Contact phenomena. I: stresses, deflections and contact dimensions for normally-loaded unlubricated elastic components



78035

CONTACT PHENOMENA. I: STRESSES, DEFLECTIONS AND CONTACT DIMENSIONS FOR NORMALLY-LOADED UNLUBRICATED ELASTIC COMPONENTS

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1.

NOTATION AND UNITS

SI

British

A

body geometry parameter, one half of minor principal relative curvature

m–1

in–1

a

contact ellipse major semi-axis, measured in X-direction

m

in

B

body geometry parameter, one half of major principal relative curvature

m–1

in–1

b

contact ellipse minor semi-axis, measured in Y-direction

m

in

bf

contact ellipse minor semi-axis for two identical bodies, one of which is of film material

m

in

Ca

non-dimensional coefficient associated with a, given by Equation (A2.3)

Cb

non-dimensional coefficient associated with b, given by Equation (A2.8)

Cf

non-dimensional coefficient associated with ( f z ) , given by max Equation (A2.24)

Cβ

non-dimensional coefficient associated with β , given by Equation (A2.13)

Cδ

non-dimensional coefficient associated with δ , given by Equation (A2.17)

D

constant in surface roughness criterion

E

Young’s modulus of elasticity

N/m2

lbf/in2

E(m)

complete elliptic integral of second kind, see Appendix A

f

direct stress within contact ellipsoid, tensile stresses positive and compressive stresses negative

N/m2

lbf/in2

maximum compressive, or Hertzian, stress

N/m2

lbf/in2

H

coefficient in expression describing body surface

m–1

in–1

h

distance between body surfaces adjacent to contact Issued November 1978 With Amendments A to D, February 1995 1

m

in

normalised direct stress, given by f/ ( f z ) m ax

f ( fz )

max

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78035 K(m)

complete elliptic integral of first kind, see Appendix A

k

material constant, given by ( 1 – σ ) / ( πE )

m2/N

in2/lbf

L

length of rectangular contact

m

in

P

total normal applied load

N

lbf

p

pressure on contact surface (always positive, see Section 4)

N/m2

lbf/in2

q

shear stress within contact ellipsoid

N/m2

lbf/in2

q

normalised shear stress, given by q / ( f z )

R

principal radius of curvature of body

m

in

s

coated body film thickness

m

in

2

m ax

X, Y, Z

3 2 1/3 calculation parameter, given by  --- Pπ ( k 1 + k 2 ) ( A + B )  4  axes of orthogonal system (see Sketch 3.1)

x, y

distances along X- and Y-axes, respectively

m

in

z

distance along Z-axis, through depth of body

m

in

β

ellipse semi-axes ratio, given by b /a

δ

decrease in separation between two points on axis of symmetry, one in each body, remote from compressed region

m

in

σ

Poisson’s ratio

σε

standard deviation of surface roughness

ω

angle between X 1 - and X 2 -axes

degree

degree

W

Subscripts film

pertains to thin film of modulus of elasticity lower than that of substrate

q

pertains to point at which principal shear stress is maximum

max

indicates a maximum

x , y, z

denote directions in which stresses act

xy , yz , zx

double suffix used in conjunction with q where first suffix gives direction of normal to plane of q and second suffix gives direction of q

β =0

pertains to rectangular contacts

β =1

pertains to circular contacts

2



78035 0

pertains to contact centre at surface

1, 2

used as first subscript pertain to bodies 1 and 2, respectively, and used as second subscript pertain to planes of curvature

Dressing ∼

2.

(inferior) indicates an approximate value

INTRODUCTION

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When two bodies, one or both of whose surfaces are curved, are brought into contact and subjected to a normal load, elastic deformation at the interface gives rise to an elliptical contact area. In some instances (for example, sphere on sphere, sphere on plate, and identical cylinders crossed at right-angles) the contact area is circular. In other cases (for example, cylinder on cylinder with parallel axes, cylinder in groove, and cylinder on plate) the ellipse major semi-axis is infinitely long in comparison with the minor semi-axis and the contact area can be considered to be rectangular. The ellipse axes ratio, β , is a function of both body geometry and the orientation of the principal radii of curvature whilst the lengths of the semi-axes are also functions of the body material properties and the applied load. The theory on which this Data Item is based has been developed from classical contact geometry and contact stress work (see Derivations 17, 18 and 21). The way in which these sources of information have been adapted for non-dimensional presentation in the figures is outlined in Appendix A. This Data Item is the first of a series that deals with the stresses and deflections of contacting bodies. The series will cover comprehensively all cases of dry and lubricated, normally- and tractively-loaded, conformal and non-conformal† contacts and will be divided into the following three major sections. (i)

Dry, normally-loaded bodies (covered by this Item).

(ii)

Dry bodies subjected to combined normal and tractive loading.

(iii)

Elastohydrodynamically lubricated bodies subjected to combined normal and tractive loading.

A subsequent Item will provide additional design guidance on component geometry, fatigue life, wear, and failure modes for several common engineering applications including cams and followers, gears and bearings. The current Item gives data for the contact dimensions, normal approach, δ , surface and sub-surface stresses (for points on the axis of symmetry) together with the depth at which the maximum shear occurs, for dry, normally-loaded elastic bodies. In order to establish the contact geometry and to evaluate the contact stresses, values of the material constants and principal radii of curvature of both bodies are required. Table 9.1 lists values of material constants for a wide range of metals, ceramics and plastics, and, providing that the curvature radii and the angle between the principal axes are known, the calculation table (Table 9.3) may be used in conjunction with Figures 1 to 6 and the equations given to determine any or all of the unknown stresses and dimensions. Table 9.5 has been included for those users who have access to a programmable calculator; solutions to contact problems can be obtained with the aid of the approximate numerical methods given. The textual equations relating to circular and long rectangular contacts are summarised, for ease of access, in Table 9.6. †

The contact is non-conformal if the principal curvatures of both bodies are of the same sign (see Sketches 3.2a to 3.2f).

3



78035 (Evaluation of quantities associated with these contacts is considerably less complex than for the general elliptical case and consequently Figures 7 and 8 may be used to determine directly values of the maximum compressive stress.) This Item may be used for both conforming and non-conforming contacting bodies within the limits stated in Section 6.6. The theory is valid for all body shapes, but the Item should not be used when the principal radius of curvature of a body is infinitely small or indeterminate (as is the case with a knife edge), or when the contact zone is relatively large (as is the case with conforming cylinders of similar interface radii).

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3.

CONTACT GEOMETRY AND NORMAL APPROACH The data plotted in the figures have been obtained from a general theory of contacting bodies of compound curvature (see Sketch 3.1). The contact zone is always bounded by an ellipse whose ratio of minor to major axes lies in the interval from zero to unity. The dimensions a and b of the contact ellipse increase with ( P )1/3 , and points on the bodies remote from the contact zone approach each other by an amount δ which varies with ( P )2/3 . For the limiting case of a very long ellipse, when the contact is effectively rectangular, the width, b, increases with ( P )1/2 . Generally, before a, b, or δ can be found, values of each part of a principal relative curvature, A and B, must be calculated (see Section 3.1).

3.1

Principal Curvatures of Body Surfaces Using the first subscript to denote the body and the second to denote the plane of curvature, R 11 and R 12 are the principal radii of curvature of the unloaded surface of body 1 at the point of contact, and R 21 and R 22 are the principal radii of curvature of the unloaded surface of body 2 at the point of contact. The reciprocals of these curvature radii are termed the principal curvatures. The constants A and B have values that depend on the magnitude of the principal curvatures and the angle, ω , between the planes in which the principal curvatures lie (see Sketch 3.1). Many engineering contact problems involve bodies with either infinite principal radii (for example, flat plates) or equal and finite principal radii (for example, spheres). Combinations of bodies and their associated contact shapes commonly met in practice are shown in Sketch 3.2 where it can be seen that the principal radii of curvature may be either positive (convex surfaces) or negative (concave surfaces). If the corresponding planes of principal curvature of the two bodies coincide (as in all of the body shapes except (c) and (d) in Sketch 3.2),

1 1 1 A = ---  -------- + -------- ,  2 R 12 R 22 1 1 1 B = ---  -------- + -------- ,  2 R 11 R 21

      

(3.1)

where the notation for the principal planes of curvature must be chosen such that B ≥ A .

4

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78035

Sketch 3.1 The principal curvature radii of bodies 1 and 2, respectively, in one principal plane of curvature are R 11 and R 21 and the principal curvature radii in the other principal plane are R 12 and R 22 . If the planes of principal curvature do not coincide (as shown in Sketch 3.1 and Sketch 3.2d then 1 1 1 1 1 1 1 1 2 1 2 A = --- -------- + -------- + -------- + -------- –  -------- – -------- +  -------- – -------- R R 4 R 11 R 12 R 21 R 22 R 12 R 22 11 21 1 1 1 1 + 2  -------- – --------  -------- – -------- cos 2 ω R  R  R R 11 12 21 22

(3.2a)

1/ 2

,

1 1 1 1 1 1 1 1 2 1 2 B = --- -------- + -------- + -------- + -------- +  -------- – -------- +  -------- – -------- R R 4 R 11 R 12 R 21 R 22 R 12 R 22 11 21 1 1 1 1 + 2  -------- – --------  -------- – -------- cos 2 ω R   R 12 R 21 R 22 11 where the notation for the principal planes of curvature must be chosen such that B ≥ A .

5

(3.2b)

1/ 2

,



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78035

Sketch 3.2 †

†

Care must be taken to ensure that B ≥ A when assigning subscripts to the radii of curvature. It can be seen that the infinite radii in Sketch 3.2c are denoted by R 12 and R 22 , whilst the infinite radius in Sketch 3.2h is denoted by R 21 . It is necessary to choose the subscripts of R in this manner to ensure that when Equations (3.1) are used, B ≥ A . Note that because pairs of principal curvature radii are coincident in both cases Equations (3.2a) and (3.2b) need not be used.

6



78035 It may be found that the line of action of the applied load does not coincide with the Z-axis of a convenient orthogonal set of axes corresponding to the obvious principal planes of curvature of the bodies. If this is the case, the equations describing both surfaces must be expressed in a new set of axes. Example 2 in Section 8.2 demonstrates how the radius of curvature of a body may be calculated at the contact point for one specific case, but a general theory of axis transformation can be found in Appendix 1 of Reference 5.

