We examine, within a simple bearing model of a knee joint that only consideres pure sliding, the effect of the presence of a small vertical hole in the load area on the fluid film properties. The calculations indicate that fluid is entrapped in such a hole, which, for constant load, causes a smaller minimal film separation of the two surfaces.

This will lower the horizontal friction, but may also bring about surface contact in high load situations

Hjorth, P.G.

King, J.

Korobeinikov, A.

Mason, J.

McKee, S.

Wilson, S.

English

Mathematics in Industry Information Service Eprints Archive

Stryker Osteonics:
Prosthetic Knee Joint
P.G. Hjorth∗,
J. King †
A. Korobeinikov‡,
J. Mason§,
S. McKee ¶
S. Wilson ‖
Problem Presented by:
C. Birkinshaw, S.Flanagan,
University of Limerick
∗Technical University of Denmark.
†University of Nottingham
‡University of Limerick
§Bristol University
¶University of Strathclyde
‖University of Strathclyde
1
1 The Problem
The human knee is a hinge-type joint comprising two major bones, the femur
and the tibia. The bone is end capped with a soft compliant substance
cartilage, which acts as a low friction bearing. Lubrication is added by the
cartilage synovial fluid, which ensures that the bearing works with full fluid
film lubrication.
Figure 1: The human knee joint.
As the knee articulates, fluid is drawn through the joint space, ensuring
that the two opposing cartilage surfaces do not come into contact. The load
applied to the knee depends upon the activity, and can be as much as 8 times
body weight. The cartilage also acts as a shock absorber, dampening the
excessive loads applied and protecting the bones from shock. Synovial fluid
is non-Newtonian, and when the joint is stationary, the fluid is almost gel-
like. The cartilage is porous, and never dries out in the healthy knee.
Figure 2: Elements of a prosthetic knee joint. The tibia is capped with a layer of
polyurethane.
In the diseased knee cartilage breaks down, and the articulation tends to a
bone-on-bone bearing. This is painful, and can be operated on to replace
the diseased joint with a prosthetic joint. See figure 2.
2
The material properties of the prosthetic joint do not simulate cartilage in
general. However, a new type of material is being considered polyurethane
(PU), which simulates cartilage more closely than the conventional materi-
als used in joint replacement. The system being considered is a PU tibial
bearing articulating against a metal femoral bearing.
The PU is not porous, but can develop full fluid film lubrication if the
conditions are correct. Many mathematical approaches ([2]-[11]) have been
adopted to explore the nature and tribology of lubrication between poroe-
lastic surfaces. The entire lubrication process is complex and involves the
simultaneous solution of the equations of fluid dynamics with the equations
of Hertian contact mechanics.
In the joints manufactured, it is necessary to have a soft, compliant PU as
the bearing material. Under load, this material is easily deformable, and
as the load increases so the contact area increases, thereby lowering the
contact stress. The soft PU bearing is supported by a harder PU backing.
The interface between the two materials is formed by fusion bonding, and
is consequently an inherent weakness in the system. The weakness is most
noticeable at the edges; when the material is loaded and the soft PU deforms,
the displaced material exerts a shear force at the interface.
The Study Group was asked
(1) To investigate the inclusion of holes / channels in the loaded area of
bearing to determine whether these features can: entrap fluid and thus sim-
ulate a porous surface, and can release fluid when the bearing is articulated
to maximise the development of a fluid film.
(2) To investigate if holes around the periphery of the bearing can reduce
the shear stress at the interface.
2 Thin Film Lubrication Theory
In thin film lubrication theory (see e.g., [1]), one considers a 2-D model of
two surfaces in relative motion, and a thin layer of fluid in between.
Let UR be the relative velocity of the two surfaces (here given to the upper
surface), and h(x) the x-dependent separation of the surfaces. See figure
3.
3
h(x)
x
z
UR
L
Figure 3: Thin film model. The fluid forms a thin layer of width h(x) between the
femur (top) and the tibia (bottom) surfaces. The curvature relative to other distances is
exaggerated. The top surface moves with a velocity UR.
The horizontal velocity field in the film is approximatly given ([1]) by
u(x, z) =
[
UR
h
− z
2µ
px
]
(h− z) (1)
where µ is the fluid viscosity, and px is the x-derivative of the pressure
p.
The overpressure p − p0 (which in the joint supports the vertical load) is
connected to the width function h(x), by:
p− p0
6µ
= UR
∫ x 1
h2(s)
ds− 2Q
∫ x 1
h3(s)
ds (2)
where UR is the relative velocity, and Q is the volume flux related to UR
by:
Q =
∫ h(z)
0
u dz =
1
2
URh− h
3
12µ
px (3)
3 A Surface cavity
Even from the lowest order equations (2) and (3) one can estimate the effect
of changes in the height function h(x). Suppose a small hole is drilled into
the PU tibia surface, as in figure 4 (b).
The small hole will be filled with fluid, since the large pressure forces in the
joint are much greater than the subtle effects like capillary forces.
4
x x
Figure 4: Two distance functions, h1 and h2, compared.
The hole causes a new height function h2(x) of the fluid. The pressure will,
for a fixed contact surface area, always ajust itself to the constant body
weight and thus be a constant. Since the hole introduces a local increase
in h, the new h2(x) will have lower values in the region where there is no
