The hydrodynamic squeeze film lubrication of the ankle joint

Transcription

1 International Journal o Mechanical Engineering and Applications 01; 1(): 4-4 Published online June 0, 01 ( doi: /j.ijmea The hydrodynamic squeeze ilm lubrication o the ankle joint Albert E. Yousi 1, Ali Amer Al-allaq 1 Proessor, Medical Engineering Department/ollege o Engineering / alnahrain University, Baghdad, Iraq Graduate student, Medical Engineering Department/ ollege o Engineering / alnahrain University, Baghdad, Iraq address: Ali_martial85@yahoo.com(A. A. Al-allaq) To cite this article: Albert E. Yousi, Ali Amer Al-allaq. The Hydrodynamic Squeeze Film Lubrication o the Ankle Joint. International Journal o Mechanical Engineering and Applications. Vol. 1, No., 01, pp doi: /j.ijmea Abstract: The main aim o the study presented in this paper is to determine the characteristics o synovial luid ilm region by modeling the human ankle joint to obtain an analytical expression or the pressure distribution, load carrying capacity, coeicient o riction and reduction o synovial ilm thickness with time o approach. Thus in order to reach a comprehensive analysis o the human ankle joint lubrication, variable behavior o synovial luid during dierent time o joint activities and the inluences o articular cartilage has to be taken into account. The behavior o the synovial luid has been assumed to be isothermal, shear thinning and non- Newtonian couple stress luid. The model o ankle joint has been taken geometrically and kinematically as a partial porous journal bearing under the action o hydrodynamic squeeze ilm lubrication. Typical geometrical and physical values o the ankle joint were acquired rom measured values reported in literature. The problem o ankle joint lubrication has been solved numerically or various couple stress luid parameters together with eect o varying the porosity o the articular cartilage. It has been shown, in the presence o porous articular cartilage, and by assuming the synovial luid to be a non- Newtonian shear thinning couple stress luid that the, pressure distribution, load carrying capacity, synovial ilm thickness with time o approach increased and a reduction coeicient o riction in the hydrodynamic squeeze action in the ankle joint resulted. Keywords: Ankle Joint, Synovial Fluid, ouple Stress Fluid, Hydrodynamic Lubrication, Human Joint Lubrication 1. Introduction The lubrication o human synovial joints has received some attention the last 0 year. The ankle joint is a very important joint because it is a high load-carrying joint in the human body. It is an essential element in maintaining posture and equilibrium o the body. Without ankles we cannot carry out important physical activities such as standing, walking or any type o sports. Thus, it is essential to study and understand the joint lubrication mechanism. The squeeze ilm phenomenon arises when the two lubricating suraces move towards each other in the normal direction and generate a positive pressure and hence support a load. This is due to the act that a viscous lubricant present between the two suraces cannot be instantaneously squeezed out when the two suraces move towards each other and this action provides a cushioning eect in bearings. The squeeze ilm lubrication phenomenon is observed in several applications such as gears, bearings, machines tools, rolling elements and automotive engines [1], dampers and human joints []. The ankle joint consists o three bones namely, the tibia, ibula, and talus. The tibia and ibula orm the mortise (socket) into which the talus its, orming the hinge joint. The talus constitutes the link between the leg and the rest o the oot. The superior articular surace is covered with cartilage or articulation with the medial malleolus portion o the tibia, the inerior surace o the tibia, and the medial surace o the ibula [], shown in Figure (1). Articular cartilage is the bearing material that lines the ends o the bones o synovial joints. Its primary unction is to reduce riction and wear at the articulations o the musculoskeletal system. The tribological properties o cartilage are intimately related to its structure and mechanical properties. The modes o lubrication in cartilage extend beyond the traditional mechanisms o luid-ilm or boundary lubrication. The purpose o this review is to describe the salient properties o articular cartilage necessary to understand the unique biotribology o diarthrodial joints [4]. artilage is a white connective tissue which is

