§11.1 Fatigue,Creep and Fracture 453 500 Tensile strength 400 300 04 dwo cycles 200 107 cycles 400 o at am0 Yield stress 100 200 300 400 500 Mean stress (am)MN/m2 Fig.11.11.Haigh diagram. by the engineer these diagrams can be used to predict the fatigue life of a component under a particular stress regime.If the reader wishes to gain further information about the use of these diagrams it is recommended that other texts be consulted. The effect of a compressive mean stress upon the life of a component is not so well documented or understood as that of a tensile mean stress but in general most materials do not become any worse and may even show an improved performance under a compressive mean stress.In calculations it is usual therefore to take the mean stress as zero under these conditions. 11.1.4.Effect of stress concentration The influence of stress concentration (see $10.3)can be illustrated by consideration of an elliptical crack in a plate subjected to a tensile stress.Provided that the plate is very large, the "theoretical stress concentration"factor K:is given by: 2A K=1+B (11.11) where“A"and“B”are the crack dimensions as shown in Fig.ll.l2. If the crack is perpendicular to the direction of stress,then A is large compared with B and hence K,will be large.If the crack is parallel to the direction of stress,then A is very small compared with B and hence K,=1.If the dimensions of A and B are equal such that the crack becomes a round hole,then K,=3 and a maximum stress of 3onom acts at the sides of the hole. The effect of sudden changes of section,notches or defects upon the fatigue performance of a component may be indicated by the "fatigue notch"or"fatigue strength reduction"factor
$11.1 \\ Yield stress- \ \ . \\ \\ \ \‘ \ \ Fatigue, Creep and Fracture 453 500 N E 400 I - bo - 300 0 0) c .- - n 200 VI r & ul 100 0 \ -\ \ Tensile strength \ \’ \ \ \ \ \ . \ Mean stress (urn) MN/mz Fig. 11.1 1. Haigh diagram. by the engineer these diagrams can be used to predict the fatigue life of a component under a particular stress regime. If the reader wishes to gain further information about the use of these diagrams it is recommended that other texts be consulted. The effect of a compressive mean stress upon the life of a component is not so well documented or understood as that of a tensile mean stress but in general most materials do not become any worse and may even show an improved performance under a compressive mean stress. In calculations it is usual therefore to take the mean stress as zero under these conditions. 1 I .I .4. Effect of stress concentration The influence of stress concentration (see 910.3) can be illustrated by consideration of an elliptical crack in a plate subjected to a tensile stress. Provided that the plate is very large, the “theoretical stress concentration” factor K, is given by: 2A Kf=l+B (11.11) where “A” and “B’ are the crack dimensions as shown in Fig. 1 1.12. If the crack is perpendicular to the direction of stress, then A is large compared with B and hence K, will be large. If the crack is parallel to the direction of stress, then A is very small compared with B and hence K, = 1. If the dimensions of A and B are equal such that the crack becomes a round hole, then Kt = 3 and a maximum stress of 3an,, acts at the sides of the hole. The effect of sudden changes of section, notches or defects upon the fatigue performance of a component may be indicated by the “fatigue notch” or “fatigue strength reduction” factor
454 Mechanics of Materials 2 $11.1 ominal stress(aoom】 Fig.11.12.Elliptical crack in semi-infinite plate. Kf,which is the ratio of the stress amplitude at the fatigue limit of an un-notched specimen, to that of a notched specimen under the same loading conditions. K is always less than the static theoretical stress concentration factor referred to above because under the compressive part of a tensile-compressive fatigue cycle,a fatigue crack is unlikely to grow.Also the ratio of K/K,decreases as K,increases,sharp notches having less effect upon fatigue life than would be expected.The extent to which the stress concentration effect under fatigue conditions approaches that for static conditions is given by the "notch sensitivity factor"q,and the relationship between them may be simply expressed by: K1-1 g=K-1 (11.12) thus q is always less than 1.See also $10.3.5. Notch sensitivity is a very complex factor depending not only upon the material but also upon the grain size,a finer grain size resulting in a higher value of g than a coarse grain size.It also increases with section size and tensile strength (thus under some circumstances it is possible to decrease the fatigue life by increasing tensile strength!)and,as has already been mentioned,it depends upon the severity of notch and type of loading. In dealing with a ductile material it is usual to apply the factor Kf only to the fluctuating or alternating component of the applied stress.Equation (11.10)then becomes: =x-(】 (11.13) A typical application of this formula is given in Example 11.3. 