EFFECTS OF CARTILAGE STIFFNESS AND VISCOSITY ON A NONPOROUS COMPLIANT BEARING LUBRICATION MODEL FOR LIVING JOINTS*+ E. F. Ruarc~t, W. A.
GLAESER,:
J. S. STRESEOH~KIand M. A. T-\u%t
Applied Solid Mechanics Section, Battclle’s Columbus Laboratories, 505 King Avenue, Columbus, OH -13201, U.S.A. Abstract - There are several possible lubrication mechanisms for living joints. One such mechanism, the squeeze film compliant bearing, is examined through the use of a computational model. The model is based on a finite element representation for the compliant surfaces. The lubricant is described by Reynold’s equation. A minimization technique provides solutions to this inherently nonlinear problem, and alleviates difficulties in numerical convergence. In earlier work, the results of this model showed good correlation with both numerical results of another model and laboratory measurements for film thickness of a compliant sphere approaching a glass plate. In this paper, the model is used as a basis for conducting a parametric study of the influence of cartilage modulus and fluid viscosity on the lubrication characteristics. A load of 445 N was applied in the studies. Results of the compliant system are compared with those for a system with rigid surfaces representing a prosthesis. Based on the results obtained by the model and the ranges of parameters considered, it was found that changes in viscosity of the lubricant had a greater influence than changes in ths cartilage modulus. The lubrication qualities of the compliant system were superior to those of the system uith rigid surfaces if the same lubricant properties were used. However, lubrication viscosities could be found such that a compliant system with a low viscosity has poorer lubrication properties than a rigid system with a higher viscosity.
ISTRODUCI-ION
The human body has more than ten major load bearing joints connecting long bones to other skeletal members. Some of these joints transfer forces equal to many times body weight and all permit a wide range of motion. This is accomplished under conditions of very low friction. For example, Mow (1969) reports that healthy joints articulate with coefficients of friction ranging from 0.005 to 0.025. However, disease can increase joint friction and severely limit joint function. In some cases, joint function is partially restored by using a joint replacement prosthesis. Current designs for prostheses include several characteristics ofnormal joint articulation. For example, prostheses have geometric compatibility with the kinematics of joint members. Implant materials are compatible with body tissue and are selected on the basis of reducing wear between roiling and sliding parts. However, many properties of the living joint are not included in current designs for joint prostheses. An example is the lubri-
* Receired 3 May 1977; in revised form31 August 1978. t This research was sponsored by the Battelle Institute Program, “Fundamental Studies of Lubrication in Living Joints”, Contract No. B-1333-1210. : Structural Mechanics and Tribology Section. $ Other possible lubrication mechanisms in living joints include boundary or soft tissue lubrication (especially under slow movement and constant loading) and “boosted lubrication” in which concentrated hyaluronic acid at the contracting surfaces increases load support. 8.M.
116-A
cation aspects of living joints. One reason lubrication
qualities are not reproduced in the replacement prostheses is that a full understanding of the basic lubrication mechanisms in living joints is not available. In principle, the existing technological understanding ofdifferent lubrication mechanisms is applicable to the problem of better understanding the lubrication mechanisms of living joints and can also increase the understanding of lubrication in prosthetic devices. Several theories are set forth in the literature to explain lubrication mechanisms of living joints. As a step toward a better understanding of the lubrication of living joints, this paper focuses on one possible mechanism, the compliant bearing lubrication me&anism_$ A computational model, which draws on existing elastohydrodynamic lubrication theory, is described and used as a means of examining the effect of different constituent properties on joint lubrication. Using a simple geometric configuration, the effects of the elastic modulus of the cartilage and the viscosity of the fluid lubricant are considered. The following sections give background information on compliant bearing lubrication and joints, a description of the computational model, a discussion of numerical results and the conclusions.
BACKGROUNDON COMPLIANT BEARISG LLBRlCATtON FOR JOINTS
After 403
the development
of elastohydrodynamic
404
E. F. RY~ICKI.W. A. GLAESER. J. S. STRESKOWSKI
(EHD) theory of lubrication as a viable engineering design tool, researchers such as bintenfass (1963) and Tanner (1966) investigated the relevance of EHD theory for lubrication of living joints. Since then, a number of studies by, for example, Dowson (1969) and Higginson and Norman (1974) have further supported the relevance of EHD theory to lubrication of living joints. There are three characteristics of a compliant bearing involving EHD lubrication. These are the existence of compliant bearing surfaces, a nearly incompressible bearing material and a lubricant. The cartilage is a compliant bearing surface with modulus values of 25.5 MN/mm” as reported by Walker er al. (1976). The cartilage is close to incompressible, as evidenced by Poisson’s ratios ofO.40, reported by Parsons and Black (1977) and 0.47 by Hori and Mockros (1976). The availability of synoviai fluid as a lubricant is an important feature of a compliant bearing lubrication. There appear to be at least two possible ways for bringing fluid to the contact area. One way is that, as the joint rotates, the unloaded bearing surface is wetted by the fluid which is then available when that surface is weight bearing. Another possible way of getting lubricant to the weight bearing area is based on the notion that in order to form a fluid film between contacting cartilage surfaces, fluid must be exuded from the surfaces to the contact zone. This type of weeping lubrication has received attention by Monsour and Mow (1976). Based on experiments, a mathematical model was developed in which synovial fluid is squeezed from the cartiiage just in advance of the moving contact. Little flow occurs in the cartilage under compression and the “starved” cartilage released from pressure In the trailing area soaks up the synovial fluid and holds it for lubrication as the contact moves back over it. However, a significant question is left unanswered, concerning the amount of fluid that must be maintained between the load bearing surfaces to achieve viable lubrication, as compared to the amount that can be delivered by squeezing the cartilage. Experiments by Harris et al. (1972), Parsons and Black (1977), and others have shown that cartilage behaves as a viscoelastic material. Experiments by Giaeser and Rybicki (1977) indicate that behavior in short time load applications is predominantly elastic. Since the cartilage has very low permeability for flow out of its surface, it might be expected to perform like a fluid filled membrane during short time impulsive loading and would be essentially incompressible. Synovial fluid is non-Newtonian and Davies and Palfrey (1969) report decreasing values in apparent viscosity for increasing shear rates. In this study, the joint is represented approximately as a squeeze film bearing having low modulus elastic compliant surfaces and separated by a viscous fluid. This simplification of the model should only be valid for joints under impulse short time load application. It would not hold for a load bearing joint at rest.
and %I.