3.2

Circular Contacts

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When the major and minor semi-axis of the ellipse are of equal length, the contact area is circular, and a = b . For this case β = 1.0 and ( A/B ) = 1.0 and Figures 1 and 2 show that the coefficients C a , C b , C β and C δ have values of unity. Equations (3.9) to (3.12) reduce to 3 Pπ ----------- ( k 1 + k 2 ) 4

( a ) β = 1.0 = ( b ) β = 1.0 =

R 11 R 21 -----------------------R 11 + R 21

1/ 3

,

β = 1.0 ,

and

( δ ) β = 1.0 =

(3.3) (3.4)

3P π ----------- ( k 1 + k 2 ) 4

2

1/ 3

R 11 + R 21 ------------------------R 11 R 21

,

Pπ ≡ 0.75 ------- ( k 1 + k 2 ). a

        

(3.5)

These equations cannot be used when any radius of curvature is zero, and are expressly for use with a sphere in contact with either a sphere, or plate, or socket. That is, when both principal radii in body 1 are equal and positive, and when both principal radii in body 2 are equal. 3.3

Rectangular Contacts When the ellipse is theoretically of infinite length and β = 0 and ( A /B ) = 0 , the major semi-axis, a, is infinite. This type of contact is exemplified by a cylinder loaded against a plate, or a groove, or another, parallel, cylinder. In this circumstance the ellipse semi-width is given by 1/2

( b )β = 0 =

R 11 R 21 P 4  --- ( k 1 + k 2 ) ------------------------ L R 11 + R 21

,

(3.6)

where ( P/L ) is the load per unit length along the contact. The elastic compression of two-dimensional bodies in contact cannot be calculated solely from the contact stresses given by the Hertz theory. Some account must be taken of the shape and size of the bodies and the way in which they are supported. However, the normal approach of the axes for a particular case of contacting parallel cylinders is given by  4 R 11 1  4 R 21 1 P P ( δ ) β = 0 = 2k 1  --- log e  ------------ – --- + 2 k 2  --- log e  ------------ – --- .  L  L  b  2  b  2

7

(3.7)



78035 When the composition of both materials is identical Equation (3.7) simplifies to  4 R 11   4 R 21 P ( δ ) β = 0 = 2k  --- log e  ------------ + log e  ------------ – 1 .  L  b   b 

(3.8)

These expressions describing the contact geometry of cylinders have been obtained from Derivation 20.

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Figures 1 and 2 cannot be used when β = 0 and ( A /B ) = 0 as some of the quantities are then infinite. Very few of the rectangular contacts occurring in practice can be represented by ellipses of infinite length. Usually the contact rectangle is relatively short (either by design or because of the practical impossibility of loading two long surfaces uniformly) and the stress distribution along the X-axis (see Sketch 4.3)) does not approximate to that of an ellipse. Short rollers in contact with either rollers or plates (of equal or unequal length) have displacement and contact dimension characteristics that cannot be evaluated accurately using the method of this Item. 3.4

Elliptical Contacts When the ratio ( A/ B ) has been calculated from Equations (3.1) or (3.2a) and (3.2b), Figures 1 and 2 may be used to find values of the coefficients C a , C b , C β and C δ , and substitution of these into Equations (3.9) to (3.12) gives the contact ellipse dimensions and their ratio, and the mutual approach of the bodies. (However, if the contact is rectangular, A /B = 0 , and Equations (3.6) and (3.7) must be used.) C a W A – 1/ 3 , a = --------------------  --- ( A + B )  B Cb W A b = --------------------  --- ( A + B )  B

(3.9)

1/ 3

,

(3.10)

A 2/3 β = C β  --- ,  B

(3.11)

2

CδW A 1/ 3 δ = --------------------  --- , ( A + B )  B

where

W =

(3.12)

3P π ----------- ( k + k ) ( A + B ) 2 1 2 4

1/ 3

.

(3.13)

Not only does the term W occur in each of Equations (3.9), (3.10) and (3.12), but it is also present in the expression for peak surface stress (see Section 4.1, Equation (4.15)). Calculation time can be saved by evaluating this term at the outset of the analysis, as demonstrated in the calculation table. The development of these equations is given in Appendix A to this Item. A semi-empirical approximate method that enables both pressures and contact dimensions to be calculated to within 1 per cent, and normal approach to within 4 per cent, is given in Table 9.5 in the form of a flow chart for use with programmable calculators.

4.

SURFACE AND SUB-SURFACE STRESSES The distribution of stress f z is a mirror-image of the pressure distribution, p, arising from the applied load, P. Stresses f x and f y arise from the applied load because of the effects of material elasticity. At the centre 8



78035 of contact, and at all points directly below the centre, the three direct stresses f x , f y and f z are also principal stresses. At all points on or directly below the centre of the contact surface the shear stresses q zx , q zy and q xy are equal to zero. Nevertheless, there exist principal shear stresses acting on planes bisecting the angle between any pair of principal planes. (On the axis of symmetry, the principal planes correspond to the ZX-, ZY- and XY-planes.) The magnitudes of the principal shear stresses are given by

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fz – fy --------------- , 2

fz – fx --------------- , and 2

f y – fx --------------- , 2

(4.1)

that is, half the difference between two principal stresses. The largest of these three principal shear stresses is denoted by ( q max ) at the centre of the contact surface 0 (see Section 4.1), and q m ax at some depth below the surface on the axis of symmetry (see Section 4.2), respectively. 4.1

Surface Stresses Once the maximum compressive stress, ( f z ) m ax , often referred to as the peak Hertzian stress, has been calculated (see Equation (4.15) and Figures 3 and 4), Figure 5 may be used to obtain non-dimensional values of the remaining direct surface stresses at the centre of the contact. However, for circular contacts involving a sphere and either a sphere or a plate or a socket, or for long rectangular contacts involving a cylinder and either a cylinder or a plate or a groove, Figures 7 and 8, respectively, may be used to determine values of ( f z ) m ax . The principal shear arising from the surface direct stresses in the X- and Z-directions is always largest and is of most interest (see Section A2.5.3); therefore normalised values of this stress, labelled ( q max ) 0 , have been plotted in Figure 6. Equations for surface stresses at the periphery of the contact are given in Appendix A.

4.1.1

Circular contacts When the major and minor semi-axes of the ellipse are of equal length, the contact area is circular and β = 1.0 and ( A/ B ) = 1.0 . For this case, the distribution of pressure, p, across the contact is represented by the ordinates of an ellipsoid of revolution (see Sketch 4.1)).

Sketch 4.1

9



78035 the distribution is given by 1/2 2

2

x +y ( p ) β = 1.0 = p 0 1 – -----------------2 a

,

(4.2)

and hence the direct stress in the Z-direction ( f z , given by –p) may be calculated at any point on the contact surface. The maximum direct stress at the surface occurs at the centre of the boundary of the ellipsoid of revolution and is given by

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1.5 P p 0 = ----------- = – ( f z ) max , 2 πa

(4.3)

which shows that this maximum stress is greater than the average stress over the contact zone by 50 per cent, and ( f z ) m ax may be evaluated if the contact dimensions are known. Figures 3 and 4 show that for circular contacts † (where ( A /B ) = 1.0 ), C f = 1.0 and Equation (4.15) reduces to 1/ 3

2

R 11 + R 21 6P ( f z ) m ax = – --------------------------------- ------------------------5 2 R 11 R 21 π ( k1 + k2 )

,

(4.4)

and this equation should be used if the contact dimensions are not known. The stresses at the centre of the contact surface are given by

and

1 ( f x ) = ( f y ) = --- ( 1 + 2σ ) ( f z ) max , 0 0 2 ( f z ) = ( f z ) max . 0

(4.5) (4.6)

Sketch 4.2 has been included in this Item to demonstrate the effect that body geometry and applied load have on the peak surface compressive stress for circular contacts. The sketch is for guidance only and should not be used to obtain values of stress which can instead be read directly from Figure 7. The broken lines on Sketch 4.2 illustrate an example for two steel spheres of radius 12.5 mm and 10 mm, under a load of 100 N. The resulting value of ( f z ) m ax is –2.0 GN/m2. Equation (4.4) can be used to evaluate the maximum compressive stress arising from circular contacts associated with identical cylinders crossed at right-angles provided that the subscripts assigned to the principal curvature radii give B ≥ A . If this is done, ( A + B ) = 1/R and Equation (4.4) becomes 1/ 3

6P ( f z ) m ax = – ---------------------------------------5 2 2 π R ( k1 + k2 ) †

See text associated with Equations (4.7) and (4.8).

10

,

(4.7)



78035

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where R is the common radius.

Sketch 4.2 Stresses in circular contacts (steel-on-steel) In comparison, the value of ( A + B ) for identical spheres (of the same radius R) is equal to 2/R, and Equation (4.4) becomes 1/ 3

24 P ( f z ) m ax = – ---------------------------------------5 2 2 π R ( k1 + k2 )

.

(4.8)

1/ 3

Thus, the stress for the spheres is ( 4 ) times greater than that for the cylinders when P, R and k are the same for both contacts. Equations (4.7) and (4.8) reveal that the maximum compressive stress in a pair of identical cylinders crossed at right-angles is equal to the stress in a pair of identical spheres of radii equal to the diameter of the cylinders. Also, the stress in the identical crossed cylinders is equal to the stress in a sphere of the same radius in contact with a plane surface. 4.1.2

Rectangular contacts If the surface of contact is a long and narrow rectangle and β = 0 and ( A /B ) = 0 , the distribution of pressure, p, across the width of the contact area is represented by the ordinates of a semi-ellipse (see Sketch 4.3). 11



78035 The distribution is given by 2 1/ 2

y ( p ) β = 0 = p 0 1 – ----2 b

,

(4.9)

and hence the direct stress in the Z-direction ( f z , given by –p) may be calculated at any point on the contact surface. The maximum direct stress at the surface occurs at the centre line of the rectangular boundary and is given by

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( p0)

β = 0

P 2 =  --- ------ = – ( f z ) max ,  L πb

(4.10)

Sketch 4.3 which shows that this maximum stress is ( 4 /π ) times as great as the average stress over the contact zone, and ( f z ) m ax may be evaluated if the contact width is known. Figures 3 and 4 cannot be used when β = 0 and ( A /B ) = 0 as some of the quantities are also zero at this point; the maximum compressive stress is instead given by  R 11 + R 21 P 1 ( ( f z ) m ax ) = –   --- ------------------------------ ------------------------  L 2 β = 0 π ( k 1 + k 2 ) R 11 R 21 

   

1/ 2

,

(4.11)

and this equation should be used if the contact width is not known. The stresses at the centre of the contact surface are given by    ( f y ) = ( f z ) = ( f z ) ma x  0 0  ( f x ) = 2σ ( f z ) max , 0

12

(4.12)



78035 Sketch 4.4 has been included in this Item to demonstrate the effect that body geometry and applied load have on the peak surface compressive stress for long rectangular contacts. This sketch is for guidance only and should not be used to obtain values of stress which can instead be read directly from Figure 8.

