CHAPTER 10 CONTACT STRESS,RESIDUAL STRESS AND STRESS CONCENTRATIONS Summary The maximum pressure po or compressive stress oc at the centre of contact between two curved surfaces is: 3P P0=-0c= 2πab where a and b are the major and minor axes of the Hertzian contact ellipse and P is the total load. For contacting parallel cylinders of length L and radii R:and R2, maximum compressive stress,oc=-0.591 R2 L△ =一P0 1 with△= 1-响+E,-1 and the maximum shear stress,tmax=0.295 po at a depth of 0.786b beneath the surface,with: P△ contact width, b=1.076 R R2 For contacting spheres of radii R and R2 maximum compressive stress, 0e=-( =-p0 maximum shear stress, rmax =0.31 po at a depth of 0.5b beneath the surface with: P△ contact width (circular) b=0.88 (+ For a sphere on a flat surface of the same material PE2 maximum compressive stress, 0c=-0.62 4R2 381
CHAPTER 10 CONTACT STRESS, RESIDUAL STRESS AND STRESS CONCENTRATIONS Summary The maximum pressure po or compressive stress a, at the centre of contact between two curved surfaces is: 3P po = -0, = - 21rab where a and b are the major and minor axes of the Hertzian contact ellipse and P is the total load. For contacting parallel cylinders of length L and radii RI and R2, - -Po LA maximum compressive stress, a, = -0.591 and the maximum shear stress, rmax = 0.295~0 at a depth of 0.7861, beneath the surface, with: contact width, PA b = 1.076 For contacting spheres of radii RI and R2 maximum compressive stress, maximum shear stress, rmax = 0.31 po at a depth of 0.56 beneath the surface with: PA contact width (circular) b = 0.88 For a sphere on ajat sulface of the same material maximum compressive stress, a, = -o.ti*$lg 38 1
382 Mechanics of Materials 2 §10.1 For a sphere in a spherical seat of the same material maximum compressive stress,oc =-0.62 PE2 R2-R12 For spur gears maximum contact stress,oc=-0.475K W m+11 with K Fwd m with W tangential driving load;Fw=face width;d pinion pitch diameter;m=ratio of gear teeth to pinion teeth. For helical gears maximum contact stress,oc = mp where mp is the profile contact ratio and C a constant,both given in Table 10.2. Elastic stress concentration factor K,= maximum stress,Omax nominal stress,Onom Fatigue stress concentration factorK Sn for the unnotched material S for notched material with S the endurance limit for n cycles of load. Notch sensitivity factor K- K,-1 or,in terms of a significant linear dimension (e.g.fillet radius)R and a material constant a 1 9= (1+a/R) max.strain at notch Strain concentration factor Ke=- nominal strain at notch Stress concentration factor Kp in presence of plastic flow is related to Ke by Neuber's rule KpKe=K好 10.1.Contact Stresses Introduction The design of components subjected to contact,i.e.local compressive stress,is extremely important in such engineering applications as bearings,gears,railway wheels and rails,cams, pin-jointed.links,etc.Whilst in most other types of stress calculation it is usual to neglect local deflection at the loading point when deriving equations for stress distribution in general bodies,in contact situations,e.g.the case of a circular wheel on a flat rail,such an assumption
382 Mechanics of Materials 2 $10.1 For a sphere in a spherical seat of the same material maximum compressive stress, a, = -0.62 For spur gears with maximum contact stress, a, = -0.475fi with W = tangential driving load; F, = face width; d = pinion pitch diameter; m = ratio of gear teeth to pinion teeth. For helical gears - maximum contact stress, a, = -C - \ifp where mp is the profile contact ratio and C a constant, both given in Table 10.2. maximum stress, amax nominal stress, anom Elastic stress concentration factor K, = S, for the unnotched material S, for notched material Fatigue stress concentration factor Kf = with S, the endurance limit for n cycles of load. Kf - 1 K, - 1 Notch sensitivity factor q = - or, in terms of a significant linear dimension (e.g. fillet radius) R and a material constant a 1 (1 + a/R) 4’ max. strain at notch nominal strain at notch Strain concentration factor K, = Stress concentration factor K, in presence of plastic flow is related to K, by Neuber’s rule K,K, = Kf 10.1. Contact Stresses Introduction The design of components subjected to contact, i.e. local compressive stress, is extremely important in such engineering applications as bearings, gears, railway wheels and rails, cams, pin-jointed-links, etc. Whilst in most other types of stress calculation it is usual to neglect local deflection at the loading point when deriving equations for stress distribution in general bodies, in contact situations, e.g. the case of a circular wheel on a flat rail, such an assumption