MODEL
A T.MM
FOR COMPLIAST LUBRiCATlON
BEARI%
A review of the literature reveals several computational approaches describing the mechanics of compliant bearing squeeze film lubrication. Huebner (1975) describes three papers based on the finite element method and iteration technique for solving lubrication problems. Iteration techniques are needed because of the inherent nonlinearity relating the film thickness to the pressure. In each reference, the compliant surfaces are modeled as an elastic deformable material and Reynoid’s equation is used to model the behavior of the lubricant. Taylor and O’Callaghan (1972) and Oh and Huebner (1973) follow a procedure that iterates on the deformation state of the elastic bearing. Pressures are calculated based on the film thickness and Reynold’s equation. The pressure distribution is applied to the elastic bearing surface to obtain a deformation. A new film thickness is obtained, based on this deformation, and the process continues until convergence of the deformation is obtained. Taylor and O’Callaghan (1972) and Oh and Huebner (1973) point out that this procedure only converges for”lightiy loaded bearings” and that steps must be taken to increment the load and bound the estimated deformation, or else the method diverges. It is also reported that this procedure does not work well for “soft” bearing materials. In another approach, Day (1974) maintains compatibility between the film thickness and deformations while formulating an initial value problem to satisfy equilibrium between the fluid and compliant material. The procedure starts with an assumed pressure distribution and marches out in time until the pressure does not change. It is reported that this procedure works well for “soft materials” but encounters difficulties for “relatively stiff’ bearing materials. In a recent paper, Rohe et ~1. (1976) present a number of analyses for EHD squeeze film problems to handle time dependent loading conditions for both soft and hard surfaces. In work reported by Bupara et al. (1973, an iterative technique similar to those described by Taylor and O’Callaghan (1972) and Huebner (1975) demonstrated that solutions were feasible for very low load levels - up to 45 N. Difficulties in obtaining solutions for higher loads were attributed to the inherent nonlinearities which could not be satisfactorily treated by iterative techniques. The model for compliant bearing squeeze film lubrication used in this study is numerically stable and is not confined to small load levels. The small load level limitation was alleviated by introducing a minimization algorithm to replace the iterative solution procedure. The basic advantage of the minimization algorithm is that at each step, a better solution is obtained, based on the value of the criterion that is being minimized. Thus, the method does not have the divergence problem associated with the direct iteration procedure. A detailed description of the model is given by
Effects of cartilage
stifftie%
1524
,270 , toordlno+e.
Fig. 1. Coxrse
finite
element
mm
mesh for simple
axisymmetric
model of the human knee joint. Rybicki er nl. (1977). Thus, only an overview is given here. Figure 1 illustrates the geometry considered. The model consists of three parts: a fluid lubricant, compliant surfaces. and a base material. The fluid behavior is governed by Reynolds equation. This means that the fluid is Newtonian, inertia effects are negligible, the film is thin enough so that the significant velocity change is across the thickness, and curvature effects of the bearing surface are neglected. The compliant bearing surfaces are modeled as a linear elastic material with a Poisson’s ratio close to 0.5. The base material is linear elastic and stiffer than the compliant surfaces. The equation governing the fluid pressure and the film thickness is
and viscosity
To obtain solutions to the problem, the system is discretized to N nodes and X - 1 intervals. Equation (1) is expressed in a discretized form, as is the stress displacement relation for the compliant surface. The fluid and the compliant surfaces are coupled through equilibrium and compatibility. Compatibility is satisfied exactly. Equilibrium is satisfied approximately at discrete locations by minimizing a function which is the sum of the squares of the differences, between the pressures of the fluid and the stress of the solid at these points. In this minimization formulation, the deformations of the compliant surface, at discrete points, are the unknowns. The governing equations for the compliant bearing surface shown in equation (2) are a iinear relation between the surface stress and dispiacements. I-lowever, equation (1). which relates the film thickness (and hence the surface deformations) to the pressure. is nonlinear. Thus, the system of equations to be solved is nonlinear. Imposing equilibrium conditions between the fluid pressure and the stress at the cartilage surface for a discrete number of locations gives a set of algebraic nonlinear equations to solve. The equations were solved by minimizing the sum of the squares of the residuals. The minimization technique used here is that described by Fletcher and Powell (1963). The method can be described as one of the conjugate gradient techniques and is basically for minimizing a function that is not subject to constraints. In the formulation of this problem, there was one constraint and the procedure was modifieo to handle it. The constraint is that the film thickness must always be greater than zero. This condition was met by starting with an acceptable solution and shortening the step size if an unacceptable solution was suggested by the method. This is a valid procedure for this particular problem because the solution cannot lie on a constraint /I = 0 since the value of the function to be minimized is infinite if II = 0 at any point. NUMERICAL
where p is the pressure in the fluid at location r, R is the outer radius, p is the viscosity, lJZ is the velocity, and h is the film thickness. The relation between the stress and the deformation of the compliant surface is given in discretized form by i"il VI I
=
[K,jl{w,l, Y.xI
(2,
.\'I,
where ci is the stress at a location denoted by i, wj denotes the displacement at point .i, and Kij denotes the stiffness matrix. The film thickness is related to the deformation of the compliant surfaces by h, = s - 2ni,
(3)
where .? defines the relative distance between the two compliant surfaces when there is no load on the system.
405
RESULTS 6OR PARAMETRIC STUDIES
Prior to undertaking the parameter studies, two different finite-element models of the cylinders were examined. The purpose was to develop a finite element model which would provide accurate results and also be economical to use. The first finite element model consisted of 21 radial nodes on the axisymmetric circular surface of each cylinder. The finite element representation of the lower cylinder is shown in Fig. 1. With this coarse model and the properties summarized in Table 1, it was found that the solutions for film thicknesses less than about 7.62 x lo-’ mm did not converge. Therefore, a refined axisymmetric model was developed, which consisted of 57 radial nodes (nonuniformly spaced) on each cylindrical surface and a total of 448 nodes throughout each cylinder. This mesh is shown in Fig. 2. along with an enlargement of a portion of this mesh. Using this refined representation.
E. F. RYBICKI,W. A. GLAESER,J. S.
STRENKOV.XKI and
X4. A. TA.CIM
IA
Fig. 2. Refined finite element mesh for simple axisymmetric model of the human knee
joint.
