Stress occurring in the friction node of elements in the total knee endoprosthesis

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T11138 Tribology and Lubrication Engineering

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Journal of Applied Mathematics and Computational Mechanics 2015, 14(2), 41-48
www.amcm.pcz.pl p-ISSN 2299-9965
DOI: 10.17512/jamcm.2015.2.05 e-ISSN 2353-0588

STRESS OCCURRING IN THE FRICTION NODE OF ELEMENTS

IN THE TOTAL KNEE ENDOPROSTHESIS

Marcin Nabrdalik

Institute of Mechanical Technology, Czestochowa University of Technology

Częstochowa, Poland

marcin@iop.pcz.pl

Abstract. The paper presents the numerical analysis of stress occurring in the friction node
of bi-condylar knee endoprostheses. The Finite Elements Method allows one to calculate
and present the stress distribution in all elements of the computing model.

Keywords: endoprosthesis, stress, FEM

1. Introduction

The main objective of the following paper is to present the analysis of stress
distribution occurring on the contact surface between sleds and flat polyethylene
insert in bi-condylar knee joint endoprosthesis. This analysis was conducted by
means of numerical calculations in ADINA System 8.6. Its aim was to find
an optimal shape of a sled that would make the stress distribution on the insert
surface as favorable as possible. The model cannot interrupt the movement biomechanics of the joint.

Such strict criteria make it very difficult to find a proper sled shape. It is
important to apply proper calculating methods, like the finite elements method.
All numerical analyses allow one to define and draw the map of stress for each
computing variant.

2. Stress analysis in the pair sled - flat insert for bi-condylar
endoprosthesis by W.LINK

The calculations were conducted for a total, bi-condylar knee endoprosthesis.
It shows reduced stress and contact stress occurring in all types of polyethylene
inserts. UHMWPE polyethylene is the weakest point of the endoprosthesis, which
is why it is important to present the reduced stress distribution in the inserts.

The formulas presented below have been applied in the analysis; formula (1)
presents a short notation of physical equations connecting together the values of
stress and strain in the 3D case, in material, isotropic, linear - elastic bodies [1, 2].

42 M. Nabrdalik

(1)
��� = 2���� + �������

Applying material constant:
`E` - Young’s modulus (elasticity)
`G` - Kirchhoff’s modulus (non-dilatational strain)
`ν` - Poisson’s coefficient

�
(2)

- = � = ������

��
(3)

- = �����������

After displacing (1) with (2) and (3), we get physical formulas as follows:

��� = 2���� + ��������� ������ = 2����� + ����� ������� = ���� ���� + ����� ������� (4)

After displacing `i`, `j`, `k` with 1, 2, 3, we get:

� � �
(5)
��� = ��� ���� + ���� ���� + ���+����� ��� = ��� ��� = 2����

� � �
(6)
��� = ��� ���� + ���� ���� + ���+����� ��� = ��� ��� = 2����

� � �
(7)
��� = ��� ���� + ���� ���� + ���+����� ��� = ��� ��� = 2����

3. The shape of contact surfaces between elements of bi-condylar knee
joint endoprosthesis

There are four types of bi-condylar knee joint endoprostheses as far as the shape
of polyethylene insert is concerned. Figure 1 presents the shapes of contact surfaces
between elements of knee joint endoprosthesis.

a) b) c) d)

Fig. 1. Shapes of contact surfaces between elements of knee joint endoprosthesis: a) surface

contact, b) point contact, c) linear contact, d) quasi-linear contact [3-5]

Stress occurring in the friction node of elements in the total knee endoprosthesis 43

The durability of the endoprosthesis depends on mechanical and tribological features
of its weakest element, which is the polyethylene insert.

4. Bi-condylar total knee endoprosthesis by W.LINK

Figure 2 presents the product by W.LINK, which is an example of bi-condylar
knee joint endoprosthesis. The femoral part copies the shape of condyles of femur
bones. The femoral part is precisely fixed with two pins into the bone.

