mechanical-properties-and-working-of-metals-and-alloys_compress

8.7 Effect of Mean Stress 335 Fig. 8.19 Goodman diagram Compression Tension (Goodman 1899) Tension UTS, Su σmax Yield stress, S0 σa σr σe or σn Range of stress Locus of mean stress 0 σmin σe or σn 45° Compression Mean stress, σm Data of completely reversed stress cycle are plotted here fatigue test of steel in axial loading indicate that the maxi- and another from the rmin point lying on the ordinate, mum allowable stress range increases with increasing the where rm ¼ 0 , to the Su point. The former one (the upper compressive mean stress up to the compressive yield stress curve) shows the variation of rmax with the mean stress, beyond which the existence of test data is limited. This rm, and the latter (the lower curve) is the variation of rmin agrees well with the fact that the compressive residual stress with rm. has a beneficial effect on fatigue life. The Goodman diagram • Further, the linear curves of rmax and rmin are extended shown in Fig. 8.19 has been constructed without carrying as dotted lines to the left side of the origin, where the out the actual experiment. In lieu of the actual test data, an mean stress is compressive, i.e. rm\\0. approximate Goodman diagram may be obtained, the method of which is described below. The experimental curves showing the variation of the maximum allowable stress range with the mean stress are not • For a completely reversed stress cycle, either the fatigue strictly linear. In fact, these straight lines shown in Fig. 8.19 strength at any given number of cycles, ðsay; N cyclesÞ, are curves, and the experimental data usually lie somewhat rn, or the fatigue limit, re, of a given material is recorded above and below the linear lines of rmax and rmin; respec- from the S–N diagram or noted from the literature. tively. Thus, the Goodman diagram shown in Fig. 8.19 is a completely safe region and represents a conservative design • On the vertical axis of the diagram, where the mean stress criterion for mean stress effects. is zero, the value corresponding to either rn or re is marked as points at two places equidistant from the ori- 8.8 Stress Fluctuation and Cumulative gin—one in the tensile (positive) side and another in the Fatigue Damage compressive (negative) side. The point in tension side represents the maximum stress, rmax, and the point in 8.8.1 Overstressing, Understressing compression side is the minimum stress, rmin, for a and Coaxing purely alternating stress cycle. Overstressing is a process where a virgin specimen is sub- • The ultimate tensile strength, Su, of the material is jected to a stress above the fatigue limit for some number of recorded from the static uniaxial tensile test or noted cycles less than that required to cause failure, and subse- from the literature. quently, the specimen is subjected to another stress level until it fractures. Experiments (Dieter et al. 1954) showed • In the tensile part of the plot, two boundary lines—one that overstressing causes a greater reduction in fatigue life. If from the horizontal and another from the vertical axes— a virgin specimen is subjected to a stress below the fatigue are drawn at a distance equal to the value of Su from the limit for a large number of cycles so that it remains unbroken origin. These boundary lines are shown as dotted lines in and if subsequently, it is subjected to a higher stress level, Fig. 8.19. The intersection of these boundary lines gives the point, Su. • Now, two straight lines are drawn—one from the rmax point lying on the ordinate, where rm ¼ 0, to the Su point

336 8 Fatigue the specimen is said to have been understressed. Either the required to reach this critical length both increase. Thus, the fatigue life or the fatigue limit of understressed specimens is speeds at which cracks propagate depend on the applied often higher than that expected for virgin specimens. The stress levels. Hence, the propagation of a coarse crack ini- localized strain hardening at sites of possible crack initiation tiated by a previous high stress will tend to slow down under is considered to be the reason for the improvements in a subsequent low stress, while the propagation of a fine fatigue properties resulting from understressing. A special crack formed by a previous low stress might be very rapid case of understressing is coaxing, which may result in a under a subsequent high stress. Further, the growth of a fatigue limit as much as 50% greater than the virgin fatigue crack under subsequent lower or higher stresses may be limit. Coaxing is a procedure where a virgin specimen is affected by the localized strain hardening at the tip of the subjected to a stress below the fatigue limit for a large crack. All these variations tend to average out if the stress number of cycles without failure and then the stress is raised levels are applied in random order, which is found in many in small increments after subjecting the specimen for a large loading spectra. It is the basis on which the linear cumulative number of cycles at each stress level, causing the fatigue fatigue damage theory has been developed. limit to increase to a great extent. An investigation (Sinclair 1952) showed that the strong coaxing effect is directly cor- The linear cumulative fatigue damage theory proposed by related with the ability of the material to undergo strain Palmgren (1924) and Miner (1945) is the most widely used ageing. The strengthening due to localized strain ageing at due to its simplicity and often referred to as the Palmgren– the tip of the crack is believed to be the reason for the Miner cumulative fatigue damage theory or simply called improvement in fatigue properties from strong coaxing. For Miner’s rule. This theory, based on the constant amplitude example, ingot iron and mild steel show great improvement test data, predicts the life of a component subjected to a in fatigue properties due to strong coaxing effect while there variable load history. According to this theory, each series of is little improvement in fatigue properties from coaxing in overstress cycles produces a certain fraction of the total heat-treated low-alloy steel, brass and aluminium alloys. fatigue damage, and when these fractions add up to unity, failure will take place. The fraction of the total fatigue 8.8.2 Cumulative Fatigue Damage damage caused by one series of overstress cycles at a