185
A full numerical solution to the problem of microelastohydrodynamic lubrication of a stationary compliant wavy layered surface firmly bonded to a rigid substrate with particular reference to human synovial joints D Dowson, CHE, FRS, FEng, FIMechE, FASME, FSTLE and Z M Jin, BSc, PhD Department of Mechanical Engineering, University of Leeds A full numerical solution procedure has been developed for the microelastohydrodynamic lubrication analysis of a stationary compliant wavy layered surjiace firmly bonded to a rigid substrate. The results obtained have been compared with those using a simplified method
adopted by Dowson and Jin ( I ) and good agreement has been obtained.
NOTATION
A
semi-width of the dry contact amplitude of the wavy surface mode non-dimensional amplitude of the wavy surface model = ao/K coefficients in the three-node differential formula in Appendix 2 coefficients defined in equation (15) layer thickness displacement coefficients calculated in Appendix 1 non-dimensional layer thickness = d / R modulus of elasticity for the compliant layer equivalent modulus of elasticity for the compliant layer = 2E/(1 - u2) film thickness central film thickness minimum film thickness for the wavy surface model minimum film thickness for the smooth surface model non-dimensional film thickness = h / R pressure distribution maximum dry contact pressure of the smooth surface model non-dimensional pressure distribution = p / E ’ bearing radius damping factors sliding velocity non-dimensional sliding velocity = qu/(E’R) load carrying capacity non-dimensional load carrying capacity = w/(E’R) coordinates cavitation boundary due to film rupture inlet boundary outlet boundary non-dimensional axial coordinate = x/d elastic deformation Th? M S w’as received un 31 March 1992 and 13 Ortiher 1992.
H01992 @ IMechE 1992
WUI
accepted fur publication on
E ~ e2 ,
y
A. A
v
4
non-dimensional elastic deformation = 6 j R tolerances for pressure and load iterations viscosity of the lubricant wavelength of the wavy surface model non-dimensional wavelength of the wavy surface model = I / d Poisson’s ratio of the compliant layer phase angle of the wavy surface model 1 INTRODUCTION
Microelastohydrodynamic lubrication has received considerable attention in the last ten years (1). Most analyses have been confined to situations of semiinfinite elastic solids and high elastic modulus materials. The problem considered by Dowson and Jin (1) was a two-dimensional compliant layered surface firmly bonded to a rigid substrate. A simplified solution procedure was developed to enable solutions to be obtained under heavily loaded conditions and for realistic rough surfaces in the normal human ankle joint. The essence of the simplified method was to split the total pressure into two parts, the first being due to the smooth surface and the second to the introduction of roughness, and was described as a perturbation solution. The corresponding elastic deformation could therefore be calculated according to simple expressions. For the pressure resulting from the smooth surface, the column model as described previously in reference (1) was adopted. However, the simplified elasticity model predicts a zero displacement where the pressure is zero. Thus, a relatively large error may be expected at the end of the contact. The purpose of the present study is to develop a complete numerical solution incorporating the full elasticity model for the compliant layered surface firmly bonded to a rigid substrate. This enables the validity of the simplified method to be evaluated. 2 PROBLEM FORMULATION, GEOMETRY AND GOVERNING EQUATIONS
The bearing model considered is shown in Fig. 1, representing the normal human ankle joint (2). The compliant layer of bearing material (representing articular
0954-4119/92 $3.00
+ .OS
Proc Instn Mech Engrs Vol 206
: D DOWSON AND 2 M JIN
186
xu
*
t 0
X
-u-
Fig. 1 An equivalent bearing model with transverse roughness for the
normal human ankle joint cartilage) of mean thickness d is firmly mounted to a rigid cylindrical substrate to form a bearing surface of radius R adjacent to a plane rigid smooth surface moving with velocity u in the x direction. The transverse roughness is assumed to be superimposed upon the stationary compliant layer, and is represented by a sinusoidal profile of amplitude a, and wavelength 3.. The effects of moving roughness have been examined by Dowson and Jin (1).
sure distribution in the lubricant film calls for the solution of an integral equation (3,4):
(3) where the kernel function K is given as follows: K(x -
r)
=
( ( 3 - 4 4 sinh(2i) - 21) cos{[(x -
A full numerical solution to the problem of microelastohydrodynamic lubrication of a stationary compliant wavy layered surface firmly bonded to a rigid substrate with particular reference to human synovial joints D Dowson, CHE, FRS, FEng, FIMechE, FASME, FSTLE and Z M Jin, BSc, PhD Department of Mechanical Engineering, University of Leeds A full numerical solution procedure has been developed for the microelastohydrodynamic lubrication analysis of a stationary compliant wavy layered surjiace firmly bonded to a rigid substrate. The results obtained have been compared with those using a simplified method
adopted by Dowson and Jin ( I ) and good agreement has been obtained.
