270 Budynas-Nisbett:Shigley's ll.Failure Prevention 6.Fatigue Failure Resulting ©The McGraw-Hill Mechanical Engineering from Variable Loading Companies,2008 Design,Eighth Edition Fatigue Failure Resulting from Variable Loading 267 Figure 6-11 80 SN bands for representative aluminum alloys,excluding wrought alloys with Sut <38 kpsi.(From R.C. luvinall,Engineering 吃 Considerations of Stress, Strain and Strength.Copyright Wrought 1967 by The McGraw-Hill Companies,Inc.Reprinted by 16 permission.] rmanent mold cast 12 10 Sand cast 8 6 103 10 10° 10* Life N,cycles (log) The ordinate of the S-N diagram is called the farigue strength Sf:a statement of this strength value must always be accompanied by a statement of the number of cycles N to which it corresponds. Soon we shall learn that S-N diagrams can be determined either for a test specimen or for an actual mechanical element.Even when the material of the test specimen and that of the mechanical element are identical,there will be significant differences between the diagrams for the two. In the case of the steels,a knee occurs in the graph,and beyond this knee failure will not occur,no matter how great the number of cycles.The strength corresponding to the knee is called the endurance limit Se,or the fatigue limit.The graph of Fig.6-10 never does become horizontal for nonferrous metals and alloys,and hence these mate- rials do not have an endurance limit.Figure 6-11 shows scatter bands indicating the S-N curves for most common aluminum alloys excluding wrought alloys having a tensile strength below 38 kpsi.Since aluminum does not have an endurance limit,normally the fatigue strength Sf is reported at a specific number of cycles,normally N=5(108) cycles of reversed stress (see Table A-24). We note that a stress cycle(N=1)constitutes a single application and removal of a load and then another application and removal of the load in the opposite direction. Thus N =means the load is applied once and then removed,which is the case with the simple tension test. The body of knowledge available on fatigue failure from N=1 to N=1000 cycles is generally classified as low-cycle fatigue,as indicated in Fig.6-10.High-cycle fatigue,then,is concerned with failure corresponding to stress cycles greater than 103 cycles. We also distinguish a finite-life region and an infinite-life region in Fig.6-10.The boundary between these regions cannot be clearly defined except for a specific material: but it lies somewhere between 10 and 10'cycles for steels,as shown in Fig.6-10. As noted previously,it is always good engineering practice to conduct a testing program on the materials to be employed in design and manufacture.This,in fact,is a requirement,not an option,in guarding against the possibility of a fatigue failure
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition II. Failure Prevention 6. Fatigue Failure Resulting from Variable Loading 270 © The McGraw−Hill Companies, 2008 Fatigue Failure Resulting from Variable Loading 267 80 70 60 50 40 35 30 25 20 18 16 14 12 10 8 7 6 5 103 104 105 106 107 108 109 Life N, cycles (log) Peak alternating bending stress S, kpsi (log) Sand cast Permanent mold cast Wrought Figure 6–11 S-N bands for representative aluminum alloys, excluding wrought alloys with Sut < 38 kpsi. (From R. C. Juvinall, Engineering Considerations of Stress, Strain and Strength. Copyright © 1967 by The McGraw-Hill Companies, Inc. Reprinted by permission.) The ordinate of the S-N diagram is called the fatigue strength Sf ; a statement of this strength value must always be accompanied by a statement of the number of cycles N to which it corresponds. Soon we shall learn that S-N diagrams can be determined either for a test specimen or for an actual mechanical element. Even when the material of the test specimen and that of the mechanical element are identical, there will be significant differences between the diagrams for the two. In the case of the steels, a knee occurs in the graph, and beyond this knee failure will not occur, no matter how great the number of cycles. The strength corresponding to the knee is called the endurance limit Se, or the fatigue limit. The graph of Fig. 6–10 never does become horizontal for nonferrous metals and alloys, and hence these materials do not have an endurance limit. Figure 6–11 shows scatter bands indicating the S-N curves for most common aluminum alloys excluding wrought alloys having a tensile strength