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I Blomecchunirs. 1972. Vol. 5,pp. 541-551. PeqamonPress. Printed inGreat Britain

A MATHE MATICAL ANALYSIS F OR INDE NTATIONTESTS OF ARTICULAR CARTILAGE*

W. C. HAYES, L. M. KEER , G. HER RMANN+ an d L. F. MOCKROSTh e Technological Institute, Northwestern University, Evanston, Ill. 60201, U.S.A.

Abstract- A m a t h e m a t i c a l model is developed for indentation tests of articular cartilage. Thecartilage, normally bonded to the subchondral bone, is modeled as an infinite elastic layerbonded to a rigid half space, and the indenter is assumed to be a rigid axisymmettic punch.The problem is formulated as a mixed boundary value problem of the theory of elasticity andsolutions are obtained for the indentation of the layer by the plane end of a rigid circularcylinder and by a rigid sphere. Subject to detailed verification with independent tests, thepresent solutions are suggested as useful for the determination of the elastic shear modulusof intact cartilage.INTRODUCTION

OSTEOARTHRITIS is a noninflammatory dis-order of synovial joints characterized bydeterioration of the articular cartilage andby formation of new subchondml and marginalbone. Since the process is particularly severein joints exposed to high compressive forces,mechanical factors are assumed to play somerole in the pathogenesis of the disease(Sokoloff, 1969). Experimental studies ofthe mechanical properties of articular cartil-age, however, have not clearly elucidated theearly development of osteoarthritis nor haveanalytical studies fully considered the roleplayed by cartilage in load transmission innormal and pathological synovial joints.

step loads noted in these investigations isan instantaneous indentation followed by atime-dependent creep to an asymptoticindentation. No significant differences werefound as a function of age in either thedeformability or recovery of adult articularcartilage. Large differences were found,however, between the deformabilities ofnormal and osteo&hritic tissue.

Indentation tests have been used inprevious investigations of the mechanicalcharacteristics of articuiar cartilage. Suchtests have examined the influence of immer-sion (Elmore er al., 1963), the chemicalcharacteristics of the immersion medium(Sokoloff, 1963), age (Sokoloff, 1966; Krako-vits, 1969a), site of indentation (Kempsonet al., 197 1; Krakovits, 1969b) and thedegenerative state of the tissue (Hirsch, 1944;Sokoloff, 1966) on cartilage deformability.The characteristic mechanical response to

The derivation of meamngful materialproperties from indentation tests of thearticular structures has been complicated bythe lack of theoretical solutions for the com-plex stress distributions arising from the layer-ed geometry. Several previous attempts toobtain elastic constants from indention testsassumed a uniaxial stress field beneath theindenter. This assumption ignores importantedge effects occurring with indenters thathave contact areas whose diameters arecomparable to the cartilage thickness. Otherdeterminations were based on solutions forthe puncture of an elastomer or on hardnesstests, both of which ignore the presence ofa layered geometry.Several investigators have applied welldefined stress fields such as axial compression(McCutchen, 1962; Camosso and Marotti,*Receitied 28 December 1970.tPresent address: Department of Applied Mechanics. Stanford University, Stanford. California 94305, U.S.A.

541


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542 W. C. HAYES et al .1962), uniaxial strain (Edwards, 1967) anduniaxial tension (Kempson et al., 1968) instudies of the mechanical behavior of speci-mens of articular car&ge. A recent investiga-tion, based on two independent creep testsof cylindrical cartilage samples, presentedcartilage constitutive relations suitable forthe viscoelastic stress analysis of synovialjoints (Hayes and MO&OS, 1971). Allexperimental methods based on the use ofstandardized cartilage samples are not,however, suitable for use on intact cartilage.Interest in the development of suitable in uivotests for cartilage properties and in themechanisms of load transmission in normaland pathological synovial joints promptedthis analytical investigation of cartilageindentation.