Table 1. Properties of the cylinder model Elastic modulus (MN/m’)
Thickness (mm)
Poisson’s ratio
Viscosity
Bone
Cartilage
Bone
Cartilage
Bone
Cartilage
20
6.895 x 10’
8.826
0.25
0.48
6.35
6.35
Table 2. Values varied in parametric studies
Case No. 1
2 3 4 5
Elastic modulus of cartilage (MN/m’)
Fluid viscosity (CP)
4.413 17.651 8.826 8.826 8.826
20 20 20 7 80
results were obtained for film thicknesses of the same order of magnitude expected in living joints. Based upon the above findings, the refined finite element model of the cylinders was chosen to study the effect of varying several properties of the knee joint model. Two parameters which characterize a diseased knee joint are the cartilage elasticity and the viscosity of the synovial fluid. These properties of the cylinder model were varied. A summary of the range of properties is given in Table 2. Unless otherwise indicated, the remaining properties used were identical to those given in Table 1. Evaluation of the model response due to change in material properties was based upon the decrease in film thickness as a function of time. To find the variations of film thickness with timi, it was assumed converged
Contact radius (mm) 12.7
that the local deformation velocity, d\c/dt, was negligible when compared to the overall approach velocity. @;dr. This is consistent with the rapid loading assumption in which the cartilage behaves elastically. Therefore, UI = dS/dt and the time may be found by integrating (l/U,) with respect to s. By using a suitable numerical integration scheme such as Simpson’s rule, 3 and the film thickness h(r) may then be calculated as a function of time. The first study involved varying the modulus of elasticity of the cartilage from 4.413 to 17.651 MN/m’ as shown in Tabie 2. Figure 3 shows half of the film thickness at R = 0 as a function of time and cartilage modulus. Here t = 0 corresponds to an undeformed film thickness s = 0.635 mm. Although only three different cartilage moduli were selected, certain trends may be identified from these results. For example, it may be seen that as time increases, the film thickness h/2 is less for higher values of the cartilage modulus. This means the time to reach a given film thickness decreases as the cartilage modulus increases. This trend continues so that for rigid cylinders, this time is a minimum. The improvement in the lubrication properties of compliant bearings noted above may be partially explained by the film thickness profile. This is shown in Fig. 4 at several times for a cartilage modulus of 8.826 MN,‘m’ and a fluid viscosity of,u = 20 cp. Due to
Effects oi cartilage stiffness and viscosity
0 Ecorliloq6
=
4.413
MN/
0 Ec,-r+ilapa
’
8.826
MN/
: 17.651
1.0
: 4 413 MN/m’
MN/m
2.0
Fig. 3. Film thickness at R = 0 as a function of time for various cartilage moduli.
the cartilage elasticity, the fluid trapped between the squeezing surfaces forms a pocket-type configuration. This shape restricts the flow of the synovial fluid from the region of high pressure so that a longer time is required co squeeze out the fluid. Note that as time increases, the relative depth of the pocket decreases, the ratio I?,,,,;,h(R=O) decreases, and the minimum film thickness moves radially outward. These trends were also observed for cartilage moduli of 17.651 and 4.413 MN/m’. When plotted as a function of time, the minimum film thickness approaches zero morequickly than the film thickness at R =0 (see Fig. 5). Note that use of the minimum film thickness as a basis of comparison between different cartilage moduli leads
/
I
I
c,,,,,,ape= 17.651 MN/&
0
Time (1). miilisec
I
I
407
/
10
2.0 Time(t). millisec
3.0
Fig. 5. Wnimum film thickness as a function of time for various cartilage moduli. to different conclusions than if the film thickness at R=O had been used (compare Figs. 3 and 5). This
difference may be explained by the fact that for a given time, the minimum film thickness occurred closer to the cylinder outer edge for higher cartilage moduli. Therefore, the occurrence of the minimum film thickness at different radii makes it a less likely basis by which to compare the time response of various cartilage moduli. The sensitivity of the film thickness response to variations in the viscosity was also investigated. The film thickness as a function of time is shown in Fig. 6 for three values of the viscosity. These three parametric values correspond to Cases 3, 4 and 5 in Table 2. In contrast to Fig. 3, it may be seen that the film thickness response is more sensitive to changes in viscosity than to changes in elastic modulus of the cartilage. It is interesting to note that a rigid bearing with a viscosity of synovial fluid (p = 20 cp) exhibits better lubrication properties than a compliant bearing with ,D= 7 cp. Finally, note that for the same viscosity the rigid bearing always displays poorer lubrication capabilities than a compliant bearing. When the trends shown in Fig. 6 are extrapolated out to dwell times (t) fodnd in actual joint performance Table 3. Estimated film thickness values extrapolated from Fig. 6 Joint model
Radius (RI.
mm
Fig. 4. Film thickness profiles for various times for EL&,. = 8.826 MN/m’ and p = 20 cp.
Rigid Compliant Compliant
Viscosity (CP)
I (msec)
h/Z (mm x lo-31
20 80 20
100 100 loo
5.08 12.07 7.62
408
E. F. RYBICKI.
W. A. GLAESER J. S.
STRENKOWSKI and M. A. TAMM
(say t = lOOmsec), the resulting hi2 values are not unreasonable for full film support for loaded joint surfaces. The extrapolations are summarized in Table 3.
DISCUSSIONS AND CONCLUSIONS
Based upon the results of the parametric studies, it is demonstrated that the lubrication process is enhanced by considering a compliant rather than a rigid bearing surface. The model gives reasonable values for film thickness over short dwell times at constant load. This suggests that a joint could experience full film lubrication by squeeze film effects, The results of this study also suggest that variations in ‘the viscosity could have a significant effect on the film thickness response. In addition, it was found that the time response of the film thickness is less sensitive to changes in cartilage moduli than fluid viscosity. However, both parameters must be considered in any model because it was found that rigid surfaces and a lubricant with a viscosity of 2Ocp provided better lubrication properties than compliant surfaces with a lower fluid viscosity. Thus, these results indicate that the “rigid” components of a total hip prosthesis performing with a high viscosity lubricant could provide better lubrication properties than more compliant surfaces with a much lower fluid viscosity. It is important to place conclusions drawn from any modeling study in proper perspective with the characteristics of the model. It is emphasized that the model has not been developed to the point of direct clinical applications. Rather it is a step toward developing a more realistic representation of joint lubrication. As previously mentioned, this model is based on elastohydrodynamic theory. The compliant surfaces are elastic and nonviscous. Viscous behavior is included in the Newtonian lubricant. Thus, effects of some of the characteristics of synovial fluid are not represented in the model at this time. It is of interest to discuss some of these characteristics and their possible effects on lubrication of the joint. Two such properties of synovial fluid are shear thinning and viscoelasticity. Shear thinning is the term used to describe a decreasing viscosity with increasing shear rate. According to Davies and Palfrey (19691, at low shear rates (ca. 1 set- I), the apparent viscosity is 100 cp. At shear rates greater than 1000 set- I, the viscosity seems to be between 5 and 10 cp. While the data on shear thinning of synovial fluid contains scatter, it does show that the viscosity approaches a constant value at higher shear rates. Since the shear rate depends on film thickness, it appears that the viscosity can change over some range of film thickness. However, the high shear rates (for thin film conditions) causes the viscosity to have a nearly constant value, and thus behave essentially as a non-thinning fluid for the range of interest. It Is of interest to note that the results of the model were for one value of viscosity for each case. However, it is possible to obtain the results for a different
Time (1). millisec
Fig. 6. Film thickness at R = 0 as a function of time for various values of viscosity.