below 38 kpsi. Since aluminum does not have an endurance limit, normally the fatigue strength Sf is reported at a specific number of cycles, normally N = 5(108) cycles of reversed stress (see Table A–24). We note that a stress cycle (N = 1) constitutes a single application and removal of a load and then another application and removal of the load in the opposite direction. Thus N = 1 2 means the load is applied once and then removed, which is the case with the simple tension test. The body of knowledge available on fatigue failure from N = 1 to N = 1000 cycles is generally classified as low-cycle fatigue, as indicated in Fig. 6–10. High-cycle fatigue, then, is concerned with failure corresponding to stress cycles greater than 103 cycles. We also distinguish a finite-life region and an infinite-life region in Fig. 6–10. The boundary between these regions cannot be clearly defined except for a specific material; but it lies somewhere between 106 and 107 cycles for steels, as shown in Fig. 6–10. As noted previously, it is always good engineering practice to conduct a testing program on the materials to be employed in design and manufacture. This, in fact, is a requirement, not an option, in guarding against the possibility of a fatigue failure
Budynas-Nisbett:Shigley's ll.Failure Prevention 6.Fatigue Failure Resulting T©The McGraw-Hill 271 Mechanical Engineering from Variable Loading Companies,2008 Design,Eighth Edition 268 Mechanical Engineering Design Because of this necessity for testing,it would really be unnecessary for us to proceed any further in the study of fatigue failure except for one important reason:the desire to know why fatigue failures occur so that the most effective method or methods can be used to improve fatigue strength.Thus our primary purpose in studying fatigue is to understand why failures occur so that we can guard against them in an optimum man- ner.For this reason,the analytical design approaches presented in this book,or in any other book,for that matter,do not yield absolutely precise results.The results should be taken as a guide,as something that indicates what is important and what is not impor- tant in designing against fatigue failure. As stated earlier,the stress-life method is the least accurate approach especially for low-cycle applications.However,it is the most traditional method,with much published data available.It is the easiest to implement for a wide range of design applications and represents high-cycle applications adequately.For these reasons the stress-life method will be emphasized in subsequent sections of this chapter. However,care should be exercised when applying the method for low-cycle applications, as the method does not account for the true stress-strain behavior when localized yielding occurs. 6-5 The Strain-Life Method The best approach yet advanced to explain the nature of fatigue failure is called by some the strain-life method.The approach can be used to estimate fatigue strengths,but when it is so used it is necessary to compound several idealizations,and so some uncertain- ties will exist in the results.For this reason,the method is presented here only because of its value in explaining the nature of fatigue. A fatigue failure almost always begins at a local discontinuity such as a notch, crack,or other area of stress concentration.When the stress at the discontinuity exceeds the elastic limit,plastic strain occurs.If a fatigue fracture is to occur,there must exist cyclic plastic strains.Thus we shall need to investigate the behavior of materials sub- ject to cyclic deformation. In 1910.Bairstow verified by experiment Bauschinger's theory that the elastic lim- its of iron and steel can be changed,either up or down,by the cyclic variations of stress.2 In general,the elastic limits of annealed steels are likely to increase when subjected to cycles of stress reversals,while cold-drawn steels exhibit a decreasing elastic limit. R.W.Landgraf has investigated the low-cycle fatigue behavior of a large number of very high-strength steels,and during his research he made many cyclic stress-strain plots.3 Figure 6-12 has been constructed to show the general appearance of these plots for the first few cycles of controlled cyclic strain.In this case the strength decreases with stress repetitions,as evidenced by the fact that the reversals occur at ever-smaller stress levels.As previously noted,other materials may be strengthened,instead,by cyclic stress reversals. The SAE Fatigue Design and Evaluation Steering Committee released a report in 1975 in which the life in reversals to failure is related to the strain amplitude As/2.4 LBairstow,The Elastic Limits of Iron and Steel under Cyclic Variations of Stress"Philosophical Transactions,Series A,vol.210,Royal Society of London,1910,pp.35-55. 3R.W.Landgraf,Cyclic Deformation and Fatigue Behavior of Hardened Steels,Report no.320.Department of Theoretical and Applied Mechanics,University of Illinois,Urbana,1968,pp.84-90. ATechnical Report on Fatigue Properties.SAE J1099.1975