§10.1 Contact Stress,Residual Stress and Stress Concentrations 383 would lead to infinite values of compressive stress (load -"zero"area infinity).This can only be avoided by local deflection,even yielding,of the material under the load to increase the bearing area and reduce the value of the compressive stress to some finite value. Contact stresses between curved bodies in compression are often termed"Hertzian"contact stresses after the work on the subject by Hertz()in Germany in 1881.This work was concerned primarily with the evaluation of the maximum compressive stresses set up at the mating surfaces for various geometries of contacting body but it formed the basis for subsequent extension of consideration by other workers of stress conditions within the whole contact zone both at the surface and beneath it.It has now been shown that the strength and load-carrying capacity of engineering components subjected to contact conditions is not completely explained by the Hertz equations by themselves,but that further consideration of the following factors is an essential additional requirement: (a)Local yielding and associated residual stresses Yield has been shown to initiate sub-surface when the contact stress approaches 1.2 oy (oy being the yield stress of the contacting materials)with so-called "uncontained plastic fow"commencing when the stress reaches 2.8 oy.Only at this point will material "escape" at the sides of the contact region.The ratio of loads to produce these two states is of the order of 350 although tangential(sliding)forces will reduce this figure significantly. Unloading from any point between these two states produces a thin layer of residual tension at the surface and a sub-surface region of residual compression parallel to the surface.The residual stresses set up during an initial pass or passes of load can inhibit plastic flow in subsequent passes and a so-called"shakedown"situation is reached where additional plastic flow is totally prevented.Maximum contact pressure for shakedown is given by Johnson(14) as 1.6 oy. (b)Surface shear loading caused by mutual sliding of the mating surfaces Pure rolling of parallel cylinders has been considered by Radzimovsky(5)whilst the effect of tangential shear loading has been studied by Deresiewicz(15),Johnson(16),Lubkin(7) Mindlin(18),Tomlinson(19)and Smith and Liu(20). (c)Thermal stresses and associated material property changes resulting from the heat set up by sliding friction.(Local temperatures can rise to some 500F above ambient). A useful summary of the work carried out in this area is given by Lipson Juvinal(21). (d)The presence of lubrication-particularly hydrodynamic lubrication-which can greatly modify the loading and resulting stress distribution The effects of hydrodynamic lubrication on the pressure distribution at contact (see Fig.10.1)and resulting stresses have been considered by a number of investigators including Meldahl(22),M'Ewen(4),Dowson,Higginson and Whitaker(23),Crook(24),Dawson(25)and
$10.1 Contact Stress, Residual Stress and Stress Concentrations 383 would lead to infinite values of compressive stress (load t “zero” area = infinity). This can only be avoided by local deflection, even yielding, of the material under the load to increase the bearing area and reduce the value of the compressive stress to some finite value. Contact stresses between curved bodies in compression are often termed “Hertzian” contact stresses after the work on the subject by Hertz(’) in Germany in 1881. This work was concerned primarily with the evaluation of the maximum compressive stresses set up at the mating surfaces for various geometries of contacting body but it formed the basis for subsequent extension of consideration by other workers of stress conditions within the whole contact zone both at the surface and beneath it. It has now been shown that the strength and load-carrying capacity of engineering components subjected to contact conditions is not completely explained by the Hertz equations by themselves, but that further consideration