par- ticular stress level is given by the ratio of the number of So far, we have discussed the generation of fatigue data overstress cycles actually endured at that stress level to the resulting from constant amplitude, constant frequency tests, virgin fatigue life at the same stress level. This ratio is called but these results cannot be really used in many practical cycle ratio, say, C. If n1; n2; n3; . . .nk represent the number of applications. Many components are subjected to a range of overstress cycles actually endured, respectively, at the stress frequencies, stress amplitudes and mean stress levels. When levels of r1; r2; r3; . . .rk and N1; N2; N3; . . .Nk are the a material is subjected to series of overstress cycles, there respective virgin fatigue life at the same overstress levels, will be accumulation of fatigue damage at each step of the then the respective cycle ratios are: overstress level. When the fatigue damage occurring at each step is added, the cumulative effect of the total fatigue C1 ¼ n1=N1; C2 ¼ n2=N2; C3 ¼ n3=N3; . . .Ck ¼ nk=Nk: damage is obtained. This is referred to as the ‘cumulative fatigue damage’. At this stage, understanding of the fatigue This theory says that when the summation of these cycle damage of a component subjected to a variable load history is essential, which will be discussed qualitatively because of ratios at different overstress levels adds up to unity, fatigue limited quantitative information. Pfailii¼¼u1kreCwi i¼ll occur, which can be expressed mathematically as It is well established that fatigue cracks are formed during 1, i.e. the first few cycles of loading, but the progress of fatigue damage, related to the propagation of cracks, depends on the n1 þ n2 þ n3 þ ÁÁÁ þ nk ¼ 1; or; Xi¼k ni ¼ 1 order in which the stress levels are applied to a specimen or a N1 N2 N3 Nk Ni component. Experiments (Sines and Waisman 1959) on i¼1 propagation of fatigue cracks at various constant-stress levels were performed until fracture takes place. Cracks are found ð8:25Þ to be fine at low stress levels and coarse at high stress levels. In this experiment, the crack length at each stress level was where measured as a function of the number of cycles. It was observed that with decreasing stress levels, the critical crack ri ith stress level, length at which fracture occurs and the number of cycles ni number of cycles applied at ri, Ni virgin fatigue life at ri and the value of Ni can be obtained from the S–N diagram, k number of stress levels in the block loading spectrum. Considering only two stress levels for simplicity, the linear cumulative fatigue damage theory has been illustrated

8.8 Stress Fluctuation and Cumulative Fatigue Damage 337 Fig. 8.20 Schematic S–N σmax diagram, illustrating graphically S – N curve the linear cumulative fatigue damage theory for loading at two log N1 stress levels B σ1 A log(N1/n1) E log n1 σ2 F log N2 C D log(N2 – n2) log N2 N2 – n2 log N 103 104 105 106 107 108 109 graphically in the S–N curve of Fig. 8.20. Suppose, at stress From (8.28), it is clear that whenever the stress is chan- level r1, where the virgin fatigue life is N1 cycles, a speci- ged, the line that represents the process on the diagram is men is subjected to n1 cycles that start from point A and always drawn parallel to the S–N curve. continue up to point B in Fig. 8.20. Now, the stress level is changed to r2, where the virgin fatigue life is N2 cycles, and A major limitation of the Palmgren–Miner cumulative then additional n2 cycles are applied to the same specimen fatigue damage theory given by (8.25) is that it is indepen- for failure to occur. According to (8.25), dent of the order in which the block loads are applied to the specimens or components, i.e. the sequence of loading does n1 þ n2 ¼ 1; or; n1 ¼ 1 À n2 ¼ N2 À n2 ð8:26Þ not affect the rule. But in reality, the amount of damage N1 N2 N1 N2 N2 accumulated depends on the order of the loading and varies nonlinearly with the number of cycles applied at each stress The ratio ðN2 À n2Þ=N2 represents the fraction of fatigue level. For example, if a high stress level followed by a low life N2 that was consumed during the first loading cycle at stress level (i.e. where r1 [ r2) is appliedPto unnotched stress r1. It is known as the damage ratio. Equation (8.26) specimens, test data generally indicate that ðni=NiÞ\\1, indicates that the fraction of fatigue life consumed during the i.e. overall fatigue life is less than that predicted by (8.25), first loading cycle is equal to the cycle ratio at stress r1, i.e. and in such cases, the Miner’s rule is non-conservative. equal to C1 ¼ n1=N1, which can be applied to any subse- Consequently, nonlinear cumulative fatigue damage models quent loading cycle regardless of the stress used. Since in the have grown (Halford 1997). Bennett (1946) has shown that S–N curve, N is plotted on a logarithmic scale, it follows increasing cycle ratio at high stress level results in a greater reduction in the fatigue limit or the cyclic life at lower stress from Fig. 8.20 that the distance BE ¼ log N1 À log n1 ¼ level. The reason may be that since at high stress levels, the logðN1=n1Þ. But according to (8.26), propagation of crack starts at a shorter time, the fatigue damage during the initial cycles at lower stress level will be logðN1=n1Þ ¼ log½N2=ðN2 À n2ފ ð8:27Þ more than normally expected because of truncation of the ¼ log N2 À logðN2 À n2Þ ¼ BE crack initiation process by the high stress level. On the other hand, when r1\\r2, i.Pe. a low stress level is followed by a At the stress level r2, let us assume the distance FC on high stress level, then ðni=NiÞ [ 1, for some alloys. Such the log scale in Fig. 8.20 represents the amount of the fati- understressing causes the fatigue limit of certain materials gue life N2, which was consumed during the first loading like iron and titanium alloys that have ability to undergo cycle at stress r1. Since the remaining life at stress r2 is n2 strain ageing to increase somewhat because of coaxing cycles, so the amount consumed was N2 À n2; hence effects. When stress levels are frequently changed with application of stresses for a fePw cycles at a time, it has been FC ¼ logðN2 À n2Þ. From Fig. 8.20 and (8.27), it can be experimentally observed that ðni=NiÞ tends to give values seen that the distance closer to 1. Thus with random loading spectra, (8.25) is expected to produce good results at several stress levels. CD ¼ log N2 À logðN2 À n2Þ ¼ logðN1=n1Þ ¼ BE ð8:28Þ