11.1.5.Cumulative damage In everyday,true-life situations,for example a car travelling over varying types of roads or an aeroplane passing through various weather conditions on its flight,stresses will not generally be constant but will vary according to prevailing conditions
454 Mechanics of Materials 2 $11.1 Nominal stress (am) Fig. 11.12. Elliptical crack in semi-infinite plate. Kf, which is the ratio of the stress amplitude at the fatigue limit of an un-notched specimen, to rhat of a notched specimen under the same loading conditions. Kf is always less than the static theoretical stress concentration factor referred to above because under the compressive part of a tensile-compressive fatigue cycle, a fatigue crack is unlikely to grow. Also the ratio of Kf/Kr decreases as K, increases, sharp notches having less effect upon fatigue life than would be expected. The extent to which the stress concentration effect under fatigue conditions approaches that for static conditions is given by the “notch sensitivity factor” q, and the relationship between them may be simply expressed by: Kf - 1 q=- Kt - 1 (1 1.12) thus q is always less than 1. See also $10.3.5. Notch sensitivity is a very complex factor depending not only upon the material but also upon the grain size, a finer grain size resulting in a higher value of q than a coarse grain size. It also increases with section size and tensile strength (thus under some circumstances it is possible to decrease the fatigue life by increasing tensile strength!) and, as has already been mentioned, it depends upon the severity of notch and type of loading. In dealing with a ductile material it is usual to apply the factor Kf only to the fluctuating or alternating component of the applied stress. Equation (1 1 .lo) then becomes: ON u,,, .F 0, = - [l- (--)I FXf (11.13) A typical application of this formula is given in Example 11.3. I1 .I .5. Cumulative damage In everyday, true-life situations, for example a car travelling over varying types of roads or an aeroplane passing through various weather conditions on its flight, stresses will not generally be constant but will vary according to prevailing conditions
§11.1 Fatigue,Creep and Fracture 455 Several attempts have been made to predict the fatigue strength for such variable stresses using S/N curves for constant mean stress conditions.Some of the predictive methods avail- able are very complex but the simplest and most well known is "Miner's Law." Miner(7)postulated that whilst a component was being fatigued,internal damage was taking place.The nature of the damage is difficult to specify but it may help to regard damage as the slow internal spreading of a crack,although this should not be taken too literally.He also stated that the extent of the damage was directly proportional to the number of cycles for a particular stress level,and quantified this by adding,"The fraction of the total damage occurring under one series of cycles at a particular stress level,is given by the ratio of the number of cycles actually endured n to the number of cycles N required to break the component at the same stress level".The ratio n/N is called the "cycle ratio"and Miner proposed that failure takes place when the sum of the cycle ratios equals unity. i.e.when En/N =1 or 0+%+3++ac=1 (11.14) If equation(11.14)is merely treated as an algebraic expression then it should be unimpor- tant whether we put n3/N3 before n1/NI etc.,but experience has shown that the order of application of the stress is a matter of considerable importance and that the application of a higher stress amplitude first has a more damaging effect on fatigue performance than the application of an initial low stress amplitude.Thus the cycle ratios rarely add up to 1,the sum varying between 0.5 and 2.5,but it does approach unity if the number of cycles applied at any given period of time for a particular stress amplitude is kept relatively small and frequent changes of stress amplitude are carried out,i.e.one approaches random loading conditions.A simple application of Miner's rule is given in Example 11.4. 