Fig. 2. Bi-condylar knee joint endoprosthesis by W.LINK [6]

The materials used in the above endoprostheses are:

`•` CoCrMo alloy - femoral and tibial part,

`•` UHMWPE polyethylene - the insert.

That is the most commonly used pair of materials for knee joint endoprostheses.
In some cases, the titanium alloy is used as well due to its high mechanical features
and because it is lighter than CoCrMo alloy.

5. Numerical model of knee joint endoprostheses

When designing a simplified numerical model of endoprosthesis, it is important
to draw a geometry of the endoprosthesis as true and similar to the real knee joint
as possible, not only as far as the shape is concerned, but also considering the anatomical range of movements.

5.1. Criteria taken for numerical calculations

The contact stress occurs between two elements pressed together with force.
They take place in certain areas and can reach quite high values even at a respectively low value clamp.

Theoretical criteria for contact stress according to `Hertz’s theory` have been
applied in the form as follows:

44 M. Nabrdalik

`•` Contacting elements are made of homogeneous isotropic materials.

`•` The surfaces are fixed in the contact area of the element with smooth and regularly curved surfaces.

`•` When subjected to load there are only slight strains in the contact area.

`•` The contact area is relatively small when compared with the surfaces of the contacting elements.

`•` On the contact area there only normal strains occur.

5.2. Numerical model of bi-condylar endoprosthesis

The numerical model is a simplified version of the original endoprosthesis,
though the sleds’ geometry has been maintained. That enables us to keep the general shape of endoprosthesis and to quite closely imagine the strain distribution on
the insert’s surface. Figure 3 presents the simplified sled model.

Fig. 3. Simplified sled model used for calculations

The flat polyethylene insert in the model has got the shape adjusted to the real
anatomical tibial bone.

Fig. 4. Simplified model of the polyethylene insert

The main objective of these calculations is to present the strain distribution on,
and right underneath the surface of the insert, while using various geometrical
options of both elements. Considering the polyethylene insert, it will be its thickness;
and considering the sled - it will be its cross-section radius. Furthermore, the sleds
might be of a different thickness which might provide us with additional information

Stress occurring in the friction node of elements in the total knee endoprosthesis 45

about the tested contact surfaces. The main dimensions of the sleds are shown
in Figure 5.

Fig. 5. Geometry of the femoral element of the endoprosthesis. Front view

Fig. 6. Geometry of the sled of knee joint endoprosthesis. Side view

Figures 5 and 6 present the geometry of a sled with the bonding element.
Constant geometrical dimensions:
L = 50 mm, L1 = 13 mm, L2 = 16 mm, H3 = 5 mm
Radiuses r1 and r2 are identical and have 14 mm
Calculations were conducted for three different cross-section radiuses of the sleds:
R1 = 17 mm, R2 = 26 mm, R3 = 35 mm. Dimensions of the polyethylene insert are
presented in Figure 7.

Fig. 7. Geometry and dimensions of polyethylene insert

```
M. Nabrdalik

```

The polyethylene inser t
A = 54 mm, B = 30 m m
The most significant is
G1 = 8 mm, G2 = 13 m
Numerical calculations
nesses of polyethylene
subjected to load F1 =
Table 1 presents comp u

t ’s dimensions are as follows:

m, C = 18 mm, r3 = 9 mm

the thickness of the insert = G:
m, G3 = 22 mm
have been conducted for 9 computing variants, u s

inserts cooperating with 3 different sleds. Each
1500 N.

u ting variants together with models’ marks:

ing 3 thickpair will be

Table 1

cal features

omputing variants of the analysed models

force F1 = 1500 N

sled R1 sled R2 sled R3

ins e rt G1 GR111 GR112 GR113

ins e rt G2 GR121 GR122 GR123

ins e rt G3 GR131 GR132 GR133

Simulations for tho s e loads were conducted with the following phys i
of the materials, place d in Table 2.