338 8 Fatigue Another limitation of this theory is that the amount of where the initial stress amplitude, ra ¼ re, at the start when damage accumulation is independent of stress amplitude or n ¼ 0, and rR is the stress amplitude at rupture when the stress level. To ensure conservative predictions with this rupture of specimen finally takes place at n ¼ nR. Since re; k theory, it is suggested that the values of virgin fatigue life at and a are constants, (8.33) after integration gives each stress level should be taken from the S–N curve for a high percentage of survival or from the lowest S–N curve \" À reÞm þ 1#rR ðrR À reÞmþ1 ¼ kaðm þ 1Þ available, e.g. the lower boundary of a scattered band. ðra m þ1 ¼ ka; or; When the stress amplitude varies continuously with the number of cycles rather than in steps, the summation sign in re (8.25) may be replaced by an integral sign, which then becomes ) rR ¼ re þ ½kðm þ 1ފ a1 1 ¼ re þ k0ai ð8:34Þ mþ1 mþ1 ZnR dn ¼ 1 where k0 ¼ ½kðm þ 1ފ1=ðm þ 1Þ, and i ¼ 1=ðm þ 1Þ. The value N of exponent i for ferrous metals has been observed to range 0 ð8:29Þ from 0.3 to 0.7. Prot (1948) developed a method of fatigue testing in 1948 nR the number of stress cycles at which rupture takes using increasing rather than constant-stress amplitudes. He place. obtained an equation similar to (8.34) but the main differ- For the condition of constant mean stress, if fatigue life, ence is that in his equation, the value of the exponent N, is a known function of the stress amplitude, ra, which is i ¼ 0:5, because he assumed a simple hyperbolic relation for again a known function of n, then the (8.29) can be solved. S–N curve with m ¼ 1. The form of Prot equation that For example, assume that a specimen is subjected to a corresponds to (8.34) is: completely reversed stress cycle, where the mean stress rR ¼ re þ k0 pffiffi ð8:35Þ rm ¼ 0, in which the stress amplitude starts from the a endurance limit, re, of the material and increases with each cycle by a constant amount, say a. Hence, the stress Investigations (Corten et al. 1954) have shown that the amplitude as defined above will be given by the following experimental data are better accommodated by the more equation: general equation, (8.34), than by the Prot equation, (8.35), and so, (8.34) is more reliable than (8.35). ra ¼ re þ an ð8:30Þ 8.9 Stress Concentration Effect where The majority of fatigue failures taking place in service life are due to the effects of stress concentration. The stress concen- n is the number of stress cycles applied to the specimen, tration can arise from any sharp change in the cross-section or and other geometrical discontinuities such as fillets, notches, holes, screw threads and keyways, etc. and also from surface a is the increase in stress amplitude per cycle. roughness and structural irregularities such as inclusions, blowholes or porosity and decarburization. Since stress Since re and a are constants, concentration is an important factor responsible for the majority of fatigue failures, it should be kept to a minimum ) dn ¼ dra ð8:31Þ level as far as possible. Most of the parts require some change a in the cross-section, but careful design can minimize or avoid stress concentrations. For example, if a sharp corner is used at Let us further assume that the S–N curve is of hyperbolic a change in section, the stress concentration developed will be form, which is mathematically represented by (8.11) the maximum, as indicated by the crowding of the elastic according to Weibull, i.e. lines of force at the sharp corner in Fig. 8.21a, but the stress concentration will be reduced to a smaller level if a filet is ðra À reÞmN ¼ k; or; N ¼ ðra k ð8:32Þ introduced as shown in Fig. 8.21b. The higher the radius of À reÞm the fillet, the lower is the stress concentration. If grooves or holes are introduced in the vicinity of the fillet, they will help Substitution of the values of dn from (8.31) and N from to distribute the stress more evenly, as indicated by the elastic (8.32) into (8.29) gives ZrR ðra À reÞmdra ka ¼ 1 ð8:33Þ re

8.9 Stress Concentration Effect 339 (a) (b) (c) (d) Elastic lines of force Fig. 8.21 Methods of reducing stress concentrations at a change in concentration by introducing a filet. Placement of grooves (c) or holes section. a Crowding of the elastic lines of force at the sharp corner (d) in the vicinity of the fillet distributes the elastic lines of force more causing to develop stress concentration. b Reduction of stress evenly, causing further reduction in the stress concentration lines of force in Fig. 8.21c, d, and thus cause further reduction number of cycles, say N cycles, rn, or the fatigue limit, re, of in the stress concentration. a member with no stress concentration to the fatigue strength As a first approximation, the severity of a particular stress at N cycles, rn0 , or the fatigue limit, re0 , of the same member with the specified stress concentration, i.e. concentration might be quantitatively evaluated by using theoretical or elastic stress concentration factor, Kt, which Kf ¼ rn or re ð8:36bÞ is defined as the ratio of the maximum stress at the tip of the rn0 r0e discontinuity to the applied nominal stress, usually com- The value of rn0 or r0e is generally determined by testing a puted using the minimum cross-section. The stress concen- specimen containing usually a ‘V’-notch or a circular notch with a purely alternating stress cycle, because Kf has been tration factor has been discussed in Sect. 9.3 of Chap. 9. defined assuming a completely reversed stress cycle, in Assuming elastic behaviour, the fatigue strength, r0n, at which the mean stress rm ¼ 0. In other types of fatigue N cycles or the fatigue limit, r0e, for a component having loading, no standard definition for Kf has been framed. geometrical discontinuity, would be expected to reduce by a When rm 6¼ 0, for design purposes in ductile materials, the factor equal to Kt, developed due to that discontinuity. If rn factor Kf should be applied only to the stress amplitude is the fatigue strength at N cycles or re is the fatigue limit determined from the S–N diagram for a component free from component and not to the mean stress component, whereas geometrical discontinuity, then for purely elastic situation: for brittle materials, Kf should also be applied to the mean r0n ¼ rn or r0e ¼ re ð8:36aÞ stress component. Note that the value of experimentally Kt Kt determined Kf is usually less than that of Kt. Variables that For example, for an infinitely wide plate having a small are found to affect the values of Kf and notched fatigue circular hole, Kt ¼ 3, according to (9.11), and if the fatigue limit of this plate in smooth, i.e. hole-free condition is strength are: 450 MPa, then the fatigue limit of this plate in the presence of this above circular hole would be ð450=3Þ MPa ¼ 70 MPa. • Type, strength and previous treatment of the material; In reality, the decrease in the fatigue strength at N cycles or • Geometry of the discontinuity, i.e. size, shape, type and the fatigue limit is less than that predicted by the magnitude of Kt, because localized yielding at the root of the disconti- severity (root radius) of the discontinuity; nuity occurs in fatigue. Yielding blunts the tip of the dis- • The type of loading; continuity and increases the root radius and reduces the notch • The magnitude of cyclic stress; root stress. Due to this, stress concentrations are reduced • The number of cycles endured. somewhat from their elastic values. The actual effectiveness of the stress concentration in reducing the fatigue limit or So, while giving a value for Kf , all these variables should fatigue strength is expressed by the fatigue strength reduction factor, Kf , also known as fatigue notch factor. This factor is be mentioned. defined as the ratio of the fatigue strength at a specified Figure 8.22 shows schematically the variation of Kf and Kt with notch root radius, qt, which is a measure of the severity of the discontinuity. When the value of qt is very large, both Kf and Kt are quite low and approach the value of

340 8 Fatigue Stress concentration factor, Kt fully ineffective and does not cause any reduction in the fatigue strength, then the notch sensitivity, q ¼ 0, while if Kf ¼ Kt, i.e. the notch exerts its full theoretical effect, then the notch sensitivity, q ¼ 1. From (8.37a), the fatigue notch factor can be described in terms of the material notch sensitivity as: Kf and Kt (same scale) Kf ¼ 1 þ qðKt À 1Þ ð8:37bÞ Fatigue notch factor, Kf where usually 1 Kf Kt. However, q for a given material is found to vary with the type and root radius ðqtÞ or severity 1.0 Notch tip radius, ρt of notch, the type of loading and the specimen size, and so it 0 is not a true material constant. The variation of fatigue notch Fig. 8.22 Schematic variation of fatigue strength reduction factor, Kf , sensitivity index, q, with the notch root radius, qt, for and elastic stress concentration factor, Kt, with notch root radius, qt materials of different tensile strength, Su, is illustrated schematically in Fig. 8.23. All curves in Fig. 8.23 start from one, where the notch effect tends to disappear. As the value the origin, i.e. at qt ¼ 0; q ¼ 0, because at qt ¼ 0; Kf ¼ 1. of qt decreases that means as the severity or the sharpness of The notch sensitivity, q, increases markedly with increasing the notch increases, Kt continuously increases, and when qt and approaches unity for large values of qt, as Kf is qt ! 0; Kt reaches a large value. On the other hand, with almost equal to Kt. From Fig. 8.23, it is further seen that the decreasing qt; Kf increases more slowly than Kt until it notch sensitivity, q, increases with increasing tensile attains a maximum value at certain low radius and thereafter strength, Su. Since materials of high strength normally pos- decreases and reaches the value of 1 at qt ¼ 0. The critical sess a little intrinsic plastic deformation capacity, so the notch radius for maximum Kf is probably related to the grain size of the material, but the exact value of critical radius and capacity for notch-tip blunting to increase the notch-tip the exact variations of Kf at smaller radii are not fully known. Therefore, notches with smaller radii, i.e. very sharp radius and thereby to reduce the stress concentration effect is notches, should not be assumed to be safe in fatigue. At low limited for high-strength materials. Hence, increasing tensile values of notch radius, it has been noted in some cases that strength or hardness of a material can lower its fatigue the value of Kt is as high as two or three times that of Kf . It indicates that as Kt increases, susceptibility to fatigue dam- performance in certain circumstances. age, measured by Kf , becomes less severe, i.e. the ratio of An empirical expression for Kf in terms of qt developed Kf =Kt decreases. It is expected that with increasing Kt, fatigue cracks would initiate more readily but might not by Neuber (1946) is: always grow to cause fracture. Hence, ‘non-propagating cracks’ found at stress concentrations beyond some critical Kf ¼ 1þ Kpt Àffiffiffiffi1ffiffiffiffiffiffiffiffi ð8:38aÞ 1 þ qN0 =qt value may be the reason for the above discrepancies between Kt and Kf . where The effect of stress concentration in fatigue can be con- qt contour radius at the tip of the notch, qN0 a material constant representing characteristic length, veniently studied by fatigue notch sensitivity index, q, which is defined as known as Neuber constant. Its value varies not only with the type of material but also with its tensile strength, Su, and heat treatment. The unit of qN0 is the same as that of qt. Combining (8.37a) and (8.38a), an equation for q relating with qt can be obtained as follows: q ¼ Kf À 1 ¼ 1 þ p1ffiffiffiffiffiffiffiffiffiffiffiffi ð8:38bÞ Kt À 1 qN0 =qt q ¼ Kf À 1 ð8:37aÞ It has been found that the value of qN0 increases as the Kt À 1 type of material is changed from a stronger and harder where 0 q 1. Equation (8.37a) represents the ratio of material like steel to a softer and ductile material like alu- effectiveness of notch in fatigue to that in a purely elastic minium alloy. Further, for the same material, the value of qN0 situation. From (8.37a), we see that if Kf ¼ 1, i.e. the notch is is found to increase with decreasing the strength level, for example, an aluminium alloy with tensile strength Su ¼ 600 MPa has a value of Neuber constant, qN0 ¼ 0:4 mm,