of the following factors is an essential additional requirement: (a) Local yielding and associated residual stresses Yield has been shown to initiate sub-surface when the contact stress approaches 1.2 ay (ay being the yield stress of the contacting materials) with so-called “uncontained plastic flow” commencing when the stress reaches 2.8 cy. Only at this point will material “escape” at the sides of the contact region. The ratio of loads to produce these two states is of the order of 350 although tangential (sliding) forces will reduce this figure significantly. Unloading from any point between these two states produces a thin layer of residual tension at the surface and a sub-surface region of residual compression parallel to the surface. The residual stresses set up during an initial pass or passes of load can inhibit plastic flow in subsequent passes and a so-called “shakedown” situation is reached where additional plastic flow is totally prevented. Maximum contact pressure for shakedown is given by Johnson(14) as 1.6 a,,. (b) Sugace shear loading caused by mutual sliding of the mating surfaces Pure rolling of parallel cylinders has been considered by Radzimo~sky(~) whilst the effect of tangential shear loading has been studied by Deresie~icz(’~), Johnson(16), Lubkid”) , Mindlin(’*), Tomlin~on(’~) and Smith and Lid2’). (c) Thermal stresses and associated material property changes resulting from the heat set up by sliding friction. (Local temperatures can rise to some 500°F above ambient). A useful summary of the work carried out in this area is given by Lipson & Juvinal(2’). (d) The presence of lubrication - particularly hydrodynamic lubrication - which can greatly modify the loading and resulting stress distribution The effects of hydrodynamic lubrication on the pressure distribution at contact (see Fig. 10.1) and resulting stresses have been considered by a number of investigators including Meldah1(22), M’Ew~~(~), Dowson, Higginson and Whitaker(23), Crook(24), Da~son(~~) and
384 Mechanics of Materials 2 §10.1 Lubricoted Dry Fig.10.1.Comparison of pressure distributions under dry and lubricated contact conditions. Scott(26).One important conclusion drawn by Dowson et al.is that at high load and not excessive speeds hydrodynamic pressure distribution can be taken to be basically Hertzian except for a high spike at the exit side. (e)The presence of residual stresses at the surface of e.g.hardened components and their distribution with depth In discussion of the effect of residually stressed layers on contact conditions,Sherratt(27) notes that whilst the magnitude of the residual stress is clearly important,the depth of the residually stressed layer is probably even more significant and the biaxiality of the residual stress pattern also has a pronounced effect.Considerable dispute exists even today about the origin of contact stress failures,particularly of surface hardened gearing,and the aspect is discussed further in $10.1.6 on gear contact stresses. Muro(28),in X-ray studies of the residual stresses present in hardened steels due to rolling contact,identified a compressive residual stress peak at a depth corresponding to the depth of the maximum shear stress-a value related directly to the applied load.He therefore concluded that residual stress measurement could form a useful load-monitoring tool in the analysis of bearing failures. Detailed consideration of these factors and even of the Hertzian stresses themselves is beyond the scope of this text.An attempt will therefore be made to summarise the essential formulae and behaviour mechanisms in order to provide an overall view of the problem without recourse to proof of the various equations which can be found in more advanced treatments such as those referred to below:- The following special cases attracted special consideration: (i)Contact of two parallel cylinders-principally because of its application to roller bear- ings and similar components.Here the Hertzian contact area tends towards a long narrow rectangle and complete solutions of the stress distribution are available from Belajef2),Foppl(3),M'Ewen(4)and Radzimovsky(5). (ii)Spur and helical gears-Buckingham(6)shows that the above case of contacting parallel cylinders can be used to fair accuracy for the contact of spur gears and whilst Walker(7) and Wellaver(8)show that helical gears are more accurately represented by contacting conical frustra,the parallel cylinder case is again fairly representative