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The broken lines on Sketch 4.4 illustrate an example for two parallel steel cylinders of radius 20 mm and 100 mm, under a distributed load of 20 kN/m. The resulting value of ( f z ) m ax is ≅ – 0.2 GN/m2.

Sketch 4.4 Stresses in long rectangular contacts (steel-on-steel) 4.1.3

Elliptical contacts The distribution of pressure, p, over the contact area is represented by the ordinates of the semi-ellipsoid shown in Sketch 4.5, constructed on the contact surface. The elliptical distributions in the XZ- and YZ-planes are sometimes referred to as Hertzian distributions.

13



78035

Sketch 4.5

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The distribution is given by 1/ 2

2

2

y x p = p 0 1 – ----- – ----2 2 a b

,

(4.13)

and hence the direct stress in the Z-direction ( f z , given by –p) may be calculated at any point on the contact surface. The maximum direct stress at the surface occurs at the centre of the ellipsoidal boundary and is given by 1.5 P p 0 = ----------- = – ( f z ) max , πab

(4.14)

which shows that this maximum stress is greater than the average stress over the contact zone by 50 per cent. (If the contact is rectangular, see Section 4.1.2.) When the ratio ( A/ B ) has been calculated, Figures 3 and 4 may be used to find a value for the coefficient C f ; substitution of this into Equation (4.15)† gives the maximum compressive stress at the interface of the bodies: 2W ( f z ) m ax = – C f ------------------------------ , 2 π ( k 1 + k2 )

(4.15) 1/ 3

where

W =

3 Pπ 2 ----------- ( k 1 + k 2 ) ( A + B ) 4

.

(4.16)

The derivation of Equation (4.15) is given in Appendix A to this Item. A semi-empirical approximate method that enables the contact stress to be estimated to within 1 per cent is given in Table 9.4 in the form of a flow chart for use with programmable calculators. The stresses at the centre of the contact surface are given by

†

If the contact ellipse dimensions are known, ( f z ) m ax may be evaluated by using Equation (4.14). Otherwise, Equation (4.15), which is incorporated in the calculation procedure of Table 9.3, or the relevant part of the flow chart in Table 9.5, must be used.

14



78035 b ( f x ) = 2σ ( f z ) max + ( 1 – 2σ ) ( f z ) max  ------------- ,  0  a + b   a  ( f y ) = 2σ ( f z ) max + ( 1 – 2σ )  -------------  , 0 a + b     ( f z ) = ( f z ) max . 0 

(4.17)

The maximum shear stresses are obtainable from Expressions (4.1).

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4.2

Sub-surface Stresses on Axis of Symmetry The sub-surface stress distribution shown in Sketch 4.6 is typical of that found in all isotropic elastic contacting bodies. All distributions illustrated in this Item are for points directly below the centre of contact.

Sketch 4.6

15



78035 The depth below the contact surface at which the maximum shear stress, q m ax , occurs is indicated by the point ( z/ b ) q . This stress has considerable influence in dictating the fatigue life of repeatedly loaded m ax bodies. However, the maximum shear stress at any depth below the surface is not given by half the difference of the same two principal stresses, f z and f y . Closer examination of the stress distribution for 0 ≤ z/b ≤ 1.5 (see Sketch 4.7) reveals that the maximum shear stress due to f z and f x is greater near the surface than that due to f z and f y . The planes on which these stresses act are orthogonal; care should be taken in choosing which stresses are to be used as failure criteria, although the shear of greatest magnitude is usually chosen. Reference 6 outlines criteria of yielding and failure for ductile materials. Table 9.2 illustrates how the sub-surface stress distribution alters as the contact ellipse varies in length and as Poisson’s ratio varies. The table gives an indication of the relative magnitude of the three direct stresses and the major shear stress for various combinations of body shapes.

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When a value for the maximum direct surface stress, ( f z ) m ax , has been calculated, Figures 5 and 6 may be used to determine the value of the greatest principal sub-surface shear stress, q m ax , and the depth below the surface, z q , at which its maximum occurs. The equations used to plot these figures are given in max Appendix A.

Sketch 4.7

5.

CALCULATION PROCEDURE The calculation table (Table 9.3) is self-explanatory. Provision is made for step-by-step calculations using equations and figures referred to in the preceding sections of this Data Item. The comment column and footnotes of the table direct the attention of the user to relevant sections, references and appendix that can or must be used at a particular stage in the calculations.

16



78035 The block diagram given by Table 9.5 is also self-explanatory; the way in which the flow chart may be adapted for use with programmable calculators is left to the user.

6.

THEORETICAL LIMITATIONS AND PRACTICAL CONSIDERATIONS The data given in this Item strictly apply under the conditions listed in the following table. The assumptions listed in the left-hand column of the table have been included, firstly, to draw attention to sections of the Item that deal with aspects of the invariably imperfect contacts arising in engineering and, secondly, to provide the user with an insight into the underlying theory and its development in subsequent Items dealing with contacting bodies.

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A number of the assumptions listed in this table are idealisations; in practice the effects of physical irregularities of the bodies and the way in which they are loaded may alter the contact characteristics significantly. Assumption

6.1

Comments

1

The contacting bodies are isotropic

Bodies may be coated with thin elastic layer (see Section 6.5). Most materials given in Table 9.1 may be considered to be isotropic.

2

The contacting bodies are linearly elastic.

Item is not applicable to inelastic materials.

3

The dimensions of the contact area are very small in comparison with the radii of the undeformed bodies in the vicinity of the contact.

Item should not be used for closely conforming bodies where contact area is large (see Section 6.6).

4

The shape of the deformed surfaces adjacent to the contact zone can be described with sufficient accuracy by a second order equation of the form 2 2 z = A x + H x y + By .

Bodies are considered to be smooth, flat or curved, and not corrugated or jagged near the contact. The effects of surface roughness are considered in Section 6.1.

5

At the contact interface, the bodies are approximately flat.

Unless both bodies are identical, the contact surface will be curved. Practical implications are not significant for this Data Item.

6

The bodies are frictionless; only normal stresses arising during contact are considered. Relative displacements in the X- and Y-directions are neglected.

Traction and surface asperities (considered in Sections 6.2 and 6.1, respectively) will give rise to non-zero shear stresses at the interface.

7

The surfaces are clean and unlubricated.

‘Clean’ means ‘free of surface debris’ and not ‘chemically clean’. See Section 6.5 on effects of thin films on bodies (perhaps solid lubricants) and Section 6.7 on effects of liquid lubricants.

Effects of Surface Roughness When elastic but dissimilar smooth bodies are brought into contact and subsequently loaded, facing points equidistant from the line of the applied load (such as points 1 and 2 in Sketch 6.1) not only approach each other normally but also displace laterally relative to one another. In this Item it is assumed that the lateral displacement, or slip, occurs freely and that there are no surface tractions. However, for rough bodies, as the applied load is increased and the contact widens to envelop the facing points, their lateral motion ceases. 17



78035

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This is because a finite friction coefficient is present due to surface asperities and molecular attraction. Thus, surface shear stresses exist, and although they do not alter the normal approach significantly, these tractions induce slip between the bodies over an extremely thin annulus just within the contact periphery. (See Reference 2 which extends classical contact theory to cater for relative lateral motion of surface points before but not after they enter the contact zone.)

Sketch 6.1 If the notation shown in Sketch 6.2 is adopted, the separation of both surfaces very near to the contact boundary( x 2 – x 1 << x 1 ) is given, for both line and circular contacts, by the approximation 1/ 2

h ≈ Dx 1 ( x 2 – x 1 ) 1 1 1 --- = -------- + -------- , R R 11 R 12

where

3/2

      

/R ,

D ≈ 1.

and

(6.1)

The roughness of the surfaces will have little influence on the contact at distances along the horizontal interface greater than h = 2 σ ε , where σ ε is the standard deviation of the combined surface roughnesses. This corresponds to a position where 1/ 2

x 2 – x 1 ≈ [ 2 Rσ ε / ( Dx 1

)]

2/3

,

as given by Derivation 21. Also the contact geometry will not be affected significantly if x 2 – x 1 is small compared with x 1 , that is, if 1/ 2

[ 2R σ ε / ( D x 1

)]

2 /3

<< x 1 ,

or,

2

R σ ε /x 1 << D /2 .

(6.2)

–3

If, for example, x 1 = 0.1 × 10 m and R = 0.01 m (with D ≈ 1 ), σ ε must be very much less than –6 0.5 × 10 m for the contact geometry to remain unchanged. Section 6.3 discusses the effects of surface roughness on the fatigue life of repeatedly loaded contacts.

