448 Mechanics of Materials 2 §11.1 abscissa,the amplitude being varied for each new specimen until sufficient data have been obtained.This results in the production of the well-known S/N curve-Fig.11.4. Cycles to faiture (N, Fig.114.Typical S/N curve fatigue life curve. In using the S/N curve for design purposes it may be advantageous to express the rela- tionship between oa and N,the number of cycles to failure.Various empirical relationships have been proposed but,provided the stress applied does not produce plastic deformation, the following relationship is most often used: o Nf=K (11.5) Where a is a constant which varies from 8 to 15 and K is a second constant depending on the material-see Example 11.1. From the S/N curve the "fatigue limit"or "endurance limit"may be ascertained.The "fatigue limit"is the stress condition below which a material may endure an infinite number of cycles prior to failure.Ferrous metal specimens often produce S/N curves which exhibit fatigue limits as indicated in Fig.11.5(a).The "fatigue strength"or"endurance limit",is the stress condition under which a specimen would have a fatigue life of N cycles as shown in Fatigue limit (op) Endurance limit〔aul IN Cycles to faiture Cycles to foilure Fig.11.5.S/N curve showing (a)fatigue limit.(b)endurance limit
448 Mechanics of Materials 2 $11.1 abscissa, the amplitude being varied for each new specimen until sufficient data have been obtained. This results in the production of the well-known SIN curve - Fig. 11.4. Cycles TO failure (N,) Fig. 1 I .4. Typical S/N curve fatigue life curve In using the S/N curve for design purposes it may be advantageous to express the relationship between a, and Nf , the number of cycles to failure. Various empirical relationships have been proposed but, provided the stress applied does not produce plastic deformation, the following relationship is most often used: a;Nf = K (11.5) Where u is a constant which varies from 8 to 15 and K is a second constant depending on the material - see Example 1 1.1. From the S/N curve the “fatigue limit” or “endurance limit” may be ascertained. The “jiztigue limit” is the stress condition below which a material may endure an infinite number of cycles prior to failure. Ferrous metal specimens often produce S/N curves which exhibit fatigue limits as indicated in Fig. 11.5(a). The ‘yarigue srrength” or “endurance limit”, is the stress condition under which a specimen would have a fatigue life of N cycles as shown in I I IN - Cycles Cycles to failure to failure Fig. I 1.5. SIN curve showing (a) fatigue limit. (b) endurance limit
§11.1 Fatigue,Creep and Fracture 449 Fig.10.5(b).Non-ferrous metal specimens show this type of curve and hence components made from aluminium,copper and nickel,etc.,must always be designed for a finite life. Another important fact to note is that the results of laboratory experiments utilising plain, polished,test pieces cannot be applied directly to structures and components without modifi- cation of the intrinsic values obtained.Allowance will have to be made for many differences between the component in its working environment and in the laboratory test such as the surface finish,size,type of loading and effect of stress concentrations.These factors will reduce the intrinsic (i.e.plain specimen)fatigue strength value thus, 听=cccl (11.6) where ow is the "modified fatigue strength"or"modified fatigue limit",oN is the intrinsic value,Kf is the fatigue strength reduction factor (see 11.1.4)and Ca,C and Ce are factors allowing for size,surface finish,type of loading,etc. The types of fatigue loading in common usage include direct stress,where the material is repeatedly loaded in its axial direction;plane bending,where the material is bent about its neutral plane;rotating bending,where the specimen is being rotated and at the same time subjected to a bending moment;torsion,where the specimen is subjected to conditions which produce reversed or fluctuating torsional stresses and,finally,combined stress conditions, where two or more of the previous types of loading are operating simultaneously.It is therefore important that the method of stressing and type of machine used to carry out the fatigue test should always be quoted. Within a fairly wide range of approximately 100 cycles/min to 6000 cycles/min,the effect of speed of testing (i.e.frequency of load cycling)on the fatigue strength of metals is small but,nevertheless,frequency may be important,particularly in polymers and other materials which show a large hysteresis loss.Test details should,therefore,always include the frequency of the stress cycle,this being chosen so as not to affect the result obtained (depending upon the material under test)the form of test piece and the type of machine used.Further details regarding fatigue testing procedure are given in BS3518:Parts 1 to 5. Most fatigue tests are carried out at room temperature but often tests are also carried out at elevated or sub-zero temperatures depending upon the expected environmental operating conditions.At low temperatures the fatigue strength of metals show no deterioration and may even show a slight improvement,however,with increase in temperature,the fatigue strength decreases as creep effects are added to those of fatigue and this is revealed by a more pronounced effect of frequency of cycling and of mean stress since creep is both stress- and time-dependent. When carrying out elevated temperature tests in air,oxidation of the sample may take place producing a condition similar to corrosion fatigue.Under the action of the cyclic stress, protective oxide films are cracked allowing further and more severe attack by the corrosive media.Thus fatigue and corrosion together ensure continuous propagation of cracks,and materials which show a definite fatigue limit at room temperature will not do so at elevated temperatures or at ambient temperatures under corrosive conditions-see Fig.11.6. 11.1.2.PiSiN curves The fatigue life of a component as determined at a particular stress level is a very variable quantity so that seemingly identical specimens may give widely differing results.This scatter
