§10.1 Contact Stress,Residual Stress and Stress Concentrations 391 but these can be expressed as a simple proportion of the Hertzian pressure po so that po can be used as a simple index of contact load severity. The contact situation is complicated under real service loading conditions by the presence of e.g.residual stresses in hardened surfaces,local yielding and associated additional residual stresses,friction forces and lubrication,thermal stresses and dynamic (including shock)load effects. The failure of brittle materials under contact conditions correlates more closely with the maximum tensile stress at the surface rather than sub-surface shear stresses,whilst for static or very slow rolling operations failure normally arises as a result of excessive plastic flow producing indentation ("brinelling")of the surface.In both cases,however,the Hertzian pressure remains a valuable design guide or reference. By far the greatest number of failures of contacting components remains the surface or sub-surface fatigue initiated type variably known as "pitting","spalling","onion-peel spalling"or "flaking".The principal service areas in which this type of failure occurs are gears and bearings. 10.1.6.Contact loading of gear teeth Figure 10.6 shows the stress conditions which prevail in the region of a typical gear tooth contact.Immediately at the contact point,or centre of contact,there is the usual position of maximum compressive stress (po).Directly beneath this,and at a depth of approximately one-third of the contact width,is the maximum shear stress tmax acting on planes at 45to the load axis.Between these two positions lies the maximum alternating or reversed shear stress talt acting on planes perpendicular and parallel to the surface.Whilst talt is numerically smaller than Tmax it alternates between positive and negative values as the tooth proceeds Contoct width Maximum surface compression -Sub-surface compression Sub-surfoce tension Contact moving in this direction Sub-sur foce compression Sub-surfoce tension Moximum sub-sur foce shear -Maximum alternoting or reversed shear Direction of rotation Fig.10.6.Stress conditions in the region of gear tooth contact
g10.1 Contact Stress, Residual Stress and Stress Concentrations 39 1 but these can be expressed as a simple proportion of the Hertzian pressure po so that PO can be used as a simple index of contact load severity. The contact situation is complicated under real service loading conditions by the presence of e.g. residual stresses in hardened surfaces, local yielding and associated additional residual stresses, friction forces and lubrication, thermal stresses and dynamic (including shock) load effects. The failure of brittle materials under contact conditions correlates more closely with the maximum tensile stress at the surface rather than sub-surface shear stresses, whilst for static or very slow rolling operations failure normally arises as a result of excessive plastic flow producing indentation ( “brinelling”) of the surface. In both cases, however, the Hertzian pressure remains a valuable design guide or reference. By far the greatest number of failures of contacting components remains the surface or sub-surface fatigue initiated type variably known as “pitting”, “spalling” , “onion-peel spalling” or “flaking”. The principal service areas in which this type of failure occurs are gears and bearings. 10.1.6. Contact loading of gear teeth Figure 10.6 shows the stress conditions which prevail in the region of a typical gear tooth contact. Immediately at the contact point, or centre of contact, there is the usual position of maximum compressive stress (PO). Directly beneath this, and at a depth of approximately one-third of the contact width, is the maximum shear stress rmax acting on planes at 45” to the load axis. Between these two positions lies the maximum alternating or reversed shear stress ?,It acting on planes perpendicular and parallel to the surface. Whilst sal, is numerically smaller than rmax it alternates between positive and negative values as the tooth proceeds Maximum surface compression Sub-surfme cMpresSlm Contoc t width Sub-surfoce tension t moving in this direction Sub-wrfoce tension Maximum alternating or reversed shear Sub-sur face compression Maximum sub-surfoce Sheor Direction of rotation Fig. 10.6. Stress conditions in the region of gear tooth contact