2 International Journal o Mechanical Engineering and Applications 01; 1(): synthesized and maintained by cells called chondrocytes.in human joints, the thickness o the articular cartilage layer varies rom 0.5 to 1.5 mm in upper extremity joints, such as the hand and the shoulder,and rom 1 to 6 mm in lower extremity joints, such as the hip, knee, and ankle [4]. Under normal conditions, articular cartilage provides low riction and wear over a lie span. It is a highly hydrated tissue, with a porosity varying rom 68 to 85 per cent in adult joints [4, 5]. Synovial luid is secreted by synovial lining cells. It plays a very important role in synovial joints. It occupies the joint cavity and lines the synovial joint, providing nutrients and removing catabolic products [6, 7]. Synovial luid is essentially a dialysate o blood plasma with the addition o long chain polymer (hyaluronic acid). It is hyaluronic acid that is responsible or the viscosity o synovial luid and i it is all removed by acetic acid precipitation, the viscosity o the luid reduces to the water [8].Normal healthy synovial luid is highly non-newtonian [8,9, 10, 11, 1], such that the viscosity reduces markedly with the increase o shear rate. In osteoarthritis the viscosity is reduced particularly at the low shear rates while in rheumatoid arthritis the viscosity is reduced even more.this means that with increasing severity o disease the viscosity reduces and becomes less dependent on the rate o shear. The synovial luid is considered as mixture o two incompressible luids: viscous (hyaluronic acid-protein macromolecular complex) and ideal (water and small solutes), only the ideal passing across pores o the articular surace, thus enabling a synovial gel ilm ormation. According to this asymptotic model, the iltration by cartilage is intensive, the luid ilm quickly becomes depleted, and a synovial gel layer develops over the greater part o the contact in the step-loaded joint. The gel can serve as a boundary lubricant i a sliding motion ollows beore a resh synovial luid gets into the contact. [ 14, 15, 16]. The long chain hyaluronic acid (HA) molecules ound as polar additives in synovial luid are characterized by two material constants ( η ) and ( µ ). Synovial luid usually exhibits a non- Newtonian shear thinning behavior. However, under high shear rates, the viscosity o synovial luid approaches a constant value not much higher than that o water [7]. Thereore a Newtonian lubricant model has oten been used or synovial luid in lubrication modeling [18] and in certain pathological cases it becomes Newtonian. The similar material constants are also ound in Stokes [19] couple-stress luid. The simplest generalization o the classical theory o luids by Stokes provides the macroscopic description o the behavior o luids containing a substructure such as lubricants with polymer additives. This has given the motivation to model synovial luid as couple-stress luid in human joints. [9, 10, 11, 0, 1] The coordinates (x) and (y) used or the description o the luid ilm pressure and the movement occurs in the (x y) plane, as shown in Figure (). It is assumed that the ankle joint is loaded in the (y) direction in the plane o symmetry by a load W (t) [N / m], occurring during standing and steady walking. Figure (1) show the human ankle joint (A) tibia bone (B) synovial ilm thickness () talus bone (D) calcaneus