Fatigue Life The total number of cycles for which a specimen sustains before failure is called fatigue (cyclic) life, denoted by N. The graph by plotting values of S a and N is called S-N curve or Wöhler diagram. The curve may be plotted as semilogarithmic or logarithmic. Vertical axis is stress amplitude (S a ) while horizontal axis is fatigue life (N). For S, linear scale is preferred, but sometimes log scale is also used. A log scale is always used for N. Fig. 11 shows S-N curves in log-log and semi-log scales for superalloy S/SAV. S-N curve is a straight line in log-log plot. However, in semilog plot, a smooth curve is fitted. Figure 11 log-log plot semi-log plot
Fatigue Life Fig. 12 shows schematic S-N curves for ferrous alloys & titanium (Curve A) and nonferrous alloys (except titanium) & nonmetallic materials (Curve B). For ferrous alloys & titanium, the curve becomes asymptotic to horizontal line (the specimen will not fail for an infinite number of cycles). The stress level at such point is called endurance (fatigue) limit, denoted by S e. It is not observed for nonferrous alloys and nonmetallic materials. Their fatigue strength is determined for a specified number of cycles. When a specimen does not fail even if the specified cycle is reached, test is stopped and the corresponding stress value is marked on the curve as runout (given by an arrow as in Curve B). The fatigue limit in such case is assumed as 5 * 10 8 Curve A: ferrous alloys & titanium Curve B: nonferrous alloys cycles for design purposes. Figure 12
Construction of S-N Curve S-N curve is obtained by conducting several rotating bending fatigue tests. In each test, the specimen is loaded to create a certain level of stress (S) and rotated until it fractures. When the specimen fractures, level of stress applied (S) and the number of cycles (N) to fracture are noted. The results of tests are represented by a point on the curve. Tests at different stress levels(from ultimate strength down to very low stress values) with corresponding cycles to fracture create multiples of test points for construction of curve. The stress level where the curve becomes asymptotic to horizontal line is determined as endurance strength (S e ) of material. 3
Regions on the idealized S-N curve (log-log scale) for steel are as follows: 1. Finite life (N < 10 6~7 ) a. Low cycle fatigue (10 0 < N < 10 3 ) b. High cycle fatigue (N 10 3 ) Mathematical Representation of S-N Curve S ut 0.8 S ut S e Low Cycle Finite Life High Cycle Infinite Life 2. Infinite life (N > 10 6~7 ) 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 In general, fatigue is considered as high cycle when the peak stresses in material are held within the elastic range while low cycle fatigue occurs when stresses are above the elastic limit. This arbitrary division may vary from material-to-materialto depending upon tensile properties. From design viewpoint, the main interest in engineering is for the high cycle region of S-N curve. However, low cycle fatigue data can be advantageous when only a short service life is required. 4
Mathematical Representation of S-N Curve Cycle of 10 0 refers to Finite Life ultimate tensile strength S ut (S ut ) while endurance strength (S e 0.5 S ut ) is obtained at 10 6 cycles. The stress amplitude is about 0.8 S ut by which high cycle fatigue starts 0.8 S ut S e Low Cycle High Cycle Infinite Life (10 3 cycles). 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 The correlation between S & N in high cycle region (10 3 N 10 6 ) can be obtained based on equation of the line (i.e. y = b x + c): log S = b log N + c S = 10 c N b or N = - / 1/ 10 c b S b where b & c are: b 1 0.8S ut = log 3 Se and c ( 0.8 ) 2 S ut = log Se 5
Example: Fatigue with Constant Amplitude and Frequency Stresses Q. A round steel shaft (having S ut = 90 kg/mm 2 and S e = 25 kg/mm 2 ) is carrying a static tensile stress of 10 kg/mm 2. The shaft is also subjected to a variable stress of ±40 kg/mm 2. a) Determine the stress components. b) Draw stress-time diagram, and specify the type of stress state. c) Calculate the fatigue life of this shaft. a) Stress components are as follows: b) This is fluctuating stress state (S m 0): S S m = S a max = 10 kg/mm 40 kg/mm 2 2 = 10 + 40 = 50 kg/mm 2 S min = 10 + ( 40) = 30 kg/mm 2 S (kg/mm 2 ) 50 10 0-30 time c) Fatigue life refers to the number of cycles to fracture, and calculated as: ( ) ( S S ) b = 1 3 log 0.8 = 0.1167 ( ) 2 ut e c = log 0.8Sut S e = 2.0984 N = S a = - c/ b 1/ b 4 10 1.791 10 cycles
Cumulative Fatigue Damage Until now, stresses with constant amplitude and frequency were considered. However, in actual case, machine elements are subjected to fatigue under varying stress amplitudes. When such stress steps are above the fatigue limit of material, it is necessary to consider the fatigue damage accumulated ateachstep. The total damage is known as cumulativefatiguedamage. Palmgren-Minerlineardamagerule PalmgrenMinerlineardamagerule is the most widely used concept due to its simplicity. According to this rule, the damage obtained at a specified stress level is a linear function of the number of cycles: d i = (n i / N i ) * D t the failure occurs when: d i = D t simplification gives: (n i / N i ) = 1 n ati th i : cycles i stress level during service d i : the damage at i th stress level N i : cycles at i th stress level on S-N curve D t : the total damage required for failure 7