18

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78035

Sketch 6.2 6.2

Effects of Combined Normal and Tangential Loading Items Nos 84017 and 94034 (References 11 and 16) deal with contact stresses and deflections for bodies subjected to combined normal and tangential loading but some notes on the influence of combined loading follow. Addition of tangential loading to normal loading can significantly alter the contact surface and sub-surface stress distributions. Although this Data Item is mainly concerned with normally-loaded bodies, the effects of combined loading are important especially when investigating the fatigue life of repeatedly loaded contacts. When both normal and tangential loads are distributed elliptically across the contact (in a Hertzian distribution, see Section 4.1.3), the maximum shear stress may occur at the contact surface and not at some distance below the surface. With parallel cylinders, it has been shown in Reference 1 that for a coefficient of friction (which is the linear proportionality between the tangential and normal loads immediately prior to slipping) of 1/3, the maximum shear stress is approximately 1.5 times that for purely normally-loaded cylinders. The position of the maximum shear stress is below the surface when there is no tangential loading, but it rises to the surface when the coefficient of friction is greater than 1/9. For simple normal loading, the direct surface stress varies across the contact from zero at the boundary to a value of p o at the centre (see Sketch 6.3a). This distribution is discussed further in Section 4.1.3. However, when tangential loading is applied, the direct surface stress variation is different (see Sketch 6.3c), and, in the example shown, the maximum range of stress is almost doubled; the direct stress, f y , is seen to vary between 0.67 p 0 , tensile and –1.20 p 0 , compressive across the contact width. It is this range of stress that is of considerable importance when the bodies are simultaneously loaded normally and rolled against each other, and gives insight into failure mechanisms involving predominantly compressive loads (see Section 6.3). Because the direct surface stress f y exhibits the largest variation in magnitude, it has been shown in Sketch 6.3 where the distribution is given for a coefficient of friction of 1/3. The curve in Sketch 6.3c is thus the result of adding 1/3 of the magnitude of the curve in Sketch 6.3b to the curve in Sketch 6.3a. 19



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78035

Sketch 6.3 Stress distributions for parallel cylinders 20

 6.3

78035 Effects of Cyclic Loading Surface roughness plays an important role in dictating the fatigue life of bodies subjected to cyclic loading. Reference 8 gives methods of measuring centre-line-average surface roughness, and provides data on the fatigue life of steels of tensile strengths between 400 and 1100 MN/m2, with roughness depths below 500 µ m , whilst subjected to alternating stresses at zero mean stress.

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Generally, failure of the surfaces of the bodies can be attributed to one of two primary causes. Firstly, the contact may experience local plastic deformation (which is usually associated with maximum shear stresses, see Section 4) and eventual failure may occur as a result of crack propagation. Secondly, surface or sub-surface cracks may lead to fatigue failure with little or no plastic deformation; after a large number of load cycles, parts of the bodies will break out causing pitting or shelling. Variations of stress direction, magnitude and sign throughout the loading cycle dictate fatigue life in this instance. Reference 10 gives guidance on the application of fracture mechanics to estimating crack growth under alternating loadings, and to estimating the residual strength of cracked components. This reference will be of use when the contacting bodies are normally-loaded in a cyclic manner rather than when the bodies are subjected to rolling and/or sliding contact. 6.4

Bodies not of Semi-infinite Proportions Providing that the assumptions, given in Section 6, referring to the relative sizes of contact and body are met, solutions obtained by using this Item will be correct within the practical limitations stated. However, because the stress function is such that the stresses within the bodies disappear at a great distance from the contact region, care must be taken to ensure that the bodies are thick enough to permit this. Although the theory is applicable to semi-infinite bodies, it can be used for any combination of spheres, ellipsoids, rollers, plates, grooves or sockets providing that

6.5

(i)

for the case of a finite length cylinder in contact with another body, the notes on the modified stress distributions and deflection in Sections 3.3 and 4.1.2 are consulted, and

(ii)

for the case of a plate of finite thickness in contact with a curved body, the thickness of the plate is greater than one fifth of the relative radius of curvature of the body; results are then accurate to within 5 per cent. (See Reference 4 for a discussion of contact between spheres and plates of any thickness.)

Bodies Coated with Thin Films If one or both bodies are coated with a thin film, say, for example, a solid lubricant, the size of the contact zone can be altered significantly (see Reference 7). For contact dimensions that are large in comparison with the film thickness, Sketches 6.4a and 6.4b illustrate how the contact width is affected by the coating. The sketches apply to coated identical spheres and rollers, respectively, where the modulus of elasticity of the film is less than that of the coated body or bodies. Further discussion of the effects of thin elastic coatings on contacting bodies can be found in Reference 2, where interface compliance, radii of no-slip and contact regions, and stress distributions for a variety of materials and geometries are examined.

21

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78035

Sketch 6.4 6.6

Limits of Conformality As outlined in Section 6, the data given in this Item are based on theory involving the assumptions that, firstly, the contact dimensions are small in comparison with the principal curvatures of the bodies, and, secondly, the bodies are approximately flat at the interface. These conditions do not apply for conforming bodies of almost equal coincident principal radii, and the Item should not be used when these radii differ by less than 10 per cent. Reference 3 examines the contact stresses and approach for a sphere pressed into a cavity of slightly larger radius. An extension of classical contact theory is developed that, by retaining the assumption that stresses and deflections are small, can be used in the same circumstances as this Item.

6.7

Effects of Presence of Lubricant Apart from the generation of a transient “pressure spike” when two bodies are initially brought into contact (the peak pressure is a function of the closing velocity of the bodies), lubrication does not significantly affect the contact stresses and dimensions of a statically-loaded system. The stress distribution may differ from that assumed if the surfaces are lubricated, but the results given in this Data Item are valid for boundary lubrication, and are approximately correct for the limiting case of very thin films in elastohydrodynamic lubrication (see References 12 to 15), that is, with low surface velocities and low lubricant viscosities.

22

 7.

78035 REFERENCES AND DERIVATION References

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The references given are recommended sources of information supplementary to that in this Item. 1.

SMITH, J.O. LIU, C.K.

Stresses due to tangential and normal loads on an elastic solid with application to some contact stress problems. J. appl. Mech., Vol. 20, No. 2, pp. 157-166, June 1953.

2.

GOODMAN, L.E.

Contact stress analysis of normally loaded rough spheres. J. appl. Mech., Vol. 29, No. 3, pp. 515-522, September 1962.

3.

GOODMAN, L.E. KEER, L.M.

The contact stress problem for an elastic sphere indenting an elastic cavity. Int. J. Solids Struct., Vol. 1, No. 4, pp. 407-415, October 1965.

4.

TU, Y.-O.

A numerical solution for an axially symmetric contact problem. J. appl. Mech., Vol. 34, No. 2, pp. 283-286, June 1967.

5.

DYSON, A.

A general theory of the kinematics of gears in three dimensions. Clarendon Press, 1969.

6.

ESDU

Criteria of yielding for ductile materials under static loading. Item No. 73004, ESDU International, London, July 1973.

7.

LEVESON, R.C.

The mechanics of elastic contact with film-covered surfaces. J. appl. Phys., Vol. 45, No. 3, March 1974.

8.

ESDU

The effect of surface roughness on the fatigue limit of steels (at zero mean stress). Item No. 74027, Engineering Sciences Data Unit, London, October 1974.

9.

GOODMAN, L.E. KEER, L.M.

Influence of an elastic layer on the tangential compliance of bodies in contact. Published in “The mechanics of the contact between deformable bodies”. Delft University Press, 1975.

10.

ESDU

Introduction to the use of linear elastic fracture mechanics in estimating fatigue crack growth rates and residual strength of components. Item No. 80036, ESDU International, London, November 1980.

11.

ESDU

Contact phenomena. II: stress fields and failure criteria in concentrated elastic contacts under combined normal and tangential loading. Item No. 84017, ESDU International, London, October 1984.

12.

ESDU

Film thickness in lubricated Hertzian contacts (EHL). Part 1: two-dimensional contacts. Item No. 85027, ESDU International, London, October 1985.

13.

ESDU

Film thickness in lubricated Hertzian contacts (EHL). Part 2: point contacts. Item No. 89045, ESDU International, December 1989.

14.

ESDU

Film thickness in lubricated Hertzian line contacts (guide to the use of computer program A9137). Item No. 91037, ESDU International, London, December 1991.

23



78035 15.

ESDU

Film thickness in lubricated Hertzian point contacts (guide to the use of computer program A9138). Item No. 91038, ESDU International, London, December 1991.

16.

ESDU

Dimensions, deflections and stresses for Hertzian contacts under combined normal and tangential loading. (Guide to use of computer program A9434.) Item No. 94034, ESDU International, London, December 1994.

Derivation

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The derivation lists selected sources that have assisted in the preparation of this Item. 17.

HERTZ, H.

On the contact of solids. On the contact of rigid elastic solids and on hardness. (Translated by Jones and Schott.) Miscellaneous papers, pp. 146-183, Macmillan, 1896.

18.

THOMAS, H.R. HOERSCH, V.A.

Stresses due to the pressure of ne elastic solid upon another. University of Illinois Engineering Experimentation Station, Bulletin No. 212, July 1930.

19.

BAYER, R.C. KU, T.C.

Handbook of analytical design for wear. Plenum Press, New York, 1964.

20.

SCHARTZ, J. HARPER, E.Y.

On the relative approach of two dimensional elastic bodies in contact. Int. J. Solids Struct., Vol. 7, pp. 1613-1626, 1971.

21.

DYSON, A.

Unpublished data, Chester, 1978.

22.

BARWELL, F.T.

Bearing systems: principles and practice. Oxford University Press, 1979.

8.

EXAMPLES

8.1

Example 1 It is required to find the contact area dimensions, normal approach, magnitudes of direct surface stresses, and the magnitude of the peak sub-surface stress and the depth below the surface at which it occurs for the crossed cylinders shown in Sketch 8.1. The effect on the maximum stresses and contact area of crossing the cylinders at right-angles is also to be investigated for each body. The cylinders of radius 25 mm and 75 mm, for bodies 1 and 2, respectively, are subjected to a normal load, P, of 1.25 kN. Cylinder 1 is of mild steel, and cylinder 2 is of brass.

24

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78035

Sketch 8.1 8.1.1

Calculation procedure The calculation procedure that follows in Table 8.1 illustrates the use of Table 9.3. The calculations, in SI units, are those pertaining to the brass cylinder. The important results are summarised in Table 8.2.