384 Mechanics of Materials 2 $10.1 Fig. 10.1. Comparison of pressure distributions under dry and lubricated contact conditions. Scott(26). One important conclusion drawn by Dowson et al. is that at high load and not excessive speeds hydrodynamic pressure distribution can be taken to be basically Hertzian except for a high spike at the exit side. (e) The presence of residual stresses at the surface of e.g. hardened components and their distribution with depth In discussion of the effect of residually stressed layers on contact conditions, She~~att(~~) notes that whilst the magnitude of the residual stress is clearly important, the depth of the residually stressed layer is probably even more significant and the biaxiality of the residual stress pattern also has a pronounced effect. Considerable dispute exists even today about the origin of contact stress failures, particularly of surface hardened gearing, and the aspect is discussed further in $10.1.6 on gear contact stresses. Muro(28), in X-ray studies of the residual stresses present in hardened steels due to rolling contact, identified a compressive residual stress peak at a depth corresponding to the depth of the maximum shear stress - a value related directly to the applied load. He therefore concluded that residual stress measurement could form a useful load-monitoring tool in the analysis of bearing failures. Detailed consideration of these factors and even of the Hertzian stresses themselves is beyond the scope of this text. An attempt will therefore be made to summarise the essential formulae and behaviour mechanisms in order to provide an overall view of the problem without recourse to proof of the various equations which can be found in more advanced treatments such as those referred to below:- The following special cases attracted special consideration: (i) Contact of tw~o parallel cylinders - principally because of its application to roller bearings and similar components. Here the Hertzian contact area tends towards a long narrow rectangle and complete solutions of the stress distribution are available from BelajeP2), F~ppl'~), M'Ewed4) and Radzimov~ky'~). (ii) Spur and helical gears - Buckingham(6) shows that the above case of contacting parallel cylinders can be used to fair accuracy for the contact of spur gears and whilst Walker(') and Wellaver@) show that helical gears are more accurately represented by contacting conical frustra, the parallel cylinder case is again fairly representative
s10.1 Contact Stress,Residual Stress and Stress Concentrations 385 (iii)Circular contact-as arising in the case of contacting spheres or crossed cylinders.Full solutions are available by Foppl(3),Huber(9)Morton and Close(10)and Thomas and Hoersch(11). (iv)General elliptical contact.Work on this more general case has been extensive and complete solutions exist for certain selected axes,e.g.the axes of the normal load. Authors include Belajef(2),Fessler and Ollerton(12),Thomas and Heorsch(1)and Ollerton(13) Let us now consider the principal cases of contact loading:- 10.1.1.General case of contact between two curved surfaces In his study of this general contact loading case,assuming elastic and isotropic material behaviour,Hertz showed that the intensity of pressure between the contacting surfaces could be represented by the elliptical (or,rather,semi-ellipsoid)construction shown in Fig.10.2. Maximum contact pressure Contact pressure distribution p along x=axis Fig.10.2.Hertizian representation of pressure distribution between two curved bodies in contact. If the maximum pressure at the centre of contact is denoted by po then the pressure at any other point within the contact region was shown to be given by x2 x2 P =po/1- ai-bi (10.1) where a and b are the major and minor semi-axes,respectively. The total contact load is then given by the volume of the semi-ellipsoid, 2 i.e. P 3xabpo (10.2) with the maximum pressure po therefore given in terms of the applied load as 3P P0= maximum compressive stress oe (10.3) 2nab