392 Mechanics of Materials 2 §10.1 through mesh giving a stress range greater than that of tmax which ranges between a single value and zero.It is argued by many that,for this reason,ralt is probably more significant to fatigue life than tmax-particularly if its depth relates closely to that of peak residual stresses or case-core junctions of hardened gears. As the gears rotate there is a combination of rolling and sliding motions,the latter causing additional surface stresses not shown in Fig.8.6.Ahead of the contact area there is a narrow band of compression and behind the contact area a narrow band of tension.A single point on the surface of a gear tooth therefore passes through a complex variety of stress conditions as it goes through its meshing cycle.Both the surface and alternating stress change sign and other sub-surface stresses change from zero to their maximum value.Add to these fatigue situations the effects of residual stress,lubrication,thermal stresses and dynamic loading and it is not surprising that gears may fail in one of a number of ways either at the surface or sub-surface. The majority of gear tooth failures are surface failures due to "pitting","spalling", "flaking","wear",etc.the three former modes referring to the fracture and shedding of pieces of various size from the surface.Considerable speculation and diverse views exist even among leading workers as to the true point of origin of some of these failures and considerable evidence has been produced of,apparently,both surface and sub-surface crack initiation.The logical conclusion would therefore seem to be that both types of initiation are possible depending on precisely the type of loading and contact conditions. A strong body of opinion supports the suggestion of Johnson(14.16)and Almen(30)who attribute contact stress failures to local plastic flow at inclusions or flaws in the material, particularly in situations where a known overload has occurred at some time prior to failure. The overload is sufficient to produce the initial plastic flow and successive cycles then extend the region of plasticity and crack propagation commences.Dawson(25)and Akaoka(31)found evidence of sub-surface cracks running parallel to the surface,some breaking through to the surface,others completely unconnected with it.These were attributed to the fatigue action of the maximum alternating (reversed)shear stress.Undoubtedly,from the evidence presented by other authors,cracks can also initiate at the surface probably producing a "pitting"type of failure,i.e.smaller depth of damage.These cracks are suggested to initiate at positions of maximum tensile stress in the contact surface and subsequent propagation is then influenced by the presence (or otherwise)of lubricant. In the case of helical gears,three-dimensional photoelastic tests undertaken by the author(32)indicate that maximum sub-surface stresses are considerably greater than those predicted by standard design procedures based on Hertzian contact and uniform loading along the contact line.Considerable non-uniformity of load was demonstrated which, together with dynamic effects,can cause maximum loads and stresses many times above the predicted nominal values.The tests showed the considerable benefit to be gained on the load distribution and resulting maximum stress values by the use of tip and end relief of the helical gear tooth profile. 10.1.7.Contact stresses in spur and helical gearing Whilst the radius of an involute gear tooth will change slightly across the width of contact with a mating tooth it is normal to ignore this and take the contact of spur gear teeth as equivalent to the contact of parallel cylinders with the same radius of curvature at the point of contact.The Hertzian eqns.(10.8)and (10.9)can thus be applied to spur gears and
392 Mechanics of Materials 2 $10.1 through mesh giving a stress range greater than that of tmax which ranges between a single value and zero. It is argued by many that, for this reason, talt is probably more significant to fatigue life than tmax - particularly if its depth relates closely to that of peak residual stresses or case-core junctions of hardened gears. As the gears rotate there is a combination of rolling and sliding motions, the latter causing additional surface stresses not shown in Fig. 8.6. Ahead of the contact area there is a narrow band of compression and behind the contact area a narrow band of tension. A single point on the surface of a gear tooth therefore passes through a complex variety of stress conditions as it goes through its meshing cycle. Both the surface and alternating stress change sign and other sub-surface stresses change from zero to their maximum value. Add to these fatigue situations the effects of residual stress, lubrication, thermal stresses and dynamic loading and it is not surprising that gears may fail in one of a number of ways either at the surface or sub-surface. The majority of gear tooth failures are surface failures due to “pitting”, “spalling”, “flaking”, “wear”, etc. the three former modes referring to the fracture and shedding of pieces of various size from the surface. Considerable speculation and diverse views exist even among leading workers as to the true point of origin of some of these failures and considerable evidence has been produced of, apparently, both surface and sub-surface crack initiation. The logical conclusion would therefore seem to be that both types of initiation