX
NSIZE=NSUR*NSUR1/2
CALL COLSOL(AKQ,DISP,MAXB,NSUR,NSIZE,NSUR1,1)
C
RETURN
END
- 174 -
C
C
C
FLOW -- FIND THE SURFACE FLOWS
C
C
C
C
SUBROUTINE FLOW(UR,US,ID,AKQ,MAXB,NSNOD,NSUR,R)
IMPLICIT REAL*8 (A-H,O-Z)
COMMON /VAR/ NG,MODEX,RBEST,DELTA
COMMON /TAPES/ IELMNT,ILOAD,IINIOU,IOUTIDISPISTATE,IKORG,IKTRI
DIMENSICN AEC(1),UR(1),US(1),MAX2(1),ID(2,1),R(1),NSNOD(1)
DO 200 IP=1,NSUR
200
C
NUM=ID(1,NSNCD(IPF)
US(IP)=US(IP)-TUR(NUM)
NSUR1=NSUR+1
NSIZE=NSUR*NSUR1/2
KTR=2
CALL COLSOL(AKQ,US,MAXB,NSUR,NSIZE,NSUR1,KTR)
C
C
C
300
CORRECT LOAD VECTOR
DO 300 IQ=1,NSUR
NUt=ID(2,NSNOD(IQ))
R(NU )=R(NUM)+US(IQ)*DELTA
RETURN,
END
- 175 -
APPENDIX B
FREQUENCY RESPONSE
- 176 -
As shown in Chapter 3, the frequency
response
of
the
cartilage layer, model led as an elastic network submerged
exhibits a characteristic 45 -degree
an incompressible flui
phase
stress and displ acement
surface
The s hift wil 1 decrease as effe cts
(displacement lags).
compressibility
the
the
between
shift
of th e fluid and/or the network become
is likely even hig her (it
solid constituents of carti lage it
would otherwise be dif ficul t to acc ount for the
of
wave
we
should
of the
The bulk modu lus of water is 2.24 GPa;
significant.
propagation we me.asured in cartilage).
until
compressibility
high
speed
The refore,
o
bulk
the uniaxial s train stiffnes
of the
effects
significant
expe ct
not
of
layer is say 0.2 GPa ( stiff ness is the ratio of the
surface
stress to the average strain).
Figure B-1 show S
chamber.
design
the
for
c onfinement
(See Tepi c [139 I for a detailed discuss ion of the
exper imental
this
advantages of
cartilage
ou r
positi oned
is
on
techniq ue).
top
of
a
A
plug
20 MHz ultrasonic
tranducer which faces the plug and the loa der on t op of
Thickness
of the plu g is
pulses
of ultrasonic
The
transducer.
specimen alone.
of
it.
measured by timi ng the r eflections
emi tted
di splac ement
from
and
receive d
by
the
is thus measured across the
The resol ution of the mea surement is better
than a micron, which at high frequencies is insuff icient for
a very precise estimate of the stiffness,
there
are
at
least
two
levels
of
but,
as
long
the surface position
- 177 -
7
iF
IF
SETao
150
S mm
CoreTE..oLLE.
OF WAT-ES. *AV
LoAý.
'Pot.OusAL.UII4AA
POO u S
WAf E k
GL.ASS COIE3MKE0T'"r
CHAMt•2W..
CA;TLA
PLUG
rGLUEZ (ASTMA E 9 0)
O-NLJ)(
Ya.AM4%\..JC.E18
Figure B-1. Cartilage Plug Confinement for
the Frequency Response Measurements
OL.3ER
- 178 -
recorded, is sufficient to determine the phase
thick-walled
confinement
chamber
was
made
shift
from
class
The
loader
(pyrex) rod which was bored and then polished.
consists
of
a
porous
porous pyr ex glass wafer,
of
only 0 .125 mm.
alumina
rod
and a thin, ver y fine
ground by hand to final
thi ckness
The result was a low permeability loader
(the fluid resistance was measured to
with
Tk6
be
1.5 x 10
Ns/m )
a smooth loading surface (the surface roughness should
be measure d in the future).
To prevent
scratching
against
the glass the loader is lined by a thin teflon sleeve.
The
load
was
whi ch
Simulator
applied
for
our
by
this
servo-controlled
purpose
served
as
Hip
a ny
load-controlled testing machine.
The ultrasonic signals were processed as
Sections 4.1.
The
load
signals
described
in
from the simulator and a
square-wave output from the function generator, used only to
determine
the
frequency,
analog-to-digital converter.
were
recorded
computing the
series
of
were
Five
sampled
to
twenty
through
full
LPS
cycles
and the phase shift was determined by simply
corresponding
the
load
displacement measurements.
coefficients
signal
and
of
of
the
the
Fourier
reconstructed
- 179 -
The results of the phase shift measurements
on
Curve 1 shows an average of many trials on
Figure B-2.
different plugs with the alumina loader
wafer;
wafer.
data
the
shown
are
on
Curve 2
without
glass
the
was obtained with the glass
Curve 1 is shifted by about a decade with respect to
Lee's [72] data (Curve 3);
Curve 2 by an additional decade.
At lower frequencies we have consistently measured shifts of
45 to 50 degrees,
as expected,
but the fall off above 0.1 Hz
persisted.
Even if
can
never
material
we had a "perfect"
be
loader,
the surface
effects
completely eliminated, because the cartilage
is not homogenous and even
if
microtomed
(as
some of our tests) will retain some surface roughness.
in
60.
50.
r--
40.
L
(0
t-0
lJ
30.
20.
10.
0.
-3.0
Figure B-2.
-1.0
LOG ( FREQUENCY [Hz3 )
0.0
Phase Shift Between Surface Displacement and Load
1.0
- 181 -
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