$10.1 Contact Stress, Residual Stress and Stress Concentrations 385 (iii) Circular contact - as arising in the case of contacting spheres or crossed cylinders. Full solutions are available by F~ppl(~), Huber") Morton and Close(") and Thomas and Hoersch(' I. (iv) General elliptical contact. Work on this more general case has been extensive and complete solutions exist for certain selected axes, e.g. the axes of the normal load. Authors include Belajef"), Fessler and Ollerton('*), Thomas and Heorsch(") and oiiert~n(~~). Let us now consider the principal cases of contact loading:- 10.1.1. General case of contact between two curved surj4aces In his study of this general contact loading case, assuming elastic and isotropic material behaviour, Hertz showed that the intensity of pressure between the contacting surfaces could be represented by the elliptical (or, rather, semi-ellipsoid) construction shown in Fig. 10.2. 2, Maximum contact COntOCt pressure distribution P along x =O OXIS Y Fig. 10.2. Hertizian representation of pressure distribution between two curved bodies in contact. If the maximum pressure at the centre of contact is denoted by po then the pressure at any other point within the contact region was shown to be given by (10.1) where a and b are the major and minor semi-axes, respectively. The total contact load is then given by the volume of the semi-ellipsoid, i.e. with the maximum pressure po therefore given in terms of the applied load as 2 3 P = -nabPo (10.2) 3P 2zab po = - = maximum compressive stress a, (10.3)
386 Mechanics of Materials 2 §10.1 For any given contact load P it is necessary to determine the value of a and b before the maximum contact stress can be evaluated.These are found analytically from equations suggested by Timoshenko and Goodier(29)and adapted by Lipson and Juvinal(21). ie. 「3PA]/3 a=m and b=n 「3PA7/3 4A 4A with △= -+-1 a function of the elastic constants E and v of the contacting bodies and ++后+ A- with R and R'the maximum and minimum radii of curvature of the unloaded contact surfaces in two perpendicular planes. For flat-sided wheels R:will be the wheel radius and R will be infinite.Similarly for railway lines with head radius R2 the value of R2 will be infinite to produce the flat length of rail. 1/2 8=(信)'+（后）+（信-）（信忘）m with the angle between the planes containing curvatures 1/RI and 1/R2. Convex surfaces such as a sphere or roller are taken to be positive curvatures whilst internal surfaces of ball races are considered to be negative. m and n are also functions of the geometry of the contact surfaces and their values are shown in Table 10.1 for various values of the term a cos-(B/A). Table 10.1. 20 3035 40 4550 5560 6570 75 80 85 90 degrees N 3.7782.7312.3972.1361.9261.7541.61114861.3781.2841.2021.1281.0611.000 0.4080.4930.5300.5670.6040.6410.6780.7170.7590.8020.8460.8930.9441000 10.1.2.Special case I-Contact of parallel cylinders Consider the two parallel cylinders shown in Fig.10.3(a)subjected to a contact load P producing a rectangular contact area of width 26 and length L.The contact stress distribution is indicated in Fig.10.3(b). The elliptical pressure distribution is given by the two-dimensional version of eqn (10.1) ie. p= -茶 (10.5) The total load P is then the volume of the prism i.e. P=πbLpo (10.6)
3 86 Mechanics of Materials 2 $10.1 For any given contact load P it is necessary to determine the value of a and b before the maximum contact stress can be evaluated. These are found analytically from equations suggested by Timoshenko and G~odier'~~) and adapted by Lipson and Juvinal(2'). i.e. with 3PA and b = n [=] a function of the elastic constants E and v of the contacting bodies and 11 with R and R' the maximum and minimum radii of curvature of the unloaded contact surfaces in two perpendicular planes. For flat-sided wheels RI will be the wheel radius and R', will be infinite. Similarly for railway lines with head radius R2 the value of R; will be infinite to produce the flat length of rail. with 11/ the angle between the planes containing curvatures l/Rl and 1/R;?. internal surfaces of ball races are considered to be negative. shown in Table 10.1 for various values of the term a = cos-'(B/A). Convex surfaces such as a sphere or roller are taken to be positive curvatures whilst rn and n are also functions of the geometry of the contact surfaces and their values are Table 10.1. a 20 30 35 40 45 50 55 60 65 70 75 80 85 90 degrees m 3.778 2.731 2.397 2.136 1.926 1.754 1.611 1.486 1.378 1.284 1.202 1.128 1.061 1.OOO n 0.408 0.493 0.530 0.567 0.604 0.641 0.678 0.717 0.759 0.802 0.846 0.893 0.944 1.OOO 10.1.2. Special case I - Contact of parallel cylinders Consider the two parallel cylinders shown in Fig. 10.3(a) subjected to a contact load P producing a rectangular contact area of width 26 and length L. The contact stress distribution is indicated in Fig. 10.3(b). The elliptical pressure distribution is given by the two-dimensional version of eqn (10.1) i.e. P = Pq/l - g The total load P is then the volume of the prism i.e. P = inbLpo (10.5) (10.6)
