Friction and Lubrication in Mechanical Design Episode 1 Part 4 docx

This chapter presents design formulae and methods for predicting the distribution of frictional forces and micro- slip over continuous or discrete contact areas between elastic bodies su

The Contact Between Smooth Surfaces 55

36

37

Mossakovski, V I., “Compression of Elastic Bodies Under Conditions of

Adhesion,” PMM, 1963, Vol 27, p 418

Pao, Y C., Wu, T S and Chiu, Y P., “Bounds on the Maximum Contact

Stress of an Indented Layer,” Trans ASME Series E, Journal of Applied

Mechanics, 1971, Vol 38, p 608

Sneddon, I N., “Boussinesq’s Problem for a Rigid Cone,” Proc Cambridge

Philosphical Society, 1948, Vol 44, p 492

Vorovich, 1 I., and Ustinov, I A., “Pressure of a Die on an Elastic Layer of

Finite Thickness,” Applied Mathematics and Mechanics, 1959, Vol 23, p 637

38

39

3

Traction Distribution and Microslip in Frictional Contacts Between Smooth Elastic Bodies

Frictional joints attained by bolting, riveting, press fitting, etc., are widely used for fastening structural elements This chapter presents design formulae and methods for predicting the distribution of frictional forces and micro- slip over continuous or discrete contact areas between elastic bodies sub- jected to any combination of applied tangential forces and moments The potential areas for fretting due to fluctuation of load without gross slip are discussed

The analysis of the contact between elastic bodies has long been of considerable interest in the design of mechanical systems The evaluation

of the stress distribution in the contact region and the localized microslip, which exists before the applied tangential force exceeds the frictional resis- tance, are important Factors in determining the safe operation of many structural systems

Hertz [l] established the theory for elastic bodies in contact under

normal loads In his theory, the contact area, normal stress distribution and rigid body approach in the direction of the common normal can be found under the assumption that the dimensions of the contacting bodies are significantly larger than the contact areas

Various extensions of Hertz theory can be found in the literature [2-151, and the previous chapter gives an overview of procedures for evaluating the area of contact and the pressure distribution between elastic bodies of arbi- trary smooth surface geometry resulting from the application of loading

56

Traction Distribution and Microslip in Frictional Contacts 57

An important class of contact problems is that of two elastic bodies which are subjected to a combination of normal and tangential forces The evaluations of the traction distribution and the localized microslip

on the contact area due to tangential loads are important factors in deter- mining the safe operation of many structural systems Several contributions are available in the literature which deal with the analytical aspects of this problem [16-191 The contact areas considered in all these studies are, how- ever, limited to either a circle or an ellipse, and a brief summary of the results of both cases is given in the following section

This chapter also presents algorithmic solutions which can be utilized for the analysis of the general case of frictional contacts Three types of interface loads are to be expected: tangential forces, twisting moments, and different combinations of them When the loads are lower than those neces- sary to cause gross slip, the microslip corresponding to these loads may cause fretting and surface cracks The prediction of the areas of microslip and the energy generated in the process are therefore of considerable interest

to the designer of frictional joints

3.2 TRACTION DISTRIBUTION, COMPLIANCE, AND ENERGY DISSIPATION IN HERTZIAN CONTACTS

3.2.1 Circular Contacts

As shown in Fig 3.1, when two spherical bodies are loaded along the common normal by a force P, they will come into contact over an area

with radius a When the system is then subjected to a tangential force

T < fP, Mindlin’s theory [ 161 for circular contacts defines the traction dis- tribution over the contact area and can be summarized as follows:

113

a* = a( ) I -;

F,, = 0 over the entire surface

F,, l$ = traction stress components at any radius p

a = radius of a circular contact area

p = (x2 + y 2 ) ' I 2 = polar coordinate of any point within the contact area

a* = radius defining the boundary between the slip and no-slip regions

Figure 3.2 illustrates the traction distribution as defined by Eqs (3.1) and (3.2) It can be seen that

a* = a for T = 0 and no microslip occurs;

U* = 0 for T = fP

and the entire contact area is in a state of microslip and impending gross slip

Traction Distribution and Microslip in Frictional Contacts 59

The deflection S (rigid body tangential movement) due to the any load

T >fP can be calculated from:

Consequently, at the condition of impending gross slip, T =fP:

(3.4)

The traction distribution and compliance for a tangential load fluctuating

between f T * (where T* c f P ) can be calculated as follows (see Figs 3.3 and

3.4):

