D Stamenovic

G M Hass

J C Smith

Joan 

M Genghini

P C Curti

R W Mazzone

S Kornblav

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Transpulmonary and Vascular Pressures," J. Appl. Physiology: Respirat. En-
viron. Exercise Physiol., Vol. 51, No. 3, 1981, pp. 739-745. 
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Surface Tension-Lung Volume Relationships," / . Appl. Physiology, Vol. 60, 
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Against Inhaled Particles," Respiration, Vol. 55, (Suppl. 1), 1989, pp. 60-67. 
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factant," Ann. Rev. Physiol., Vol. 47, 1985, pp. 753-763. 
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Contact of Layered Elastic Spheres as a Model of Joint 
Contact: Effect of Tangential Load and Friction 
A. W. Eberhardt1,2, J. L. Lewis1, L. M. Keer2 
Introduction 
The deterioration of articular cartilage associated with os-
teoarthritis is characterized by vertical cracking initiating at 
the cartilage surface [1], as well as horizontal splitting at the 
tidemark [2]. Animal trauma models show horizontal splitting 
at the tidemark [3], and in recent impact studies, the authors 
have observed vertical cracks which intitate in the calcified 
cartilage and underlying bone, and with higher loads, at the 
cartilage surface. 
It is feasible that tensile stresses are an essential element in 
impact induced cracks. An accurate model of articulating con-
tact therefore should predict tension along planes at which 
these cracks are observed. In a previous study [4], the authors 
modeled the joint system as normal, frictionless contact of 
two single layered elastic spheres. The results showed tension 
only for small contact areas, leading to the present extension 
which includes a frictional load between the contacting sur-
faces, to determine whether the additional loading induces 
crack initiation. 
Early studies on the response of elastic bodies to tangential 
loading include Mindlin [5] and Hamilton and Goodman [6]. 
Tangential loading of layered materials was analyzed by Good-
man and Keer [7], O'Sullivan and King [8], and Keer et al. 
[9]. The current model is an extension of [9] which assumes 
elastic, homogeneous and isotropic materials with complete 
slip between the contacting surfaces. The effects of interstitial 
fluid flow on the constitutive behavior of cartilage are not 
considered, therefore this analysis is restricted to short-time 
loads where these effects may be assumed negligible. A com-
plete discussion on the justification of the linear elastic model 
is presented in [4]. 
Fig. 1 Analytical model of joint contact 
pressive force, Pz, which produces a contact area of radius, 
a,assumed small compared with R, a tangential force Px=fPz, 
in the positive x direction, and restraining moments, M=PXR, 
as shown in Fig. 1. It is assumed that the normal pressure 
profile is unaltered by the presence of Px and M. The for-
mulation for a normal load was presented in [4, 9, 10], and a 
detailed description of the formulation for the tangential load-
ing is given in [10]. Here a brief qualitative summary is pro-
vided. 
The axisymmetric (normal load) and asymmetric (shear load) 
displacement equations for an elastic material are expressed 
in terms of harmonic stress potentials in cylindrical coordi-
nates. The stress potentials are Hankel transforms which allow 
for the determination of the necessary coefficients in the trans-
formed domain, using the matching conditions at the layer/ 
substrate interface. By prescribing the normal Hertzian dis-
placement at the contacting surfaces, a Fredholm integral equa-
tion is developed for the governing potential function and 
relative displacement between the two spheres. Using the de-
termined potential function, the normal contact pressure and 
the resulting stress and displacement fields for the normal load 
may be calculated. The tangential traction is prescribed as the 
normal pressure multiplied by a coefficient of friction, f, 
thereby assuming complete slip between contacting surfaces. 
The appropriate coefficients are obtained using the interfacial 
matching conditions, and the resulting stress field for the tan-
gential loading may be determined and results superposed with 
those from the normal load. 
Formulation and Solution 
The formulation and solution are based on that presented 
in [7]. Two identical elastic spheres, with identical elastic layers 
(radius, R, and layer thickness, h), are subjected to a com-
'Dept. of Orthopaedic Surgery, University of Minnesota, Minneapolis, MN 
55455. 
2Dept. of Civil Engineering, Northwestern University, Evanston, IL 60208. 
Contributed by the Bioengineering Division of THE AMERICAN SOCIETY OF 
MECHANICAL ENGINEERS. Manuscript received by the Bioengineering Division 