§10.1 Contact Stress,Residual Stress and Stress Concentrations 387 2b P Moximum contoct pressure p (a)Contocting porollel cylinders (b)Pressure (stress)distribution Fig.10.3.(a)Contact of two parallel cylinders;(b)stress distribution for contacting parallel cylinders. and the maximum pressure or maximum compressive stress 2p P0=0c= xbL (10.7) The contact width can be related to the geometry of the contacting surfaces as follows:- P△ b=1.076 (10.8) 】 giving the maximum compressive stress as: P 0c=-p0=-0.5911 R R2 (10.9) LA (For a flat plate R2 is infinite,for a cylinder in a cylindrical bearing R2 is negative). Stress conditions at the surface on the load axis are then: 02=0e=一P0 0y=一p0 ox=-2vpo (along cylinder length) The maximum shear stress is: tmax=0.295p%s0.3p0 occurring at a depth beneath the surface of 0.786 b and on planes at 45 to the load axis. In cases such as gears,bearings,cams,etc.which (as will be discussed later)can be likened to the contact of parallel cylinders,this shear stress will reduce gradually to zero as the rolling load passes the point in question and rise again to its maximum value as the next
$10.1 Contact Stress, Residual Stress and Stress Concentrations (01 Contacting parollel cylinders Maximum contact pressure P, (b) Pressure (stress) distribution 387 Fig. 10.3. (a) Contact of two parallel cylinders: (b) stress distribution for contacting parallel cylinders. and the maximum pressure or maximum compressive stress (10.7) The contact width can be related to the geometry of the contacting surfaces as follows:- (10.8) giving the maximum compressive stress as: 0, = -p~ = -0.591 (10.9) LA (For a flat plate R2 is infinite, for a cylinder in a cylindrical bearing R2 is negative). Stress conditions at the surface on the load axis are then: uy = -Po ax= -2vw (along cylinder length) The maximum shear stress is: occurring at a depth beneath the surface of 0.786 b and on planes at 45" to the load axis. In cases such as gears, bearings, cams, etc. which (as will be discussed later) can be likened to the contact of parallel cylinders, this shear stress will reduce gradually to zero as the rolling load passes the point in question and rise again to its maximum value as the next
388 Mechanics of Materials 2 §10.1 load contact is made.However,this will not be the greatest reversal of shear stress since there is another shear stress on planes parallel and perpendicular to the load axes known as the "alternating"or "reversing"shear stress,at a depth of 0.5 b and offset from the load axis by 0.85 b,which has a maximum value of 0.256 po which changes from positive to negative as the load moves across contact. The maximum shear stress on 45planes thus varies between zero and 0.3 po (approx)with an alternating component of 0.15 po about a mean of 0.15 Po.The maximum alternating shear stress,however,has an alternating component of 0.256 Po about a mean of zero- see Fig.10.4.The latter is therefore considerably more significant from a fatigue viewpoint. 03p 05b below surface 020 1 b below surface At surface 01p 具=max contact pressure 02 26=contac时wMdh 03p b 46-36-26062b36 46 Distancey from load axis Fig.10.4.Maximum alternating stress variation beneath contact surfaces. N.B.:The above formulae assume the length of the cylinders to be very large in comparison with their radii.For short cylinders and/or cylinder/plate contacts with widths less than six times the contact area (or plate thickness less than six times the depth of the maximum shear stress)actual stresses can be significantly greater than those estimated by the given equations. 10.1.3.Combined normal and tangential loading In normal contact conditions between contacting cylinders,gears,cams,etc.friction will be present reacting the sliding (or tendency to slide)of the mating surfaces.This will affect the stresses which are set up and it is usual in such cases to take the usual relationship between normal and tangential forces in the presence of friction viZ. F=uR or g=upo where g is the tangential pressure distribution,assumed to be of the same form as that of the normal pressure.Smith and Liu20)have shown that with such an assumption: (a)A shear stress now exists on the surface at the contact point introducing principal stresses which are different from ox,oy and o:of the normal loading case. (b)The maximum shear stress may exist either at the surface or beneath it depending on whether u is greater than or less than 1/9 respectively