Previous theoretical analyses of the loadresponse of the a3ticuIar structures have beenhampered by the lack of appropriate con-stitutive relations for cartilage and complicatedby the presence of the layered geometry.Zarek and Edwards (1963) used the Hertzsolution for the contact of elastic spheres inan attempt to correlate the observed fibrilIarultrastructure with the predicted distributionof tensile stresses. Burstein (1968) calculatedstress distributions and displacements inthree-layered spherical and cylindricalgeometries under the action of axiallysymmetric surface tractions but did notconsider the case of two contacting surfaces.The present investigation considers theindentation mechanics of an infinite elasticlayer bonded to a rigid half-space as a modelfor the layered geometry of cartilage andsubchondral bone. The analysis is formulatedas a mixed boundary value problem of thetheory of elasticity. It is based on the tech-nique of Lebedev and Ufliand (1958) forthe indentation by an axially symmetricindenter of an unbonded layer resting on arigid half-space. The present investigationexploits the Lebedev and Ufliand solution forthe case of a bonded layer indented by theplane end of a rigid circular cylinder or bya rigid sphere.

ANALYTICAL CONSIDERATIONSThe elastic contact problem is formulatedby considering he equilibrium of an infiniteelastic layer resting on an immovable rigidhalf-space. The layer deforms under the actionof a rigid axisymmetric punch pressed normalto the surface by an axial force P (Figs. 1 and2). Shear tractions between punch and layerare assumed negligible and the layer isassumed to adhere to the half-space at thesulfacez=h.Under these assumptions the problem isrepresented mathematically by a mixedboundary value problem satisfying the field

equations of the linear theory of elasticityfor homogeneous, isotropic materials. Thedisplacement equations are written as(1-2V)v%l+v(vu) =o (1)

in which body forces and inertial effects areneglected, u is the displacement vector,v is Poissons ratio, and V is the gradientoperator.The boundary conditions at the surfaceI= 0 are mixed with respect to normaltraction and displacement, the shear stressbeing zero over the entire surface. At z = h,the adhesion condition requires the displace-ments to be prescribed as zero. In cylindricalcoordinates, (r, 8, z), the boundary conditionsareu, = 00 -q(r) O=SrGa,z=O

a


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A MATHEMATICAL ANALYSIS 543

I p

20he5-- iwo___!Fig. 1. Geometry for the plane-ended cylindrical indenter.expresses the axisymmetric shape of theindenter (Timoshenko and Goodier, 1970)*,and the radius of the contact region is a.The solution of the axisymmetric contactproblem is conveniently achieved in terms ofthe Boussinesq-Papkovitch potential func-tions for the representation of the componentsof the displacement vector, i.e.

in which @, and a, are harmonic functions inthe layer, 0 s z s h, and G is the elasticshear modulus.Normal and tangential stress componentsare given in terms of the displacementpotentials as

and

The harmonic functions in equations (4) and(5) are written in the forma+,=_faA(h)shh(h-z)

+B(X)chh(h-z)]Jo(Ar) dh

a+= oa[C(A)shA(h-zz)+D(h)chA(h-z )]JoW dx, (6)

in which A, B, C, and D are functions of Ato be determined from the boundary con-ditions (2) and (3) and Jo(x) is the Besselfunction of order zero.By using boundary conditions (3) withequations (4) and (6), A and B may be deter-mined in terms of C and D asA(h) =-h C(A)- (3A4V)(h)

andB(A) = -hD(A). (7)

The third condition of (2) with equations (5)and (6) leads toD(A) = ( 1 - 2~) shAh - Ahchhh2(1 -v)chhh+hhshhh (). ()

The remaining conditions are the first two ofequations (2) which lead to the following dualintegral equationsI a0 C(A)J,(Ar) dA = 0

( r > a ) (9)I a C(A)M(A)J,,(Ar) dA-0 - &" " - - " WI

( r a ) , ( 1 0 )! P

R +

h

8

2 0_ _ Y_ _ _ - _ _ _

f - -

Fig. 2. Geometry for the spherical indenter.

*The assumptions of the theory used are given in detail in Timoshenko, S. P. and Goodier, J. N. Theory ofElm-ricir~. 3rd ed.. McGraw Hill Book Company. 1970, pp. 409-413.