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition II. Failure Prevention 6. Fatigue Failure Resulting from Variable Loading © The McGraw−Hill 271 Companies, 2008 268 Mechanical Engineering Design Because of this necessity for testing, it would really be unnecessary for us to proceed any further in the study of fatigue failure except for one important reason: the desire to know why fatigue failures occur so that the most effective method or methods can be used to improve fatigue strength. Thus our primary purpose in studying fatigue is to understand why failures occur so that we can guard against them in an optimum manner. For this reason, the analytical design approaches presented in this book, or in any other book, for that matter, do not yield absolutely precise results. The results should be taken as a guide, as something that indicates what is important and what is not important in designing against fatigue failure. As stated earlier, the stress-life method is the least accurate approach especially for low-cycle applications. However, it is the most traditional method, with much published data available. It is the easiest to implement for a wide range of design applications and represents high-cycle applications adequately. For these reasons the stress-life method will be emphasized in subsequent sections of this chapter. However, care should be exercised when applying the method for low-cycle applications, as the method does not account for the true stress-strain behavior when localized yielding occurs. 6–5 The Strain-Life Method The best approach yet advanced to explain the nature of fatigue failure is called by some the strain-life method. The approach can be used to estimate fatigue strengths, but when it is so used it is necessary to compound several idealizations, and so some uncertainties will exist in the results. For this reason, the method is presented here only because of its value in explaining the nature of fatigue. A fatigue failure almost always begins at a local discontinuity such as a notch, crack, or other area of stress concentration. When the stress at the discontinuity exceeds the elastic limit, plastic strain occurs. If a fatigue fracture is to occur, there must exist cyclic plastic strains. Thus we shall need to investigate the behavior of materials subject to cyclic deformation. In 1910, Bairstow verified by experiment Bauschinger’s theory that the elastic limits of iron and steel can be changed, either up or down, by the cyclic variations of stress.2 In general, the elastic limits of annealed steels are likely to increase when subjected to cycles of stress reversals, while cold-drawn steels exhibit a decreasing elastic limit. R. W. Landgraf has investigated the low-cycle fatigue behavior of a large number of very high-strength steels, and during his research he made many cyclic stress-strain plots.3 Figure 6–12 has been constructed to show the general appearance of these plots for the first few cycles of controlled cyclic strain. In this case the strength decreases with stress repetitions, as evidenced by the fact that the reversals occur at ever-smaller stress levels. As previously noted, other materials may be strengthened, instead, by cyclic stress reversals. The SAE Fatigue Design and Evaluation Steering Committee released a report in 1975 in which the life in reversals to failure is related to the strain amplitude ε/2. 4 2 L. Bairstow, “The Elastic Limits of Iron and Steel under Cyclic Variations of Stress,” Philosophical Transactions, Series A, vol. 210, Royal Society of London, 1910, pp. 35–55. 3 R. W. Landgraf, Cyclic Deformation and Fatigue Behavior of Hardened Steels, Report no. 320, Department of Theoretical and Applied Mechanics, University of Illinois, Urbana, 1968, pp. 84–90. 4 Technical Report on Fatigue Properties, SAE J1099, 1975.