hole, keeping the integral over the 1/h2(x) terms in the pressure integrals
equation (2) constant.
Since thin-film theory predicts that the pressure is of order µUL/h2 while
the tangential stresses are of order µU/h, horizontal forces in this model are
smaller than vertical forces by a factor of order h/L.
Thus, a hole (and even more a collection of small holes) will lead to a smaller
minimal value of h(x). A smaller h means less tangential stress, in other
words less friction in the bearing. On the other hand, a too small value of
h brings the hard surfaces even closer to undesired contact.
4 Poroelasticy
A number of studies of the properties of poroelastic systems like the knee
joint has been performed; see e.g., recent works [10] and [11], where a sim-
plified model of joint articulation in the presence of both a non-Newtonian
synovial fluid and poroelastic surfaces have been studied.
5
Even though a PU surface is not porous, it is interesting to compare the con-
clusion of the previous section with results from more complex calculations
involving poroelastic surfaces with varying degrees of roughness.
We therefore briefly outline, following [11], an extension of the model to
a poroelastic system, albeit without the feature of a couple-stress synovial
fluid. The properties of such a fluid seems nevertheless relevant for the
system and should be considered in further studies.
In figure 5, the tibia base is now poroelastic, meaning that the material
has elastic properties as well as the ability to entrain fluid in a microscopic
structure of canals, so that one should consider both a material displacement
field U, and the average fluid field V inside the poroelastic in addition to
the ‘real’ fluid field (u, v) outside the poroelastic.
h(x)
x
z
UR
L
Figure 5: A poroelastic problem.
The distance function h now consists of two terms, h(x) = h0(x) + hs,
where hs describes a random variable with zero mean and a deviation which
characterizes the roughness of the surface.
Using a statistical average, and scaling with the natural length scale L, one
can obtain a dimensionless quantity C which is a measure of the relative
roughness of the poroelastic surface.
The fluid field is considered to be incompresible:
∂u
∂x
+
∂v
∂y
= 0
as well as satisfying
µ
∂2u
∂y2
=
∂p
∂x
and for the presure p
∂p
∂y
= 0
6
In the poroelastic region, the material displacement field U
ρm
∂2U
∂t2
= divσm − µ
k
(
∂U
∂t
−V
)
In the fluid region, we U, Fluid field V
ρf
∂2V
∂t2
= divσf +
µ
k
(
∂U
∂t
−V
)
where σm and σf are stress tensors. The pressure P in the pores will satis-
fiy
∇2P = 0
Boundary conditions for the velocity field will be that at z = 0:
u = 0, v = −vn, ∂
2u
∂z2
= 0
At z = h0:
u = 0, v = −vn − dh0dt ,
∂2u
∂z2
= 0 (4)
In [11], it is found that such a system has a spatial pressure distribution
which depends on the roughness parameter C. See figure 6.
x
L0
rough (large C)
smooth (small C)
Figure 6: Dimensionless overpressure p¯− p0 across the joint for small and large values
of the dimensionless roughness parameter C.
A load-carrying capacity may be defines as
W¯ =
∫ L
0
p¯− p0 dx (5)
7
C
L0
Figure 7: Dimensionless load W capacity as a function of the dimensionless roughness
parameter C.
It is established in [11] that the load capacity is an increasing function of
the roughness parameter C. It is established in [11] that the load capacity
is an increasing function of the roughness parameter C.
5 Conclusion
We have examined, within a simple bearing model of a knee joint, the effect
on the fluid film properties of the presence of a small vertical hole in the
load area. The calculations indicates that
Fluid is entrapped in such a hole (in a porous medium, fluid can also be
released, and may thus be ‘pumped’ in and out, so the porous layer can act
as a reservoir of fluid).
In a simple model, the presence of the hole will, for constant load, cause
a smaller minimal film separation of the two surfaces. This will lower the
horizontal friction, but may also bring about surface contact in high load
situations.
We note that this result, that small holes gives better lubrication is con-
sistent with [11] and also with recent results examining both slide and roll
[12].
The model is incomplete in that it consideres pure sliding, not the com-
bined effect of sliding and rolling, and also does not consider the elastic
deformation of the PU surface. Also, we did not consider the second ques-
tion of the effect of holes to reduce sheer stresses at the soft PU - hard PU
interface.
8
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(1991).
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(2007).
9


An analysis of the squeeze-lm lubrication mechanism for articular cartilage,

Asymptotic behaviour of thin poroelastic layers,

Couple stresses in fluids,

Grootenboer: Articular contact in a three-dimensional model of the knee,

Kudenatti: An analysis of rough poroelastic bearings with reference to lubrication mechanism of synovial joints

Linear ow and deformation in a poroelastic disk with a free surface, Trans.

Mathematical Models in the Applied Sciences, Cambridge Texts in Applied Mathematics

On the fundamental fluid transport mechanics through normal pathological articular cartilage during function,

Stochastic model for hydrodynamic lubrication of rough surfaces,

Surface roughness effects on squeeze lm poroelastic bearings

Thin-film Lubrication of Dented Surfaces Tribology Transcations (STLE Annual Meeting,

Weakly Nonlinear deformation of a thin poroelastic layer with a free surface, Trans.

Stryker Osteonics: Prosthetic Knee Joint

Stryker Osteonics: Prosthetic Knee Joint

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