TABLE 8.1 Step

Parameter

Value

Units

(1)

P

1.25 × 103

N

(2)

Material, body 1

Mild steel

–

(3)

E1

207

GN/m2

SI

(4)

σ1

0.3

–

(5)

k1

1.4 × 10–12

m2/N

(6)

Material, body 2

Brass

–

(7)

E2

101

GN/m2

British

To calculate quantities below, follow bars a, b, β δ (f ) qm a x

Comments

z max

Applied normal load

Values from Table 9.1

(8)

σ2

0.35

–

(9)

k2

2.77 × 10–12

m2/N

(10)

R11

∞

m

(11)

R12

25 × 10–3

m

(12)

R21

∞

m

(13)

R22

7.5 × 10–3

m

(14)

ω

40

degree

(15)

(10)–1 – (11)–1

–40

m–1

(16)

(12)–1 – (13)–1

–13.33

m–1

(17)

cos 2 (14)

0.174

–

(18)

(15)2 + (16)2

1.78 × 103

m–2

Gives ( 1 / R 11 – 1 / R 12 ) 2 + ( 1 / R 21 – 1 / R 22 ) 2 Gives 2 ( 1 / R 11 – 1 / R 12 ) ( 1 / R 21 – 1 / R 22 ) cos 2 ω

Values from Table 9.1

Principal curvature radii

If ω = 0 ° , see Equations (3.1) Gives ( 1 / R 11 – 1 / R 12 ) Gives ( 1 / R 21 – 1 / R 22 ) Gives cos 2ω

(19)

2(15)(16)(17)

185.6

m–2

(20)

[(18) + (19)]1/2

44.34

m–1

Gives square root of sum of above two expressions

(21)

(10)–1 + (11)–1 + (12) –1 + (13)–1

53.33

m–1

Gives ( 1 / R 11 + 1 / R 12 + 1 / R 21 + 1 / R 22 )

(22)

(21) – (20)

8.993

m–1

Gives 4A

(23)

(21) + (20)

97.67

m–1

Gives 4B

(24)

(22)/(23)

0.092

–

Gives A/B

(25)

[(22) + (23)]/4

26.67

m–1

4.15

× 10–12

Gives (A + B)

m2/N

Gives (k1 + k2)

(26)

(5) + (9)

(27)

0.75π(1)(26)(25)2

8.69 × 10–6

–

Gives 0.75 P π ( k 1 + k 2 ) ( A + B ) 2

(28)

(27) 1/3

2.056 × 10–2

–

Gives W (Equation (3.13))

25



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TABLE 8.1 (29)

Ca

1.11

–

(30)

Cb

1.16

–

From Figure 1 or 2 using value of A/B from (24). If A/B = 0, see Equations (3.6) and (3.7). If A/B = 1.0 continue with calculation procedure, or see Equations (3.3) and (3.5).

(31)

Cβ

1.04

–

(32)

Cδ

1.67

–

(33)

Cf

0.78

–

(34)

(29)(28)(24)–1/3/(25)

1.896 × 10–3

m

(35)

(30)(28)(24)1/3/(25)

0.404 × 10–3

m

Gives b. (Equation (3.10)

(36)

(31)(24)2/3

0.213

–

Gives β. (Equation (3.11))

(37)

(32)(28) 2(24) 1/3/(25)

11.95 × 10–6

m

(38)

–2(33)(28)/ [ π 2 (26) ]

–783 × 106

N/m2

(39)

qm ax

0.314

–

(40)

(39)(38)

246 × 106

N/m2

(41)

( z / b )q m a x

0.74

–

(42)

(41)(35)

0.299 × 10–3

m

Gives ( z q )

(43)

( f x ) m ax

0.753

–

From Figure 5, as comment (39)

(44)

(43)(38)

–590 × 106

N/m2

(45)

(f )max

0.944

–

(46)

(45)(38)

–739 × 106

N/m2

(47)

( q m a x )0

0.125

–

(48)

|(47)(38)|

979 × 106

N/m2

y

From Figure 3 or 4 using value of A/B from (24). If A/B = 0, see Section 4.1.2. Gives a. (Equation (3.9))

Gives δ. (Equation (3.12)) Gives ( f z )m a x . (Equation (4.15)) From Figure 6 using value of A/B from (24) and value of σ from (4) or (8) according to which body is being analysed Gives q m a x (maximum sub-surface shear stress) From Figure 6, as comment (39)

Gives ( f x )

max

m ax

(depth of maximum shear)

(direct stress at surface)

From Figure 5, see comment (39) Gives ( f y ) m ax (direct stress at surface) From Figure 5, see comment (39) Gives ( q m a x ) 0 (maximum shear stress at surface)

26



78035 When the cylinders are crossed at 90° to each other, the stresses and contact area change as shown in Table 8.2. TABLE 8.2 Cylinders crossed at 40°

Cylinders crossed at 90°

Contact shape

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A/ B = 0.092 a = 1.896 mm b = 0.404 mm β = 0.213

Ellipse dimensions

Normal approach

Stresses (body 1)

A /B = 0.333

δ = 11.95 × 10

–3

mm

–3

mm

( f z ) max = – 0.94 GN/m2

q max = 0.247 GN/m2

q max = 0.299 GN/m2

0

= 0.129 GN/m2

( f z ) max = – 0.78 GN/m2 q max = 0.246 GN/m2 ( q max )

8.2

δ = 14.94 × 10

( f z ) max = – 0.78 GN/m2 ( q max )

Stresses (body 2)

a = 1.142 mm b = 0.551 mm β = 0.483

0

Depth of maximum shear (body 1)

zq

max

Depth of maximum shear (body 2)

zq

max

= 0.098 GN/m2

( q max )

0

= 0.128 GN/m2

( f z ) m ax = – 0.947 GN/m2 q max = 0.299 GN/m2 ( q max )

0

= 0.299 mm

zq

max

= 0.299 mm

zq

max

= 0.095 GN/m2 = 0.353 mm = 0.358 mm

Example 2 This example of a wheel on a rail (shown in Sketch 8.2) has been chosen to illustrate a case where axes transformation is required. The geometry of contact appears, at first sight, to be that of a narrow disc on a cylinder, but, because the wheel tyres are coned to prevent the flanges from making contact with the rail head, the contact geometry is that of a cone and a cylinder (see Sketch 8.3). The wheel and rail are steel, the applied load is 100 kN, the wheel is coned at an angle of 2.5° and has a radius of 0.5 m. It is required to find the contact dimensions, approach, and maximum compressive stress if the rail head radius is 0.3 m.

27

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78035

Sketch 8.2 8.2.1

Calculation procedure Referring to Sketch 8.4, the wheel and rail contact each other at point T and the resultant force is directed along line TU if friction is neglected. It is clear that the nominal wheel radius, r , which is the radius of a circle of revolution passing through point T, does not coincide with the radius in the plane of the normal load. From curvature theory, the required radius of curvature, R 12 , is obtainable from 2

3/2

[ ( f ′( y ) ) + 1 ] R = R 12 = ------------------------------------------ . f ″( y )

Sketch 8.3

28

(8.1)



78035 The section on TU shown in Sketch 8.4 indicates the position of the required curvature within the ellipse of major and minor semi-axes l and m , respectively. The equation of the ellipse is 2

2

y x 1 = ----- + ------- , 2 2 m l

from which

2  x  y = m  1 – ----- 2  l 

1/2

,

2 – 1/2

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mx dy x  f ′ ( y ) = ------ = – -------  1 – ----- 2 2 dx l  l 

,

2

and

d y lm -. f ′′ ( y ) = --------- = – -----------------------------2 2 2 3 / 2  dx l –x  

                

(8.2)

Substitution of f ′( y ) and f ″( y ) into Equation (8.1), and putting x = l , yields 2

m R 12 = ------- . l

(8.3)

Sketch 8.4 From geometrical considerations of Sketch 8.4, it can be seen that r sin ( 90 + θ ) l = ----------------------------------- , sin ( 90 – 2θ ) and

m = r + n tan θ ,

where

n = l sin θ .

      

(8.4)

29



78035 As the wheel is coned at an angle θ = 2.5° , Equations (8.4) yield l = 1.00286 r and m = 1.00191 r which, when substituted into Equation (8.3), gives 2

Thus,

m R 12 = ------- = 1.00092 r . l R 12 = 0.50048 , R 21 = ∞ , R 11 = ∞ ,

R 22 = 0.3 m,

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and the angle between the principal curvature planes, ω , is 90°. Use of the calculation procedure (Table 9.3) or the flow chart (Table 9.5) shows that, for an applied load of 100 kN, a b β δ and

= = = =

7.50 mm, 5.34 mm, 0.71 , 0.103 mm

( fz ) = – 1.19 GN/m2. m ax

30

 9.

78035 TABLES

TABLE 9.1 Values of Poisson’s Ratio, Young’s Modulus of Elasticity, and Material Parameter, k SI Body material ABS† (Acrilonitrilebutadienestyrene) unfilled‡ Acrylic polymers† (e.g. Perspex) Aluminium (inc. anodised)

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Aluminium alloys Aluminium bronze Alumina (high density, sintered) Beryllium

σ

E × 10

–9

(N/m2)*

British k × 10

12

(m2/N)*

E × 10

–6

(lbf/in2)*

k × 10

9

(in2/lbf)*

0.35–0.4

1.65–2.7

169–99

0.24–0.392

1160–680

0.38

2.79–3.0

97.6–90.8

0.405–0.435

672–626

0.33

68.9

4.11

10

28.4

0.33–0.35

44.8–75.8

6.33–3.68

6.5–11

43.6–25.4

0.25

103

2.90

15

20

0.25–0.31

258–366

1.16–0.786

37.4–53.1

7.69–5.62

0.024–0.030

276–303

1.15–1.04

40–44

7.95–7.23

Beryllium-copper

0.34

131

2.15

19

14.8

Brass (70/30)

0.35

101

2.77

14.7

19.1

Bronze, sintered

0.34

110

2.55

16

17.6

Cadmium

0.44

68.9

3.72

10

25.7

Cast iron

0.21–0.30

96.5–145

3.15–2.0

14–21

21.7–13.8

Chromium

0.30

248

1.17

36

8.05

Constantan

0.33

162.4

1.75

23.56

12.04

Copper

0.30

110–117

2.63–2.47

16–17

18.1–17.0

Copper-nickel

0.34

68.9

4.08

10

28.2

Duralumin

0.35

70.8

3.95

10.3

27.12

Glass (flint)

0.25

55–69

5.43–4.32

8–10

37.3–29.8

Gold

0.42

68.9

3.80

10

26.2

Invar

0.26

148

2.01

21.4

13.9

Iron

0.30

207

1.40

30

9.66

0.30

207

1.40

30

9.66

0.41–0.45

15.2

17.5

2.2

120

Iron, sintered Lead Magnesium

0.25

41.4

7.21

6

49.7

Magnesium alloy

0.25

44.8

6.66

6.5

45.9

Molybdenum

0.3–0.32

275–296

1.05–0.97

40–43

7.24–6.64

Monel

0.25–0.32

168

1.77–1.70

24.4

12.2–11.7

Nickel Nickel alloy Nylon† (type 66, ‘dry’)‡ Phosphor-bronze

0.31

207

1.28

30

8.83

0.29–0.31

186–220

1.57–1.31

28–33

10.4–8.72

0.40

3.2

83.6

0.464

576

0.34–0.38

110

2.56–2.48

16

17.6–17.0

Platinum

0.39

166

1.63

24

11.3

Polyethylene†

0.35

0.1–0.75

2790–372

0.014–0.11

20 000–2540

Polypropylene†**

0.4

1.03

260

0.149

1790

Polystyrene†‡

0.35

2.76–4.14

101–67.5

0.4–0.6

698–466

0.41–0.42

2.7–3.0

98.1–87.4

0.39–0.435

679–603

PVC† (Polyvinylchloride) unplasticised‡ For footnotes see end of Table.