Budynas-Nisbett:Shigley's ll.Failure Prevention 6.Fatigue Failure Resulting ©The McGraw-Hil Mechanical Engineering from Variable Loading Companies,2008 Design,Eighth Edition Fatigue Failure Resulting from Variable Loading 269 Figure 6-12 Ist reversal ,3d True stress-true strain hysteresis .5th loops showing the first five stress reversals of a cyclic softening material.The graph is slightly exaggerated for clarity.Note that the slope of the line AB is the modulus of elasticity E.The stress range is △a,△ep is the plastic-strain range,.and△Se is the elastic strain range.The total-strain range is △g=△Ep+△Ee Figure 6-13 o A loglog plot showing how the fatigue life is related to the true-strain amplitude for 10 hot-rolled SAE 1020 steel. (Reprinted with permission 1.0 from SAE J1099_200208 0N 2002 SAE Intemational.] Plastic strain Total strain 1.0 10 Elastic strain 10 10 10 102 103 104 103 10 Reversals to failure,2N The report contains a plot of this relationship for SAE 1020 hot-rolled steel:the graph has been reproduced as Fig.6-13.To explain the graph,we first define the following terms: Fatigue ductility coefficient s is the true strain corresponding to fracture in one re- versal (point A in Fig.6-12).The plastic-strain line begins at this point in Fig.6-13. Fatigue strength coefficient of is the true stress corresponding to fracture in one reversal (point A in Fig.6-12).Note in Fig.6-13 that the elastic-strain line begins at o/E. Fatigue ductility exponent c is the slope of the plastic-strain line in Fig.6-13 and is the power to which the life 2N must be raised to be proportional to the true plastic- strain amplitude.If the number of stress reversals is 2N,then N is the number of cycles
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition II. Failure Prevention 6. Fatigue Failure Resulting from Variable Loading 272 © The McGraw−Hill Companies, 2008 Fatigue Failure Resulting from Variable Loading 269 4th 2d 1st reversal 3d 5th A B Δ Δp Δe Δ Figure 6–12 True stress–true strain hysteresis loops showing the first five stress reversals of a cyclicsoftening material. The graph is slightly exaggerated for clarity. Note that the slope of the line AB is the modulus of elasticity E. The stress range is σ , εp is the plastic-strain range, and εe is the elastic strain range. The total-strain range is ε = εp + εe. The report contains a plot of this relationship for SAE 1020 hot-rolled steel; the graph has been reproduced as Fig. 6–13. To explain the graph, we first define the following terms: • Fatigue ductility coefficient ε F is the true strain corresponding to fracture in one reversal (point A in Fig. 6–12). The plastic-strain line begins at this point in Fig. 6–13. • Fatigue strength coefficient σ F is the true stress corresponding to fracture in one reversal (point A in Fig. 6–12). Note in Fig. 6–13 that the elastic-strain line begins at σ F /E. • Fatigue ductility exponent c is the slope of the plastic-strain line in Fig. 6–13 and is the power to which the life 2N must be raised to be proportional to the true plasticstrain amplitude. If the number of stress reversals is 2N, then N is the number of cycles. 100 10–4 10–3 10–2 10–1 100 101 102 103 104 105 106 Reversals to failure, 2N Strain amplitude, Δ/2 ' F c 1.0 b 1.0 ' F E Total strain Plastic strain Elastic strain Figure 6–13 A log-log plot showing how the fatigue life is related to the true-strain amplitude for hot-rolled SAE 1020 steel. (Reprinted with permission from SAE J1099_200208 © 2002 SAE International.)
Budynas-Nisbett:Shigley's ll.Failure Prevention 6.Fatigue Failure Resulting I©The McGraw-Hil 273 Mechanical Engineering from Variable Loading Companies,2008 Design,Eighth Edition 270 Mechanical Engineering Design Fatigue strength exponent b is the slope of the elastic-strain line,and is the power to which the life 2N must be raised to be proportional to the true-stress amplitude. Now.from Fig.6-12.we see that the total strain is the sum of the elastic and plastic components.Therefore the total strain amplitude is half the total strain range 2 (a) The equation of the plastic-strain line in Fig.6-13 is △sp=Er2Nf (6-1) 2 The equation of the elastic strain line is △ee =E(2N (6-2) 2 E Therefore,from Eg.(a),we have for the total-strain amplitude △e=车(2NP+Er2Ny 2 (6-3) E which is the Manson-Coffin relationship between fatigue life and total strain.3 Some values of the coefficients and exponents are listed in Table A-23.Many more are included in the SAE J1099 report. Though Eg.