are possible depending on precisely the type of loading and contact conditions. A strong body of opinion supports the suggestion of J~hnson(’~*’~) and Almen(30) who attribute contact stress failures to local plastic flow at inclusions or flaws in the material, particularly in situations where a known overload has occurred at some time prior to failure. The overload is sufficient to produce the initial plastic flow and successive cycles then extend the region of plasticity and crack propagation commences. Da~son(*~) and Aka~ka(~’) found evidence of sub-surface cracks running parallel to the surface, some breaking through to the surface, others completely unconnected with it. These were attributed to the fatigue action of the maximum alternating (reversed) shear stress. Undoubtedly, from the evidence presented by other authors, cracks can also initiate at the surface probably producing a “pitting” type of failure, i.e. smaller depth of damage. These cracks are suggested to initiate at positions of maximum tensile stress in the contact surface and subsequent propagation is then influenced by the presence (or otherwise) of lubricant. In the case of helical gears, three-dimensional photoelastic tests undertaken by the author(32) indicate that maximum sub-surface stresses are considerably greater than those predicted by standard design procedures based on Hertzian contact and uniform loading along the contact line. Considerable non-uniformity of load was demonstrated which, together with dynamic effects, can cause maximum loads and stresses many times above the predicted nominal values. The tests showed the considerable benefit to be gained on the load distribution and resulting maximum stress values by the use of tip and end relief of the helical gear tooth profile. 10.1.7. Contact stresses in spur and helical gearing Whilst the radius of an involute gear tooth will change slightly across the width of contact with a mating tooth it is normal to ignore this and take the contact of spur gear teeth as equivalent to the contact of parallel cylinders with the same radius of curvature at the point of contact. The Hertzian eqns. (10.8) and (10.9) can thus be applied to spur gears and
§10.1 Contact Stress,Residual Stress and Stress Concentrations 393 for typical steel elastic constant values of v=0.3 and E=206.8 GN/m2,the maximum contact stress becomes Oe =-po =-0.475K MN/m2 (10.21) where K-7 d with W=tangential driving load =pinion torque:pinion pitch radius Fw face width d pinion pitch diameter m ratio of gear teeth to pinion teeth;the pinion taken to be the smaller of the two mating teeth. For helical gears,the maximum contact stress is given by 0e=-P0=- (10.22) where K is the same factor as for spur gears mp is the profile contact ratio C is a constant the values of mp and C being found in Table 10.2,for various helix angles and pressure angles. Table 10.2.Typical values of C and mp for helical gears. Pressure Spur 15°Helix 30°Helix 45°Helix angle C mp C mp C mp C mp 14号 0.546 2.10 0.528 2.01 0.473 1.71 0.386 126 173 0.502 1.88 0.485 1.79 0.435 1.53 0.355 1.13 20° 0.474 1.73 0.458 1.65 0.410 1.41 0.335 1.05 25° 0.434 1.52 0.420 1.45 0.376 1.25 0.307 0.949 10.1.8.Bearing failures Considerable care is necessary in the design of bearings when selecting appropriate ball and bearing race radii.If the radii are too similar the area of contact is large and excessive wear and thermal stress(from frictional heating)results.If the radii are too dissimilar then the contact area is very small,local compressive stresses become very high and the load capacity of the bearing is reduced.As a compromise between these extremes the radius of the race is normally taken to be between 1.03 and 1.08 times the ball radius. Fatigue life tests and service history then indicate that the life of ball bearings varies approximately as the cube of the applied load whereas,for roller bearings,a 10/3 power relationship is more appropriate.These relationships can only be used as a rough "rule of
$10.1 Contact Stress, Residual Stress and Stress Concentrations 393 Pressure angle for typical steel elastic constant values of u = 0.3 and E = 206.8 GN/m2, the maximum contact stress becomes uc = -po = -0.475& MN/m* (10.21) Spur 15" Helix 30" Helix 45" Helix C MI, C mIJ C mi7 C mIJ where 14:' 17i0 20" 25" with W = tangential driving load = pinion torque + pinion pitch radius Fw = face width d = pinion pitch diameter m = ratio of gear teeth to pinion teeth; the pinion taken to be the smaller of the two mating teeth. For helical gears, the maximum contact stress is given by 0.546 2.10 0.528 2.01 0.473 1.71 0.386 1.26 0.502 1.88 0.485 1.79 0.435 1.53 0.355 1.13 0.474 I .73 0.458 I .65 0.410 1.41 0.335 1.05 0.434 1.52 0.420 1.45 0.376 1.25 0.307 0.949 (10.22) where K is the same factor as for spur gears mp is the profile contact ratio C is a constant the values of mp and C being found in Table 10.2, for various