31



78035 TABLE 9.1 Values of Poisson’s Ratio, Young’s Modulus of Elasticity, and Material Parameter, k (Continued) SI

Body material PTFE†‡

σ 0.53

E × 10

–9

(N/m2)*

British k × 10

0.35–0.59

12

(m2/N)*

654–388

E × 10

–6

(lbf/in2)*

0.05–0.085

k × 10

9

(in2/lbf)*

4580–2690

(Polytetrafluoroethylene) Rubber

0.5

4.14

57.7

0.6

398

Silicon carbide

0.19

241–428

1.27–0.72

35–62

8.77–4.95

Silicon nitride

0.27

172–310

1.72–0.95

25–45

11.8–6.56

Silver

0.37

68.9

3.99

10

27.5

Steel, mild

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Steel, stainless

0.3

207

1.40

30

9.40

0.28–0.30

190–215

1.54–1.35

27.6–31.2

10.6–9.28

Steel, tool

0.29

212

1.38

30.8

9.47

Steel, 0.75 per cent C

0.30

201

1.37

29.2

9.92

Tantalum

0.35

179–193

1.56–1.45

26–28

10.7–9.98

Tin

0.36

49.9

1.31

7.24

38.3

0.34–0.4

107

2.63–2.50

15.5

18.2–17.3

0.33

117–121

2.42–2.34

17–17.5

16.7–16.2

Titanium Titanium alloy Titanium carbide

0.2–0.21

372–414

0.82–0.74

54–60

5.66–5.07

Tungsten

0.28

411

71.4

59.6

49.2

Tungsten carbide

0.22

565–607

0.54–0.50

82.88

3.69–3.44

Vanadium

0.36

127–135

2.18–2.05

18.4–19.6

15.1–14.1

Zinc

0.25–0.28

90–105

3.32–2.79

13–15

23.0–19.6

Zirconium

0.37–0.41

89.6–96.5

3.07–2.74

13–14

21.1–18.9

These values are approximate and may vary considerably depending on the constituents and condition of the material. For footnotes see end of Table. *

These values of E and k have been multiplied by the powers of 10 shown in the column headings. Thus, the values of the quantities 9 9 –12 –12 6 6 in the first row of this table are 0.35 to 0.4, 1.65 × 10 to 2.7 × 10 , 169 × 10 to 99 × 10 , 0.24 × 10 to 0.392 × 10 , and –9 –9 1160 × 10 to 680 × 10 .

†

Many plastics contain fillers which, when combined with certain manufacturing processes, result in materials that behave anisotropically and not (as the theory in this Item demands) isotropically. The values given for E, σ and k are functions of temperature, strain rate and, in some cases, humidity; therefore these values are not necessarily typical.

‡

Tensile modulus, 100 second test, 1 per cent strain, at 23°C.

**

Tensile modulus, 100 second test, 1 per cent strain, at 20°C.

32



78035

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TABLE 9.2 Variation of Sub-surface Stresses with β and σ at points on Axis of Symmetr

Note: Sub-surface stress distributions for β < 0.01 do not change significantly from those shown for β = 0.01 , and thus the table has been truncated at this value. The curves of Q do not represent the maximum shear stress at all depths below the contact centre because, just below the surface, ( 1/2 ) ( f – f ) > ( 1/2 ) ( f z – f ) , see Section 4.2. z x y

33



78035 TABLE 9.3

Step

Value

Units SI

British

(1)1,3

P

N

lbf

(2)2,5

Material, body 1

–

–

(3)

E1

GN/m2

(4)

σ1

–

(5)

k1

m2/N

(6)2,5

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Parameter

To calculate quantities below, follow bars ( fz ) a, b, β δ q max

Comments

m ax

Applied normal load

lbf/in2 –

Values from Table 9.1

in2/lbf

Material, body 2

–

(7)

E2

GN/m2

–

(8)

σ2

–

(9)

k2

m2/N

(10)4,8

R11

m

in

(11)4,8

R12

m

in

(12)4,8

R21

m

in

(13)4,8

R22

m

(14)4

ω

degree

(15)6

(10)–1 – (11)–1

(16)6

lbf/in2 Values from Table 9.1 in2/lbf

Principal curvature radii

in degree

If ω = 0 ° , see Equations (3.1)

m–1

in–1

Gives ( 1 / R 11 – 1 / R 12 )

(12)–1 – (13)–1

m–1

in–1

(17)

cos 2 (14)

–

(18)

(15)2 + (16)2

m–2

in–2

Gives ( 1 / R 11 – 1 / R 12 ) 2 + ( 1/ R 21 – 1 / R 22 ) 2

Gives ( 1 / R 21 – 1 / R 22 ) Gives cos 2ω

(19)

2(15)(16)(17)

m–2

in–2

Gives 2 ( 1 / R 11 – 1 / R 12 ) ( 1 / R 21 – 1 / R 22 ) cos 2 ω

(20)

[(18) + (19)]1/2

m–1

in–1

Gives square root of sum of above two expressions

(21)

(10)–1 + (11)–1 + (12)–1 + (13)–1

m–1

in–1

Gives ( 1 / R 11 + 1 / R 12 + 1 / R 21 + 1 / R 22 )

(22)

(21) – (20)

m–1

in–1

Gives 4A

(23)

(21) + (20)

m–1

in–1

Gives 4B

(22)/(23)

–

–

Gives A/B

(25)

[(22) + (23)]/4

m–1

in–1

Gives (A + B)

(26)

(5) + (9)

m2/N

(27)

0.75π(1)(26)(25)2

(24)11

in2/lbf

Gives (k1 + k2)

–

–

Gives 0.75 P π ( k 1 + k 2 ) ( A + B ) 2

(27)1/3

–

–

Gives W (Equation (3.13))

2

Ca

–

–

(30)12

Cb

–

–

(31)12 (32)11,1

Cβ

–

–

2

Cδ

–

–

(33)12

Cf

–

–

(28) (29)11,1

From Figure 1 or 2 using value of A/B from (24). If A/B = 0, see Equations (3.6) and (3.7). If A/B = 1.0 continue with calculation procedure, or see Equations (3.3) and (3.5.) From Figure 3 or 4 using value of A/B from (24). If A/B = 0, see Section 4.1. Gives a. (Equation (3.9))

(34)

(29)(28)(24)–1/3 /(25)

m

in

(35)

(30)(28)(24)1/3/(25)

m

in

Gives b. (Equation (3.10))

(36)

(31)(24)2/3

–

–

Gives β. (Equation (3.11))

(37)7

(32)(28)2(24)1/3/(25)

m

(38)9,12

–2(33)(28)/ [ π 2 (26) ]

N/m2

qm ax

–

(39)(38)

N/m2

(39) (40)10

in

Gives δ. (Equation (3.12))

lbf/in2

lbf/in2

Gives ( f z )ma x . (Equation (4.15)) From Figure 6 using value of A/B from (24) and value of σ from (4) or (8) according to which body is being analysed Gives q m a x (maximum sub-surface shear stress)

–

( z / b )q m a x

–

–

From Figure 6, as comment (39)

(42)10

(41)(35)

m

in

(43)

( f x ) m ax

–

Gives ( z q ) m a x (depth of maximum shear) From Figure 5, as comment (39)

(44)9

(43)(38)

N/m2

lbf/in2

Gives ( f x ) m a x (direct stress at surface)

(45)

(f )max

–

(46)9

(45)(38)

N/m2

lbf/in2

Gives ( f y ) m a x (direct stress at surface)

lbf/in2

Gives ( q m a x ) 0 (maximum shear stress at surface)

(41)

y

(47)

(qmax )0

–

(48)9

|(47)(38)|

N/m2

From Figure 5, see comment (39)

From Figure 6, see comment (39)

For footnotes see Table 9.4

34



78035

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TABLE 9.4 Footnotes for Table 9.3 1

See Section 6.3 for the effects of combined normal and tangential loading

2

See Section 6.1 for the effects of surface roughness

3

See Section 6.3 for the effects of cyclic loading

4

See Section 6.4 if bodies are not semi-infinite

5

See Section 6.5 if bodies are coated with thin film

6

See Section 6.6 for concave and convex, conforming bodies

7

See Section 3 on contact geometry

8

See Section 3.1 for information on principal curvatures

9

See Section 4.1 on surface stresses

10

See Section 4.2 on sub-surface stresses

11

See Sections 3.3 and 4.1.2 on rectangular contacts (A/B = 0)

12

Contact geometry, approach and maximum compressive stress may be evaluated using the flow chart, Table 9.5

35

78035

 TABLE 9.5 Flow Chart for use with Programmable Calculators

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This procedure allows the user to estimate contact dimensions (to within 1 per cent), normal approach (to within 4 per cent) and contact stress (to within 1 per cent) when body size, orientation, material and load are known. (The expressions in steps 13 to 16 give approximate solutions for β and the complete elliptic integrals; these approximations will usually yield less accurate results in steps 18 to 22 than are obtainable from Table 9.3.)