bone [0].. Human ankle Joint Geometry Model The human ankle joint can be represented kinematically and geometrically by cylindrical suraces[9,1,14], where the motion is allowed only in the sagittal plane [9, 14, and 15]. The coupling stress model is assumed by two ininite rigid circular cylinders (subchondral bone) in the internal contact (a cylinder encased in a cylindrical cavity), covered with thin layer (articular cartilage) o uniorm thickness, the lower (talar) articular surace is supposed stationary while the upper (tibial) surace is assumed to have pure squeeze motion. Figure () shows the physical and geometry coniguration o the ankle joint (partial porous journal bearing).. Governing Equation The basic ield equations o the micropolar luid were developed by Eringen [] and Stokes [19] and later adopted by Prakash and Sinha[]. The basic equations or the low

3 6 Albert E. Yousi et al.: The Hydrodynamic Squeeze Film Lubrication o the Ankle Joint o micro polar luid in the ilm region in vectorial orm are []. dv 4 p V V ρ = + µ η (1) dt The steady-state laminar equation o low o the luid ilm region in artesian coordinates is the continuity equation: u u + = 0 Under the usual assumptions o luid ilm lubrication applicable to thin ilms,the equation o motion o an incompressible couple stress luid within the ilm region[9, 10,4] the equation(1)becomes:- p 4 = µ u η u 4 () () The ratio ( η µ ) is a dimensional square length and hence characterizes the chain length. l η = (4) µ The low o couple stress luid in a porous matrix is governed by the modiied Darcy law, which accounts or the polar eects [4]. as q q = ( u, v ) and φ = p µ (1 β) φh θ µ (1 β ) ( η µ ) l β = = φ φ p v 0 ( ) = h u dy 0 The relevant boundary conditions or the velocity components are: i. At the porous journal surace y = 0 u = 0, u = 0 ii. At the boundary surace u = 0, u (5) (6), and V = v (7) y = h = 0 and V = v θ (8) The solution o equation () subject to the boundary conditions (6) and (7) and applying sine and cosine hyperbolic expressions to get inal orm o velocity in artesian coordinates as :- y h cosh( ) (, ) = 1 p { ( ) + [1 l ]} µ cosh( h) l u x y y y h l Integrating equation () across the luid ilm and utilizing boundary conditions (7) and (8) and expressions in (6) and (9) in the modiied Reynolds equation as applicable to squeeze action is obtained in the orm 1φH 0 p {[ ( h, l) + ( )] } = 1v θµ (1 β) Where unction ( h, l ) is given as:- h l h l h l h l (, ) 1 4 = + tanh( ) (9) (10).1. Squeeze Film Pressure o the Porous Journal Bearing Representing the Ankle Joint Introducing the ollowing non-dimensional quantities o the governing equation o pressure or neat presentation as ollows:- p p =, h = h = 1 ε cos( θ), µ R ( d ε / dt) X θ =, l = l, φ φ = R, H = H R = R, ψ = kh e, ε = Applying the above dimensionless equations in equation (10).Thus the inal dimensionless orm o the modiied Reynolds equation becomes: {[ (, ) ] p } 1 cos( θ) (1 β) θ 1H φ h l + = θ The boundary conditions or the luid ilm pressure are: p 0 at π (11) = θ = ± (1) dp 0 at θ 0 dθ = = (1) Integrating equation (11) with boundary conditions (1), (1) and inserting the two limits (θ, π ) to obtain inal orm o pressure distribution: ( π 4 θ ) 1φ H 1 (1 ) l p ( θ) = (( 1 + ε) 1( 1 + ε) l ( ) 4l tanh( ε)) β (14).. Load arrying apacity o the Porous Partial Journal Bearing Representing Ankle Joint The load carrying capacity o the porous bearing in a squeeze action is obtained by integrating the ilm pressure equation acting on the surace o the journal shat. The load