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544 W. C. HAYES et al.in which

1M(A) =5; (3 - 4v)shhhchhh - Ah(Ah)*+4(1-v)*+(3-4v)sh*Ah 1and the function, C, is related to C by

1=() =x E AhshAh + 2 ( 1- v)chAh(Ah)2+4(1-v)2+(3-4v)sh2Ah 1

The objective of the analysis is to reducethe dual integral equations (9) and (10) toa form more amenable to numerical analysis.First, the function C is written in the formC(A) = A fO= (t) cos Ardt, (13)

which, when put into (9), automaticallysatisfies that equation. Equation (10) istreated by Grst isolating the half-space solutionin it as

in which

in which

andF(x) =$(O)+x~f(XsinB) dB]

The determination of cpt) from (16) resultsin a complete solution to the problem sinceall coefficients A, B, C, and D are given interms of this auxiliary function by means of(13).The distribution of normal stress beneaththe indenter may be given directly in termsofv(t) as

Integrating this expression over the area ofthe contact region gives a relation for thetotal applied force P,P = 27r _/; p(t) dr. (20)

Equation (20) allows the determination of

By putting (13) into (14) and following the the magnitude of the displacement of thestandard reduction procedures given ~ by center of the indenter, oo, under the action ofLebedev and Ufliand and later workers the a given load P. In the case of an indenter withequations may be written as a Fredholm a non-plane base, the continuity condition onintegral equation of the second kind with normal stress in the plane z = 0 provides thesymmetric kernel: additional relation

Iomp(t)[H(t+x)+H(t-x)] dt q(a) = 0. (21)In this case, equations (20 and 21) are used= F(x) (0 s x s a) (16) for the determination of the displacement w.


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A MATHEMATICAL ANALYSIS 545of the indenter and of the radius a of the Table 1. Values of K for the plane-ended cylindricalcontact region. indenterPlane-ended cylindrical indenter

For the geometry shown in Fig. 1, q(r) = 0.Introducing the dimensionless variables,

a/h v=O.30 Y =0*35 Y = 040 v-O-45 Y = 0.500.2 1.207 1,218 1.232 I.252 1.281o-4 1.472 I.502 l-542 1.599 1,683O-6 l-784 l-839 1.917 2.031 2-2110.8 2.124 2.211 2.337 2.532 2.8551.0 2.480 2603 2.789 3.085 36091.5 3-m 3.629 3-996 4-638 5.9702.0 4-335 4.685 5.271 6.380 9.0692.5 5.276 5.754 6.586 8.265 13.003-o 6-218 6.829 7-923 IO-26 17.863.5 7.160 7906 9.274 12.32 23.744-O 8.100 8.983 10.63 14.45 30.755-O 9.976 11.13 13.35 18.80 48.476-O 11&I 13-27 16.07 23.23 71.757.0 13.70 15.41 18.79 27.69 101.278-O 15.55 17.53 21.49 32.15 137.7

and CY Ah, equation ( 16) becomesW(5) =1 l - I w~T)[K(T 0 + K(7- 01 dr, (23)lr 0in which, from ( 15) and ( 17),K(u) =

(3 -4 u )s h a t P-[a (l+a)+$ -l--v)21[(a )* +4(1 -v)] +(3 -4v)Stia

X cos [a(a/h)ul da. (24)Numerical methods are used for the solutionof the integral equation (23) with the sym-metric kernel given by (24). At a given valueof the parameters a/h and u, the functionK(U) is calculated in the interval 0 c u =S2.The integral in (23) is then replaced by itsapproximation obtained from the trapezoidalintegration formula. The determination of thefunction w (7) is thus reduced to the solutionof a system of linear algebraic equations(Kopal, 1955). The relation between theapplied force, P, and the displacement, oo,is found from (20) and assumes the dimension-less form

* (25)Numerical values for o,(7) thus allow com-pUmiOn Of K at gk% VdUeS Of he parametersa/h and v. Values of K for a range of a/h aregiven in Table 1 for 1= 0.30, 0.35, 0.40.O-45 and 0.50. These K values may be

used to calculate relationships betweenthe load P and the indenter displacement W,for an indenter of radius a on a layer ofthickness h, shear modulus G and Poissonsratio v.Spherical indent&

For the spherical indenter geometry ofFig. 2, the function %(T) = 1212R, in whichR is the radius of the indenter. Introducingthe dimensionless variables of (22), theintegral equation ( 16) becomes