8.9 Stress Concentration Effect 341 Fig. 8.23 Schematic variation of Notch-sensitivity index, q1.0 fatigue notch sensitivity index, q, Su1 with notch root radius, qt, for materials of different tensile Su2 strength, Su Su3 Tensile strength, Su1> Su2 > Su3 0 0 Notch root radius, ρt while that with Su ¼ 150 MPa has q0N ¼ 2 mm. For large of one cycle to the unnotched endurance limit re divided by notches with large values of qt; Kf approaches Kt, as seen Kf at 107 cycles for ferrous metals or at 108 cycles for from (8.38a) and the value of q is almost equal to one, as non-ferrous metals. seen from (8.38b). For sharp notches with small values of 8.10 Size Effect qt; Kf ( Kt (8.38a) and the value of q is very low (8.38b) It has been found in most cases that fatigue strengths of for soft ductile metals with high values of q0N, although for small-diameter specimens tested in the laboratory usually stronger metals having low values of q0N, both Kf and q are show greater values than those of large-diameter parts used higher. In general, hard metals are usually more notch sen- in service. The phenomenon of the size dependency of the test results is called the size effect, and it is one of the most sitive than softer metals. important problems encountered in fatigue applications. Since the size of the test bar controls the fatigue strength, it A somewhat similar empirical formula relating q with qt would be erroneous to predict the fatigue performance of developed by Peterson (1974) is: large-diameter parts in service from the laboratory test results on small-diameter specimens. Large parts in service q ¼ 1 þ 1 ¼ Kf À 1 ð8:39Þ often fail due to stress concentrations. Fatigue strengths of ðq0P =qt Þ Kt À 1 service parts of large diameter are difficult to determine precisely from geometrically similar laboratory specimens of where qP0 is another material characteristic length and will be large diameter because it is usually impossible to duplicate designated as Peterson constant. For aluminium alloys, qP0 is the same stress concentration, residual stress and metallur- estimated to be 0.635 mm, whereas an empirical relationship gical structure throughout the cross-section of prepared between the tensile strength Su and q0P is found to exist for laboratory specimens as those present in service parts. Fur- steels as follows (Web Site 1): ther, the usual capacities of fatigue testing machines are limited for conducting experiments with large-diameter qP0 ¼ 0:025420S7u 01:8 ð8:40Þ specimens. where the unit of Su is MPa and that of qP0 is mm. When the diameter of a fatigue specimen is increased, The fatigue strength r0n or the fatigue limit r0e of the there is an increase in the volume or surface area of the specimen. Since the weak points that cause fatigue failures notched component can be determined from (8.36b) by are usually at the surface, some investigator assumes that the increase in surface area is responsible for the decrease in taking the unnotched fatigue strength rn or the unnotched fatigue strength or fatigue limit for larger sized parts. fatigue limit re from S–N diagram, provided Kf is known for the given conditions. Finally, a suitable factor of safety is applied to rn0 or r0e to determine the design stress. In the absence of data, the semilogarithmic plot for S–N curve of the notched material may approximately be made by draw- ing a straight line from the ultimate tensile strength at a life

342 8 Fatigue Further, more important is that for smooth or notched Let us assume that the critical distance from the outer specimens loaded in bending or torsion or for axially loaded notched specimens, an increase in diameter usually decrea- surface of the specimen is Dr, where the flexural stress r ses the stress gradient and increases the volume of material decreases to the minimum stress r0, so that which is subjected to high stress levels. It has been strongly confirmed that the size effect is related to the stress gradient r0 ¼ MB ðr À DrÞ ð8:42Þ existing in the specimen. I When either smooth or notched specimens are subjected Dividing (8.41) by (8.42), we get to bending or torsion, it is more commonly observed that fatigue strengths and fatigue limits decrease with increasing r ¼ r r ¼ 1 À 1 or r ¼ 1 À r0 ð8:43Þ diameter of the specimens. This size effect is also observed r0 À Dr ðDr=rÞ ðDr=rÞ when notched specimens are subjected to axial tension– compression loading, but no size effect is noted in axial In case of axially loaded smooth specimen, as the entire loading of smooth specimens (Phillips and Heywood 1951). cross-section is stressed equally, i.e. r ¼ r0, so no size effect In the above all cases, where a size effect is noted, a stress is expected as seen from (8.43). However according to (8.43), gradient exists in the specimen. Let us compare the case of bending and axial loading of unnotched specimen of same a size effect is expected to exist in small-sized specimens that cross-section. In bending, the smaller the diameter of the test piece, the higher is the stress gradient and the smaller is the possess a large stress gradient, i.e. in smooth or notched small volume of material enduring high stress levels. Conversely, no stress gradient exists in axially loaded smooth specimen specimens loaded in bending or torsion or in axially loaded and the entire cross-section. i.e. a larger volume of material notched small specimens. In these cases, since r [ r0, so the in the same size of specimen is subjected equally to high fatigue strengths are greater. Finally, it is evident from (8.43) stress levels. As a result, a higher value of fatigue strength is observed in bending than in axial loading. From a statistical that the size effect vanishes for large-sized specimens since point of view, the larger the highly stressed volume of r ) Dr, i.e. ðDr=rÞ is negligible, which makes r0 to approach material, the greater is the probability of finding a weak point r. The existence of stress gradient seems to be more rational or an imperfection that would cause failure at a lower stress. criterion of size effect than the change in surface area due to Let us consider an estimation of the specimen size effect the change in diameter of specimen. in fatigue testing, as given by Findley (1972). Large-diameter specimens having shallow stress gradients 8.11 Surface Effects and Surface Treatments show lower fatigue strengths. This fact leads us to assume that fatigue crack will initiate when cyclic slip develops over Fatigue cracks almost always are nucleated at the surface of a certain critical depth from the outer surface of the speci- a component. Some of the factors responsible for initiation men requiring a minimum value of stress, say r0, then slip