4 International Journal o Mechanical Engineering and Applications 01; 1(): carrying capacity is there given by:- π W = p( θ). R.cos( θ) (15) π Let the dimensionless load carrying capacity be: W W = (16) ( ) µ R d ε dt π W = P ( θ) cos( θ) dθ (17) π The terms p ( θ ), p ( θ).cos( θ ) is diicult to deal with by usual mathematical methods, and it cannot be obtained by direct integration. Thus the use o a solution method such as (power series, Gaussian Quadrature and Simpson method) is appropriate.in this research the power series was used and both terms have been substituted with the well itting approximate unction which has been substituted with power series, then integrating the output resulting rom the power series to get the dimensionless load. All mathematical analyses and output resulting curves were done by computer program called wolram Mathematica (8.0). Then equation (17) becomes:- 4 W = π (180 π + π ) 180(( 1 + ε) 1( 1 + ε) ( ) 4 tanh( )) (18) 1 φ H l l 1 ε (1 β ) l.. Squeeze Time-ilm Thickness Relationship The most important characteristics o the squeeze ilm bearings is the squeeze ilm time, i.e., the time required or reducing the initial ilm thickness ( h ) to ( h o) (minimum ilm thickness). The response time o the squeeze ilm is one o the signiicant actors or the deep understanding o biological bearings. The response time is the time that will elapse or a squeeze ilm to be reduced to some minimum permissible level., The ilm thickness at any time (t) can be obtained by integrating equation (18) or the given load []. ( U h p τ = µ + ) h µ (1) Thus, the equation o riction orce in a dimensionless orm is [8, 9]:- F ( 1 h p ) dθ h θ 1 = + 0 () Substitute or ( p ) rom equation (14) in equation () obtain the dimensionless riction orce. The coeicient o riction can be calculated rom the relation [8, 9]:- = F () W The result o equation () can be seen in appendix (B). 4. Results and Discussion 4.1. Result The lubrication o the human ankle joint object is very diicult i not impossible to study in vivo. For this reason the need or a numerical study arises, and the selection o values or a good estimation to get the best analysis results. The parameters were chosen based on previously selected values use in research, specializing with problem to reach to practical parameters that are close to reality such as the dimensions o ankle joint, articular cartilage thickness and permeability o porous region. The selected parameter values are listed in table (1) Table (1) the numerical values o the parameters involved [9, 1, 5]. Parameters Numerical values units Radial learance () (m) Dimensionless couple stress length ( l ) t 1 = W dε (19) h The detailed result o equation (19) can be seen in appendix (A)..4. oeicient o Friction The riction orce can be obtained by integrating the shear stress around the journal surace, as ollows [19]. τ = µ ( u ) y h η( u ) = = y h (0) Eccentricity ratio parameter ( ε ) Permeability o the cartilage matrix (φ ) Tibia and Talus cartilage thickness Viscosity o synovial luid( µ ) ( m ) (m) 0.01 Pa.s

5 8 Albert E. Yousi et al.: The Hydrodynamic Squeeze Film Lubrication o the Ankle Joint Figure () shows the variation o dimensionless pressure ( p ) with angle ο ( θ ) or dierent couple stress length parameters ( l ), φ = 10 m and ε = Figure (7) shows the variation dimensionless pressure ( p ) with angle ο ( θ ) or dierent Permeability value o the cartilage matrix (φ ) with l = 0., ε = 0.5. Figure (4) shows the variation dimensionless pressure ( p ) with angle ο ( θ ) or dierent eccentricity ratios ( ε ), l 18 = 0. andφ = 10 m. Figure (8) shows the variation dimensionless load ( W ) with eccentricity ratio ( ε ) or dierent couple stress length parameters ( l ), φ = m Figure (5) shows the variation o dimensionless pressure ( p ) with couple stress length parameters ( l ) and eccentricity ratios ( ε ), or θ = 0 18 and φ = 10 m. ο Figure (9) shows the variation dimensionless load ( W ) with dierent Permeability value o the cartilage matrix ( φ ) at ε = 0.6, l = 0 (Newtonian) Figure (6) shows the variation dimensionless pressure ( p ) with angle ( θ ο ) at Permeability value o the cartilage matrix φ = 0mwith l = 0., 0.5 ε =. Figure (10) Shows the variation dimensionless Squeeze Time ( t ) with dimensionless ilm thickness ( h ) or dierent couple stress length parameters.