388 Mechanics oj Materials 2 $10.1 load contact is made. However, this will not be the greatest reversal of shear stress since there is another shear stress on planes parallel and perpendicular to the load axes known as the “alrernating” or “reversing” shear stress, at a depth of 0.5 b and offset from the load axis by 0.85 b, which has a maximum value of 0.256 po which changes from positive to negative as the load moves across contact. The maximum shear stress on 45” planes thus varies between zero and 0.3 po (approx) with an alternating component of 0.15 po about a mean of 0.15 PO. The maximum alternating shear stress, however, has an alternating component of 0.256 po about a mean of zero - see Fig. 10.4. The latter is therefore considerably more significant from a fatigue viewpoint. I Distance y from load axis Fig. 10.4. Maximum alternating stress variation beneath contact surfaces. N.B.: The above formulae assume the length of the cylinders to be very large in comparison with their radii. For short cylinders and/or cylindedplate contacts with widths less than six times the contact area (or plate thickness less than six times the depth of the maximum shear stress) actual stresses can be significantly greater than those estimated by the given equations. 10.1.3. Combined normal and tangential loading In normal contact conditions between contacting cylinders, gears, cams, etc. friction will be present reacting the sliding (or tendency to slide) of the mating surfaces. This will affect the stresses which are set up and it is usual in such cases to take the usual relationship between normal and tangential forces in the presence of friction viz. F=pR or q=ppo where q is the tangential pressure distribution, assumed to be of the same form as that of the normal pressure. Smith and Lid2’) have shown that with such an assumption: (a) A shear stress now exists on the surface at the contact point introducing principal stresses (b) The maximum shear stress may exist either at the surface or beneath it depending on which are different from a,, uy and uz of the normal loading case. whether p is greater than or less than 1/9 respectively
§10.1 Contact Stress,Residual Stress and Stress Concentrations 389 (c)The stress range in the y direction is increased by almost 90%on the normal loading value and there is also a reversal of sign.A useful summary of stress distributions in graphical form is given by Lipson and Juvinal(21). 10.1.4.Special case 2-Contacting spheres For contacting spheres,eqns.(10.9)and (10.8)become Maximum compressive stress (normal to surface) P「1. 172 0c=-p0=-0.621 △2R1 +R2】 (10.10) with a maximum value of Oc =-1.5P/na2 (10.11) Contact dimensions (circular) P△ a=b=0.88 (10.12) 「1 11 As for the cylinder,if contact occurs between one sphere and a flat surface then R2 is infinite, and if the sphere contacts inside a spherical seating then R2 is negative. The other two principal stresses in the surface plane are given by: (1+2) 0x=0y= 2P (10.13) For steels with Poisson's ratio v=0.3 the maximum shear stress is then: tmax±0.31po (10.14) at a depth of half the radius of the contact surface. The maximum tensile stress set up within the contact zone occurs at the edge of the contact zone in a radial direction with a value of: (1-2) Utmax 3P0 (10.15) The circumferential stress at the same point is equal in value,but compressive,whilst the stress normal to the surface is effectively zero since contact has ended.With equal and opposite principal stresses in the plane of the surface,therefore,the material is effectively in a state of pure shear. The maximum octahedral shear stress which is also an important value in consideration of elastic failure,occurs at approximately the same depth below the surface as the maximum shear stress.Its value may be obtained from eqn(8.24)by substituting the appropriate values of ox,oy and o:found from Fig.10.5 which shows their variation with depth beneath the surface. The relative displacement,e,of the centres of the two spheres is given by: e=nP(G+)'(民+》 (10.16)