viscosity without rerunning the analysis. The reason is that the product ,LJU,in equation (1) is a single quantity and thus the solution for doubling the viscosity to 2/l can be obtained by halving the velocity to l/ZU, and the time to t/2. For example, the film thickness results shown in Fig. 6 for 80 cp occur at times which are four times the value for the 20 cp case. Concerning viscoelasticity, some investigators have detected viscoelastic behavior in synovial fluid. All liquids are to some degree viscoeiastic. However, the effect is so small that it is ignored in conventional hydrodynamic calculations. If synovial fluid possesses greater than normal sensitivity. to shear rate, the viscoelastic property would influence film thickness in the joint. The result would be a thicker film. Under squeeze film conditions, impact would produce high shear rates and thus an elastic response from the liquid which will decay with time. Therefore, the pressure pocket will be established with a given separation distance between the load supporting surfaces. Thus, the time at which practical separation distance is maintained will be longer .than that predicted from viscous effects alone. These effects of viscoelastic behavior would tend to enhance the lubrication process. However, an important unanswered question is whether synovial fluid has significant viscoelastic properties to effect the lubrication process. Measurements with a Weissenberg rheogoniometer were made by Davies and Palfrey (1969) and have detected viscoelastic effects in synovial fluids. The elasticiry values were quite small, on the order of a maximum of 10 N/m’. Thus, it appears that the rates of applied load in normal joint performance are enough to require consideration of viscoelastic effects. However, the
Effects of cartiiage stiffness and viscosity
properties of synovial fluid are not yet well understood and funher experimental work on physical properties of synovial fluid are needed before the significance of non-Newtonian properties can be confidently applied to the theory of joint lubrication. Some comment on the time frame in Figs. 3,5 and 6 is appropriate. These times are on the order of milliseconds. Time intervals for high load applications in human joints are on the order of tenths ofa second. It is anticipated that if the time is extended to one tenth of a
second, a reasonable film thickness will remain as the trends in Figs. 3 and 5 indicate. The results presented here are the trends for short time periods in the beginning of the load cycle. The model presented here displays some of the characteristics of lubrication in living joints. It is recalled that in earlier work by Rybicki er al. (1978). this model showed good comparisons with results obtained from another numerical approach (Bupara er al., 1975) and with laboratory data for compliant bearing lubrication between a rubber sphere and a glass plate (Roberts and Tabor, 1971). Since the model is based on a finite element formulation, it has the flexibility to handle a range of geometries including surface curvature and properties that make it a candidate tool for understanding the lubrication in living joints and improving lubrication in joint replacement devices.
REFERENCES Bupara. S. S., Rybicki. E. F. and Glaeser, W. A. (1975) An examination of the compliant bearing model of lubrication for joints. Proc. 3rd Annual New England Bioengineering Conf. Medford, Massachusetts. Davies, D. V. and Palfrey, J. (1969) Physical properties of svnovial fluid. Lubrication and Weur in Joints (Edited by kright, V., pp. 20-28. Lippincotf, Philadelphia. _ Day, C. P. (1974) Transient elastohydrodynamic lubrication by finite elements, M. S. Thesis. Cornell University. Ithaca. New York. Dowson, D. (1969) Lubrication in human joints. Lubricorion and Wear in Joints (Edited by Wright, V.), pp. 9-14. Lippincott, Philadelphia. Dintenfass, L. (1963) Lubrication in synovial joints: a theoretical analysis. J. Bone Jr Surg. (Al45, 1241-1256. Fletcher, R. and Powell, ,M.J. D. f 1963) A rapidly convergent descent method for minimization. Comput. J. 6, 163-168. Glaeser, W. A. and Rybicki, E. F. Deformation and elastic recovery of fresh cartilages. To be published.
409
Harris, E. D.. Parker. H. G., Radin, E. L. and Krane. S. M. (1972) Effects of proteolytic enzymes in structural and mechanical properties of cartilage. krhriris Rheum. 15, 497-503.
Hayes, W. C., Keer; L. M., Herrman, G. and Mockros. L. F. (1972) A mathematical analysis for indentation tests of articular cartilage. J. Biomech. 5, 541-551. Higginson, G. R. and.Norman, R. (1974) The lubrication of porous elastic solids with reference to the functioning of human joints. J. mech. Engng Sci. 16, 250-257. Hori, R. Y. and Mockros, L. F. (1976) Indentation test of human articular cartilage. /. Biomech. 9. 259. Huebner. K. H. (1975) The Finite EIemenr Merhod for Engineers, Chapter 8. Wiley. New York. ,MacConaill, M. A. (1967) Basic anatomy of weight bearing Proc. Lubricarion and Wear in Licitu~and Arr~~cial Human loinrs, pp. l-7. Institute of Mechanical Engineers.
joints.
London. Mansour. 1. M. and Mow, V. C. (1976) The permeability of articular cartilage under compressive strain and at high pressures. J. Bone Jr Surg. (A)58, 506-516. Mow, V. C. (1969) The role of lubrication in biomechanical joints. Trans. Am. Sot. mech. Engrs. J. Lubr. Technol. 91, 320-328. Mow, V. C., Lai, W. M. and Radler, I. (1974) Some surface
characteristics of articular cartilage. J. Biomech. 5, 449-456. Nowinsky. J. L. (1971) Bone articulations as systems of poroelastic bodies in contact. AIAA J. 9, 62-67. Oh, K. P. and Huebner, K. H. (1973) Solution of the elastohydrodynamic journal bearing problem (ASME) Paper 672-LUB-26). Trans. Am. Sot. mrch. Engrs. J. Lubr. Technol. (F)95, 342. Parsons, J. R. and Black, 1. (1977) The viscoelastic shear behavior of normal rabbit anicular cartilage. J. Biomech. 10.21-29. Roberts, A. D. and Tabor, D. (1971) The extrusion of liquids between highly elasticsolids. Proc. R. Sot. (A)X.S. 323-345. Rohde, S. M., Whicker, D. and Browne, A. L. (1976) Dynamic analysis of elastohydrodynamic squeeze films. Truns. Am. Sot. mech. Enars. J. Lubr. Technol. (F) 98. 401-407. Rybicki, E. F., S&enkowski, J.. Tamm, h. K. and Glaeser, W. A. (1978) A finite element model for compliant bearing lubrication using a minimization algorithm. WEAR 47, 279-292.
Sellar. P. C., Dowson, D. and Wright, V. (197 1) The rheology of synovial fluid. Rheol. Acra 10. 2-7. Serifini-Fracassini, A. and Smith, J. W. (1974) The Srrucrure and Biochemisrry of Carrilage. Churchill Livingstone. London. Tanner, R. I. (1966) An alternative mechanism for the lubrication of synovial joints. Phys. Med. Biol. 11, 119. Taylor, C. and O’Callaghan, J. F. (1972) A numerical solution ofrheelastohydrodynamic lubrication problem using finite elements. J. mech. Engng Sci. 14, 229. Walker,T. W., Graham. J. D. and Mills, R. H. (1976) Changes in the mechanical behaviour of the human femoral head associated with arthritic pathologies. J. Biomech. 9, 615-624.