helix angles and pressure angles. Table 10.2. Typical values of C and mp for helical gears. I I 1 1 I L I I I I 10.1.8. Bearing failures Considerable care is necessary in the design of bearings when selecting appropriate ball and bearing race radii. If the radii are too similar the area of contact is large and excessive wear and thermal stress (from frictional heating) results. If the radii are too dissimilar then the contact area is very small, local compressive stresses become very high and the load capacity of the bearing is reduced. As a compromise between these extremes the radius of the race is normally taken to be between 1.03 and 1.08 times the ball radius. Fatigue life tests and service history then indicate that the life of ball bearings varies approximately as the cube of the applied load whereas, for roller bearings, a 10/3 power relationship is more appropriate. These relationships can only be used as a rough "rule of
394 Mechanics of Materials 2 §10.2 thumb",however,since commercially produced bearings,even under nominally similar and controlled production conditions,are notorious for the wide scatter of fatigue life results. As noted previously,the majority of bearing failures are by spalling of the surface and most of the comments given in $10.1.6 relating to gear failures are equally relevant to bearing failures. 10.2.Residual Stresses Introduction It is probably true to say that all engineering components contain stresses (of variable magnitude and sign)before being subjected to service loading conditions owing to the history of the material prior to such service.These stresses,produced as a result of mechanical working of the material,heat treatment,chemical treatment,joining procedure,etc.,are termed residual stresses and they can have a very significant effect on the fatigue life of components.These residual stresses are "locked into"the component in the absence of external loading and represent a datum stress over which the service load stresses are subsequently superimposed.If,by fortune or design,the residual stresses are of opposite sign to the service stresses then part of the service load goes to reduce the residual stress to zero before the combined stress can again rise towards any likely failure value;such residual stresses are thus extremely beneficial to the strength of the component and significantly higher fatigue strengths can result.If,however,the residual stresses are of the same sign as the applied stress,e.g.both tensile,then a smaller service load is required to produce failure than would have been the case for a component with a zero stress level initially;the strength and fatigue life in this case is thus reduced.Thus,both the magnitude and sign of residual stresses are important to fatigue life considerations,and methods for determining these quantities are introduced below. It should be noted that whilst preceding chapters have been concerned with situations where it has been assumed that stresses are zero at zero load this is not often the case in practice,and great care must be exercised to either fully evaluate the levels of residual stress present and establish their effect on the strength of the design,or steps must be taken to reduce them to a minimum. Bearing in mind that most loading applications in engineering practice involve fatigue to a greater or less degree it is relevant to note that surface residual stresses are the most critical as far as fatigue life is concerned since,almost invariably,fatigue cracks form at the surface.The work of $11.1.3 indicates that whilst tensile mean stresses promote fatigue crack initiation and propagation,compressive mean stresses are beneficial in that they impede fatigue failure.Compressive residual stresses are thus generally to be preferred (and there is not always a choice of course)if fatigue lives of components are to be enhanced.Indeed, compressive stresses are often deliberately introduced into the surface of components,e.g. by chemical methods which will be introduced below,in order to increase fatigue lives. There are situations,however,where compressive residual stress can be most undesirable; these include potential buckling situations where compressive surface stresses could lead to premature buckling failure,and operating conditions where the service loading stresses are also compressive.In the latter case the combined service and residual stresses may reach a sufficiently high value to exceed yield in compression and produce local plasticity on the first cycle of loading.On unloading,tensile residual stress "pockets"will be formed and