of most fatigue failure at the surface are: can occur only in the outer fibres of the specimen, where the applied stress, say r, is greater than the minimum stress r0. • Surface is the source for the development of many stress In bending, the applied stress, r, at the outer fibre of the concentrations, such as surface scratches, machining specimen of circular cross-section is [see (4.14)]: marks, dents and fillets, etc. r ¼ MBr ð8:41Þ • The surface roughness caused by corrosive attack and I oxidation also acts as stress raiser. • For bending and torsion types of loading, the maximum stress develops at the surface. • The formation of surface discontinuities, such as intru- sions and extrusions, caused by cyclic slip leads to the formation of actual fatigue cracks. where Since fatigue properties are very sensitive to surface conditions, great changes in property are possible to achieve r flexural stress developed at the outer fibre of the by certain surface treatments that would locally modify the specimen, surface of fatigue specimen or component. A number of surface treatments that have been developed may be divided MB bending moment applied to the specimen, into three broad categories: I moment of inertia of the cross-sectional area about its • Mechanical treatments, such as shot peening, surface neutral axis, cold rolling, grinding and polishing; r distance from the neutral axis to the outermost fibre of the specimen, i.e. radius of circular specimen.

8.11 Surface Effects and Surface Treatments 343 • Thermal treatments, such as flame- and condition that any changes in the fatigue strength of the induction-hardening heat treatments, heat softening and surface material will greatly alter the fatigue performance of thermal contraction techniques; and the material. • Surface coatings, such as case carburizing, nitriding and Case hardening by either carburizing or nitriding of steel electroplating. parts considerably improves their fatigue strengths due to the formation of harder and stronger surfaces. Steel parts to be Again, the surface treatments mainly produce changes in carburized are usually heated at 850–950 °C for about the 8–15 h in a carbonaceous atmosphere to form a harder carbide-rich layer of about 0.8–2.5 mm deep from the sur- • Stress concentrations at the surface or surface roughness; face, while in nitriding, usually alloy steel parts are heated • Tensile as well as fatigue strength of the surface metal; between 500 and 650 °C for a period of about 10–40 h in an • Residual stress condition of the surface. anhydrous ammonia atmosphere where nitrogen reacts with the nitride-forming elements within the alloy steel to form a The above changes coupled with the surface treatments harder nitrided layer of about 0.5 mm deep from the surface. will now be discussed in the following sections. In both cases, the tensile strength as well as the fatigue strength increases within the carburized case and the nitrided 8.11.1 Surface Roughness and Treatment case. The extent of strengthening depends on the depth of the case hardening and the diameter or thickness of the part. As the surface is a potential source of weakness, the con- dition of surface is very important. A rough surface can act Similar to carburizing and nitriding, flame- and as a strong stress raiser and reduce the fatigue strength by as induction-hardening heat treatments of steel parts may also much as 15–20%. The type of surface preparation greatly improve their fatigue strengths. In these heat treatments, the affects the fatigue life and fatigue strength of the member. surface of steel parts is preferentially heated into the For example, the median fatigue life of a ground and pol- austenite-phase region so that only thin layers at the surface ished specimen with a better surface finish may be around 9– are transformed to homogeneous austenite phase and the 10 times greater than that of a lathe-formed specimen having core regions remain untransformed. Thereafter, steel parts a poor surface finish. The surfaces of all fatigue specimens are quenched rapidly from the austenite-phase field so that and members must therefore be carefully prepared by slow austenite present at those thin surface layers is transformed grinding and polishing operations which will eliminate into hard, untempered martensite phase. Since the hardness coarse scratches and damaged surfaces without introduction and tensile strength of this martensitic layer are increased of any new residual stresses. Specimens and members are noticeably, the fatigue strength is also improved. To heat a smoothly polished in such a way that the finely formed steel part, flame of oxy-hydrogen or oxy-acetylene torch is scratches acting as stress raisers are oriented parallel with the usually used in flame hardening, whereas in induction direction of the applied principal tensile stress. Because hardening, a high-frequency alternating current passing specimens with such orientation of fine polishing scratches through a water-cooled copper induction coil that surrounds show the best properties in fatigue test. Such carefully pol- the part to be treated is generally used. The depth of the ished specimens are known as ‘par bars’, which are usually hardened (martensitic) zone for a given steel part in flame used as fatigue test pieces in laboratory. The stress con- hardening depends on the flame intensity, heating time or centration produced by surface irregularities or roughness is speed of flame travel and in induction hardening on the relaxed by plastic flow which can be carried out by cold frequency of alternating current. working the specimen. Each of these above treatments—carburizing, nitriding, 8.11.2 Surface Properties and Treatment flame hardening and induction hardening—not only improves fatigue strength by forming higher strength mate- Fatigue strength of a material is normally dependent on its rial on the surface but also develops a favourable compres- tensile strength, i.e. higher the tensile strength, higher is the sive residual stress in the hardened surface layer which is fatigue strength. Hence, anything that increases or decreases beneficial to fatigue properties. The beneficial effect of the tensile strength of the surface material will also, compressive residual stresses developed in surface layer has respectively, increase or decrease the fatigue strength at the been discussed in the next section. surface. Since fatigue failure depends so much on the surface One of the most widely used mechanical treatment for improvement of fatigue properties involves the use of shot peening. In this technique, fine steel or cast iron shots (balls) ranging from about 0.18–4.45 mm in diameter are projected against the surface to be treated at high velocities (perhaps