GLAESER,:
J. S. STRESEOH~KIand M. A. T-\u%t
Applied Solid Mechanics Section, Battclle’s Columbus Laboratories, 505 King Avenue, Columbus, OH -13201, U.S.A. Abstract - There are several possible lubrication mechanisms for living joints. One such mechanism, the squeeze film compliant bearing, is examined through the use of a computational model. The model is based on a finite element representation for the compliant surfaces. The lubricant is described by Reynold’s equation. A minimization technique provides solutions to this inherently nonlinear problem, and alleviates difficulties in numerical convergence. In earlier work, the results of this model showed good correlation with both numerical results of another model and laboratory measurements for film thickness of a compliant sphere approaching a glass plate. In this paper, the model is used as a basis for conducting a parametric study of the influence of cartilage modulus and fluid viscosity on the lubrication characteristics. A load of 445 N was applied in the studies. Results of the compliant system are compared with those for a system with rigid surfaces representing a prosthesis. Based on the results obtained by the model and the ranges of parameters considered, it was found that changes in viscosity of the lubricant had a greater influence than changes in ths cartilage modulus. The lubrication qualities of the compliant system were superior to those of the system uith rigid surfaces if the same lubricant properties were used. However, lubrication viscosities could be found such that a compliant system with a low viscosity has poorer lubrication properties than a rigid system with a higher viscosity.
ISTRODUCI-ION
The human body has more than ten major load bearing joints connecting long bones to other skeletal members. Some of these joints transfer forces equal to many times body weight and all permit a wide range of motion. This is accomplished under conditions of very low friction. For example, Mow (1969) reports that healthy joints articulate with coefficients of friction ranging from 0.005 to 0.025. However, disease can increase joint friction and severely limit joint function. In some cases, joint function is partially restored by using a joint replacement prosthesis. Current designs for prostheses include several characteristics ofnormal joint articulation. For example, prostheses have geometric compatibility with the kinematics of joint members. Implant materials are compatible with body tissue and are selected on the basis of reducing wear between roiling and sliding parts. However, many properties of the living joint are not included in current designs for joint prostheses. An example is the lubri-
* Receired 3 May 1977; in revised form31 August 1978. t This research was sponsored by the Battelle Institute Program, “Fundamental Studies of Lubrication in Living Joints”, Contract No. B-1333-1210. : Structural Mechanics and Tribology Section. $ Other possible lubrication mechanisms in living joints include boundary or soft tissue lubrication (especially under slow movement and constant loading) and “boosted lubrication” in which concentrated hyaluronic acid at the contracting surfaces increases load support. 8.M.
116-A
cation aspects of living joints. One reason lubrication
qualities are not reproduced in the replacement prostheses is that a full understanding of the basic lubrication mechanisms in living joints is not available. In principle, the existing technological understanding ofdifferent lubrication mechanisms is applicable to the problem of better understanding the lubrication mechanisms of living joints and can also increase the understanding of lubrication in prosthetic devices. Several theories are set forth in the literature to explain lubrication mechanisms of living joints. As a step toward a better understanding of the lubrication of living joints, this paper focuses on one possible mechanism, the compliant bearing lubrication me&anism_$ A computational model, which draws on existing elastohydrodynamic lubrication theory, is described and used as a means of examining the effect of different constituent properties on joint lubrication. Using a simple geometric configuration, the effects of the elastic modulus of the cartilage and the viscosity of the fluid lubricant are considered. The following sections give background information on compliant bearing lubrication and joints, a description of the computational model, a discussion of numerical results and the conclusions.
BACKGROUNDON COMPLIANT BEARISG LLBRlCATtON FOR JOINTS
After 403
the development
of elastohydrodynamic
404
E. F. RY~ICKI.W. A. GLAESER. J. S. STRESKOWSKI
(EHD) theory of lubrication as a viable engineering design tool, researchers such as bintenfass (1963) and Tanner (1966) investigated the relevance of EHD theory for lubrication of living joints. Since then, a number of studies by, for example, Dowson (1969) and Higginson and Norman (1974) have further supported the relevance of EHD theory to lubrication of living joints. There are three characteristics of a compliant bearing involving EHD lubrication. These are the existence of compliant bearing surfaces, a nearly incompressible bearing material and a lubricant. The cartilage is a compliant bearing surface with modulus values of 25.5 MN/mm” as reported by Walker er al. (1976). The cartilage is close to incompressible, as evidenced by Poisson’s ratios ofO.40, reported by Parsons and Black (1977) and 0.47 by Hori and Mockros (1976). The availability of synoviai fluid as a lubricant is an important feature of a compliant bearing lubrication. There appear to be at least two possible ways for bringing fluid to the contact area. One way is that, as the joint rotates, the unloaded bearing surface is wetted by the fluid which is then available when that surface is weight bearing. Another possible way of getting lubricant to the weight bearing area is based on the notion that in order to form a fluid film between contacting cartilage surfaces, fluid must be exuded from the surfaces to the contact zone. This type of weeping lubrication has received attention by Monsour and Mow (1976). Based on experiments, a mathematical model was developed in which synovial fluid is squeezed from the cartiiage just in advance of the moving contact. Little flow occurs in the cartilage under compression and the “starved” cartilage released from pressure In the trailing area soaks up the synovial fluid and holds it for lubrication as the contact moves back over it. However, a significant question is left unanswered, concerning the amount of fluid that must be maintained between the load bearing surfaces to achieve viable lubrication, as compared to the amount that can be delivered by squeezing the cartilage. Experiments by Harris et al. (1972), Parsons and Black (1977), and others have shown that cartilage behaves as a viscoelastic material. Experiments by Giaeser and Rybicki (1977) indicate that behavior in short time load applications is predominantly elastic. Since the cartilage has very low permeability for flow out of its surface, it might be expected to perform like a fluid filled membrane during short time impulsive loading and would be essentially incompressible. Synovial fluid is non-Newtonian and Davies and Palfrey (1969) report decreasing values in apparent viscosity for increasing shear rates. In this study, the joint is represented approximately as a squeeze film bearing having low modulus elastic compliant surfaces and separated by a viscous fluid. This simplification of the model should only be valid for joints under impulse short time load application. It would not hold for a load bearing joint at rest.
and %I.
