Stress analysis and optimization of a new intramedullary
fixation system, to be used for finger-joint prostheses
Citation for published version (APA):
Huiskes, H. W. J., & van Heck, J. G. A. M. (1978). Stress analysis and optimization of a new intramedullary fixation system, to be used for finger-joint prostheses. (DCT rapporten; Vol. 1978.010). Technische Hogeschool Eindhoven.
Document status and date: Published: 01/01/1978
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Preface
This report t r e a t s a theoretical s t r e s s analysis of a new intramedullary stem fixation system, t o be used i n finger-joint prostheses. This prosthesis was designed by Dr. Peter S. Walker and Mr. David Greene, Howmedica Inc.,
Rutherford N.J., U.S.A.
The a n a l y s i s was carried o u t using Finite Element Methods; a computer oriented
method, wel suited f o r a r b i t r a r i l y shaped structures of several materials, under various loading conditions. A computer system FEMSYS, based on these methods and developed a t the computer center of the Eindhoven University o f Technology, was p u t a t our disposal t h r o u g h the assistance o f Ir. J. Banens.
On a grant from Howmedica Inc., a student a s s i s t a n t ( J . van Heck) could be
appointed t o take care o f the data handling. He found h o s p i t a l i t y a t the Dept. o f Applied Mechanics, Mechanica: 5ng. faculty, Eindhoven University of
Technology (Chairman: Prof.Dr.Ir.
J.D.
Janssen). The computer program'Triquamesh' an i n p u t generating program for the FEMSY-system, developed
i n this department (Schoofs, v. Beukering and S l u i t e r ; 19781, was gratefully
used.
R. Huiskes
Report no. RH-78-21, Dept. o f Orthopaedic Surgery, University o f Nijmegen, The Netherlands.
Of t h i s report, nothing may be copied o r Gsed i n any way, w i t h o u t permission of the author.
Contents Preface 1. Introduction 1.1. Problem definition 1 . 2 Assumptions 1.3 Parameters 1.4 General considerations 2. Method o f solution 2 . 1 2 . 2 i n e axisymmetric Fourier-element 2.3 2 . 4 Presentation o f output 2.5
Fini t e Element Methods
El ement meshes and boundary condi t i ons
T h e modelling o f the sliding interfaces
-.
3. Results -^I_ 3.1 The s t r e s s distribution 3.2 Conclusion 3.3 3.4 3 . 5Stress distribution with a fixed stem
The influence of the stem length
The influence o f the p l u g properties
4. Discussion and conclusions
References 11 16 19 21 40 42 52 54 60 66
- I -
1. Introduction
1.1. Problem definition
The s t r u c t u r e t o be analysed ( f i g . 1 .
stem, t h a t s l i d e s into a plug, which i s fixed i n t o the medullar cavity of
consists of a s t r a i g h t , round metal
the bone. bone I Plug I- 7 pse os-tke s i s
I
i
I d i sL;-~1 - - - - __ I 27 -*
----f i p , , l . l The structure t o be analyzed.
7 110
I
The stem has a broad shoulder, s i t t i n g on the plug. A gap e x i s t s between the plug and the proximal rim of the bone. Forces and couples may be applied on the semi -round head of the prosthesis.
I t i s the object
of
t h i s investigation t o evaluate the s t r e s s d i s t r j b u t i o n i nthe d i f f e r e n t materials and a t the interfaces, on a r b i t r a r y loading of the head; second, t o study the influence o f choices of geometrical parameters of
the stem (two d i f f e r e n t stem lengths) and material properties of the p l u g
( t h r e e d i f f e r e n t materials: Delrin, UHWM Poly Ethylene, S i l a s t i c ) ; and t h i r d
t o develop c r i t e r i a f o r design optimization, based on the analyses r e s u l t s .
1.2. Assumptions
The three components a r e assumed
t o
be a x i s p e t r i c in geometry. The mechanical properties of the three materials a r e assumed t o be linear e l a s t i c , isotropic and homogeneous-2
-
Friction i s assumed t o be zero between the s l i d i n g stem and the p l u g . The plug i s assumed t o be fixed t o the bone, i n tension as well as in shear.
The local e f f e c t s of the load introduction on the head a r e assumed t o have disappeared near the stem (Saint Vinant's principle).
Due t o the s l i d i n g stem, there i s no torsional load.
1.3. Parameters
i n the analysis, two d i f f e r e n t stem lengths are considered: L = 10 mm and
i = i 6 mm; other geometrical parmeters reinain constant.
As a consequenceof the above mentioned assumptions, any loading can be
simulated combining three d i f f e r e n t u n i t loading s i t u a t i o n s ( f i g . 1 . 2 . ) :
Axial force ( Z ) transverse force ( X ) and bending couple ( M ) .
-
fig.l.2 Loading on the joint.Although the r e s u l t s ( s t r e s s e s ) will be proportional t o the loading, so t h e i r absolute values do not matter i n the analysis, the loading parameters a r e given more or less real values. I t i s assumed t h a t the fixation'systen! would have t o function i n the metacarpo-phalangeal j o i n t . Values f o r the loads are based on measurements by Chao (1977), f i g . 1.3.
The values he gives a r e much higher t h a n those given by Berme e t al ( 1 9 7 7 ) , f o r normal j o i n t s and a l s o higher t h a n values resulting when the finger t i p
-3
flexion-extension
u n d e r
compression
-
s h e a r
forces
ntorsion in extension
lateral
moment in
m
5.ûNm
f
3.1Nm
m
2.32Nn-i
di
sP
ract
i
on
metacarpal
.stem
Qf
'fig.i.3 The different forces which can act on a prosv
thesis, as a basis f o r a test program. The
values given are the average maxima of nor-
- 4
-
forces i n normals, as measured by Berme (19771, a r e m u l t i p l i e d by factors f o r MP-prostheses, as given by Chao e t al. (1976). The values a r e compared
i n Table I .