For 0 ≤ A ⁄ B ≤ 2 ×10

–6

, see equations in Sections 3 and 4.1.2.

36



78035 TABLE 9.6 Equations for Circular and Rectangular Contacts Parameter

Circular contacts

A/ B

1.0

β

1.0 1/ 3

R 11 R 21 3P π ----------- ( k 1 + k 2 ) -----------------------R 11 + R 21 4

a

b

Same as for a above

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1/ 3

3 Pπ ----------- ( k 1 + k 2 ) 4

δ

2

R 11 + R 21 ------------------------R 11 R 21 1/ 3

( f z ) m ax

–

R 11 + R 21 2 ------------------------R 11 R 21

6P --------------------------------5 2 π ( k1 + k2 )

Parameter

Rectangular contacts*

A/ B

0

β

0

a

Theoretically infinite 1/2

b

δ

R 11 R 21 P 4  --- ( k 1 + k 2 ) ------------------------ L R 11 + R 21 4 R 11 4 R 21 P 2  --- k log e ------------ + log e ------------ – 1  L b b 1/2

( f z ) m ax

*

R 11 + R 21 P 1 –  --- ------------------------------ ------------------------ L 2 π ( k 1 + k 2 ) R 11 R 21

It is important that the notes relating to rectangular contacts (see Sections 3.3 and 4.1.2) be read, and that the meaning of “rectangular” be understood.

37

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 101 9 8 7

6

5

Cb 4

Cδ 3

38

Cβ

For data in this region, see Figure 2 2

Ca

10

2

3 4

-6 6 810−6

2

3 4

-5 6 810−5

2

3 4

-4 6 810−4

2

3 4

-3 6 810−3

2

3 4

-2 6 810−2

2

3 4

-1 6 810−1

A B

FIGURE 1 CONTACT DIMENSIONS, ELLIPSE RATIO, AND APPROACH COEFFICIENTS

100 2

3 4

6 8 10 10

Circular contacts

78035

−7 -7

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 1.7 Cδ 1.6

1.5

1.4

1.3

39 1.2

Cb Ca

1.1

Cβ

0.2

0.3

0.4

0.5

0.6

0.7

A B

FIGURE 2 CONTACT DIMENSIONS, ELLIPSE RATIO AND APPROACH COEFFICIENTS

0.8

0.9

1

Circular contacts

78035

1.0 0.1

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 1.00 0.90 0.80 0.70 For data in this region, see Figure 4

0.60 0.50 0.40

0.30 Cf

40

0.20

0.10 0.09 0.08

-7 10−7

2

3 4

-6 6 810−6

2

3 4

-5 6 810−5

2

3 4

-4 6 810−4

2

3 4

-3 6 810−3

2

3 4

-2 6 810−2

2

A B

FIGURE 3 MAXIMUM DIRECT SURFACE STRESS COEFFICIENT

3 4

-1 6 810−1

0.06 2

3 4

6 8 10 10

Circular contacts

78035

0.07

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 1.0

0.9

Cf

41 0.8

0.2

0.3

0.4

0.5

0.6

A B

FIGURE 4 MAXIMUM DIRECT SURFACE STRESS COEFFICIENT

0.7

0.8

0.9

1

Circular contacts

78035

0.75 0.1

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0.5

0.5 0.45 0.4

0.35 σ 0.3 0.25 0.2

0.45

1.0

0.9

0.8

0.4

0.7

0.35

0.6

0.3

0.5

0.25

42 0.4

0.2 σ

0.3 Upper set of curves is fy

max

Lower set of curves is fx

0.2

max

These maximum principal stresses occur at contact surface 0.1

Rectangular contacts

-6 10−6

2

3 4

-5 6 810−5

2

3 4

-4 6 810−4

2

3 4

-3 6 810−3

2

3 4

-2 6 810−2

A B

FIGURE 5 DIRECT STRESSES AT CONTACT CENTRE

2

3 4

-1 6 810−1

0.0 2

3 4

6 8 10 10

Circular contacts

78035

0

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 1.0

This set of curves is for z rb q max

σ 0.5 0.45

0.9

0.4 0.35 0.3 0.25 0.2

This set of curves is for qmax

0.7

0.6

43

σ 0.5 0.45 0.4 0.35 0.3 0.25 0.2

This set of curves is for (qmax)0

0.8

0.5

0.4

σ 0.3

0.2 0.25 0.3

0.2

0.35 0.4

0.1

0.45

0 Rectangular contacts

10

2

3 4

−5 -5

6 810

2

3 4

−4 -4

6 810

2

3 4

-3 6 810−3

2

3 4

-2 6 810−2

2

A B

FIGURE 6 SHEAR STRESSES AND DEPTH OF MAXIMUM SHEAR

3 4

-1 6 810−1

0.0 2

3 4

6 8 10 10

Circular contacts

78035

0.5 −6 -6

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 1011 5 + 1010

10−3 This graph gives data for steel-on-steel contacts. To obtain stresses for other materials, multiply value of fz by ksteel 2/3

2 + 1010

max

kother Data are valid for alike materials only.

1010



5 + 109

−2

10

2 + 109 Radius, R21 and R22 (m) 109 5 + 108

10−1

44

2 + 108

Maximum compressive stress, fz (N/m2) max

108



1

Non-conforming contacts

5 + 107 −1 −10

"

−10−1

2 + 107



−2

Conforming contacts

7

!

10 10

5 + 106

−10 -10 -3 10−3

-2 10−2

-1 10 10−1 10 Radius, R11 and R12 (m)

2

101

2 + 106 102 1

101

102

103

Load P (N)

FIGURE 7 CIRCULAR CONTACTS (SPHERE AND EITHER SPHERE, OR PLATE, OR SOCKET)

104

105

78035

102

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1011 5+1010 10−3

2+1010

This graph gives data for steel-on-steel contacts. To obtain stresses for other materials, multiply by ksteel 1/2 value of fz

1010

max

5+109

kother Data are valid for alike materials only. 10−2



2+109 109

Radius, R21 (m)

5+108

10−1

45

2+108 Non-conforming contacts

!

1

−2

−10

max

108 5+107

!

−1

−1

−10

10

!

!

Maximum compressive (N/m2) stress, fz

Conforming contacts

2+107

107

2

10

5+10

6

2+106

−102 −10

106

-3 10−3

-2 10−2

-1 10−1

10 10 Radius, R11 (m)

101

2 3 2 10 1010

104

105

106

107

Load per unit length, P/L (N/m)

FIGURE 8 LONG RECTANGULAR CONTACTS (CYLINDER AND EITHER CYLINDER, OR PLATE, OR GROOVE)

108

78035





78035

APPENDIX A EQUATIONS FOR CONTACT GEOMETRY AND STRESSES

A1.

ADDITIONAL NOTATION AND UNITS

E(m)

SI

British

m

in

complete elliptic integral of second kind, given by E(m) =

π/2

∫0

2

2

1/2

( 1 – m sin θ )

dθ

E ( γ , m ) incomplete elliptic integral of second kind, given by γ

∫0 ( 1 – m

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E ( γ , m) =

2

2

sin θ )

1/ 2

dθ

F ( γ , m ) incomplete elliptic integral of first kind, given by γ

∫0 ( 1 – m

F(γ, m) = K(m)

2

2

sin θ )

– 1/ 2

dθ

complete elliptic integral of first kind, given by K(m) =

π/2

∫0

2

2

( 1 – m sin θ )

– 1/ 2

dθ

m

calculation parameter, given by Λ/a

V

calculation parameter, given by 2

2 2

1/ 2

[ ( 1 + ζ ) /( 1 + ζ β ) ]

–1

γ

calculation parameter, given by cot

ζ

non-dimensional depth, given by z/b

Λ

calculation parameter, given by ( a – b )

( βζ )

2 1/ 2

2

Subscripts

A2.

a

denotes stresses at ends of contact ellipse major axis

b

denotes stresses at ends of contact ellipse minor axis

p

denotes stresses at periphery of contact ellipse

NOTES The equations given in Sections A2.1 to A2.4 are those that have been used to calculate data for inclusion in this Item. (The equations in Section A2.5 have been provided for users who may wish to investigate contact stresses at points on the contact surface other than at the centre.) The method by which the elliptic integrals have been evaluated and the data subsequently prepared is given in Derivation A1.

46

 A2.1

78035 Contact Dimensions and Axes Ratio It can be shown that (see Derivation A2) ( k1 + k2 ) E ( m ) ( A + B ) = 3 P ------------------------ ------------- , 3 2 β 2a

(A2.1)

which can be rearranged such that –1 / 3

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4(A + B) a = C a ----------------------------------3Pπ ( k 1 + k 2 )

A ---  B

–1 / 3

,

(A2.2)

1/3

where

2E ( m ) ----------------2 πβ

Ca =

A ---  B

1/ 3

(A2.3) 1/ 3

and, as

W =

3 2 --- P π ( k 1 + k 2 ) ( A + B ) 4

,

(A2.4)

C a W A –1 / 3 . a = --------------------  --- ( A + B )  B Using the substitution β

2

2

(A2.5)

2

= b / a in Equation (A2.1),

3P ( k 1 + k 2 ) ( A + B ) = ------------------------------- E ( m ) β , 3 2b

(A2.6)

which can be rearranged so that – 1/ 3

4(A + B) b = C b ----------------------------------3 Pπ ( k 1 + k 2 )

A ---  B

1/ 3

,

(A2.7)

1/3

where

Cb =

2E ( m )β --------------------π

A ---  B

– 1/ 3

,

(A2.8) 1/ 3

and, as

W =

3 2 --- Pπ ( k 1 + k 2 ) ( A + B ) 4 C b W A 1/ 3 . b = --------------------  --- ( A + B )  B

47

,

(A2.9)

(A2.10)



78035 Division of Equation (A2.10) by Equation (A2.5) yields Cb A 2/3 b --- = ------  --- , C a  B a

(A2.11)

which is (because C β = C b / C a ) the same as A β = C β  ---  B

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so that

A2.2

2/3

,

(A2.12)