P

Figure 3.3 Traction distribution for decreasing tangential load T -= T *

h* = inner radius of slip region

U* = inner radius of slip region at the peak tangential load T*

Traction Distribution and Microslip in Frictional Contacts 61

Figure 3.4 Hysteresis loop

where

3P

yo = maximum contact pressure = -

2na2

The deflection can be calculated from:

Sd = deflection for decreasing tangential load

The frictional energy generated per cycle due to the load fluctuation can therefore be calculated from the area of the hysteresis loop as:

A similar theory was developed by Cattaneo [17] for the general case of

Hertzian contacts where the pressure between the two elastic bodies occurs over an elliptical area Cattaneo’s results for the traction distribution in this case can be summarized as follows:

on slip region

F,, = 0 for the entire surface

where

a, b = major and minor axes of an elliptical contact area

a*, b* = inner major and minor axes of the ellipse defining the boundary between the slip and no-slip regions

This section presents a computer-based algorithm for the analysis of the traction distribution and microslip in the contact areas between elastic

Traction Distribution and Microslip in Frictional Contacts 63

bodies subjected to normal and tangential loads The algorithm utilizes a modified linear programming technique similar to that discussed in the previous chapter It is applicable to arbitrary geometries, disconnected con- tact areas, and different elastic properties for the contacting bodies The analysis assumes that the contact areas are smooth and the pressure distri- bution on them for the considered bodies due to the normal load is known beforehand or can be calculated using the procedures discussed in the previous chapter

3.3.1 Problem Formulation

The following nomenclature will be used:

x, y = rectangular coordinates of position

U , v = rectangular coordinates of displacement in the x- and y-directions respectively

E = Young’s modulus

v = Poisson’s ratio

G = modulus of rigidity

P = applied normal force

T = applied tangential force

f = coefficient of friction

N = number of discrete elements in the contact grid

Fk = discretized traction force in the direction of the tangential force

uk = discretized displacement force in the direction of the tangential

F,., F,: = rectangular components of traction on a contact area

at any point k

force at any point k

force at point k

point k

ylk = displacement slack variables in the direction of the tangential

yuc = force slack variables in the direction of the tangential force at

The contact area is first discretized into a finite number of rectangular grid elements Discrete forces can be assumed to represent the distributed shear traction over the finite areas of the mesh Since the two bodies in contact

obey the laws of linear elasticity, the condition for compatibility of defor- mation can therefore be stated as follows:

uk = p

uk < p

in the no-slip region

in the slip region where the difference between the rigid body movement /? and the elastic deformation uk at any element in the slip region is the amount of slip The constraints on the traction values can also be stated as:

Fk <f’Pk

Fk =fPk

in the no-slip region

where

Fk = the discretized traction force in the x direction at any point k

Pk = the discretized normal force at any point k

f’ = the coefficient of friction

The condition for equilibrium can therefore be expressed as:

Introducing a set of nonnegative slack variables Y l k and Y 2 k , Eqs (3.6) and

(3.7) can be rewritten as follows:

in the no-slip region

in the slip region

Fk 4- Y2k = f p k

in the no-slip region

in the slip region

(3.9)

(3.10)

Since a point k must be either in the no-slip region or in the slip region,

therefore:

Traction Distribution and Microslip in Frictional Contacts 65

3.3.2 General Model for Elastic Deformation

Since both bodies are assumed to obey the laws of elasticity, the elastic deformation uk at a point k is a linear superposition of the influences of

all the forces 4 acting on a contact area Accordingly:

(3.12)

j = 1

where

a y = the deformation in the x-direction at point k due to a unit force at point j

The discrete contact problem can now be formulated in a form similar to

that given in Chapter 2 as:

The problem can be restated in a form suitable for solution by a modified

linear program as follows:

2N+1

Minimize z,

i= I

according to Brand's rule

START = s+l Replace the rth basic variable

by the 8th variable with Jordan exchange by pivoting on a,,

(a)

FiQure 3.5 (a) Flow chart for the analysis algorithm (b) Initial table

Traction Disfribution and Microslip in Frictional Contacts 67

where E : a cost coefficient vector of length N

2, = first N artificial variable vector

Z2 = next N artificial variable vector

-

22N+1 = artificial variable for the equilibrium equation

The above problem can be solved as a linear programming problem [20] with a modification of the entry rule Suppose the entering variable is chosen as Y,,