(6-3)is a perfectly legitimate equation for obtaining the fatigue life of a part when the strain and other cyclic characteristics are given,it appears to be of lit- tle use to the designer.The question of how to determine the total strain at the bottom of a notch or discontinuity has not been answered.There are no tables or charts of strain concentration factors in the literature.It is possible that strain concentration factors will become available in research literature very soon because of the increase in the use of finite-element analysis.Moreover,finite element analysis can of itself approximate the strains that will occur at all points in the subject structure.7 6-6 The Linear-Elastic Fracture Mechanics Method The first phase of fatigue cracking is designated as stage I fatigue.Crystal slip that extends through several contiguous grains,inclusions,and surface imperfections is pre- sumed to play a role.Since most of this is invisible to the observer,we just say that stage I involves several grains.The second phase,that of crack extension,is called stage II fatigue.The advance of the crack (that is,new crack area is created)does produce evi- dence that can be observed on micrographs from an electron microscope.The growth of 5J.F.Tavernelli and L.F.Coffin,Jr.,"Experimental Support for Generalized Equation Predicting Low Cycle Fatigue."and S.S.Manson,discussion,Trans.ASME.J.Basic Eng..vol.84,no.4.pp.533-537. See also,Landgraf,Ibid. For further discussion of the strain-life method see N.E.Dowling,Mechanical Behavior of Materials, 2nd ed.,Prentice-Hall,Englewood Cliffs.NJ.,1999,Chap.14
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition II. Failure Prevention 6. Fatigue Failure Resulting from Variable Loading © The McGraw−Hill 273 Companies, 2008 270 Mechanical Engineering Design • Fatigue strength exponent b is the slope of the elastic-strain line, and is the power to which the life 2N must be raised to be proportional to the true-stress amplitude. Now, from Fig. 6–12, we see that the total strain is the sum of the elastic and plastic components. Therefore the total strain amplitude is half the total strain range ε 2 = εe 2 + εp 2 (a) The equation of the plastic-strain line in Fig. 6–13 is εp 2 = ε F (2N) c (6–1) The equation of the elastic strain line is εe 2 = σ F E (2N) b (6–2) Therefore, from Eq. (a), we have for the total-strain amplitude ε 2 = σ F E (2N) b + ε F (2N) c (6–3) which is the Manson-Coffin relationship between fatigue life and total strain.5 Some values of the coefficients and exponents are listed in Table A–23. Many more are included in the SAE J1099 report.6 Though Eq. (6–3) is a perfectly legitimate equation for obtaining the fatigue life of a part when the strain and other cyclic characteristics are given, it appears to be of little use to the designer. The question of how to determine the total strain at the bottom of a notch or discontinuity has not been answered. There are no tables or charts of strain concentration factors in the literature. It is possible that strain concentration factors will become available in research literature very soon because of the increase in the use of finite-element analysis. Moreover, finite element analysis can of itself approximate the strains that will occur at all points in the subject structure.7 6–6 The Linear-Elastic Fracture Mechanics Method The first phase of fatigue cracking is designated as stage I fatigue. Crystal slip that extends through several contiguous grains, inclusions, and surface imperfections is presumed to play a role. Since most of this is invisible to the observer, we just say that stage I involves several grains. The second phase, that of crack extension, is called stage II fatigue. The advance of the crack (that is, new crack area is created) does produce evidence that can be observed on micrographs from an electron microscope. The growth of 5 J. F. Tavernelli and L. F. Coffin, Jr., “Experimental Support for Generalized Equation Predicting Low Cycle Fatigue,’’ and S. S. Manson, discussion, Trans. ASME, J. Basic Eng., vol. 84, no. 4, pp. 533–537. 6 See also, Landgraf, Ibid. 7 For further discussion of the strain-life method see N. E. Dowling, Mechanical Behavior of Materials, 2nd ed., Prentice-Hall, Englewood Cliffs, N.J., 1999, Chap. 14.