6 International Journal o Mechanical Engineering and Applications 01; 1(): Figure (11) Shows the variation dimensionless Squeeze Time ( t ) with dierent Permeability value o the cartilage matrix at h = 0.5, l = 0 (Newtonian) Figure (15) shows the variation o the dimensionless coeicient o riction ( ) with eccentricity ratio (ε ) or dierent value o the couple stress length parameters ( l ) 4.. Discussion Figure (1) shows the variation dimensionless Squeeze Time ( t ) with dimensionless ilm thickness ( h ) or dierent values o permeability with l = 0. Figure (1) shows the variation o the dimensionless coeicient o riction ( ) with eccentricity ratio ( ε ) or dierent couple stress length parameters ( l ),φ = 10 m 18 Figure (14) shows the dimensionless coeicient o riction ( ) with dierent Permeability value o the cartilage matrix (φ ) at ε = 0.6, l = 0 (Newtonian) The model, on the basis o the Stokes micro-continuum theory, takes into account the non- Newtonian nature o the synovial luid considering the eects o couple stresses on the squeeze ilm lubrication o the ankle joint. The ilm pressure was solved using a modiied Reynolds equation or a non-newtonian couple stress luid. It was applied to predict the pressure distribution and load-carrying capacity or lubricated system o human ankle joint in squeeze action. As the value o couple stress parameter ( l ) or the luid used approaches zero as shown in igures (),(4)and (8), the bearing characteristics predicted by the present analysis approach the Newtonian-lubricant case. This is the agreement with results reported [9, 14, 4]. According to the results obtained, the inluence o couple stress eects on the bearing characteristics is physically signiicant. It ound through that, the eect o large permeability values reduce the dimensionless pressure distribution, dimensionless load-carrying capacity, time o approach and increase the coeicient o riction as compared to the corresponding normal size o porosity. In contrast, the lowest pore size leads to the increase in the amount o load and this explains the inluence o osteoarthritis disease on the eiciency o ankle joint unction, where the cartilage is less sti in both compression and shear, and luid lows more easily through the tissue in joints with osteoarthrosis. This implies greater displacement o osteoarthrotic cartilage than normal (decreased stiness) and a greater rate o deormation (increased permeability) [6] as shown in igures (6), (7), (9), (11), (1), and (14). The inverse proportionality relationship between squeeze time and ilm thickness has been seen in the journal bearing model o ankle joint lubrication. At the initial time, the ilm thickness has its maximum value and Vice versa, when time elapses, the ilm thickness is reduced until reaching a minimum value, as shown in igure (10).The results o coeicient o riction explain the inversely proportional relationship between eccentricity ratio and its value. When the joint moves, the ilm thickness decreases leading to a decrease in coeicient o riction value between articular

7 40 Albert E. Yousi et al.: The Hydrodynamic Squeeze Film Lubrication o the Ankle Joint cartilages,as shown in igure(1). This decrease in coeicient o riction depends on a many parameters such as eccentricity ratio (ε ), couple stress length parameter ( l ) and permeability o porous articular cartilage. The values obtained in igure (1) are close to the values o reported in reerence [7], which specializes in the study o the rictional properties o articular cartilage. 