MODEL
A T.MM
FOR COMPLIAST LUBRiCATlON
BEARI%
A review of the literature reveals several computational approaches describing the mechanics of compliant bearing squeeze film lubrication. Huebner (1975) describes three papers based on the finite element method and iteration technique for solving lubrication problems. Iteration techniques are needed because of the inherent nonlinearity relating the film thickness to the pressure. In each reference, the compliant surfaces are modeled as an elastic deformable material and Reynoid’s equation is used to model the behavior of the lubricant. Taylor and O’Callaghan (1972) and Oh and Huebner (1973) follow a procedure that iterates on the deformation state of the elastic bearing. Pressures are calculated based on the film thickness and Reynold’s equation. The pressure distribution is applied to the elastic bearing surface to obtain a deformation. A new film thickness is obtained, based on this deformation, and the process continues until convergence of the deformation is obtained. Taylor and O’Callaghan (1972) and Oh and Huebner (1973) point out that this procedure only converges for”lightiy loaded bearings” and that steps must be taken to increment the load and bound the estimated deformation, or else the method diverges. It is also reported that this procedure does not work well for “soft” bearing materials. In another approach, Day (1974) maintains compatibility between the film thickness and deformations while formulating an initial value problem to satisfy equilibrium between the fluid and compliant material. The procedure starts with an assumed pressure distribution and marches out in time until the pressure does not change. It is reported that this procedure works well for “soft materials” but encounters difficulties for “relatively stiff’ bearing materials. In a recent paper, Rohe et ~1. (1976) present a number of analyses for EHD squeeze film problems to handle time dependent loading conditions for both soft and hard surfaces. In work reported by Bupara et al. (1973, an iterative technique similar to those described by Taylor and O’Callaghan (1972) and Huebner (1975) demonstrated that solutions were feasible for very low load levels - up to 45 N. Difficulties in obtaining solutions for higher loads were attributed to the inherent nonlinearities which could not be satisfactorily treated by iterative techniques. The model for compliant bearing squeeze film lubrication used in this study is numerically stable and is not confined to small load levels. The small load level limitation was alleviated by introducing a minimization algorithm to replace the iterative solution procedure. The basic advantage of the minimization algorithm is that at each step, a better solution is obtained, based on the value of the criterion that is being minimized. Thus, the method does not have the divergence problem associated with the direct iteration procedure. A detailed description of the model is given by
Effects of cartilage
stifftie%
1524
,270 , toordlno+e.
Fig. 1. Coxrse
finite
element
mm
mesh for simple
axisymmetric
model of the human knee joint. Rybicki er nl. (1977). Thus, only an overview is given here. Figure 1 illustrates the geometry considered. The model consists of three parts: a fluid lubricant, compliant surfaces. and a base material. The fluid behavior is governed by Reynolds equation. This means that the fluid is Newtonian, inertia effects are negligible, the film is thin enough so that the significant velocity change is across the thickness, and curvature effects of the bearing surface are neglected. The compliant bearing surfaces are modeled as a linear elastic material with a Poisson’s ratio close to 0.5. The base material is linear elastic and stiffer than the compliant surfaces. The equation governing the fluid pressure and the film thickness is
and viscosity
To obtain solutions to the problem, the system is discretized to N nodes and X - 1 intervals. Equation (1) is expressed in a discretized form, as is the stress displacement relation for the compliant surface. The fluid and the compliant surfaces are coupled through equilibrium and compatibility. Compatibility is satisfied exactly. Equilibrium is satisfied approximately at discrete locations by minimizing a function which is the sum of the squares of the differences, between the pressures of the fluid and the stress of the solid at these points. In this minimization formulation, the deformations of the compliant surface, at discrete points, are the unknowns. The governing equations for the compliant bearing surface shown in equation (2) are a iinear relation between the surface stress and dispiacements. I-lowever, equation (1). which relates the film thickness (and hence the surface deformations) to the pressure. is nonlinear. Thus, the system of equations to be solved is nonlinear. Imposing equilibrium conditions between the fluid pressure and the stress at the cartilage surface for a discrete number of locations gives a set of algebraic nonlinear equations to solve. The equations were solved by minimizing the sum of the squares of the residuals. The minimization technique used here is that described by Fletcher and Powell (1963). The method can be described as one of the conjugate gradient techniques and is basically for minimizing a function that is not subject to constraints. In the formulation of this problem, there was one constraint and the procedure was modifieo to handle it. The constraint is that the film thickness must always be greater than zero. This condition was met by starting with an acceptable solution and shortening the step size if an unacceptable solution was suggested by the method. This is a valid procedure for this particular problem because the solution cannot lie on a constraint /I = 0 since the value of the function to be minimized is infinite if II = 0 at any point. NUMERICAL
where p is the pressure in the fluid at location r, R is the outer radius, p is the viscosity, lJZ is the velocity, and h is the film thickness. The relation between the stress and the deformation of the compliant surface is given in discretized form by i"il VI I
=
[K,jl{w,l, Y.xI
(2,
.\'I,
where ci is the stress at a location denoted by i, wj denotes the displacement at point .i, and Kij denotes the stiffness matrix. The film thickness is related to the deformation of the compliant surfaces by h, = s - 2ni,
(3)
where .? defines the relative distance between the two compliant surfaces when there is no load on the system.
405
RESULTS 6OR PARAMETRIC STUDIES
Prior to undertaking the parameter studies, two different finite-element models of the cylinders were examined. The purpose was to develop a finite element model which would provide accurate results and also be economical to use. The first finite element model consisted of 21 radial nodes on the axisymmetric circular surface of each cylinder. The finite element representation of the lower cylinder is shown in Fig. 1. With this coarse model and the properties summarized in Table 1, it was found that the solutions for film thicknesses less than about 7.62 x lo-’ mm did not converge. Therefore, a refined axisymmetric model was developed, which consisted of 57 radial nodes (nonuniformly spaced) on each cylindrical surface and a total of 448 nodes throughout each cylinder. This mesh is shown in Fig. 2. along with an enlargement of a portion of this mesh. Using this refined representation.
E. F. RYBICKI,W. A. GLAESER,J. S.
STRENKOV.XKI and
X4. A. TA.CIM
IA
Fig. 2. Refined finite element mesh for simple axisymmetric model of the human knee
joint.