274 Budynas-Nisbett:Shigley's ll.Failure Prevention 6.Fatigue Failure Resulting ©The McGraw-Hil Mechanical Engineering from Variable Loading Companies,2008 Design,Eighth Edition Fatigue Failure Resulting from Variable Loading 271 the crack is orderly.Final fracture occurs during stage III fatigue,although fatigue is not involved.When the crack is sufficiently long that KI Kie for the stress amplitude involved,then Ki is the critical stress intensity for the undamaged metal,and there is sudden,catastrophic failure of the remaining cross section in tensile overload (see Sec.5-12).Stage III fatigue is associated with rapid acceleration of crack growth then fracture. Crack Growth Fatigue cracks nucleate and grow when stresses vary and there is some tension in each stress cycle.Consider the stress to be fluctuating between the limits of omin and omax,where the stress range is defined as Ao omax-omin.From Eq.(5-37)the stress intensity is given by KI=Boa.Thus,for Ao,the stress intensity range per cycle is △Ki=B(cmax-Omin)√πa=B△a√ra (6-4) To develop fatigue strength data,a number of specimens of the same material are tested at various levels of Ao.Cracks nucleate at or very near a free surface or large discon- tinuity.Assuming an initial crack length of a;,crack growth as a function of the num- ber of stress cycles N will depend on Ao,that is,AK1.For AKI below some threshold value (AKIth a crack will not grow.Figure 6-14 represents the crack length a as a function of N for three stress levels (Ao)3>(Ao)2>(Ao)1.where (AK1)3> (AK)2>(AK1)1.Notice the effect of the higher stress range in Fig.6-14 in the pro- duction of longer cracks at a particular cycle count. When the rate of crack growth per cycle,da/dN in Fig.6-14,is plotted as shown in Fig.6-15,the data from all three stress range levels superpose to give a sigmoidal curve.The three stages of crack development are observable,and the stage II data are linear on log-log coordinates,within the domain of linear elastic fracture mechanics (LEFM)validity.A group of similar curves can be generated by changing the stress ratio R=omin/omax of the experiment. Here we present a simplified procedure for estimating the remaining life of a cycli- cally stressed part after discovery of a crack.This requires the assumption that plane strain Figure 6-14 The increase in crack length a from an initial length of a as a function of cycle count for (aK3/ (△K2 (AK) three stress ranges,【△al3> |△al2>(△ah- da dN Log N Stress cycles N
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition II. Failure Prevention 6. Fatigue Failure Resulting from Variable Loading 274 © The McGraw−Hill Companies, 2008 Fatigue Failure Resulting from Variable Loading 271 the crack is orderly. Final fracture occurs during stage III fatigue, although fatigue is not involved. When the crack is sufficiently long that KI = KIc for the stress amplitude involved, then KIc is the critical stress intensity for the undamaged metal, and there is sudden, catastrophic failure of the remaining cross section in tensile overload (see Sec. 5–12). Stage III fatigue is associated with rapid acceleration of crack growth then fracture. Crack Growth Fatigue cracks nucleate and grow when stresses vary and there is some tension in each stress cycle. Consider the stress to be fluctuating between the limits of σmin and σmax, where the stress range is defined as σ = σmax − σmin. From Eq. (5–37) the stress intensity is given by KI = βσ√πa. Thus, for σ, the stress intensity range per cycle is KI = β(σmax − σmin) √πa = βσ√πa (6–4) To develop fatigue strength data, a number of specimens of the same material are tested at various levels of σ. Cracks nucleate at or very near a free surface or large discontinuity. Assuming an initial crack length of ai, crack growth as a function of the number of stress cycles N will depend on σ, that is, KI. For KI below some threshold value (KI)th a crack will not grow. Figure 6–14 represents the crack length a as a function of N for three stress levels (σ )3 > (σ )2 > (σ )1, where (KI)3 > (KI)2 > (KI)1. Notice the effect of the higher stress range in Fig. 6–14 in the production of longer cracks at a particular cycle count. When the rate of crack growth per cycle, da/d N in Fig. 6–14, is plotted as shown in Fig. 6–15, the data from all three stress range levels superpose to give a sigmoidal curve. The three stages of crack development are observable, and the stage II data are linear on log-log coordinates, within the domain of linear elastic fracture mechanics (LEFM) validity. A group of similar curves can be generated by changing the stress ratio R = σmin/σmax of the experiment. Here we present a simplified procedure for estimating the remaining life of a cyclically stressed part after discovery of a crack. This requires the assumption that plane strain Log N Stress cycles N Crack length a a ai (ΔKI )3 (ΔKI )2 (ΔKI )1 da dN Figure 6–14 The increase in crack length a from an initial length of ai as a function of cycle count for three stress ranges, (σ) 3 > (σ) 2 > (σ) 1.