+K(r--_5)] d+& (26)in which K(u) is defined as in (24). Theunknown function w (5) may be written

o(S) = 0, (5) +&w?(I). (27)in which ~~(5) is the solution to (23) andw2 5) satisfiesW?(5) =-ri+; i w~(~)[K(~+[)+ZC(T--~)] d7. (28)I,


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546 W. C. HAYES et al.Using (27) the integral equation (26) is satis- S O.tied identically. The solution for the sphericalindenter geometry thus reduces to the deter-mination of the function or(e) satisfying (28)with symmetric kernel as given by equation(24). Numerical techniques outlined for thecase of the cylindrical indenter are used tosolve equation (28). The continuity condition,o(l)=O,onnormalstressattheboundaryofthe contact region allows solution for thedimensionless parameter

x bJ _-w(l)06R wz(l) (3)which relates the displacement % to the radiusof the contact region. With x determined,the relation between the displacement, 00,and the load, P, is given from (20)

K = w f O(T) dT. (30)0 0Values of x and K for a range of a/h are givenin Table 2 for Y= 0.30, 0.35, 0.40, 0.45,and O-50. These values of x and K may beused to calculate relationships between theload P and the indenter displacement o. andcontact region radius Q for an indenter of rad-ius R on a layer of thickness h, shear modulusG, and Poissons ratio Y.

RICSULTSThe present theory is a linear theory. As aresult, the predicted indentation by a plane-

ended indenter, with its fixed contact area,is directly proportional to the load. Becauseof the iinite extent of the load applicator,however, the indentation for a given appliednormal traction depends on the area aspectratio, u/h. The ratio (wo/h)/(P/GuZ) is plottedagainst a/h for five values of Poissons ratioin Figs. 3 and 4. The amount of indentationfor any given applied normal stress depends,quite markedly, on a/h for u/h values less thanunity. Fig. 3 shows the indentation depend-ence on u/h and shows the approach to

Area Aspect Ratio, u/hFig. 3. Nondimensional indentation of the plane-endedcylindrical indenter for a large mnge of area-aspect ratio.limiting values as u/h becomes large. Forindenters large compared to the layer thick-ness, the edge effects are negligible and theproblem is merely the compression of a thinlayer by a wide plate. Fig. 4 shows details ofthe behavior for 0 < u/h < 1, the usefulpractical range for indentation tests.Unlike the flat indenter, the contact areafor the spherical indenter depends on theload. Thus, the indentation of a sphericalindenter is not linearly proportional to load.Figure 5 shows the relative approach, oo/h,of spherical indenters as a function of a loadfactor PIGRP. The effect of Poissons ratio isshown for an indenter with R/h = 10. Theeffect of relative indenter curvature, R/h,is shown for v = 0.45. The results indicate,among other things, that the indentation forany particular load is less when applied witha large radius indenter than when appliedwith a small radius indenter.For the spherical indenter, calculationswere made of stresses at the interface betweenthe elastic layer and the rigid half space,z = h, and radial displacements at the surface,z = 0. The nondimensional normal stress,a,,/G, at the interface, z = h, is plotted as afunction of the nondimensional radial co-ordinate, r/u, in Fig. 6 for v = O-45. Asexpected, the normal stress reaches a maxi-mum at the axis of symmetry and diminishesasymptotically to zero outside the contact


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0.06

Area Aspect Ratio,o/hFig. 4. Nondimensional indentation of a plane-endedcylindrical indenter for area-aspect ratios between zeroand unity.

I The nondimensional shear stress, a,/G,

Load Factor, F/GRFig. 5. Relative approach as a function of load factor forthe spherical indenter.

region. For given values of Poissons ratioand area aspect ratio, the normal stress at theinterface increases with decreasing R/h ratio.Increasing a/h, while maintaining R/h fixed,markedly increases the maximum normalstress at the interface.

at the interface, I = h, is plotted as a functionof r/a in Fig. 7. As required by symmetry.the shear stress vanishes at r/a = 0. The shearstress reaches a maximum near the radius ofthe contact region and then diminishes to zerooutside the region. Shear stress at this inter-face is a feature of the present solution thatis different than the Lebedev and Ufliand(1958) solution. They assumed no shearingstresses at the interface. As with normalstress, shear stress increases with decreasingRlh and with increasing alh.