Chao, 1976 and Taken f o r the
Berme, 1977 analysis Chao, 1977 Bernie, 1977
z
( N ) 628 186 280 314 628 54 180 314--
1 616 M (Nmm) 2320--
Table f Maxinuril j o i ~ t forces.
The mechanical material properties of the materials, as used i n the analyses, are summed u p i n t a b l e
11.
~~ ~
Young’s M2dulus (E) Poisson’s r a t i o ( v ) 6To.2 (N/mm2)
N/m ( 1 ) (11 Yield strength metal (Co.Cr.Mo) 200,000 0.30 450 (2) delrin UHMWPE Si 1 asti c 4,000 1,200 10 0.35 0.38 0.45 cum ) Bone 14,000 0.30
Table i 1 Material properties ( ( 1 ) : Walker, 1978; ( 2 ) : Walker, 1977;
- 5
-
1.4. General considerations
The model is applied with a cylindric coordinate system (r, y , Z) (fig. 1.4.);
the tangential direction, described by the angle 9 , is being denoted by ' t i e Displacement of a point (r, 9 , 21 inside the structure, is being described
by a radial
(u,),
a tangential(ut)
and an axial (u. z ) component (fig.l.5).X
1
d i __ ____ stal.
fig.1.5 Displacements
fig.1.4 Cylindric coordinate
system
Stress in a point o f the structure is being described by the stress tensor4 that has 9 components (fig.1.6). (Timoshenko and Goodier,l970). The notation
is such, thatsigma's ( U ) are normal stresses on a plane indicated by the in-
dex ( D ~ : Normal stress on an r = constant plane), tension is positive, com- pression negative; tau's ( T ) are shear stresses, plane and direction indicated
by the indices ( T ~ ~ : shear stress on r = constant plane
in
tangential direction). The stress tensor is symmetric, SO that: T~~ =T.^^;
-crZ-
- 'zr and T~~ = T ~ ~ ,which means that only 6 independent components remain to be considered.
As was already indicated: On an arbitrarily sectioned plane
inside the structure, 3 stress components are to be considered (fig.1-71, 2 shear stresses and one normal stress.
- 6
-
fig.l.6.: The nine components of t h e stress tensor
fiE.l.7 s t r e s s e s on arbi- trarily sectioned plane..
-
fig.1.8 s t r e s s e s on coordinate planes,- 7
-
i s
If this plane an r = constant, 4 = constant or z = constant plane, then
these 3 stress components coincide w i t h 3 o f the 9 components of the stress tensor ( f i g . 1.8.).
In the structure to be analyzed, the interface-planes are either r = constant
(stem-plug interface), r = constant b y approximation (plug-bone interface) or z = constant (shoulder-plug interface), so the stress components can be used
without transformation.
The 3 stress components on, for instance, the stem-plug interface plane
( r = constant plane) are ar (normal stress), -rrZ (axial shear stressj and
T r t (tangential shear stress) (fig.l.9).
contineouc discontineous I ., -
....
f ' i g . l . 2 Stress components on and near the stemrpìug contact planes (interface).
From the 'action = reaction'-law i t follows, that these interface stress com-
ponents have t o be contineous across the interface, so that ( f i g . l . 9 ) :
o r stem 3 0 r bone r zstem 'r zbone rtstem rtbone
-
T = T TThe other s t r e s s components near the interface (G
,
CJ,
T t 2 tzbe continews, i n f a c t have t o be discontineous since t h e i r related s t r a i n components a r e contineous and the E-moduli of the materials a r e d i f f e r e n t , so that:
1
do n o t have t o t stem+
't bone t stem # ''2 bone tzstem+
Ttzbone (TA t the interface ( r = constant) (1
z
A t the shoulder-plug contact (z = constant plane) the contineous interface
s t r e s s components are: oZ, r
zr
and T ~ ~ .If two materials a r e able t o s l i d e r e l a t i v e t o each other inside the structure, thmboth the interface shear s t r e s s e s have t o be zero, whiïe tne normal stress i s e i t h e r equa! t o zero o r negative (Compression).
To judge the s t r e s s s i t u a t i o n i n a point of the s t r u c t u r e one can e i t h e r study the separate s t r e s s somponents or combine them t o , what i s called an'equivalent s t r e s s ' . Several c r i t e r i a may be found in l i t e r a t u r e ;
val ent s t r e s s according t o Maxwell -Huber-Hencky-v .Mises (Koi t e r
,
1960) ehere i s used the equi-
This formula follows from the c r i t e r i a t h a t a material will f r a c t u r e as the specific def ormati on energy reaches a certain 1 evel ; t h i s 1 evel bei ng dependent
on the strength of the material, T h i s O
eq
the strenth of the considered material, as evaluated i n t e n s i l e or bending may then d i r e c t l y be compared w i t h
t e s t s and
I t should cal mater l i t t l e i s
given i n the l i t e r a t u r e .
be k e p t i n mind t h a t this c r i t e r i o n has a l s l i k e s t e e l . Nothing is known about known about p.iastics.