510.1 Contact Stress, Residual Stress and Stress Concentrations 389 The stress range in the y direction is increased by almost 90% on the normal loading value and there is also a reversal of sign. A useful summary of stress distributions in graphical form is given by Lipson and Juvinal(21). 10.1.4. Special case 2 - Contacting spheres For contacting spheres, eqns. (10.9) and (10.8) become Maximum compressive stress (normal to surface) with a maximum value of a, = - 1.5~lna~ Contact dimensions (circular) a = b = O.8tl3 jE (10.10) (10.11) (10.12) As for the cylinder, if contact occurs between one sphere and a flat surface then R2 is infinite, and if the sphere contacts inside a spherical seating then R2 is negative. The other two principal stresses in the surface plane are given by: (1 + 2u) 2 a, = by = -___ Po (10.13) For steels with Poisson’s ratio v = 0.3 the maximum shear stress is then: tmax 10.31~0 (10.14) The maximum tensile stress set up within the contact zone occurs at the edge of the at a depth of half the radius of the contact surface. contact zone in a radial direction with a value of (1 - 2u) 3 utmax = ___ Po (10.15) The circumferential stress at the same point is equal in value, but compressive, whilst the stress normal to the surface is effectively zero since contact has ended. With equal and opposite principal stresses in the plane of the surface, therefore, the material is effectively in a state of pure shear. The maximum octahedral shear stress which is also an important value in consideration of elastic failure, occurs at approximately the same depth below the surface as the maximum shear stress. Its value may be obtained from eqn (8.24) by substituting the appropriate values of a,, a,, and a, found from Fig. 10.5 which shows their variation with depth beneath the surface. The relative displacement, e, of the centres of the two spheres is given by: e = 0.77 J P2 (i1 - + - iJ2(d+&) (10.16)
390 Mechanics of Materials 2 §10.1 Stress 04 0 04P 2 b.contact semi- Oxis Tu=Ty P。mox.contact pressure Fig.10.5.Variation of stresses beneath the surface of contacting spheres. For a sphere contacting a flat surface of the same material R2=oo and E=E2=E. Substitution in egns.(10.10)and (10.16)then yields maximum compressive stress 0c=-0.62 PE2 4R子 (10.17) and relative displacement of centres P2 e=1.54 2E2R1 (10.18) For a sphere on a spherical seat of the same material 72 oe=-0.62PE2 R2-R (10.19) RR2 P2 R2-R1 with e=1.54 2E2 (10.20) R1R2」 For other,more general,loading cases the reader is referred to a list of formulae presented by Roark and Young(33). 10.1.5.Design considerations It should be evident from the preceding sections that the maximum Hertzian compressive stress is not,in itself,a valid criteria of failure for contacting members although it can be used as a valid design guide provided that more critical stress states which have a more direct influence on failure can be related directly to it.It has been shown,for example,that alternating shear stresses exist beneath the surface which are probably critical to fatigue life
390 Mechanics of Materials 2 Stress $10.1 Fig. 10.5. Variation of stresses beneath the surface of contacting spheres. For a sphere contacting a flat surj-ace of the same material R2 = 00 and El = E2 = E. Substitution in eqns. (10.10) and (10.16) then yields maximum compressive stress U, = -0.62’ - /:; (10.17) and relative displacement of centres For a sphere on a spherical seat of the same material (10.18) ( 10.19) with (10.20) For other, more general, loading cases the reader is referred to a list of formuIae presented by Roark and Young(33). 10.1.5. Design considerations It should be evident from the preceding sections that the maximum Hertzian compressive stress is not, in itself, a valid criteria of failure for contacting members although it can be used as a valid design guide provided that more critical stress states which have a more direct influence on failure can be related directly to it. It has been shown, for example, that alternating shear stresses exist beneath the surface which are probably critical to fatigue life