394 Mechanics of Materials 2 $10.2 thumb”, however, since commercially produced bearings, even under nominally similar and controlled production conditions, are notorious for the wide scatter of fatigue life results. As noted previously, the majority of bearing failures are by spalling of the surface and most of the comments given in $ 10.1.6 relating to gear failures are equally relevant to bearing failures. 10.2. Residual Stresses introduction It is probably true to say that all engineering components contain stresses (of variable magnitude and sign) before being subjected to service loading conditions owing to the history of the material prior to such service. These stresses, produced as a result of mechanical working of the materia!, heat treatment, chemical treatment, joining procedure, etc., are termed residual stresses and they can have a very significant effect on the fatigue life of components. These residual stresses are “locked into” the component in the absence of external loading and represent a datum stress over which the service load stresses are subsequently superimposed. If, by fortune or design, the residual stresses are of opposite sign to the service stresses then part of the service load goes to reduce the residual stress to zero before the combined stress can again rise towards any likely failure value; such residual stresses are thus extremely beneficial to the strength of the component and significantly higher fatigue strengths can result. If, however, the residual stresses are of the same sign as the applied stress, e.g. both tensile, then a smaller service load is required to produce failure than would have been the case for a component with a zero stress level initially; the strength and fatigue life in this case is thus reduced. Thus, both the magnitude and sign of residual stresses are important to fatigue life considerations, and methods for determining these quantities are introduced below. It should be noted that whilst preceding chapters have been concerned with situations where it has been assumed that stresses are zero at zero load this is not often the case in practice, and great care must be exercised to either fully evaluate the levels of residual stress present and establish their effect on the strength of the design, or steps must be taken to reduce them to a minimum. Bearing in mind that most loading applications in engineering practice involve fatigue to a greater or less degree it is relevant to note that surface residual stresses are the most critical as far as fatigue life is concerned since, almost invariably, fatigue cracks form at the surface. The work of $1 I .1.3 indicates that whilst tensile mean stresses promote fatigue crack initiation and propagation, compressive mean stresses are beneficial in that they impede fatigue failure. Compressive residual stresses are thus generally to be preferred (and there is not always a choice of course) if fatigue lives of components are to be enhanced. Indeed, compressive stresses are often deliberately introduced into the surface of components, e.g. by chemical methods which will be introduced below, in order to increase fatigue lives. There are situations, however, where compressive residual stress can be most undesirable; these include potential buckling situations where compressive surface stresses could lead to premature buckling failure, and operating conditions where the service loading stresses are also compressive. In the latter case the combined service and residual stresses may reach a sufficiently high value to exceed yield in compression and produce local plasticity on the first cycle of loading. On unloading, tensile residual stress “pockets” will be formed and
§10.2 Contact Stress,Residual Stress and Stress Concentrations 395 these can act as local stress concentrations and potential fatigue crack initiation positions. Such a situation arises in high-temperature applications such as steam turbines and nuclear plant,and in contact load applications. Whilst it has been indicated above that tensile residual stresses are generally deleterious to fatigue life there are again exceptions to this "rule",and very significant ones at that! It is now quite common to deliberately overload structures and components during proof testing to produce plastic flow at discontinuities and other stress concentrations to reduce their stress concentration effect on subsequent loading cycles.Other important techniques which involve the deliberate overloading of components in order to produce residual stress distribution favourable to subsequent loading cycles include"autofrettage"of thick cylinders (see $3.20(a))."overspeeding"of rotating discs (see $3.20(b))and pre-stressing of springs (sees3.8). Whilst engineers have been aware of residual stresses for many years it is only recently that substantial efforts have been made to investigate their magnitudes and distributions with depth in components and hence their influence on performance and service life.This is probably due to the conservatism of old design procedures which generally incorporated sufficiently large safety factors to mask the effects of residual stresses on component integrity. However,with current drives for economy of manufacture coupled with enhanced product safety and reliability,design procedures have become far more stringent and residual stress effects can no longer be ignored.Principally,the designer needs to consider the effect of residual stress on structural or component failure but there is also need for detailed consideration of distortion and stability factors which are also closely related to residual stress levels. 