proven
t s val
t o be valid f o r teckni- d i t y f o r bone and
The above mentioned nine component s t r e s s tensor describes a 3-dimensional
s t r e s s s t a t e ; i f plane s t r e s s i s assumed, the s t r e s s s i t u a t i o n can be described
- 9 -
Although i t would be possible t o use a 2-dimensional model f o r the structure
t o be analyzed, i t has been shown t h a t such a model i s not very well suited t o study the s t r e s s d i s t r i b u t i o n i n such a s t r u c t u r e , especially in the p l u g ,
a t the interfaces and i n the proximal side of the bone (Huiskes; 1978a)
Because local e f f e c t s from the load introduction on the head may be neglected, the loads can be assumed t o be distributed over a c i r c l e ( f i g . l . 1 0 ) .
fig.l.lC Modeling of t h e loads ( 2 , X and NI) as 9-dependend
f u n c t i o n s , f, o r f,i w i t h amplitudes
?,
andpz,
The d i s t r i b u t i o n follows from:
Axial force: f, =
fz
(N/radian), w i t h Z = 2{?,.
d+
Couple:
Transverse force: f r = fr.cos+ (N/radian), w i t h : X
=qfr.
OO n 2
f e = ?,.cos 4 (N/radianj, w i t h : M = 2/f,.rb:Cos +.d+
cos +.d4
o n 2
I t can be shown t h a t i n t h i s case the s t r e s s components can be described by
-
- / I
-
2. Method of solution
2 . 1 . Finite Element Methods
Although analytical theoretical s t r e s s analyses methods ;iiay be used in certain circumstances, F i n i t e Element Methods (Ziekiewicz, 1977) a r e most suited f o r relatively complicated structures l i k e the one t o be analyzed.
I n using t h i s method, the structure i s devided into a certain type of elements;
the elements a r e connected in discrete points, called nodal points. A t these nodal points bowidary conditions, e.g. forces o r displacements, may be
speci f i ed.
I n every element a displacement f i e l d i s assumed as function o f ine coordinates.
This f i e l d i s expressed in the displacements o f the nodal points o f the element. Using a minimal energy principle, a relation i s found between the nodal forces and the nodal displacements. This r e l a t i o n , a s e t of 3n l i n e a r equations ( n being the number o f nodal points) i s solved u s i n g a computer. Stresses a r e calculated fron: the displacement f i e l d , u s i n g compatibil i t y relations and Hooke'c law.
The accilracy of the r e s u l t s may be increased by increasing the number o f e7 ements.
For the calculations reported here, a Finite Element Method computer system was used (Banens, 1978).
The system uses as i n p u t : Characterization of the element; nodal point coordinates; nodal point locations ( r e l a t i v e t o the elements) ; material properties of each element; prescri bed displacements or nodal forces ; data concerning the connection of nodal points.
O u t p u t of tne system: Displacements and s t r e s s e s i n each nodal p o i n t .
2 . 2 . The axisymmetric Fourier-element
The computer-time and -memory needed hold limitations f o r the number of elements used. If t h e structure i s t o be analyzed f o r d i f f e r e n t values of the parameters,
i t i s advisable t o choosea ' l i t t l e time and memory consuming element.
An axisymmetric construction can accurately be devided i n t o ring elements. If a n axisymmetric load i s applied t o the construction, then every section containing the axis will deform in
a n
identical way; consequently then onlyone section has t o be studied. For non-axisymmetric loading t h i s i s not t r u e . However, i f appl ied 1 oads displacements
,
s t r a i n s and s t r e s s e s a r e expanded in Fourier s e r i e s , then the same single section can be used f o r every term of the load expansion,and s t r e s s e s a r e calculated term by term (Wilson, 1965; Zienkieuicz, 1977; Huiskes 1974, 1977a).
-
12
-
In t h i s way computer time i s reduced tremendously compared t o a r b i t r a r y 3-D elements, while the r e s u l t s f o r an axisymmetric structure a r e most accurate.
For the structure t o be analyzed here, only one term of the Fourier s e r i e s
hzs t o be used (see chapter
1).
The s t r e s s amplitudes, defined i n chapter 1,are eval uated.
The 6 node Fourier element i s shown i n f i g . 2 . 1 .
f i g . 2 . i. : the used
ring
element with cylindrical
coordinate system
2.3. El ement meshes and boundary condi tions
As explained before, only one section containing the axis has t o be divided in elements. This division i s carried o u t automatically by a computer program (Triquamesh; Schoofs, v.Beukering and Sluiter1978).
Three l e v e l s of refinement were tri;-ld(fig. 2 . 2 . ) . The coarse mesh would n o t be accurate enough, the f i n e mesh would use too much computer time and memory. The one i n between was chosen t o be used in the calculations. The structure consists of 3 substructures ( f i g . 2 . 3 . ) , connected by t h e i r nodal points a t the interfaces.
Fig. 2.4. shows again the used element mesh, together w i t h the mesh f o r the longer stern. Also the boundary conditions a r e given: The s t r u c t u r e i s fixed a t the d i s t a l side ; loads (amplitudes) a r e applied on nodal point 6.
m r. 9 hi W
..
I I
I
.. 1 1 .. .. ' .. .. ... .. ... .. ,.. .. ... .." ..' <. .. .. .. .. .. .. .. .. .. .. .. .. .7
.. .. .. .. .. .. .. .- --7
.-
I
2.4. Presentation of output
Three displacement amplitudes and 6 stress amplitudes in each nodal point are calculated and printed out, for every loading case.
In addition two different kinds of plots are drawn:
1. In the longitudinal section of the 3 substructures the lines o f equal stress values, in a certain range, of every stress amplitude (
6r,0t,$Z,+rt,
I rz yftz) (fig. 2.5.)