11.1.6.Cyclic stress-strain Whilst many components such as axle shafts,etc.,have to withstand an almost infinite number of stress reversals in their lifetime,the stress amplitudes are relatively small and usually do not exceed the elastic limit.On the other hand,there are a growing number of structures such as aeroplane cabins and pressure vessels where the interval between stress cycles is large and where the stresses applied are very high such that plastic deformation may occur.Under these latter conditions,although the period in time may be long,the number of cycles to failure will be small and in recent years interest has been growing in this"low cycle fatigue”. If,during fatigue testing under these high stress cycle conditions,stress and strain are continually monitored,a hysteresis loop develops characteristic of each cycle. Figure 11.13 shows typical loops under constant stress amplitude conditions,each loop being displaced to the right for the sake of clarity.It will be observed that with each cycle, because of work hardening,the width of the loop is reduced,eventually the loop narrowing to a straight line under conditions of total elastic deformation. The relationship between the loop width W and the number of cycles N is given by: W=AN- (11.15) where A is a constant and h the measure of the rate of work hardening
$11.1 Fatigue, Creep and Fracture 455 Several attempts have been made to predict the fatigue strength for such variable stresses using S/N curves for constant mean stress conditions. Some of the predictive methods available are very complex but the simplest and most well known is “Miner’s Law.” Mined7) postulated that whilst a component was being fatigued, internal damage was taking place. The nature of the damage is difficult to specify but it may help to regard damage as the slow internal spreading of a crack, although this should not be taken too literally. He also stated that the extent of the damage was directly proportional to the number of cycles for a particular stress level, and quantified this by adding, “Thefraction of the total damage occurring under one series of cycles at a particular stress level, is given by the ratio of the number of cycles actually endured n to the number of cycles N required to break the component at the same stress level”. The ratio n/N is called the “cycle ratio” and Miner proposed that failure takes place when the sum of the cycle ratios equals unity. i.e. when Xn/N = 1 or nl n2 n3 - + - + -+.as+ etc = 1 N1 N2 N3 (11.14) If equation (1 1.14) is merely treated as an algebraic expression then it should be unimportant whether we put n3/N3 before nl/N1 etc., but experience has shown that the order of application of the stress is a matter of considerable importance and that the application of a higher stress amplitude first has a more damaging effect on fatigue performance than the application of an initial low stress amplitude. Thus the cycle ratios rarely add up to 1, the sum varying between 0.5 and 2.5, but it does approach unity if the number of cycles applied at any given period of time for a particular stress amplitude is kept relatively small and frequent changes of stress amplitude are carried out, i.e. one approaches random loading conditions. A simple application of Miner’s rule is given in Example 11.4. 11.1.6. Cyclic stress-strain Whilst many components such as axle shafts, etc., have to withstand an almost infinite number of stress reversals in their lifetime, the stress amplitudes are relatively small and usually do not exceed the elastic limit. On the other hand, there are a growing number of structures such as aeroplane cabins and pressure vessels where the interval between stress cycles is large and where the stresses applied are very high such that plastic deformation may occur. Under these latter conditions, although the period in time may be long, the number of cycles to failure will be small and in recent years interest has been growing in this “low cycle fatigue”. If, during fatigue testing under these high stress cycle conditions, stress and strain are continually monitored, a hysteresis loop develops characteristic of each cycle. Figure 11.13 shows typical loops under constant stress amplitude conditions, each loop being displaced to the right for the sake of clarity. It will be observed that with each cycle, because of work hardening, the width of the loop is reduced, eventually the loop narrowing to a straight line under conditions of total elastic deformation. The relationship between the loop width W and the number of cycles N is given by: w AN-^ (11.15) where A is a constant and h the measure of the rate of work hardening