NOTATION
A
semi-width of the dry contact amplitude of the wavy surface mode non-dimensional amplitude of the wavy surface model = ao/K coefficients in the three-node differential formula in Appendix 2 coefficients defined in equation (15) layer thickness displacement coefficients calculated in Appendix 1 non-dimensional layer thickness = d / R modulus of elasticity for the compliant layer equivalent modulus of elasticity for the compliant layer = 2E/(1 - u2) film thickness central film thickness minimum film thickness for the wavy surface model minimum film thickness for the smooth surface model non-dimensional film thickness = h / R pressure distribution maximum dry contact pressure of the smooth surface model non-dimensional pressure distribution = p / E ’ bearing radius damping factors sliding velocity non-dimensional sliding velocity = qu/(E’R) load carrying capacity non-dimensional load carrying capacity = w/(E’R) coordinates cavitation boundary due to film rupture inlet boundary outlet boundary non-dimensional axial coordinate = x/d elastic deformation Th? M S w’as received un 31 March 1992 and 13 Ortiher 1992.
H01992 @ IMechE 1992
WUI
accepted fur publication on
E ~ e2 ,
y
A. A
v
4
non-dimensional elastic deformation = 6 j R tolerances for pressure and load iterations viscosity of the lubricant wavelength of the wavy surface model non-dimensional wavelength of the wavy surface model = I / d Poisson’s ratio of the compliant layer phase angle of the wavy surface model 1 INTRODUCTION
Microelastohydrodynamic lubrication has received considerable attention in the last ten years (1). Most analyses have been confined to situations of semiinfinite elastic solids and high elastic modulus materials. The problem considered by Dowson and Jin (1) was a two-dimensional compliant layered surface firmly bonded to a rigid substrate. A simplified solution procedure was developed to enable solutions to be obtained under heavily loaded conditions and for realistic rough surfaces in the normal human ankle joint. The essence of the simplified method was to split the total pressure into two parts, the first being due to the smooth surface and the second to the introduction of roughness, and was described as a perturbation solution. The corresponding elastic deformation could therefore be calculated according to simple expressions. For the pressure resulting from the smooth surface, the column model as described previously in reference (1) was adopted. However, the simplified elasticity model predicts a zero displacement where the pressure is zero. Thus, a relatively large error may be expected at the end of the contact. The purpose of the present study is to develop a complete numerical solution incorporating the full elasticity model for the compliant layered surface firmly bonded to a rigid substrate. This enables the validity of the simplified method to be evaluated. 2 PROBLEM FORMULATION, GEOMETRY AND GOVERNING EQUATIONS
The bearing model considered is shown in Fig. 1, representing the normal human ankle joint (2). The compliant layer of bearing material (representing articular
0954-4119/92 $3.00
+ .OS
Proc Instn Mech Engrs Vol 206
: D DOWSON AND 2 M JIN
186
xu
*
t 0
X
-u-
Fig. 1 An equivalent bearing model with transverse roughness for the
normal human ankle joint cartilage) of mean thickness d is firmly mounted to a rigid cylindrical substrate to form a bearing surface of radius R adjacent to a plane rigid smooth surface moving with velocity u in the x direction. The transverse roughness is assumed to be superimposed upon the stationary compliant layer, and is represented by a sinusoidal profile of amplitude a, and wavelength 3.. The effects of moving roughness have been examined by Dowson and Jin (1).
sure distribution in the lubricant film calls for the solution of an integral equation (3,4):
(3) where the kernel function K is given as follows: K(x -
r)
=
( ( 3 - 4 4 sinh(2i) - 21) cos{[(x -