This was done for the transverse force only.
f i g . 2 . 5 : L i n e s of e q u a l stress i n p a r t of a s u b s t r u c t u r e , for a c e r t a i n l o n g i t u - d i n a l s e c t i o n ( e x a m p l e ) .
It should be kept
in
mind, that the stress amplitudes may be interpreted as stresses in the following way (see also fig. 2 . 6 @ ) .For the transverse force and the couple:
a - a
u and? rz as ar, at’ aZ and ‘crZ
in
the section 9 =o
Or, “t,
z
frt arid ftz as
For the axiai force:
i n the section 4 = 90’
‘rt and ‘tz
2. Displacement- and stress-amplitudes
in
nodal points laying on certain lines are plotted as function o f the location of the points on the line (see fig.2.6. ).
Also the equivalent stresses on each line in the section
plotted in this way.
' I
-
t o
-
The l a s t method gives 7 graphs f o r every l i n e , so 42 graphs f o r each loading case, 1 2 6 graphs f o r each calculation.
From analyses of comparable structures i t has been established t h a t not a l l
the s t r e s s components in a l l the materials are important f o r an evaluation o f t h e s t r e s s s i t u a t i o n .
The s i g n i f i c a n t components a r e summed up in table 111.
--
Material
1
ine Si g n i f i cant s t r e s s Number o f graphs component axial load. trans.load.i
1
3 4 n h A eq o r o Stem 1 OZ Plug 2 o z ’ atzand 5 3 3 4 Bone 4 3 I n t e r faces 112 3 1 4 2 2 2 3 3 3 t o t a l 16 21Table
Ui
Significant s t r e s s components.Further i t was established t h a t the r e s u l t s from the transverse force and from the couple give, in general, t h e same type of information, so the s t r e s s s i t u a t i o n of the s t r u c t u r e can be studied using 37 graphs.
I t s h o u l d be remarked a t t h i s point t h a t not a l l the graphs f o r a l l the calcula- tions were plotted in the r i g h t way. Some f a i l e d because of detailed technical or numerical problems. Only in the cases where the specific f a i l e d graphs were r e a l l y needed,it was decided t o r u n the program again.
W do hope t h a t the reader will not develope the “graph-aches’’ from a l l the e graphs t h a t will be presented. The F i n i t e Element Method gives much information about t h e stress-distribution and there i s no way of going around t h e f a c t t h a t t h i s d i s t r i b u t i o n i s 3-dimensional and has t o be described by 6 parameters i n each point.
I t was f e l t t h a t i n f u r t h e r reduction of the data (e.g. presentation of only equivalent s t r e s s e s ) important information would have been thrown away,whále leaving o u t o f certain
,
possible in principle, would do no right t o t h e time and work invested.I t c e r t a i n l y will need some practice for a reader without a background in
applied mechanics t o f i n d - - h i s way through
our utmost t o make the interpretation as easy as possible.
the presented r e s u l t s . We have done
2.5. The modeling of the sliding-interfaces
A special problem i s the modeling of the loose interface between stem a n d plug.
I n the FEMSYS program, substructures a r e coupled together by coupling
the three degrees of freedom
(ur,
u t anduz
)of t h e i r interface nodal p o i n t s (see f i g . 2 . 7 . ) .a. b. 6.
fig.2.7.: Coupling of interface degrees of freedom (b); if ut and ut' are un- coupled, thglthe 2 materials may slide in circumferential direction relative to each other (a); if uz and u'z are uncoupled, then in axial direction.
If a l l 3 degrees of freedom a r e l e f t uncoup loose, a l s o w i t h respect t o compression.
If only
ut
and uz a r e l e f t uncoupled, thened, then the interface i s completely
he materials may s1 ide r e l a t i v e t o each other in a l l directions, b u t there i s s t i l l a connection in the direction perpendicular t o the surfaces, resulting i n compression o r tension.
FEMSYS has the option, t h r o u g h an i berative process, t o model selective coupling;
f o r instance t o couple
u r
only i f a negative s t r e s s (compression) i s present. With this option i t i s possible t o model s l i d i n g interfaces quite accurately.- 2 0
-
Since the displacements on a "nodal ring" a r e represented by:
= ' G r cos @
,
ut = Ct s i n + and uz = UZ cos+Such a r i n g may e i t h e r be model ea
(sliding o f the materials t o each other) t h i s i s no problem, since the sliding will occur on the whole interface. For u r t h i s will be a problem, since one s i d e
of the nodal ring may be in tension and the other one i n compression; meaning t h a t the option of the i t e r a t i v e process can not be used.
coupled or uncoupled. For u t and uz
The e f f e c t of different modeling of loose interfaces i s presently being investi- gated, u s i n g 3-dimensional models, comparable t o the model studied here, 2-
dimensional mode? s and analytical methods.
Based on f i r s t r e s u l t s of t h i s study, the following procedure was chosen t o
rn~de! the slidin3 stem
-
plug interface, f o r transverse loading:u t and uz a r e uncoupled a t the interface (so there will be no shear s t r e s s e s , t h i s proved t o De very important).
u r i s coupled a t the interface and the s t r e s s amplitude r e s u l t s near the i n t e r f a c e i n the plug a r e interpreted i n the following way:
71 IT CT = 2 . 8 .cosqj
-
-
r r 2 <$ <- 2z
<+ <- 2 i f û < O r TT"
o= o
r a = o r o = 2 . 8 .cos+ r rThe only s t u d y t h a t took loose interfaces into account, was published by
Svesnsson e t al e (1977) in a 2-dimensional model. Beside t h e circumstance t h a t
in such a model the 3-0 stress d i s t r i b u t i o n i s not accurately evaluated, he could only couple o r un'couple nodal points as a whole; this means t h a t i f
compressive s t r e s s e x i s t s , then also shear stresses e x i s t . Their conclusion
was, t h a t t h e "loose" interface d i d n o t have significant influences on the magnitudes of the maximum s t r e s s e s , b u t d i d change the s t r e s s d i s t r i b u t i o n .