Table 2

Mechanical fe a tures and density of materials used for endoprostheses [ 7 ]

Young’s modulus

Poisson’s
coefficient ν

`E` [MPa]

CoCrMo

UHMWPE

Young’s modulus Poisson’s Density

`E` [MPa] coefficient ν `ρ` [kg/m `3` ]

2.1·10 `5` 0.29 8300

1000 0.4 960

6. The results of nu m erical calculation conducted
with finite eleme n ts method and ADINA System 8.6.

p roved that the stresses in endoprosthesis are conce n

All the calculations p
underneath the contact s
surface [6]. Figures 8 a

n trated right

the insert’s

lations.

s urfaces of both elements, particularly underneat h

nd 9 present exemplary results of numerical calc u

Fig. 8. Model GR111 - reduced stress distribution on the surface of the polyethyle n

Fig. 8. Model GR111 -

e insert

```
       Stress occurri n g in the friction node of elements in the total knee endoprosthesis

```

Fig. 9. Mo d el GR113 - reduced stress distribution on the insert’s surfa c e

7. Conclusions

The lowest values  - f
GR123 - 28,91 MPa a n
sled was 35 mm. The
GR111 - 38,85 MPa a n
smallest and = 17 mm.
radius and insert’s thic k
insert.

f reduced strain have been gained with the compu t

d GR133 - 26,85 MPa where the cross-section r a

highest values of the reduced strain occurre d

n d GR112 - 36,93 MPa where the sled’s radi u

Figure 10 presents the influence of the sled’s c r

ing variant:

a dius of the

in models

u s was the
r oss-section

olyethylene

ness on the value of generated stress in the flat p

8mm

13mm

22mm

17 26 35

Sled's cross section radius [mm]

Fig. 10. The in fl uence of the sled’s cross-section radius and insert’s thickness

on the v alue of generated stress in the flat UHMWPE insert

1. The key issue is th
prosthesis and the v
the values of gener a
2. The knee joint endo p
optimal the contact

e geometry of the contact between both eleme n
alue of the cross-section radius of the sled, whic h

ts of endo
h influences

mm assures
fortunately,

ted strains.

p rosthesis sled with the cross-section radius of 35

strain distribution in the polyethylene insert. U n

48 M. Nabrdalik

this geometry of the pair: sled - insert, decreases the mobility of the knee after
implantation.
3. The thickness of the insert slightly influences the values of strains and displacement in it.
4. In all the considered numerical variants the stress values generated by the sleds
have not crossed the yield point of UHMWPE, and the prevalent issue here are
the tribological wear processes.
5. Numerical calculations were conducted in order to define the optimal geometry
of the friction node of the knee joint endoprosthesis. To design a knee joint
endoprosthesis of the best construction, it is important to complete all numerical
analysis with empirical research on prototype of endoprosthesis conducted on
a knee joint simulator.

References

[1] Zienkiewicz O.C., Metoda elementów skończonych, Wyd. Arkady, Warszawa 1972.

[2] Prospekt firmy W.LINK.

[3] Będziński R., Biomechanika inżynierska, Oficyna Wydawnicza Politechniki Wrocławskiej,

Wrocław 1997.

[4] Gierzyńska-Dolna M., Biotribologia, Wydawnictwo Politechniki Częstochowskiej, Częstochowa

2002.

[5] Gierzyńska-Dolna M., Kubacki J., Specyfika zużycia endoprotez stawu biodrowego i kolanowe
go, Mat. II Sympozjum Inż. Ortop. i Prot. IOP’99, Białystok 1999, 45-51.

[6] Nabrdalik M., Use of the finite element method in the analysis of load of polyethylene inserts

of knee joint endoprostheses, Scientific Research of the Institute of Mathematics and Computer
Science 2012, 1(11).

[7] Marciniak J., Biomateriały, Wydawnictwo Politechniki Śląskiej, Gliwice 2002.

[8] Zakrzewski M., Zawadzki J., Wytrzymałość materiałów, PWN, Wrocław 1975.

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