Table 1. Properties of the cylinder model Elastic modulus (MN/m’)
Thickness (mm)
Poisson’s ratio
Viscosity
Bone
Cartilage
Bone
Cartilage
Bone
Cartilage
20
6.895 x 10’
8.826
0.25
0.48
6.35
6.35
Table 2. Values varied in parametric studies
Case No. 1
2 3 4 5
Elastic modulus of cartilage (MN/m’)
Fluid viscosity (CP)
4.413 17.651 8.826 8.826 8.826
20 20 20 7 80
results were obtained for film thicknesses of the same order of magnitude expected in living joints. Based upon the above findings, the refined finite element model of the cylinders was chosen to study the effect of varying several properties of the knee joint model. Two parameters which characterize a diseased knee joint are the cartilage elasticity and the viscosity of the synovial fluid. These properties of the cylinder model were varied. A summary of the range of properties is given in Table 2. Unless otherwise indicated, the remaining properties used were identical to those given in Table 1. Evaluation of the model response due to change in material properties was based upon the decrease in film thickness as a function of time. To find the variations of film thickness with timi, it was assumed converged
Contact radius (mm) 12.7
that the local deformation velocity, d\c/dt, was negligible when compared to the overall approach velocity. @;dr. This is consistent with the rapid loading assumption in which the cartilage behaves elastically. Therefore, UI = dS/dt and the time may be found by integrating (l/U,) with respect to s. By using a suitable numerical integration scheme such as Simpson’s rule, 3 and the film thickness h(r) may then be calculated as a function of time. The first study involved varying the modulus of elasticity of the cartilage from 4.413 to 17.651 MN/m’ as shown in Tabie 2. Figure 3 shows half of the film thickness at R = 0 as a function of time and cartilage modulus. Here t = 0 corresponds to an undeformed film thickness s = 0.635 mm. Although only three different cartilage moduli were selected, certain trends may be identified from these results. For example, it may be seen that as time increases, the film thickness h/2 is less for higher values of the cartilage modulus. This means the time to reach a given film thickness decreases as the cartilage modulus increases. This trend continues so that for rigid cylinders, this time is a minimum. The improvement in the lubrication properties of compliant bearings noted above may be partially explained by the film thickness profile. This is shown in Fig. 4 at several times for a cartilage modulus of 8.826 MN,‘m’ and a fluid viscosity of,u = 20 cp. Due to
Effects oi cartilage stiffness and viscosity
0 Ecorliloq6
=
4.413
MN/
0 Ec,-r+ilapa
’
8.826
MN/
: 17.651
1.0
: 4 413 MN/m’
MN/m
2.0
Fig. 3. Film thickness at R = 0 as a function of time for various cartilage moduli.
the cartilage elasticity, the fluid trapped between the squeezing surfaces forms a pocket-type configuration. This shape restricts the flow of the synovial fluid from the region of high pressure so that a longer time is required co squeeze out the fluid. Note that as time increases, the relative depth of the pocket decreases, the ratio I?,,,,;,h(R=O) decreases, and the minimum film thickness moves radially outward. These trends were also observed for cartilage moduli of 17.651 and 4.413 MN/m’. When plotted as a function of time, the minimum film thickness approaches zero morequickly than the film thickness at R =0 (see Fig. 5). Note that use of the minimum film thickness as a basis of comparison between different cartilage moduli leads
/
I
I
c,,,,,,ape= 17.651 MN/&
0
Time (1). miilisec
I
I
407
/
10
2.0 Time(t). millisec
3.0
Fig. 5. Wnimum film thickness as a function of time for various cartilage moduli. to different conclusions than if the film thickness at R=O had been used (compare Figs. 3 and 5). This
difference may be explained by the fact that for a given time, the minimum film thickness occurred closer to the cylinder outer edge for higher cartilage moduli. Therefore, the occurrence of the minimum film thickness at different radii makes it a less likely basis by which to compare the time response of various cartilage moduli. The sensitivity of the film thickness response to variations in the viscosity was also investigated. The film thickness as a function of time is shown in Fig. 6 for three values of the viscosity. These three parametric values correspond to Cases 3, 4 and 5 in Table 2. In contrast to Fig. 3, it may be seen that the film thickness response is more sensitive to changes in viscosity than to changes in elastic modulus of the cartilage. It is interesting to note that a rigid bearing with a viscosity of synovial fluid (p = 20 cp) exhibits better lubrication properties than a compliant bearing with ,D= 7 cp. Finally, note that for the same viscosity the rigid bearing always displays poorer lubrication capabilities than a compliant bearing. When the trends shown in Fig. 6 are extrapolated out to dwell times (t) fodnd in actual joint performance Table 3. Estimated film thickness values extrapolated from Fig. 6 Joint model
Radius (RI.
mm
Fig. 4. Film thickness profiles for various times for EL&,. = 8.826 MN/m’ and p = 20 cp.
Rigid Compliant Compliant
Viscosity (CP)
I (msec)
h/Z (mm x lo-31
20 80 20
100 100 loo
5.08 12.07 7.62
408
E. F. RYBICKI.
W. A. GLAESER J. S.
STRENKOWSKI and M. A. TAMM
(say t = lOOmsec), the resulting hi2 values are not unreasonable for full film support for loaded joint surfaces. The extrapolations are summarized in Table 3.
DISCUSSIONS AND CONCLUSIONS
Based upon the results of the parametric studies, it is demonstrated that the lubrication process is enhanced by considering a compliant rather than a rigid bearing surface. The model gives reasonable values for film thickness over short dwell times at constant load. This suggests that a joint could experience full film lubrication by squeeze film effects, The results of this study also suggest that variations in ‘the viscosity could have a significant effect on the film thickness response. In addition, it was found that the time response of the film thickness is less sensitive to changes in cartilage moduli than fluid viscosity. However, both parameters must be considered in any model because it was found that rigid surfaces and a lubricant with a viscosity of 2Ocp provided better lubrication properties than compliant surfaces with a lower fluid viscosity. Thus, these results indicate that the “rigid” components of a total hip prosthesis performing with a high viscosity lubricant could provide better lubrication properties than more compliant surfaces with a much lower fluid viscosity. It is important to place conclusions drawn from any modeling study in proper perspective with the characteristics of the model. It is emphasized that the model has not been developed to the point of direct clinical applications. Rather it is a step toward developing a more realistic representation of joint lubrication. As previously mentioned, this model is based on elastohydrodynamic theory. The compliant surfaces are elastic and nonviscous. Viscous behavior is included in the Newtonian lubricant. Thus, effects of some of the characteristics of synovial fluid are not represented in the model at this time. It is of interest to discuss some of these characteristics and their possible effects on lubrication of the joint. Two such properties of synovial fluid are shear thinning and viscoelasticity. Shear thinning is the term used to describe a decreasing viscosity with increasing shear rate. According to Davies and Palfrey (19691, at low shear rates (ca. 1 set- I), the apparent viscosity is 100 cp. At shear rates greater than 1000 set- I, the viscosity seems to be between 5 and 10 cp. While the data on shear thinning of synovial fluid contains scatter, it does show that the viscosity approaches a constant value at higher shear rates. Since the shear rate depends on film thickness, it appears that the viscosity can change over some range of film thickness. However, the high shear rates (for thin film conditions) causes the viscosity to have a nearly constant value, and thus behave essentially as a non-thinning fluid for the range of interest. It Is of interest to note that the results of the model were for one value of viscosity for each case. However, it is possible to obtain the results for a different
Time (1). millisec
Fig. 6. Film thickness at R = 0 as a function of time for various values of viscosity.