A check must be made to see if the Ya corresponding Y1, is in the basis and if

the Y2.$ is not in the basis, Y1, is free to enter the basis If Yr, is in the basis, then

it must be in the leaving row, r, for Y l , to enter the basis If Y2.$ is not in the

leaving row, r, Y 1 , cannot enter the basis and a new entering variable must be chosen The same logic can be applied when the chosen entering variable is YZs

The flow chart for the algorithm utilizing linear programming with modified entry rule is shown in Fig 3.5

It is assumed for the circular and elliptical contacts that the surface of contact is very small compared to the radii of curvature of the bodies; therefore, the solution obtained for semi-infinite bodies subjected to point loads can be employed Accordingly, the influence coefficients, akj, can be expressed as follows [21, 221:

EXAMPLE 1: Circular Hertzian Contact with Similar Materials The

first application of the developed algorithm is finding the traction dis- tribution over the contact area of a steel sphere of 1 in radius on a steel half space The normal load is taken as 21601bf, the tangential load is 1441bf and the coefficient of friction is 0.1 A grid with 80 elements is

used in this case to approximate the circular contact A comparison

between Mindlin's theory, which is discussed in Section 3.2 (solid line) and the numerical results (symbol s) obtained by the modified linear

program is shown in Fig 3.6 and good agreement can be seen The rigid body movement (0.66196 x 10-4 in.) was also found to compare favorably with Mindlin's prediction (0.67 139 x 10-4 in.) with a deviation of 1.41 O/O

EXAMPLE 2: Circular Hertzian Contact with Different Materials The

contact of steel sphere of I in radius on a rubber half space is considered The material constants used are as follows:

Tract ion Disi r ibu t ion and Microslip in Frictional Contacts

the same material as compared with Mindlin's theory Traction distribution on the circular contact between two bodies of

A normal load of 2701bf, a tangential load of 361bf, and a coefficient of friction of 0.2 are used in this case

As shown in Fig 3.7, the traction distribution (symbol s) shows good

agreement with Mindlin's theory (solid line) The rigid body movement of

the rubber half space (0.39469 x 10-3in.) was found to be 30 times that of the steel sphere (0.13156 x 10-4in.) and both agree well with Mindlin's

prediction with a 2.29% deviation when a grid with 80 elements was used

EXAMPLE 3: Elliptical Hertzian Contact Four cases were investigated

in this example with different ratios between the major and minor axes

Figure 3.7 Traction distribution on the circular contact between two bodies of differrent materials as compared with Mindlin’s theory

using different rectangular grid elements, as shown in Table 3.1 A Hert-

zian-type pressure distribution was assumed in all cases

The results, which are plotted in Figs 3.8 to 3.1 1, respectively, show good agreement with Cattaneo’s theory [ 171 The rectangular grid elements were used in order to save conveniently in computer storage If a square grid element had been used, better correlation would have been obtained The tangential force is applied in the direction of the a-axis in all cases

Table 3.1 The Four Elliptical Hertzian Contact Cases

Contact area Number of grids Resulting figure

x 103

[i-]

Normalized Distance

Figure 3.8 Traction distribution on the elliptical contact as compared with

Cattaneo’s theory (a/b = 2)

x 103

0.0 0.2 0.4 0.6 0.8 1 .o

Figure 3.9 Traction distribution on the elliptical contact as compared with

Cattaneo’s theory (a/b = 0.5)

Traction Distribution and Microslip in Frictional Contacts 73

EXAMPLE 4: Square Contact Area on Semi-Infinite Bodies with Uniform Pressure Distribution A hypothetical square contact area between two steel bodies with uniform contact pressure of 10,000 psi and a coefficient

of friction, f = 0.12, is discretized with 100 square grid elements The equal traction contours are shown in Figs (3.12) and (3.13) for a tangen- tial force, T = 8001bf and 10001bf, respectively The development of the slip region with increasing tangential load and the rigid body movement is shown in Figs 3.14 and 3.15, respectively

EXAMPLE 5: Discrete Contact Area on Semi-Infinite Bodies Two dis-

connected square areas of the same size (0.6in x 0.6in) on semi-infinite steel bodies are in contact with uniform pressures assumed on each con- tact region The centroids of the two squares are placed 1.Oin apart

i Traction Contour

contact area with T = 8001bf

Contour plot of traction distribution on a uniformly pressed square