March 9, 1990; revised manuscript received November 20, 1990. 
Results and Discussion 
. The results indicate that the tangential surface traction most 
significantly affects the resulting radial and shear stresses. Cal-
culations for Figures 2-4 were performed using the material 
properties for short-time loading of the "normal" cartilage/ 
bone system as described in [4], with JX\ = 1.5 MPa, v\ = 0.475, 
Lt0 = 580 MPa, and j>0
 = 0.3. Figure 2 shows the nondimen-
sionalized radial stress at the surface due to the friction load 
for values of the ratio of contact radius to layer thickness, 
a/h = 1, 3, and 5, at 6 = ir/2 (see Fig. 1). The values shown are 
normalized with/= 1.0, so that actual stress values are those 
Journal of Biomechanical Engineering FEBRUARY 1991, Vol. 113 / 107 
Copyright © 1991 by ASME
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Pz/(7ia2) 
H a/h = 1 
® a/h = 3 
+ a/h = 5 
Fig. 2 Normalized radial surface stress \T„ l{Pxfaa
2)], lor f = 1 , as func-
tion of a/h. 
Trr 
Pz/fra2) 
0.00-
-0.25 -
-0.50 -
-0.75 -
-1.00 -
-1.25-
/ . i s -s - r * * * 1 1 
mf D surface 
j S ^ • interface 
1 i • i ' i « " 
0.0 2.0 
r/a 
Fig. 3 Normalized radial stress [rn l(P,ha
2)] at surface and interface 
due to normal and shear load, for a/h = 2 and r = 0.1 
tzr 
Pz/(Jia2) 
-0.0 -
-0.1 -
-0 .2 -
-0.3 -
-0.4 -
s. 
V y 
v ^ v ' • 
nv-mz^* 
* * * * 
norm 
norm + .1 tan 
1.0 2.0 
r/a 
Fig. 4 Normalized shear stress [T„ /(Pz/?ra
2)] at layer-substrate interface 
due to normal and shear load, for a/h = 2 and f = 0.1 
shown multiplied by the actual/< 1. The figure illustrates that 
tensile stresses induced by the friction load are greatest near 
the edge of contact, decrease and become less peaked with 
increasing a/h. Actual a/h values in "normal" joint contact 
are uncertain, however, using Fuji film, Chin-Purcell [11] 
measured the contact area of impacted canine patellas, showing 
a/h values of approximately 2. Figure 3 shows the predicted 
normalized radial stress at the surface and in the substrate at 
the layer-substrate interface along 6 = ir/2, for a/h = 2.0, due 
to the combined normal and shear loads for/= 0.1. The figure 
illustrates that tension is induced at the surface near the edge 
of contact while no tension appears in the substrate. The same 
calculations, for a/h = 5 and /= 0.1, show no tension in the 
layer or substrate, indicating that compressive radial stresses 
induced by the normal load, which increase as a/h increases, 
eliminate tensile radial stresses induced by the shear load. This 
implies that if the impact induced cracks are due to friction, 
the contact areas must be small, and/or/must be large, ap-
preciably larger than those values (0.01 < / < 0.04) measured 
for cartilage under normal loads and sliding velocities [12]. 
Significant increases of/values have been observed for long 
load times [13], however it is unknown whether friction is 
elevated during high impact loads. 
Surface radial stresses were also calculated for the same 
system with either a 20 fold increase in cartilage stiffness as 
seen in impact studies [14] or a 20 fold decrease in bone stiff-
ness, as might occur in osteoporosis. Changes in results were 
insignificant. 
Figure 4 shows the normalized shear stress at the interface 
along 0 = 71-/2, for a/h = 2.0, /=0 .1 , for the normal load and 
the combined normal and shear loading. The figure shows that 
the magnitude of the interfacial shear stress increases with 
increasing/, increasing the possibility of interfacial separation. 
Vertical cracking at the interface due to shear is also a pos-
sibility. 
The results of this analysis suggest that surface cracks ob-
served after joint impact are a result of tension due to either 
(i) an unusually high coefficient of friction, (ii) small contact 
areas, or (iii) factors not accounted for in this model, such as 
large deformations, cartilage anisotropy, interstital fluid flow 
and/or joint geometry. Further work is necessary to assess the 
importance of each of these features. 
Acknowledgments 
The authors gratefully acknowledge support from NIH grant 
#AR07555. 
References 
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of Human Synovial Joints-Part 1: Natural Joints," J. Lubrication Tech., 1975, 
pp. 369-376. 
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108 / Vol. 113, FEBRUARY 1991 Transactions of the ASME 
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Elastic Tissue II. A Study of the Elasticity and Tensile Strength of Elastic Tissue Isolated from the Human Aorta

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Elastic Tissue II. A Study of the Elasticity and Tensile Strength of Elastic Tissue Isolated from the Human Aorta

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