The nondimensional radial displacement,u,lh, at the surface, z = 0, is plotted as afunction of r/a in Fig. 8. For given values ofPoissons ratio and area-aspect ratio, thesolution predicts negative displacementswithin most of the contact region and positivedisplacements outside the contact region.At large radial distances the displacementsdiminish asymptotically to zero. Thesedisplacements increase with decreasing R/h


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548 W. C. HAYES et al.Table 2. Values of K and x for the spherical indenter

Y = o-30 Y = o-35 v=o*40 Y= 0.45 v=o*SOa/h K X K X K X K X K X

O- 0 4 . 04809O- 0 6 06891048 0-69750.1 o-7061o-2 O-75200.3 0*8031o-4 O-8594O-5 0.9209O-6 O-98720.7 l-0580.8 1.1330.9 l-2101-O l-2911.25 l-5031.50 l-7231.75 1.9492.00 2.1792.25 2,4122.50 26472.75 2.8833GO 3.121

1.0241.036l-0491.061l-1261.1921.2561.317l-3721.4221467l-5071.542l-6131667l-708l-7401.7661.7881.8051.820

O-6816O-690204990o-7080o-7564O-8105O-8705o-9363l-0081.084l-1651.250l-339l-5711.8162-0692.3272.5892.8553.1243.394

1.025 O-68261.038 0.69171.051 0.7010l-064 0*71061.133 O-7622l-202 O-82041.270 o-88541.333 0.9572l-392 1a036I.445 l-1211.492 1.211l-534 1.307l-571 14071.646 l-673i - 7 0 2 l-957I.745 2.2541,778 2.561l-805 2.8771.826 3.1991.844 3.527l-858 3.860

1.026l-0401*OS4l-0681,1411.2151.2881.3561.419l-4771.5281a5731.6141.696l-7571*&x051.841l-871l-894l-913l-928

0.6838 1.0280.6936 l-0430.7037 1.0580.7140 1.073o-770 1 1.1520.8339 1.2330+060 1.312o-9866 1.388I.076 l-458l-173 1.5221.278 1.5801.390 1.6321.509 1a678l-831 1.7752.184 1.8492.564 l-9072.967 l-9543.391 1.9923.834 2.0234.294 2.0474.770 2.068

o-6855O-6963o-70730.71870.78100.8530o-93551 ow1.1351.2521.381I.5221a6742*1022.5973.1613.7974.5075.2%6.1697.130

l-0311.0471.0631*0801.1671.258l-3481.4351.5161.5921.6621.725I.784I.9112.0172.1092.1892.2602.3252.3842.438

ratio.Radial strainst the. cartilage surfacemay be predicted from the results shown inFig. 8 by differentiation. The solution showscompressive strains within the center portionof the contact region, tensile strains in theremaining part of the contact region and thenear part of the outside region, and com-pressive strains at further radial distances.This result is qualitatively similar to thepredictions of the Hertz theory for indentationof an elastic half-space by a rigid sphericalindenter.

DISCUSSIONArticular cartilage is a viscoelastic materialand any dynamic analysis must treat it as-such.Nevertheless, the present theory, whichassumes the material is elastic, is useful intwo limiting cases. Creep tests of cartilageindicate an instantaneous elastic responsefollowed by creep over several minutes toan asymptotic deformation. The presentelastic theory for indenter-loaded layers maybe useful in predicting the instantaneous

response to a step load and in predicting the

asymptotic deformation. Both applicationsshould be, of course, conflned to small loadsto stay within the limits of the small-strainassumption.The present theory may be qualitativelycompared with some of the previously re-ported results of indentation tests of cartilage.For example, Elmore ,ef al. ( 1963) appliedconstant unit loads of I.37 X lo6 dynes/cm2to bovine patellar cartilage using circular-plane indenters of various diameters. Inden-tations were measured 18 minutes afterapplication of the load, i.e. after most of thecreep should have occurred. They observedlarger indentations with the larger diameterindenters. Unfortunately, they did not reportthe cartilage layer thickness and the valuesof a/h cannot be computed. Their indentershad radii ranging from 0.565-1.382 mm. In allprobability the cartilage would have been2 mm or more thick and their a/h values wouldall have been less than unity. As shown inFigs. 3 and 4, the larger indenters should haveproduced the larger indentations. This effectis relevant to the interpretation of previous