10.2.1.Reasons for residual stresses Residual stresses generally arise when conditions in the outer layer of a material differ from those internally.This can arise by one of three principal mechanisms:(a)mechanical processes,(b)chemical treatment,(c)heat treatment,although other mechanism are also discussed in the subsequent text. (a)Mechanical processes The most significant mechanical processes which induce surface residual stresses are those which involve plastic yielding and hence"cold-working"of the material such as rolling,shot- peening and forging.Practically all other standard machining procedures such as grinding. turning,polishing,etc.,also involve local yielding (to a lesser extent perhaps)and also induce residual stresses.Reference should also be made to $3.9 and $3.10 which indicate how residual stresses can be introduced due to bending or torsion beyond the elastic limit. Cold working Shot peening is a very popular method for the introduction of favourable compressive residual stresses in the surface of components in order to increase their fatigue life.It is a process whereby small balls of iron or steel shot are bombarded at the component surface at high velocity from a rotating nozzle or wheel.It is applicable virtually to all metals and all
$10.2 Contact Stress, Residual Stress and Stress Concentrations 395 these can act as local stress concentrations and potential fatigue crack initiation positions. Such a situation arises in high-temperature applications such as steam turbines and nuclear plant, and in contact load applications. Whilst it has been indicated above that tensile residual stresses are generally deleterious to fatigue life there are again exceptions to this “rule”, and very significant ones at that! It is now quite common to deliberately overload structures and components during proof testing to produce plastic flow at discontinuities and other stress concentrations to reduce their stress concentration effect on subsequent loading cycles. Other important techniques which involve the deliberate overloading of components in order to produce residual stress distribution favourable to subsequent loading cycles include “autofrettage” of thick cylinders (see $3 .%(a)), “overspeeding” of rotating discs (see 53.20(b)) and pre-stressing of springs (see $3.8). Whilst engineers have been aware of residual stresses for many years it is only recently that substantial efforts have been made to investigate their magnitudes and distributions with depth in components and hence their influence on performance and service life. This is probably due to the conservatism of old design procedures which generally incorporated sufficiently large safety factors to mask the effects of residual stresses on component integrity. However, with current drives for economy of manufacture coupled with enhanced product safety and reliability, design procedures have become far more stringent and residual stress effects can no longer be ignored. Principally, the designer needs to consider the effect of residual stress on structural or component failure but there is also need for detailed consideration of distortion and stability factors which are also closely related to residual stress levels. 10.2.1. Reasons for residual stresses Residual stresses generally arise when conditions in the outer layer of a material differ from those internally. This can arise by one of three principal mechanisms: (a) mechanical processes, (b) chemical treatment, (c) heat treatment, although other mechanism are also discussed in the subsequent text. (a) Mechanical processes The most significant mechanical processes which induce surface residual stresses are those which involve plastic yielding and hence “cold-working” of the material such as rolling, shotpeening and forging. Practically a11 other standard machining procedures such as grinding, turning, polishing, etc., also involve local yielding (to a lesser extent perhaps) and also induce residual stresses. Reference should also be made to 53.9 and $3.10 which indicate how residual stresses can be introduced due to bending or torsion beyond the elastic limit. Cold working Shot peening is a very popular method for the introduction of favourable compressive residual stresses in the surface of components in order to increase their fatigue life. It is a process whereby small balls of iron or steel shot are bombarded at the component surface at high velocity from a rotating nozzle or wheel. It is applicable virtually to all metals and all