344 8 Fatigue 60–61 m s−1) (Sines and Waisman 1959). Another important surface layer is added to an externally applied tensile stress mechanical treatment used commercially is the cold rolling on that surface, the surface of the part, from which most of surface of parts with contoured rollers. The fatigue cracks initiate, is subjected to a lower value of tensile above-mentioned both mechanical treatments cause some stress and thus the probability of fatigue failure at that sur- increases in the tensile strength and also in the fatigue face of the part is reduced. Hence, compressive residual strength of the material at and near the surface due to strain stresses existing in the surface layers of the part are con- hardening that results from plastic deformation of surface sidered to be beneficial for fatigue, while surface tensile materials during these processes. Particularly in low-strength residual stresses are detrimental to fatigue performance. alloys having high work-hardening capacity, strain harden- ing of surface materials contributes to a higher fatigue Let us consider a beam whose surface has been treated strength associated with the higher tensile strength. The with shot peening and then subjected to bending fatigue. The aforementioned beneficial effect caused by either shot linear distribution of externally applied elastic bending peening or surface cold rolling is of secondary importance, stresses in the beam without any residual stress is shown in and the primary beneficial effect of either of these processes Fig. 8.24a. Figure 8.24b shows a typical distribution of only is to develop favourable compressive residual stress in the residual stress pattern, as expected to be produced by shot surface layer, which is discussed in the next section. peening of the beam. The condition of the material at and near the surface undergoes a number of changes by the 8.11.3 Surface Residual Stress and Treatment action of shot peening (Sherrett 1966). The foremost important contribution of shot peening is the development of The most effective manner of improving fatigue properties is a thin layer of compressive residual stress, whose depth of probably to produce a favourable compressive residual stress penetration from the surface is about one-quarter to one-half pattern at the surface. Residual stresses can be considered as of the shot diameter. Since the shot peening increases the locked-in stresses present in a component which experiences volume of the surface layers by plastic deformation process, no external force. Residual stresses arise when there are the elastic materials surrounding the surface layers force the non-uniform volume changes in the material from the sur- permanently deformed peened regions back towards their face to the centre, which might occur due to thermal gradient or non-uniform plastic deformation from the surface to the (a) σmax Tension centre, or due to localized-phase transformation, or any other reasons. If the volume of the surface material is higher than MB MB the rest of the material, then the surface materials tend to stretch the interior materials having lower volumes and thus Compression σmax induce tensile residual stresses in the inner regions, while the inner regions produce compressive residual stresses in the (b) σR surface layers to balance the stresses over the cross-section of the part. The higher volume in the surface material may σR occur due to more thermal expansion at the surface having higher temperature than the interior during rapid heating (c) σmax + σR particularly for thicker sections, or due to plastic deforma- tion in tension at the surface (the centre is elastically MB MB deformed), or due to some phase transformation only at the surface, e.g. austenite to martensite transformation in steel σmax + σR during rapid cooling. Similarly, if the volume of the surface material is less than the rest of the material, then the surface Fig. 8.24 a Linear distribution of externally applied elastic bending materials experience tensile residual stresses and the interior stresses in a beam without any residual stress. b A typical distribution materials compressive residual stresses. This may happen of only residual stress pattern, as expected to be produced by shot due to more thermal contraction at the surface having lower peening of the beam. c Schematic distribution of stresses in the beam temperature than the interior during fast cooling particularly due to algebraic summation of externally applied elastic bending for thicker sections, or due to plastic deformation in com- stresses and the residual stresses pression at the surface (the centre is elastically deformed), or due to some phase transformation only at the surface, e.g. austenitization of plain carbon steel while heating it. Note that when a compressive residual stress developed in the

8.11 Surface Effects and Surface Treatments 345 initial dimensions, and thus, compressive residual stresses surface regions, which improve fatigue properties. Similarly, are induced in the peening regions which must be balanced in flame- and induction-hardening heat treatments of steel, or by tensile residual stresses over the interior of the when shallow-hardening steel is drastically quenched, the cross-section. The magnitude of the maximum compressive austenite to martensite-phase transformation in the surface residual stress can reach about one-half of the yield strength layers of the treated part involves an expansion in volume. of material depending on the hardness, type and diameter of The untransformed interior regions restrain this expansion shot, velocity and pressure of shot stream, and duration of and develop favourable residual compressive stresses in the the peening process. As a result, higher strength alloys are hardened surface layers, which contribute to the improve- benefitted more than the weaker ones by the peening pro- ment in fatigue properties. Similar to shot peening, the cess. The distribution of stresses in the beam resulting from fatigue crack initiation site shifts