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::b Q

Fig. 6

0.00

-a05

-0.10


Fig. 7. Nondimensional shear stress at the interface,z = h, for the spherical indenter.. ,

0.003

Nondimensional normal stress at the interface,- = h, for the spherical indenter..indentation tests that, by assuming a uniaxial

Jo.cQ2

stress field beneath the indenter, neglectedthis relationship between area aspect ratio andrelative indentation.

P- 0.001Hirsch (1944), in an early study, applied SC4

step loads and measured the initial indentationof cartilage of the tibia1 plateau. Using aplane-ended indenter, he measured theinitial deformation of a I.9 mm thick layer of .ooocartilage for three different step loads.According to the present theory, the deforma-tions with a plane-ended indenter should have

-1

been linearly prooortional to the load. Hirsch - 0.w 1 1did not observe deformations proportionalto load. His loads, however, caused indenter Fig. 8. Nondimensional radial displacement at thesurface, ,- = 0, for the spherical indenter.


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5 5 0 W. C. HAYES et al.displacements that were 12, 16, and 2 1 percent of the cartilage thickness and theoretical-experimental comparison is not justified.These fmite displacements exceed the limitsof the theory and may exceed the linear rangefor cartilage elasticity.Using spherical indenters of 10 and 30 mmradii, Hirsch measured the initial elasticindentations at step loads of 490 X 10s dynes.Larger indentations were noted with the10 mm indenter. The curves of Fig 5 predictthat at a given value of the load, P, theindentation will be larger for the indenter ofsmaller radius. Also, using a 30 mm radiusspherical indenter Hirsch measured theindentation for six levels of load.Unfortunately, quantitative comparisonsbetween the present theory and the measure-ments of Elmore et al. (1963) and of Hirsch(1944) are not possible. The shear modulus,Poissons ratio, and thickness of the cartilagetested are required for a quantitative com-parison. The material pmperties are notknown for either experimental study andElmore et al. did not report cartilage layerthickness. All these quantities would have tobe measured independently if a definitive com-parison is to be made. A previous study(Hayes and Mockross 1971),however, showedthat for the initial elastic response in a creeptest the shear modulus and Poissons ratiowere 4.1 X 10 dynes/cm* and O-42, espective-ly, for healthy cartilage and 5.8x IO*dyn/cm* and O-39, respectively, for somewhatdegenerated cartilage. Selecting values forthe material constants somewhere in thisrange gives good quantitative agreementbetween the present theory and the tests ofHirsch. In as much as the material propertiesin this comparison were simply chosen togive a good fit (i.e. not independently deter-mined), the agreement must be consideredsubjective. Nevertheless, this preliminarycomparison is interesting.If, indeed, the theory is verified with teststhat include an independent measure of thecartilage material properties, the indenter

solution may provide a theoretical basis fora quantitative determination of the elasticshear modulus of cartilage in situ. The earlierstudy (Hayes and Mockross 1971) indicatesa strong dependence of shear modulus,measured at low loads, on the degree ofosteoarthritic degeneration. The measuredvalues of Poissons ratio, on the other hand,indicate a fairly narrow range. Thus, theresults of an indenter test and an assumedvalue for Poissons ratio could be used withtheory to estimate the elastic shear modulusand, consequently, the degenerative conditionof the in situ cartihge.The spherical indenter solution reportedhere may also be examined as a preliminarymodel for loading in normal and artificialsynovial joints. To do so, it is important tocompare the assumptions upon which theanalysis is based with the characteristics ofthe biological tissues involved. The assumption of the rigidity of the supporting half-spaceis. not overly restrictive considering experi-mental data for the elastic modulii of cartilageand of cortical and cancellous bone. Such