viscosity without rerunning the analysis. The reason is that the product ,LJU,in equation (1) is a single quantity and thus the solution for doubling the viscosity to 2/l can be obtained by halving the velocity to l/ZU, and the time to t/2. For example, the film thickness results shown in Fig. 6 for 80 cp occur at times which are four times the value for the 20 cp case. Concerning viscoelasticity, some investigators have detected viscoelastic behavior in synovial fluid. All liquids are to some degree viscoeiastic. However, the effect is so small that it is ignored in conventional hydrodynamic calculations. If synovial fluid possesses greater than normal sensitivity. to shear rate, the viscoelastic property would influence film thickness in the joint. The result would be a thicker film. Under squeeze film conditions, impact would produce high shear rates and thus an elastic response from the liquid which will decay with time. Therefore, the pressure pocket will be established with a given separation distance between the load supporting surfaces. Thus, the time at which practical separation distance is maintained will be longer .than that predicted from viscous effects alone. These effects of viscoelastic behavior would tend to enhance the lubrication process. However, an important unanswered question is whether synovial fluid has significant viscoelastic properties to effect the lubrication process. Measurements with a Weissenberg rheogoniometer were made by Davies and Palfrey (1969) and have detected viscoelastic effects in synovial fluids. The elasticiry values were quite small, on the order of a maximum of 10 N/m’. Thus, it appears that the rates of applied load in normal joint performance are enough to require consideration of viscoelastic effects. However, the
Effects of cartiiage stiffness and viscosity
properties of synovial fluid are not yet well understood and funher experimental work on physical properties of synovial fluid are needed before the significance of non-Newtonian properties can be confidently applied to the theory of joint lubrication. Some comment on the time frame in Figs. 3,5 and 6 is appropriate. These times are on the order of milliseconds. Time intervals for high load applications in human joints are on the order of tenths ofa second. It is anticipated that if the time is extended to one tenth of a
second, a reasonable film thickness will remain as the trends in Figs. 3 and 5 indicate. The results presented here are the trends for short time periods in the beginning of the load cycle. The model presented here displays some of the characteristics of lubrication in living joints. It is recalled that in earlier work by Rybicki er al. (1978). this model showed good comparisons with results obtained from another numerical approach (Bupara er al., 1975) and with laboratory data for compliant bearing lubrication between a rubber sphere and a glass plate (Roberts and Tabor, 1971). Since the model is based on a finite element formulation, it has the flexibility to handle a range of geometries including surface curvature and properties that make it a candidate tool for understanding the lubrication in living joints and improving lubrication in joint replacement devices.
REFERENCES Bupara. S. S., Rybicki. E. F. and Glaeser, W. A. (1975) An examination of the compliant bearing model of lubrication for joints. Proc. 3rd Annual New England Bioengineering Conf. Medford, Massachusetts. Davies, D. V. and Palfrey, J. (1969) Physical properties of svnovial fluid. Lubrication and Weur in Joints (Edited by kright, V., pp. 20-28. Lippincotf, Philadelphia. _ Day, C. P. (1974) Transient elastohydrodynamic lubrication by finite elements, M. S. Thesis. Cornell University. Ithaca. New York. Dowson, D. (1969) Lubrication in human joints. Lubricorion and Wear in Joints (Edited by Wright, V.), pp. 9-14. Lippincott, Philadelphia. Dintenfass, L. (1963) Lubrication in synovial joints: a theoretical analysis. J. Bone Jr Surg. (Al45, 1241-1256. Fletcher, R. and Powell, ,M.J. D. f 1963) A rapidly convergent descent method for minimization. Comput. J. 6, 163-168. Glaeser, W. A. and Rybicki, E. F. Deformation and elastic recovery of fresh cartilages. To be published.
409
Harris, E. D.. Parker. H. G., Radin, E. L. and Krane. S. M. (1972) Effects of proteolytic enzymes in structural and mechanical properties of cartilage. krhriris Rheum. 15, 497-503.
Hayes, W. C., Keer; L. M., Herrman, G. and Mockros. L. F. (1972) A mathematical analysis for indentation tests of articular cartilage. J. Biomech. 5, 541-551. Higginson, G. R. and.Norman, R. (1974) The lubrication of porous elastic solids with reference to the functioning of human joints. J. mech. Engng Sci. 16, 250-257. Hori, R. Y. and Mockros, L. F. (1976) Indentation test of human articular cartilage. /. Biomech. 9. 259. Huebner. K. H. (1975) The Finite EIemenr Merhod for Engineers, Chapter 8. Wiley. New York. ,MacConaill, M. A. (1967) Basic anatomy of weight bearing Proc. Lubricarion and Wear in Licitu~and Arr~~cial Human loinrs, pp. l-7. Institute of Mechanical Engineers.
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London. Mansour. 1. M. and Mow, V. C. (1976) The permeability of articular cartilage under compressive strain and at high pressures. J. Bone Jr Surg. (A)58, 506-516. Mow, V. C. (1969) The role of lubrication in biomechanical joints. Trans. Am. Sot. mech. Engrs. J. Lubr. Technol. 91, 320-328. Mow, V. C., Lai, W. M. and Radler, I. (1974) Some surface
characteristics of articular cartilage. J. Biomech. 5, 449-456. Nowinsky. J. L. (1971) Bone articulations as systems of poroelastic bodies in contact. AIAA J. 9, 62-67. Oh, K. P. and Huebner, K. H. (1973) Solution of the elastohydrodynamic journal bearing problem (ASME) Paper 672-LUB-26). Trans. Am. Sot. mrch. Engrs. J. Lubr. Technol. (F)95, 342. Parsons, J. R. and Black, 1. (1977) The viscoelastic shear behavior of normal rabbit anicular cartilage. J. Biomech. 10.21-29. Roberts, A. D. and Tabor, D. (1971) The extrusion of liquids between highly elasticsolids. Proc. R. Sot. (A)X.S. 323-345. Rohde, S. M., Whicker, D. and Browne, A. L. (1976) Dynamic analysis of elastohydrodynamic squeeze films. Truns. Am. Sot. mech. Enars. J. Lubr. Technol. (F) 98. 401-407. Rybicki, E. F., S&enkowski, J.. Tamm, h. K. and Glaeser, W. A. (1978) A finite element model for compliant bearing lubrication using a minimization algorithm. WEAR 47, 279-292.
Sellar. P. C., Dowson, D. and Wright, V. (197 1) The rheology of synovial fluid. Rheol. Acra 10. 2-7. Serifini-Fracassini, A. and Smith, J. W. (1974) The Srrucrure and Biochemisrry of Carrilage. Churchill Livingstone. London. Tanner, R. I. (1966) An alternative mechanism for the lubrication of synovial joints. Phys. Med. Biol. 11, 119. Taylor, C. and O’Callaghan, J. F. (1972) A numerical solution ofrheelastohydrodynamic lubrication problem using finite elements. J. mech. Engng Sci. 14, 229. Walker,T. W., Graham. J. D. and Mills, R. H. (1976) Changes in the mechanical behaviour of the human femoral head associated with arthritic pathologies. J. Biomech. 9, 615-624.