- 2 1
-
3. Results
3.1. The stress distribution
On the following pages
,
the results (stresses) for the "reference" calculationare presented:
Stem length: 10 mm; P l u g material: Delrin; sliding stem.
Figs. 3.1 t h r o u g h 3.14 give stress components as calculated f o r loading case Z ,
near and a t the interfaces. Stress values are given in N/mm
.
The graphs are directly traced from the computer plots; values between nodal
p o i n t s are
1
inearly interpolated.2
Figs. 3.15 t h r o u g h 3.35 present the stress components for loading cases X
(straight lines) and M (intercepted lines). In this case the stresses are not
independent o f
+
: The iseatisn UP the stress cmnponent i s I n d i r ~ t e dfigures
.
I t should be kept in mind, t h a t the stresses were calculated for
values : Z = 314 (NI
X = 314 (NI
M = 1,616 (Nmm)
the stresses are linear with the loads, so results of different
may be found by simple multiplication.
the
oad
i n the
1 oad
ng values
I n those graphs t h a t present norma7 stress components (contrarily t o shear
stress components), the maximum tensile yield strength
material
,
i s indicated, i f this value f i t s within the stress scale.(Y.s.) of the specific
Figs. 3.36 through 3.41 present plots of iss-stress lines in longitudinal sections o f the 6 independent stress components; lines of equal stresses are plotted
in a section, t o be interpreted as the section o f maximum stress for t h a t
component.
The choice of values, for which the lines of equal stress (hereafter called
i s o ' s ) are plotted, was chosen before-hand. Because this plotting program has
only been run once, the choice of the is0 values i s not optimal. Plotted
are i s o ' s higher then a certain value in absolute sense; these values are indica-
-22.
-
LEGEND AND SEQUENCES OFTHE GRAPHS
.
Stem-plug interface Plug,at shoulder interface.
Plug,near the stem 2 l a _ k S t r e s s component Stressin
H/mmt
-Loading case,Bone edge region
.
Plug-bone Bone,nearinterface the plug
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Stress in N/mm
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Stress in N/mm
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3.2. Concl us ion
The results should be interpreted i n r e l a t i o n with the strength o f the materials and also with the applied loads.
The stem: The introduction o f the axial force holds no problem f o r the stern ( f i g . 3.1). Stress concentrations are found i n the curve, where the stern i s connected to the head.
Stresses resulting from introducing the transverse force are quite high ( f i g s .
3.15 and 3.16) and may be higher then the strength o f the material. It should
be noted from f i g . 3.15 (axial stress) and 3.16 (equivalent stress) that the
axial stress i s p r a c t i c a l l y equal t o the equivalent stress, indicating that
the stem i s behaving accord-ing t o beam theory.
High stress concentrations are found near the stem-head connection ( f i g s . 3.36 through 3.41). Especially interesting i s f i g . 3.40, showing the shear stress
; T~~ equals i w û a t the sten-plug interface, as i t s h ~ l i ! d , but i s
quite high inside the stem. This i s because the steiii hàs to c a r r y a71 the shear
stress frm the trafisverse force ( f i g . 3.42).
I n the model, ’load i s also introduced from the shoulder t o the plug. If t h i s
shoulder does not r e s t on the plug anymore f o r same reason i n reality, then al SO
the f u l l moment froni the transverse force and the couple i s carried by the stem.
In t h i s case the axial stress i n the stem w i l l be very high (see f i g . 3.42).
z r
T r z = , r
The stewplug interface: O f the interface stresses (ur, -crt9
only
resulting from the transverse loading and couple has significant value plug however.
ar
( f i g . 3.17)
2
. The maximum value (
,+
-
80 N/mm ) w i l l not harm the stem; could harm theThe plug: Y i e l d strength values mentioned i n the graphs f o r the plug are those o f Delrin. On introduction o f the axial force, the stress situation i s not very serious i n the plug, although high stress concentrations e x i s t near the bone edge contact. ( f i g s . 3.6, 3.7, 3.8, 3.9 and 3.10). The maximum equivalent stress
i s about 25 N/mm ( f i g . 3.81, which i s quite high for t h i s material. Also the stresses near the stem-head connection are high ( f i g . 3.14), up t o about the same order o f magnitude. As can be e a s i l y seen from f i g . 3.14, the outer part o f the plug on which the shoulder rests, has no real load carrying function.
On introduction o f the transverse force, again high stress concentrations are seen near the same regions, the bone contact and the contact a t the stem-head connection
( f i g s . 3.36 through 3.42); i n the l a t t e r part up t o about 120 N/mm
.
Stress components near the interfaces are also very high f o r t h i s material. ( f i g s . 3.24 through 3.34). The maximum equivalent stress hereis appr. 80 N/nm2 ( f i g . 3.27) The stresses on the distale side are somewhat lower, but s t i l l i n the order o f magnitude o f 30 N/mm
.
2
2
- 4 2
-
The p l u g - b o n e interface: The introduction of the axial force gives no problem
t o the plug-bone interface, except ofcourse f o r the said concentrations near the bone edge ( f i g s . 3.36 through 3.41). The shear s t r e s s ( T ~ reaches a maximum ~ ) of appr. 6 N/mm
.