an assumption, however, precludes investi-gating the stress field in the subchondralregion, an investigation of interest in atheoretical analysis of the subchondral andmarginal changes occurring in osteoarthritis.The assumption of the elasticity of thecartilage also precludes investigation of itsviscoelastic response but again this is notoverly restrictive under short term loadingconditions as in normal gait. The assumptionof the rigidity of the indenter is a more seriousrestriction, since, in such a case, all deforma-tions are assumed to occur in the layer. In thenormal joint, both contacting condyles displaythe layered geometry and deformations inthe contact region are shared. The rigidindenter assumption results in sufficientlylarge normalized indentations at low valuesof joint normal force to violate the smalldisplacement requirements of a linear theory.Results for the stress and displacement fieldsare thus only valid for joint forces on the order


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A MATHEMATICAL ANALYSIS 551of 10 kponds. Therefore, although thesolution applies directly to low-level loadingof joints in which one component has beenreplaced prosthetically, it should be regardedas a preliminary model for the loading innormal synovial joints. Major improvementsin the applicability of the analysis can beexpected with the consideration of a deform-able indenter, a layered, elastic half-space andthe viscoelastic characteristics of the articularcartilage.Acknowledgment-W. C. Hayes was supported duringthis investigation by Training Grant GM 00874 from theNational Institutes of Health.

REFERENCESBurstein, A. H. (1968) Elastic analysis of condylarstructures. Ph.D. Thesis, New York University.Camosso, M. E. and Marotti, G. (1962) The mechanicalbehavior of articular cartilage under compressive stress.J. Bone Jt Surg. 44A, 699-709.Edwards, J. (1967) Physical characteristics of articularcartilage. Proc. I&n Mech. Engrs 181(3), 16-24.Ehnore, S. M.. Sokoloff. L.. Norris. G. and Carmeci, P.(1963) Nature of imperfect eiasticity of articularcartilage. J. appl. Physiol. 18,393-396.Hayes, W. C. and Mockros, L. F. (1971) Viscoelastic

constitutive relations for human articular cartilage.J. appl. Physiol. 31,562-568.Hirsch, C. (1944) A contribution to the pathogenesis ofchondromalacia of the patella. Acta chir. scund.Supp. 83.Kempson, G. E., Freeman, M. A. R. and Swanson, S. A.V. (1968) Tensile properties of attic&r cartilage.Nature, Land. 220, 1127-l 128.Kempson, G. E., Freeman, M. A. R. and Swanson, S. A.V. (197 1) The determination of a creep modulus for

articular cartilage from indentation tests on the humanfemoral head. J. Biomechanics 4,239-250.Kopal, 2. (1955) Numerical Analysis, 556 pp., Wiley,New York.Krakovits, G. (1969a) Die elastizitat der gelenksknotpel.Anat.Anz. 124,113-119.Krakovits, G. (1969b) Bestimmung des elastizititsmodulsder knorpelschicht auf femurkiipfen mit hilfe der

kugeldruckprobe. Anat. Anz. 124,1X-166.Lebedev, N. N. and Ufliand, I. A. (1958) Axisymmetriccontact problem for an elastic layer. PMM 22,442-450.McCutchen, C. W. (1962) The frictional properties ofanimal joints. Weor 5, l-17.Sokoloff, L. (1963) Elasticity of articular cartilage: effectof ions and viscous solutions. Science 144,1055-1057.Sokoloff, L. (1966) Elasticity of aging cartilage. FedernProc. 25,1089-1095.Sokoloff, L. (1969) The Biology of Degenerative JointDisease, 162 pp., University of Chicago Press,Chicago.Timoshenko, S. P. and Goodier, J. N. (1970) Theory ofElusticity. 3rd Edn, pp. 409-413, McGraw-Hill,New York.Zarek, I. M. and Edwards, J. (1963) The stress-structurerelationship in articular cartilage. Med. Electron. Biol.Engng 1.497-507.NOMENCLAlWRE

radius of contact regionelastic shear moduluswtilage thicknessapplied normal forceradial coordiiteaxial coordinate displacement vectornondimensional parameterPoissons rationormalized radial coordinatenormaiized radii coordinateBoussinesq-Papkovitch potential functionsnondimensional parametershape function for axisymmetric indentorindentation of cartilage layercomponents of stress tensor

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