Example: Cumulative Fatigue Damage (using S-N curve) Q. Based on given S-N curve, calculate the remaining life of a part at the stress amplitude of 50 kg/mm 2 after subjected to the following alternating stresses: 66 kg/mm 2 45 kg/mm 2 for 2.5 * 10 4 cycles for 1.5 * 10 5 cycles 72 kg/mm 2 for 10 4 cycles A. For the given stress amplitudes, the corresponding number of cycles (N) are determined from curve: No. S (given) n (given) N (from curve) n i / N i 1 66 2.5 * 10 4 10 5 0.25 2 45 1.5 * 10 5 7 * 10 5 0.21 3 72 10 4 7 * 10 4 0.14 = 0.60 S 3, N 3 S 1, N 1 S 2, N 2 For 50 kg/mm 2 : N = 3 * 10 5 cycles (from the curve) Remaining life : 1 0.6 = 0.4, which corresponds to 0.4 * (3 * 10 5 ) = 1.2 * 10 5 cycles
Example: Cumulative Fatigue Damage (using formula) Q. Calculate the remaining life of a round steel bar (having S ut = 75 kg/mm 2 and S e = 25 kg/mm 2 ) at the stress amplitude of 42 kg/mm 2 after subjected to the given alternating stresses: 35 kg/mm 2 for 1.5 * 10 4 cycles 2 4 30 kg/mm 2 for 5 * 10 4 cycles 40 kg/mm 2 for 10 4 cycles A. For the given stress amplitudes, Thus, the total damage is calculated as follows: the corresponding number of No. S (given) n (given) N (from formula) n i / N i cycles (N) are determined using 1 35 1.5 * 10 4 7 * 10 4 0.21 following equations: 2 30 5 * 10 4 2.4 * 10 5 0.21 1 0.8S b log ut 3 40 10 4 2.5 * 10 4 0.41 = 3 S = e 0.83 ( 0.8 ) 2 S ut For 42 kg/mm 2 : N = 1.7 * 10 4 cycles (from formula) c = log Se Remaining life : 1 0.83 = 0.17, corresponding to N = - / 1/ 10 c b S b 0.17 * (1.7 * 10 4 ) = 2.9 * 10 3 cycles
Interpretation of Fatigue Fatigue is influenced by many factors. S-N curves are approximations to represent fatigue behaviour of specimens at laboratory conditions. Fatigue life of an actual part varies considerably from laboratory tests. Followings are important considerations: 1. Statistical nature of fatigue: When identical specimens are tested at the same stress level, their fatigue lives are generally not the same, but scatter at a great deal. S-N curve representsastatisticalaverage of the test results. Thereby, a modifying factor (M r ) is used to modify S-N curve for different probabilities (reliabilities) of failure, denoted by P (Fig. 13): Figure 13 M r = 1 0.08 P 10
Interpretation of Fatigue 2. Effect of surface quality: This factor predominantly affects the fatigue behaviour. Microscopic irregularities on the part surface which are not visible to naked eye (e.g. rough surface after machining, a decarburized layer, corrosion pits, inclusions and gas blowholes, etc.) will trigger the fatigue failure. Modifying factor for surface quality (M s ) Figure 14 is used to modify fatigue strength of a part machined with specific process (Fig. 14). Fatigue life increases with decrease in surface roughness. Machining is detrimental to fatigue life due to formation of tensile residual stresses in the near-surface area. Cold working causes relaxation of stress concentration, so better fatigue properties. Finishing and heat treatment operations are also beneficial by forming compressive residual stresses to improve fatigue life. a : mirror polished b : ground c : honed d : machined e : hot-rolled f : corroded in tap water g : as-forged h : corroded in salt water 11
Interpretation of Fatigue 3. Size effect: Experiments have shown that the fatigue results depend strongly upon specimensize, which is one of the most important problems in fatigue applications. The general observation is that fatigue strength of large parts may be considerably lower than that of small specimens. This may be due to the fact that materials become more heterogeneous with increasing size, which makes it impossible to prepare specimens retaining the nominal properties of specified material. Also, the capacities of testers are limited to conduct experiments with large parts. 4. Method of testing: S-N curves obtained by three methods of fatigue testing differ appreciably (Fig. 15). The ranking based oncurvesfromthehighesttothelowest: highest to 1) alternating bending test2) rotating bending test3) push-pull test Most S-N curves are produced by the rotating bending test due to its simplicity. Figure 15 12
Modified (Actual) Fatigue Strength Since fatigue properties of materials are easily influenced by many factors (as mentioned previously), S-N curve obtained from laboratory tests must be related to real-life design conditions. Therefore, the endurance strength of Actual endurance strength material obtained by laboratory tests is modified with following factors always has lower value than laboratory result (Fig. 16). (having values of < 1.0): Figure 16 S e = k a * k b * k c * k d * k e * k f * S e S e : actual endurance strength of material S e : endurance strength in laboratory conditions k a : surface quality factor k b : size factor k c : reliability (probability) factor k d : temperature factor k e : stress concentration factor k f : miscellaneous factor 13
Endurance Ratio Endurance limit of a material can usually be related to its tensile strength. Ratio of endurance limit (fatigue strength) to tensile strength is known as endurance ratio, that is used to predict fatigue behaviour of materials. 14