( f i g . 3.10).Upon introduction of the transverse force this shear s t r e s s reaches a value
of 50 N/mm
i s not very high ( a p p r . 2 N/mm2, f i g . 3.29). The normal s t r e s s on t h i s interface reaches appr. 35 N/mm ( r r , f i g . 3.28).
2
2
( f i g . 3.30). The other interface shear s t r e s s component ( - r r t )
2
The bone: Assuming t h a t the bone in i t s natural s t a t e behaves approximately according t o beam theory, the most s i g n i f i c a n t s t r e s s component will then be the axial normal s t r e s s (O,). F i g s . 3.13 and 3.?2 present t h i s s t r e s s
component a s calculated f o r the "natural
''
bone without implant, using beam theory; identical loading as in the FEM-calculationsse s t r e s s values match f a i r l y wel 1 with the "unnatural
''
values, except on the p r o -xima! side, where the s t r e s s e s a r e 2 t o 3 times as high.
i s assumed. As can be seen, the-
Also ofcourse the Interface s t r e s s e s ( T ~ , , T r t , a r ; f i g s . 3.4, 3 , 8 , 3.28, 3.2gI and 3.30) a r e quite unnatural f o r the bone; their values a r e very h i g h f o r the transverseloading case, especially near the bone edge (see also s t r e s s concentrations f i g s . 3.3 6 through 3.41). Also the circumferential s t r e s s ( U
mich is'unnatural'high near the edge ( s e e f i g s . 3.11 and 3.311, may be of impor- tante.
I t i s d i f f i c u l t t o use c r i t e r i a f o r the bone material, however; since too l i t t l e i s known about the influences of d i f f e r e n t s t r e s s components i n bone.
3.3. Stress d i s t r i b u t i o n w i t h a fixed stem
In t h i s chapter the influence of the s l i d i n g aspect of the stem i s being evaluated by comparing the calculated r e s u l t s f o r t h i s case w i t h r e s u l t s from the case in which the stem was modeled as being fixed.
Figs. 3.43 through 3.51 present comparisons of some stress-components as
calculated with a s l i d i n g stem and as calculated with a fixed stem, f o r the axial force.
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; '
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Ofcourse, f o r the axial force, the sliding stem has no mechanical function and i s unstressed ( f i g . 3.43). I n the case o f the fixed stem, the stem stresses reach maximum values o f appr. 60 Nmm', f o r t h i s loading. A shear stress o f
maximum value appr. 6 N/mm w i l l be present a t the stem-plug interface. ( f i g . 3.44). As could be expected, the stresses i n the plug near the stem-plug interface
are higher with the fixed stem ( f i g . 3.45); except f o r the proximal side. Near the plug-bone interface the stress components are smaller f o r the fixed
2
stem, as can be seen i n f i g s . 3.48 and 3.49.
The stress situation i n the bone i s better f o r the fixed stem ( f i g s . 3.50 and
3.51).
Aso the normal stress on the shoulder-interface i s less f o r the f i x e d stem ( f i g . 3.46); ofcourse i n t h i s case there w i l l be shear stresses a t t h i s inter- face ( f i g . 3 . 4 7 1 , these are however, not too high.
F i g s . 3.52 through 3.60 present comparisons o f the same cases f o r the transverse
1 oadi na.
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int.
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.
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bo.
Stresses in the stem will be higher in t h e s l i d i n g case ( f i g . 3.52).
Stem-plug i n t e r f a c e s t r e s s e s ( f i g s . 3.55 and 3.56) a r e lesslofcourse, except f o r
CI the normal s t r e s s a t the contact surfaces ( f i g . 3.53 and 3.54); a s a r e s u l t
a l s o the equivalent s t r e s s in the plug, near the stem interface i s higher f o r the s l i d i n g case ( f i g . 3.57). The s t r e s s e s near and a t the plug-bone
a r e somewhat less in the sliding stem case ( f i g . 3.58).
The s t r e s s s i t u a t i o n s in the bone a r e approximately equal f o r the two cases ( f i g s . 3.59 and 3.60).
r’
interface
Figs. 3.61 and 3.62 give the Tines of equal s t r e s s in the stem, f o r the transverse loading case and a fixed stem.
Fig.s. 3.63 and 3.64 give magnitude and directions of principal s t r a i n s in a longitudinal plain, also f o r the transverse loading and the fixed stem.
O M
b
4
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fig.3.66: Magnitude and direction of principle strains in the plug
int.
In general i t can be concluded t h a t there is a significant difference between
both cases, especially w i t h reference t o stresses in the stem and a t the interfaces.
But also the s t r e s s s i t u a t i o n i n the bone on introduction of t h e axial force d i f f e r s considerably.
In using these r e s u l t s t o f i n d c r i t e r i a f o r a design, one should weigh the strength o f t h e interfaces against the strengths of the d i f f e r e n t materials.
In f u r t h e r calculations (long stem, UHMWPE-and s i l a s t i c p l u g )
t h a t the stem was fixed i n the p l u g , for reasons o f efficiency and economy (computer space and time). The comparison, presented i n t h i s chapter, can be usea t o extrapolate t h e results sf these further analyses t o t h e case o f the
i t was assumed
sliding stem.
3.4. The influence of the stem length
The only possible reason f o r eióngätiny the ctex WOU!^! be t o enlarge the i n t e r - face surface, as t o lower those interface stress components that t r a n s f e r the load t o the p l u g . These s t r e s s components a r e the shear s t r e s s -rrz f o r the axial load; the normal s t r e s s ar and the shear s t r e s s e s -crt f o r the transverse load. As could be concluded from other analyses of intramedullary systems (HLtiskes
1 9 7 7 a ) , these components may be lowered by elongation of the stem u p t o a
certain length. The value of this certain length i s dependent on other
properties of the structure. If the stem i s made longer than t h i s optimal length, then this h a a negative influence.