A –2 / 3 . C β = β  ---  B

(A2.13)

Normal Approach of Bodies It can be shown that (see Derivation A2) 3P δ = ------- ( k 1 + k 2 ) K ( m ) . 2a

(A2.14)

Equations (A2.2) and (A2.3) can be written – 1/ 3

a =

1/3

2E(m) ----------------2 πβ

4(A + B) ----------------------------------3P π ( k 1 + k 2 )

,

(A2.15)

which, when substituted into Equation (A2.14) and rearranged, gives 1/3

3P π 4(A + B ) δ = C δ ----------- ( k 1 + k 2 ) ----------------------------------4 3 Pπ ( k 1 + k 2 )

A ---  B

1/3

,

(A2.16)

1/3

2

where

4β C δ = K ( m ) -------------------2 π E(m)

A ---  B

– 1/ 3

,

(A2.17)

1/3

and, as

W =

3 --- Pπ ( k 1 + k 2 ) ( A + B ) 2 4

,

(A2.18)

2

CδW A 1/3 δ = --------------------  --- . ( A + B )  B

(A2.19)

48

 A2.3

78035 Maximum Direct Surface Stress Multiplication of Equation (A2.2) by Equation (A2.7) produces the result –2 / 3

4(A + B) ab = C a C b ----------------------------------3 Pπ ( k 1 + k 2 )

,

(A2.20)

into which substitution of Equations (A2.3) and (A2.8) gives 1/3

–2 / 3

2

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ab =

Now,

4(A + B) ----------------------------------3Pπ ( k 1 + k 2 )

4[ E(m )] ------------------------2 π β

.

(A2.21)

3P ( fz ) = – ------------- , m ax 2πab

(A2.22)

which, combining Equations (A2.21) and (A2.22) gives 2 /3

3P 4(A + B) ( fz ) = – C f ------- ----------------------------------m ax 2π 3Pπ ( k 1 + k 2 )

,

(A2.23)

1/ 3

2

where

Cf =

π β ------------------------2 4[E(m)]

,

(A2.24) 1/ 3

and, as

W =

3 2 --- Pπ ( k 1 + k 2 ) ( A + B ) 4

,

(A2.25)

2W ( fz ) = – C f ------------------------------ . m ax 2 π ( k1 + k2 ) A2.4

(A2.26)

Sub-surface Stresses on Axis of Symmetry The non-dimensionalised direct stresses on the axis of symmetry within either body are given by (see Derivation A2) 1  E ( γ , m ) 2 ζ ( σ – β 2 ) + F ( γ , m )2ζ β 2 ( 1 – σ ) fx = --------------------2  (1 – β )

(A2.27)

+ β ( βV + 2 σ ) – ( β + 2σV )  , 

49



78035 1  E ( γ , m ) 2 ζ ( 1 – σ β 2 ) –F ( γ , m ) 2 ζ β2 ( 1 – σ ) f y = --------------------2  (1 – β ) + β ( 1 – 2σ + 2σ βV ) + V ( V and

f z = –  V ( 1 + β 2 ζ 2 )  

(A2.28) –2

– 2 ) 

–1

.

(A2.29)

When the contact is circular, the non-dimensionalised direct stresses are given by

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1 ( f x ) β = 1.0 = ( f y ) β = 1.0 = ( 1 + σ ) ( ζ γ – 1 ) + --- sin2 γ 2 and

2

( f z ) β = 1.0 = sin γ .

(A2.30) (A2.31)

When the contact is rectangular, the non-dimensionalised direct stresses are given by 2 1/ 2

( f x ) β = 0 = 2σ [ ( 1 + ζ )

A2.5

(A2.32)

2

2 1/ 2

and

– ζ] ,

[(1 + ζ ) – ζ] ( f y ) β = 0 = ---------------------------------------------2 1/ 2 (1 + ζ )

(A2.33)

1 - . ( f z ) β = 0 = --------------------------2 1/ 2 (1 + ζ )

(A2.34)

Stresses on Surface of Contact The following sections give equations for the calculation of the stresses at the periphery of the contact and at the ends of the contact ellipse; the maximum principal shear stress at the surface may be found from Expression (A2.46). Equations (A2.35) to (A2.45) are based on work from Derivation A3.

A2.5.1 Stresses at contact periphery The direct stresses (which are principal stresses at the ends of the contact ellipse, see Section A2.5.2) at the periphery of the contact zone are obtainable from

and

2  1 – 2σ x y Λy + xΛ – --- tan– 1 ------- , ( f x ) p = β  ----------------- 1 – ------- log e a-------------------2 2 Λ 2Λ 2 1 – β  b a – xΛ

(A2.35)

( fy )p = –( fx )p

(A2.36)

( fz )p = 0 .

(A2.37)

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78035 The shear stresses at the periphery of the contact ellipse are obtainable from

and

2  1 – 2σ y x – 1 Λy a + x Λ – --- tan ------( q xy ) = β  ----------------- ------- log e -------------------p 2 2 Λ 2  1 – β  2Λ b a – xΛ

(A2.38)

( q xz )

(A2.39)

p

= ( q yz ) = 0 . p

A2.5.2 Stresses at ends of contact ellipse

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At the ends of the major semi-axis, where x = ± a and y = 0 , the principal stresses are obtainable from  1 – 2σ a +Λ ( fx )a = β  ----------------- 1 – ------- log e a-------------2 Λ 2 a–Λ 1 – β 

and

,

( fy ) = – ( fx ) a a ( f z ) = ( q xy ) = ( q yz ) = ( q zx ) = 0 . a a a a

(A2.40) (A2.41) (A2.42)

At the ends of the minor semi-axis, where y = ± b and x = 0 , the principal stresses are obtainable from

and

  1 – 2σ b –1 Λ ( fx ) b = β  ----------------- 1 – ---- tan ---- ,  2 Λ b 1 – β 

(A2.43)

( f y ) = –( f x ) b b

(A2.44)

( f z ) = ( q xy ) = ( q yz ) = ( q zx ) = 0 . b b b b

(A2.45)

A2.5.3 Maximum values of principal shear stresses It can be shown that the largest principal shear stress (see Expressions (4.1)) on the surface of contact is ( q max ) when 0 < β ≤ 0.458 . Non-dimensionalised values of ( q m ax ) are plotted in Figure 6. For all o o larger values of β , the greatest principal shear stress occurs at the ends of the ellipse major semi-axis and is given by 1 --- [ ( f x ) – ( f y ) ] . a a 2 Thus and

1 ( q max ) > --- [ ( 0 2 1 ( q max ) < --- [ ( 0 2

   f x ) – ( f y ) ] when 0.458 < β ≤ 1.0 .  a a  f x ) – ( f y ) ] when 0 < β ≤ 0.458 , a a

51

(A2.46)

 DERIVATION

A1.

ROWLAND, H.S.

Complete and incomplete elliptic integrals of the first and second kinds: numerical evaluation. ESDU Memor. No. 29, 1978.

A2.

CLIFTON, C.J.

Derivation of stress, deflection and contact geometry expressions for point-loaded bodies. ESDU Memor. No. 30, 1978.

A3.

LUR’E, A.I.

Three-dimensional problems of the theory of elasticity. Interscience Publishers, John Wiley and Sons, Inc. 1964.

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A3.

78035

52



78035 THE PREPARATION OF THIS DATA ITEM The work on this particular Item was monitored and guided by the Non-conformal Contact Working Party which has the following membership:

Prof. F.T. Barwell* Prof. D. Dowson Mr A. Dyson Mr W. Lauder Dr B.J. Roylance

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*

– – – – –

University of Hong Kong University of Leeds Shell Research Ltd, Chester University of Strathclyde University College of Swansea

Corresponding Member

and which was appointed by the Mechanisms Committee, the Stress Analysis and Strength of Components Committee, and the Tribology Steering Group, on whose behalf the Working Party acted. The Item was accepted for inclusion in the Mechanisms Sub-series by the Mechanisms Committee which first met in 1975 and now has the following membership:

*

Chairman Mr D. Bastow

– Consulting Engineer

Members Prof. D.H. Chaddock Mr R.W. Davies Mr T.H. Davies Dr G. Druce Prof. F. Freudenstein* Mr J. Rees Jones Mr J.E. Reeve Dr M.R. Smith Mr F.E. Taylor

– – – – – – – – –

Independent Molins Ltd, London University of Technology, Loughborough University of Surrey Columbia University, NY, USA Liverpool Polytechnic Manifold Indexing Ltd, London University of Newcastle-upon-Tyne National Coal Board, Burton-on-Trent.

Corresponding Member

53



78035

ESDU Copyright material. For current status contact ESDU.

The Item was accepted for inclusion in the Stress and Strength Sub-series by the Stress Analysis and Strength of Components Committee which first met in 1964 and now has the following membership:

Chairman Mr A.J. Batchelor

– Imperial Chemical Industries Ltd

Vice-Chairman Mr J.V. Vint

– C.A. Parsons and Company Ltd

Members Mr H.L. Cox Dr M.S.G. Cullimore Mr C.E. Day Prof. J.P. Duncan Mr D. Duval Dr R.B. Heywood Dr L.C.Laming Dr R.M.G. Meek Dr J Spence Dr C.E. Turner

– – – – – – – – – –

Independent University of Bristol Independent University of British Columbia Foster Wheeler Power Products Ltd A. Macklow-Smith Ltd Imperial College of Science and Technology National Engineering Laboratory University of Strathclyde Imperial College of Science and Technology.

The Item was accepted for inclusion in the Tribology Sub-series by the Tribology Steering Group which first met in 1973 and now has the following membership:

*

Chairman Mr M.J. Neale

– Michael Neale & Associates

Members Prof. D. Dowson Dr M. Godet* Dr D. Summers-Smith Dr W.H. Roberts Dr W.F. Wilcock*

– – – – –

University of Leeds INSA, France Imperial Chemical Industries Ltd National Centre of Tribology Mechanical Technology Inc., NY, USA.

Corresponding Member

The work on this Item was carried out in the Mechanisms Group of ESDU. The member of staff who undertook the technical work involved in the initial assessment of the available information and the construction and subsequent development of the Item was Mr C.J. Clifton

– Executive Engineer, Head of Mechanisms.

54