5. onclusions On the basis o the results presented, the ollowing conclusions can be drawn: The long chain o hyaluronic acid molecules existing in the synovial luid gives the motivation or assuming the synovial luid as a Stokes ouple stress luid. The eect o micropolar luid provides an increased pressure, load carrying capacity, squeeze ilm time and decrease coeicient o riction as compared to the corresponding Newtonian case. The synovial luid ilm thickness between two the articular cartilages reduces gradually with continuous application o the load. The degree o this reduction depends on many reasons such as, type o daily activities [standing, walking, running], healthy o synovial luid and time o applying load on the ankle joint. Radial clearance parameter is important actor to determine eatures o partial journal bearing o ankle joint because o its direct eect on the computational o all physical parameters such as luid pressure, load applied, coeicient o riction and luid ilm thickness variation with squeeze time. Joints with normal cartilage give high load capacity and low coeicient o riction as compared with degenerating cartilage (Osteoarthritis), which is a desirable aspect in typical synovial joint lubrication. The eect o large porosity parameter in articular cartilage causes reduction in load carrying capacity and time o approach to minimum ilm thickness as compared with small porous size. Nomenclature gradient operator radial clearance, m e eccentricity, m ε =( e ) eccentricity ratio. φ Permeability o the cartilage matrix, m φ φ = ( ) dimensionless permeability o the cartilage matrix. φh ψ = permeability parameter. µ viscosity o synovial luid, pa.s η material constant responsible or couple stress property. η l = µ couple stress parameter, m l = ( l ) dimensionless couple stress parameter. u, v components o luid velocity in x, y, directions, respectively, m/s V h h c h =( ) H velocity vector synovial ilm thickness, mm dimensionless ilm thickness. porous layer thickness, mm l β = ( ) ratio o microstructure size to pore size. φ p pressure in the porous region, pa p dimensionless pressure. W load carrying capacity per unit length o the bearing, N/m W dimensionless load capacity. F dimensionless rictional orce. ρ density o synovial luid, g/cm t response time taken by journal centre to move romε =0 to ε, sec. 1 t dimensionless response time. R Appendix (A) radius o the journal, mm coeicient o riction. The output result o equation (19) 4 ( 1 + h) π (180 π + π ) 4 t = (9 60 l + 88 l 1φ H 560(1 1 l + ( ) + 4 l tanh( 1 ) (1 β ) l 1 φ H 1 φ H 1φ H 1 φ H + h (4 + 4 l ( )) + 5( ) 48 l ( ) + ( ) (1 β) (1 β) (1 β) (1 β) 1 φ H 4 1 φ H + h(7 1 l + ( )) 4 l( l + h( l ( )) (1 β ) (1 β ) 1 φ H 1 φ H 1 φ H + h ( l ( )) ( ) 6 l ( + 4( )) tanh( 1 ) (1 β ) (1 β ) (1 β ) l 4 1 φ H 1 φ H + 1 l ( l + 96 l + h ( l ( )) ( ) (1 β) (1 β) φ H 1φ H tanh( ) 4 l (1 1 l + 7 l + ( ) + h (1 1 l + ( )) l (1 β) (1 β) 4 1 φ H 1 + h(1 1 l + 7 l + ( ))tanh( ) ) (1 β) l

8 International Journal o Mechanical Engineering and Applications 01; 1(): Appendix (B) The output result o equation () 1 H 1 180(( 1 + ε) 1( 1 + ε) l ( ) 4 l tanh( ) (1 β) l = 4 π (180 π + π ) φ ε [ 1 ε 1 φ H 6( 1 + ε) ( 1 ) 1( 1 ) ( ) 4 tanh( 1 1 ε ε l l ε ε + + ) (1 β) l ε 1 φ H ( 1 + ε) 1( 1 + ε) l ( ) 4 l tanh( 1ε ) (1 β ) l Reerences [1] Naduvinamani N. B. (010), Santosh S., Micropolar Fluid Squeeze Film Lubrication o Finite Porous Journal Bearing, Proceedings o the 1th Asian ongress o Fluid Mechanics 17-1 December, Dhaka, Bangladesh,pp [] Lin J. R., Liao W. H., Hung. R., (004) The eects o couple stresses in the squeeze ilm characteristics between a cylinder and a plane surace, Journal o Marine Science and Technology, Vol. 1, No., pp [] Berish Strauch, Han-Liang Yu, (006) Atlas o Microvascular Surgery: Anatomy and Operative Techniques, Second Edition, New York, Thieme,pp [4] Ateshian G. A., Hung. T., (006) The natural synovial joint: properties o cartilage, Journal Engineering Tribology, Vol. 0 Part J, pp [5] Van.. Mow, Rik Huiskes (005) Basic orthopaedic biomechanics and mechano-biology, Third Edition, Lippincott Williams & Wilkins, Philadelphia, pp [6] Rekha B., Shukla A. K., (000) Rheological Eects o Synovial Fluid on Nutritional Transport, Tribology Letters 9(-4), pp. -9. [7] Jing Liang, (008) Investigation o Synthetic and Natural Lubricants, PHD Thesis, Raleigh, North arolina, North arolina State University, pp [8] Dumbleton J.H., (1981) The Lubrication o Natural Joints, Tribology o Natural and Artiicial Joints, Series, Elsevier, pp [9] Ruggiero A., Gòmez E., D'Amato R,( 01), Approximate closed-orm solution o the synovial luid ilm orce in the human ankle joint with non-newtonian lubricant, Tribology International, 57,pp [10] Bujurke N.M., Ramesh B. Kudenatti, (006) An analysis o rough poroelastic bearings with reerence to lubrication mechanism o synovial joints, Applied Mathematics and omputation 178,pp [11] Singh S.P., hadda G.., Sinha A. K., (1988) A model or micropolar luid ilm mechanism with reerence to human joint,indian journal pure application mathematical ] 19(4),pp [1] Ogston A. G., Stanier j. E. (1950), On the State o Hyaluronic Acid in Synovial Fluid, biochemistry journal, 46, pp [1] Medley J. B., Dowson D., Wright V., (1984) Transient elastohydrodynamic lubrication models or the human ankle joint, Medical Engineering & Physics limited Vol. 1 No., pp [14] Ruggiero A., Gòmez E., D'Amato R, (010) Approximate analytical model or the squeeze-ilm lubrication o the human ankle joint with synovial luid iltrated by articular cartilage, Tribology Letters,41(),pp.7 4. [15] Hlavacek M,( 009) Lubrication o the Human Ankle Joint in Running, International Review o Mechanical Engineering,, pp [16] Hlavacek M.. (000) Squeeze-ilm lubrication o the human ankle joint with synovial luid iltrated by articular cartilage with the supericial zone worn out. Journal o Biomechanics, vol., pp [17] ooke A. F., Dowson D., Wright V., (1978) The Rheology o synovial luid and some potential synthetic lubricants or degenerate synovial joints Engineering in Medicine, vol. 7, no., pp [18] Bujurke N. M., Ramesh B. Kudenatti,( 006) An analysis o rough poroelastic bearings with reerence to lubrication mechanism o synovial joints, Applied Mathematics and omputation 178,pp [19] Vijay.K. Stokes (1966), ouple stresses in luids, Physics o Fluids, Vol. 9,pp [0] Walicki E., Walika A. (1998) Mathematical modelling o some biological bearings. Smart Materials and Structures, novel materials paper presented at 4th ESSM and nd MIMR conerence, vol. 9, pp [1] Bujurket N.M., Jayaraman G.,( 198) The inluence o couple stresses in Squeeze ilms, international Journal Mechanics Vol. 4, no. 6, pp [] Prakash J., Sinha P.,( 1976) A Study O Squeezing Flow In Micropolar Fluid Lubricated Journal Bearings, wear, vol. 8, pp [] Eringen A.. (1966), Theory o micropolar luids, Journal o Mathematical and Mechanics. vol. 16, pp.1. [4] Naduvinamani N. B., Hiremath P.S., Syeda Tasneem Fathima, (005), On the squeeze ilm lubrication o long porous journal bearings with couple stress luids. Industrial Lubrication and Tribology, vol. 57, pp.1-0. [5] Medley J.B., Dowson D., Wright V. (198) Surace geometry o the human ankle joint, Medical Engineering & Physics limited, Vol. 1. No. 1, pp [6] arol A. Oatis Kinesiology, (008), The Mechanics and Pathomechanics o Human Movement, Second Edition, Lippincott Williams and Wilkins, Philadelphia, pp [7] Mccutchen. W., (196) The Frictional Properties o Animal Joints, wear, 5, pp

9 4 Albert E. Yousi et al.: The Hydrodynamic Squeeze Film Lubrication o the Ankle Joint [8] Mokhiamer U. M., rosby W. A., El Gamal H. A.,( 1999) A study o a journal bearing lubricated by luid with couple stress considering the elasticity o the liner, Wear, 4, pp [9] Bujurke N.M., Naduvinamani N. B., Jayaraman G.,( 1991) Theoretical modelling o poro-elastic slider bearings lubricated by couple stress luid with special reerence to synovial joints Appl. Math. Modeling, Vol. 15, pp [0]