$11.1 Fatigue, Creep and Fracture 449 Fig. 10.5(b). Non-ferrous metal specimens show this type of curve and hence components made from aluminium, copper and nickel, etc., must always be designed for a finite life. Another important fact to note is that the results of laboratory experiments utilising plain, polished, test pieces cannot be applied directly to structures and components without modification of the intrinsic values obtained. Allowance will have to be made for many differences between the component in its working environment and in the laboratory test such as the surface finish, size, type of loading and effect of stress concentrations. These factors will reduce the intrinsic (i.e. plain specimen) fatigue strength value thus, (11.6) where oh is the “modified fatigue strength” or “modified fatigue limit”, ON is the intrinsic value, Kf is the fatigue strength reduction factor (see $ 11.1.4) and C, Cb and C, are factors allowing for size, surface finish, type of loading, etc. The types of fatigue loading in common usage include direct stress, where the material is repeatedly loaded in its axial direction; plane bending, where the material is bent about its neutral plane; rotating bending, where the specimen is being rotated and at the same time subjected to a bending moment; torsion, where the specimen is subjected to conditions which produce reversed or fluctuating torsional stresses and, finally, combined stress conditions, where two or more of the previous types of loading are operating simultaneously. It is therefore important that the method of stressing and type of machine used to carry out the fatigue test should always be quoted. Within a fairly wide range of approximately 100 cycles/min to 6000 cycledmin, the effect of speed of testing (i.e. frequency of load cycling) on the fatigue strength of metals is small but, nevertheless, frequency may be important, particularly in polymers and other materials which show a large hysteresis loss. Test details should, therefore, always include the frequency of the stress cycle, this being chosen so as not to affect the result obtained (depending upon the material under test) the form of test piece and the type of machine used. Further details regarding fatigue testing procedure are given in BS3518: Parts 1 to 5. Most fatigue tests are carried out at room temperature but often tests are also carried out at elevated or sub-zero temperatures depending upon the expected environmental operating conditions. At low temperatures the fatigue strength of metals show no deterioration and may even show a slight improvement, however, with increase in temperature, the fatigue strength decreases as creep effects are added to those of fatigue and this is revealed by a more pronounced effect of frequency of cycling and of mean stress since creep is both stressand time-dependent . When carrying out elevated temperature tests in air, oxidation of the sample may take place producing a condition similar to corrosion fatigue. Under the action of the cyclic stress, protective oxide films are cracked allowing further and more severe attack by the corrosive media. Thus fatigue and corrosion together ensure continuous propagation of cracks, and materials which show a definite fatigue limit at room temperature will not do so at elevated temperatures or at ambient temperatures under corrosive conditions - see Fig. 11.6. I I .I .2. PISIN curves The fatigue life of a component as determined at a particular stress level is a very variable quantity so that seemingly identical specimens may give widely differing results. This scatter
450 Mechanics of Materials 2 §11.1 (p)apnilldwo 《a】Without corrosion ssa/IS (b】with corrosion 102 104 106 100 Cycles to foilure (N,) Fig.11.6.The effect of corrosion on fatigue life.S/N Curve for (a)material showing fatigue limit;(b)same material under corrosion conditions. arises from many sources including variations in material composition and heterogeneity, variations in surface finish,variations in axiality of loading,etc. Mean stress【g.1-0 E p.0.9 (90%chonce of foilure) p0.5 (50%chonce of foilure) ssaJiS p0.I (10%chonce of foilure) Cycles to foilure (N) Fig.11.7.P/S/N curves indicating percentage chance of failure for given stress level after known number of cycles (zero mean stress) To overcome this problem,a number of test pieces should be tested at several different stresses and then an estimate of the life at a particular stress level for a given probability can be made.If the probability of 50%chance of failure is required then a P/S/N curve can be drawn through the median value of the fatigue life at the stress levels used in the test. It should be noted that this 50%(p =0.5)probability curve is the curve often displayed in textbooks as the S/N curve for a particular material and if less probability of failure is required then the fatigue limit value will need to be reduced