Although there appeared t o be some differences i n the s t r e s s distributions of
b o t h s t r u c t u r e s (short stem, 10 mm and long stem, 16 mm), these were not
very s i g n i f i c a n t . In general i t can be concluded t h a t the longer stem i s c e r t a i n l y not worse then the short stem, b u t there i s l i t t ? e benefit f o r the mentioned interface s t r e s s components, as shown i n f i g , 3.65, 3.66 and 3.67.
int.
4
int.
The i n t e r f a c e s t r e s s -crZ (transverse force) is higher f o r the longer stem
( f i g . 3.68). This i s due t o the higher bending moment a t the d i s t a l t i p ,
resulting from the force.
int.
-5%
-
If the stem i s s l i d i n g in the p l u g ,
zero, ofcourse. In t h i s case the longer stem would be more beneficial for the rather h i g h normal interface s t r e s s u r ( f i g . 3.173
t h i s s t r e s s component ( T ~ would be ~ )
3.5. The influence of the p l u g properties
The r e s u l t s presented upto now r e l a t e t o the Delrin plug ( E = 4000 N/mm $
I> = 0.35).
Calculations were also performed simulating an UHMWPE ( E = 1200 N/mm and a S i l a s t i c (E= lON/nd,v = 0.45) plug.
2
2
v = 0.38)
The UHlill.I?E-pl iicj compured t a the De? ris?-pl ug:
Because the UHMWPE i s more f l e x i b l e then the Delrin, the stem will bent more inside the f i r s t material. T h i s r e s u l t s i n much higher stem s t r e s s e s especially f o r the transverse loading ( f i g . 3.75).
The difference wiii be even mort. ûulspcker?, if the stem i s ableto s l i d e i n the
P1 u9.
Aso because of the greater f l e x i b i l i t y , the interface shear stresses will be
smoothed o u t better across t h e surfaces, using the UHMWPE ( f i g s . 3.70, 3.71 and 3.78), this means t h a t t h i s material would benefit more Prom u s i n g a longer stem.
Further differences a r e present, b u t n o t so important. Because the plug has l i t t l e e f f e c t on t h e overall s t i f f n e s s , the deformation i s in essence being prescribed by the bone so t h a t due t o t h e lower modulus o f e l a s t i c i t y , s t r e s s e s in the
UHMWPE plug a r e in general lower ( f i g s . 3.72 and 3.81).
Differences are also due t o the higher Poisson’s r a t i o : t h i s e f f e c t i s treated in the next part ( Silastic p l u g ) .
The S i l a s t i c p l u g compared t o the DeIrin-plug.
Because t h e modulus of e l a s t i c i t y of S i l a s t i c i s many times lower then the Delrin and UHMWPE, the above mentioned e f f e c t s would be even more outspoken.
As can be seen in f i g . 3.’75, the s t r e s s e s in the stem upon transverse loading will be appr. 4 times as h i g h , u s i n g the S i l a s t i c . Ofcourse, because more load i s being introduced through the stem, l e s s load i s being introduced across the
shoulder ; normal s t r e s s e s a t the shoul der interface a r e therefor much 1 ower
,
using the S i l a s t i c a s can be seen i n f i g . 3.84.Because of t h e high Poissons r a t i o (almost incompressible) of the S i l a s t i c , the s t r e s s e s in the plug will be very high f o r a l l loading cases ( f i g s . 3.70,
3.73, 3 * 7 6 , 3.77 and 3.79). The interface shear s t r e s s e s will be a l i t t l e lower, however ( f i g s . 371, 3.78 and 3.80).
The s t r e s s distribution in the S i l a s t i c plug, i s such, t h a t the equivalent s t r e s s o are smaller then those in the Delrin and UHMWPE plugs ( f i g s . 3.72 and 3,131).
Because of t h e h i g h Poissons ratio,there i s also a marked influence on the circum- ferential stress("Hoop"stress) i n the bones using the S i l a s t i c plug ( f i g s . 3.74
and 3 . 8 2 ) , a l s o here the equivalent s t r e s s is smaller ( f i g . 3.83). As can be seen in most g r a p h s , s t r e s s concentrations especially on the distal side in the p l u g will be higher, using t h e S i l a s t i c .
Interpreting these r e s u l t s i t should be kept in mind t h a t the strengths o f the materials a r e different (Delrin : 70 N/mm2; UHMWPE : 2 1 N/mm and S i l a s t i c : In the order of 10 N/mm ) . 2 2 int. int. 3.
If
3.71I I
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4. Discussion and conclusions
The r e s u l t s should be interpreted carefully and it should be kept in mind t h a t several assumptions underly the model on which the calculations a r e based. The materials were assumed t o be l i n e a r e l a s t i c , isotropic and homogeneous; although this i s only t r u e by approximation, i t will give a f a i r description o f r e a l i t y , especially w i t h respect t o the technical materials. Bone has proven t o show a more complicated mechanical behaviour; as was concluded from
coiiiparable studies, however t h a t t h i s assumption leads t o acceptable r e s u l t s . (Huiskes, v.Heugten and Slooff 1976; Huiskes 1978a).
TLP. S I K geometry was approximated by an axisymmetric model e Again t h i s assumption
has t h e most consequences f o r the bone; as was shown i n t h e studies mentioned, also t h i s assumption will lead t o acceptable r e s u l t s , i f the real geometry
o f the bone i s not too wild.