456 Mechanics of Materials 2 §11.1 Stroin Fig.11.13.Cyclic stress-strain under constant stress conditions-successive loading loops displaced to right for clarity.Hysteresis effects achieved under low cycle,high strain (constant stress amplitude)fatigue. 9 Strain(e】 Piostic strain range △ep Fig.11.14.Cyclic stress-strain under constant strain amplitude conditions. If instead of using constant stress amplitude conditions,one uses constant strain amplitude conditions then the form of loop is indicated in Fig.11.14.Under these conditions the stress range increases with the number of cycles but the extent of the increase reduces with each cycle such that after about 20%of the life of the component the loop becomes constant
456 Mechanics of Materials 2 $11.1 Fig. 11.13. Cyclic stress-strain under constant stress conditions - successive loading loops displaced to right for clarity. Hysteresis effects achieved under low cycle, high strain (constant stress amplitude) fatigue. Plastic strain ' I ronga Ac, LFig. 1 1.14. Cyclic stress-strain under constant strain amplitude conditions. If instead of using constant stress amplitude conditions, one uses constant strain amplitude conditions then the form of loop is indicated in Fig. 11.14. Under these conditions the stress range increases with the number of cycles but the extent of the increase reduces with each cycle such that after about 20% of the life of the component the loop becomes constant
§11.1 Fatigue,Creep and Fracture 457 If now a graph is drawn (using logarithmic scales)of the plastic strain range against the number of cycles to failure a straight line results (Fig.11.15).From this graph we obtain the following equation for the plastic strain range Asp which is known as the Coffin-Manson Law.(8) △Ep=K(Wr)）-b (11.16) Log N, Fig.11.15.Relationship between plastic strain and cycles to failure in low cycle fatigue. The value of b varies between 0.5 and 0.6 for most metals,whilst the constant K can be related to the ductility of the metal.Equation (11.16)can also be expressed as: N (11.17) where D is the ductility as determined by the reduction in area r in a tensile test. ie. D=() In many applications,the total strain range may be known but it may be difficult to separate it into plastic and elastic components;thus a combined equation may be more useful. △e1=△ee+△ep Where As,Ase and Asp stand for total,elastic and plastic strain ranges respectively. Relationships between Asp and N are given above but Ase may be related to N by the following modified form of Basquin's Law.9) A,=35x管×N (11.18) If a graph is plotted(Fig.11.16)of strain range against number of cycles to failure,it can be seen that the beginning part of the curve closely fits the slope of Coffin's equation while the latter part fits the modified Basquin's equation,the cross-over point being at about 105
01 1.1 Fatigue, Creep and Fracture 457 If now a graph is drawn (using logarithmic scales) of the plastic strain range against the number of cycles to failure a straight line results (Fig. 11.15). From this graph we obtain the following equation for the plastic strain range Asp which is known as the Coffin-Manson Law .@) A&, = K(Nf)-b (11.16) \ Log ” Fig. 11.15. Relationship between plastic strain and cycles to failure in low cycle fatigue. The value of b varies between 0.5 and 0.6 for most metals, whilst the constant K can be related to the ductility of the metal. Equation (1 1.16) can also be expressed as: -b AE, = (z) (1 1.17) where D is the ductility as determined by the reduction in area r in a tensile test. i.e. D=l, (-) 1 1-r In many applications, the total strain range may be known but it may be difficult to separate it into plastic and elastic components; thus a combined equation may be more useful. AE~ = AE~ + Where AE~, AE, and stand for total, elastic and plastic strain ranges respectively. Relationships between A&, and Nf are given above but AE, may be related to Nf by the following modified form of Basquin’s Law.f9) CTS Ace = 3.5 x ~ x NT0.lz E (1 1.18) If a graph is plotted (Fig. 1 1.16) of strain range against number of cycles to failure, it can be seen that the beginning part of the curve closely fits the slope of Coffin’s equation while the latter part fits the modified Basquin’s equation, the cross-over point being at about lo5