450 Mechanics of Materials 2 $11.1 102 lo4 106 108 io1O Cycles to failure (N,) Fig. 11.6. The effect of corrosion on fatigue life. S/N Curve for (a) material showing fatigue limit; (b) same material under corrosion conditions. arises from many sources including variations in material composition and heterogeneity, variations in surface finish, variations in axiality of loading, etc. Mean stress luml=O - - bo V u - 0'05 150% chonce of foilure) - a s v1 u 0) foilure I Cycles to foilure IN,) Fig. 11.7. P/S/N curves indicating percentage chance of failure for given stress level after known number of cycles (zero mean stress) To overcome this problem, a number of test pieces should be tested at several different stresses and then an estimate of the life at a particular stress level for a given probability can be made. If the probability of 50% chance of failure is required then a P/S/N curve can be drawn through the median value of the fatigue life at the stress levels used in the test. It should be noted that this 50% (p = 0.5) probability curve is the curve often displayed in textbooks as the S/N curve for a particular material and if less probability of failure is required then the fatigue limit value will need to be reduced
§11.1 Fatigue,Creep and Fracture 451 11.1.3.Effect of mean stress If the fatigue test is carried out under conditions such that the mean stress is tensile (Fig.11.3),then,in order that the specimen will fail in the same number of cycles as a similar specimen tested under zero mean stress conditions,the stress amplitude in the former case will have to be reduced.The fact that an increasing tensile mean stress lowers the fatigue or endurance limit is important,and all S/N curves should contain information regarding the test conditions (Fig.11.8). :) m0 MN/m2 R,4-10 正 apn ldwo om25 MN/m2 R,-0.75 m:50 MN/m2 R、-0.50 Cycles to foilure (N,) Cycles to failure (N, Fig.11.8.Effect of mean stress on the S/N curve expressed in alternative ways. A number of investigations have been made of the quantitative effect of tensile mean stress resulting in the following equations: Goodman() da =ON -( (11.7) Geber(2) 1-(=)月 (11.8) Soderberg(3) 1-（,月 (11.9) where oN =the fatigue strength for N cycles under zero mean stress conditions. o the fatigue strength for N cycles under condition of mean stress om. ors tensile strength of the material. oy =yield strength of the material. The above equations may be shown in graphical form (Fig.11.9)and in actual practice it has been found that most test results fall within the envelope formed by the parabolic curve of Geber and the straight line of Goodman.However,because the use of Soderberg gives an additional margin of safety,this is the equation often preferred-see Example 11.2. Even when using the Soderberg equation it is usual to apply a factor of safety F to both the alternating and the steady component of stress,in which case eqn.(11.9)becomes: =%(1-x (11.10)
911.1 Fatigue, Creep and Fracture 45 1 11 .I .3. Effect of mean stress If the fatigue test is carried out under conditions such that the mean stress is tensile (Fig. 11.3), then, in order that the specimen will fail in the same number of cycles as a similar specimen tested under zero mean stress conditions, the stress amplitude in the former case will have to be reduced. The fact that an increasing tensile mean stress lowers the fatigue or endurance limit is important, and all S/N curves should contain information regarding the test conditions (Fig. 11.8). VI - ?! "l C W E c u 0 2 H C t R, *-0 75 R, =-0 50 I I Cycles to failure (N,) Cycles to failure (N,) Fig. 1 1.8. Effect of mean stress on the SIN curve expressed in alternative ways. A number of investigations have been made of the quantitative effect of tensile mean stress resulting in the following equations: Goodman(') a, = UN (11.7) S~derberg(~) a, = UN [I - (z)] (11.8) (11.9) where ON = the fatigue strength for N cycles under zero mean stress conditions. a, = the fatigue strength for N cycles under condition of mean stress a,. CTTS = tensile strength of the material. cry = yield strength of the material. The above equations may be shown in graphical form (Fig. 11.9) and in actual practice it has been found that most test results fall within the envelope formed by the parabolic curve of Geber and the straight line of Goodman. However, because the use of Soderberg gives an additional margin of safety, this is the equation often preferred - see Example 1 1.2. Even when using the Soderberg equation it is usual to apply a factor of safety F to both the alternating and the steady component of stress, in which case eqn. (1 1.9) becomes: (1 1.10)
452 Mechanics of Materials 2 §11.1 Goodman g Geber Soderberg gy Mean stres5igm】 Fig.11.9.Amplitude/mean stress relationships as per Goodman.Geber and Soderberg. 600 _Tensile strength 500 400 1 300 cycles 1 0 200 cycles 6 Yield stress 100 100 200300 400 500600 Meon stress (MN/m2) =100 IO7 cycles 0￥ cycles -200 -300 -400 Fig.11.10.Smith diagram. The interrelationship of mean stress and alternating stress amplitude is often shown in diagrammatic form frequently collectively called Goodman diagrams.One example is shown in Fig.11.10,and includes the experimentally derived curves for endurance limits of a specific steel.This is called a Smith diagram.Many alternative forms of presentation of data are possible including the Haigh diagram shown in Fig.11.11,and when understood
452 Mechanics of Materials 2 $11.1 0 Pr Mean stress (urn) Fig. 11.9. Amplitude/mean stress relationships as per Goodman. Geber and Soderberg. 400 t I C I / / Meon stress (MN/m') io' cycles 10' cycles 5 -200 -400 -300 t Fig. 11 .lo. Smith diagram. The interrelationship of mean stress and alternating stress amplitude is often shown in diagrammatic form frequently collectively called Goodman diagrams. One example is shown in Fig. 11.10, and includes the experimentally derived curves for endurance limits of a specific steel. This is called a Smith diagram. Many alternative forms of presentation of data are possible including the Haigh diagram shown in Fig. 11.1 1, and when understood