The used theories are based an the asstiinptiori t h a t displurementc are small
compared t o the dimensions of the s t r u c t u r e ; this wil1 perhaps not be t r u e anymore in the case of the S i l a s t i c plug, so t h a t geometrically nonlinearities would
have t o be taken into account; s t r e s s e s wouid be higher, i n t h i s case, f o r the transverse force.
The magnitudes of the loads were estimated, using l i t e r a t u r e values. The magni- tudes used a r e probably somewhat on the h i g h side. Howevers t h i s gives no
principle limitations to the r e s u l t s , since these a r e ~ r ~ ~ o r t i ~ ~ a l with the magnitudes o f the loads. Any loading situation can d i r e c t l y be evaluated using the s t r e s s e s as presented.
The most d i f f i c u l t aspect of the s t r u c t u r e t o model i s t h e connection between t h e material s o
The pl ug-bone-interface was assumed t o be fixed. There e x i s t s much uncertáin-
t y about what happens a t t h i s interface. Probably t h i s assumption wil1 hold
for the shear s t r e s s e s and i t was shown t h a t these have the most influence on the s t i f f n e s s of the s t r u c t u r e and the s t r e s s d i s t r i b u t i o n .
The stem-plug interface was modeled as being a b l e $ s l i d e and also a s fixed.
Both these s i t u a t i o n s were propably modeled well enough; however, in r e a l i t y
f r i c t i o n will e x i s t between the materials, related t o the contact pressure, so parts o f the interface will be without shear s t r e s s b u t also some parts
w i l l be a b l e t o t r a n s f e r shear s t r e s s e s . Especially when u s i n g the s i l a c t i c
plug, the f r i c t i o n may be q u i t e h i g h upon combined axial ancl t r a n s v e r s e
loading ( s e e also f i g . 3.70).
-I! -
connections i s presently being studied, using different FEM and analytical models.
As f o r the r e s u l t s wesent?U kei-: o w could do hest by considvino hot t h o sliding stem mod+ls as well as the fixed stem model.
Looking a t t h e r e s u l t s one can conclude i n t h e f i r s t place t h a t i f the loading parameters were chosen in accordance w i t h r e a l i t y , a t l e a s t some- what of the r i g h t order o f magnitude, then the S i l a s t i c plug would be much
t o o weak.
Since there i s l i t t l e difference between the r e s u l t s f o r the Delrin-plug and
the UHMWPE plug, the f i r s t would be the best choice, because of i t s better
strength properties. As was shown, a l s o the lower Poisson's r a t i o of t h i s material i s favourable.
As t o the l e n g t h of the stem, i t can e a s i l y be concluded t h a t there will be very l i t t l e difference. The use of the longer stem has a small advantage, for t h i s s e t of parameters.
If the interfctces could be made s t r o n g enough, then there would be no use for the application of a sliding stem, reckoning with the loading cases considered here. Ofcourse in the case of a non sliding stem also torsional
the j o i n t loading could occur; probably i t i s possible however, t o design
in such a way t h a t r o t a t i o n a b o u t a longitudinal axis i s not be strained, a t l e a s t not f u l l y .
Another loading case t h a t might occur u s i n g a fixed stem i s the "pull o G t " of the stem upon impingement of the a
f i g . 1.3). The consequences of t h i s loading case evaluated by negative interpretation of the resu case, as presented here.
n g con-
axial see also joining bones
f o r the s t r e s s e s can be t s of the axial loading
As t o the geometry of stem and p l u g , several remarks can be made.
I n the f i r s t place the plug could be made much shorter without any
consequence.
The outer p a r t of the plugs shoulder appears t o have no function; a t the contact point with the bone rim, h i g h s t r e s s concentrations occur.
I t would be advantageous i f there would be no gap between the shoulder of the p l u g and the bone, so force would be d i r e c t l y transfered from the shol;lder t o the bone. This would not only diminish the s t r e s s concentrations in bone and p l u g , b u t also lower the stresses in the
-
69 -
4 4
This design perhaps has a disadvantage compared t o t h e original design rela- ted t o the procedure of f i x a t i o n : using t h i s design the tapered form of the bone would have t o be q u i t e exactg while the original one would have some clearance due t o the gap.
This disadvantage disappears i f the plug i s made i n two p a r t s , as i s shown i n f i g . 4.3.
fig.4.3: Design of the p l u g in two parts.
Although "pull out" forces and torsional loading dc not occur i n theory, using a s l i d i n g stem, due t o f r i c t i o n t h i s type o f l o a d i n g cou7
in real i t y . I t would therefor be advisable t o give the plug-bone interface some resistance t o movement i n t h a t direction, f o r instance by applying a p r o f i l e on the plug ( f i g . 4.
ingrowth ( f i g . 4 . 5 ) .
and a transverse screw o r room f o r bone
-&--
If the stem would n o t be designed as being able t o s l i d e , then i t would be advisable t o give i t a tapered form ( f i g . 4 . 6 ) .
fig.4.5: Proposal for optimal design, if a f i x e d stem is used.
The r e s u l t s , presented here, were interpreted assuming t h a t the f i x a t i o n system would be used for a finger prosthesis. Ofcourse the r e s u l t s have also general value for comparable structures. Also this system could be used f o r o t h e r j o i n t prostheses.
The design changes presented here, ~ h Q ~ l d be l o o k upon as clenera1 i d e a s s ; some analyses should be done t o evaluate the actual d ~ ~ ~ ~ s ~ o n s *