from the surface of the algebraic summation of externally applied elastic bending component to the case–core boundary region in carburizing, stresses and the residual stresses produced by shot peening nitriding, flame- and induction-hardening treatments, process is shown in Fig. 8.24c. Note that the maximum because (1) the peak tensile stress is displaced to the case– tensile stress at the surface of the beam, which is the most core boundary, and (2) the intrinsic strength of the core is damaging part of the applied stress range, is reduced by an substantially lower than that associated with the case amount equal to the magnitude of the surface compressive material. residual stress. The peak tensile stress shifts to the boundary between peened- and unpeened-region in the interior of the The role of compressive residual stress in improving beam. The value of this peak tensile stress depends on the fatigue performance is more effective for cases where a high distribution of residual stress as well as on the applied stress stress gradient is present in a fatigue component than cases gradient. Hence, the fatigue crack initiation site shifts from where no stress gradient exists. A steep stress gradient exists the surface of the beam to the region of the peak tensile in smooth or notched fatigue components loaded in bending stress, i.e. peened–unpeened boundary region. Note that shot or torsion or in axially loaded notched fatigue components. peening process roughens the surface by producing small So, these components undergoing any of the treatments, ‘dimples’, which would act as countless stress raisers and such as carburizing, nitriding or shot peening, which pro- have a harmful effect on fatigue life. So to be on the safe duce surface residual compressive stresses show greater side, the shot-peened surface of the component is carefully improvement in fatigue properties than axially loaded polished to reduce surface roughness and to achieve addi- smooth fatigue components that have also undergone the tional improvement in fatigue properties. Shot peening is same treatment. However, thermal activation or plastic particularly applied to mass-produced components of quite deformation may change the residual stress distributions. small size. Shot peening can be used on almost any surface Application of high stresses in the low-cycle fatigue region except an inner surface and is therefore a versatile process. or periods of overloading in the fatigue cycle may cause plastic deformation, which in turn leads to rapid fading of The main contribution of shot peening and surface cold residual stress. So, the influence of residual stresses on the rolling in the improvement of fatigue properties is to develop fatigue properties is little at high applied stress levels, a favourable compressive residual stress in the surface layer, whereas surface compressive residual stress is highly bene- but the compressive residual stress produced by surface cold ficial in the high-cycle portion of the fatigue life associated rolling can penetrate deeper than that produced by shot with lower stress levels, particularly near the fatigue limit. peening. Moreover, surface cold rolling does not roughen the surface of the part. It is particularly applied to large parts Other methods of introducing surface compressive possessing surfaces of rotation, such as bearing surface of residual stresses include heat softening and thermal con- railroad axles, threaded steel bolts and fillets of crankshafts. traction techniques. Heat softening is applied to rubber tire Similar to shot peening, subsurface initiation of fatigue crack treads. When the tread is bent, tensile stress is developed at is possible in a surface-rolled component. It is important to its outer convex surface. Now, steam jets are applied to heat note that excessive shot peening or surface rolling may the outer convex surface, as shown at B in Fig. 8.25. As the damage the surface and the improvement in fatigue prop- surface is softened by heat, it flows and the tensile stress is erties may not result from these mechanical treatments. relieved. When the tread is released, the stresses in the inner Hence, it is required to establish the proper conditions of surface spring it back from the bent configuration to its producing the optimum distribution of residual stresses. original size and shape, as shown at A in Fig. 8.25. The outer surface of the rubber tire is now subjected to a favourable Introduction of carbon in carburizing, or nitrogen in ni- compressive residual stresses, which make the rubber tire triding, causes volumetric expansion in carbide-rich or less susceptible to fatigue. In thermal contraction technique, nitrided surface layers. The interior regions of carburized or the material is first heated to produce a uniform expansion nitrided parts tend to resist this expansion and produce throughout its cross-section and then the surface is cooled favourable compressive residual stresses in the hardened rapidly than the interior. The surface contracts and produces

346 8 Fatigue A • Electroplating of the surface of steel. Nickel and chro- Original unbent rubber tire treads mium plating of steel produce surface tensile residual stresses and impair the fatigue resistance of steel, while E A softer cadmium, zinc, lead or tin plating has little effect T M on the fatigue strength of steel. S • Quenching of deep-hardening steel, where the steel part may be hardened up to its centre, i.e. the transformation B of austenite to martensite occurs throughout the cross-section, produces poor fatigue strength, because Tire treads subjected to this treatment results in an unfavourable tensile residual bending force stress field at the surface, which may persist at low-tempering temperatures. Fig. 8.25 Compressive residual stress developed by application of heat softening on rubber tire treads (Courtesy Prof. K. Biswas, IIT • Decreasing the frequency of alternating current during Kharagpur) induction hardening of steel, which may lead to austen- itization of the whole cross-section of the part instead of a compressive stress in the warm interior, which is softer and only at the surface layers. Upon quenching, austenite to thus easily yields to maintain equilibrium over the whole martensite transformation occurs throughout the cross-section. After the surface has cooled, the interior cools cross-section, which produces unfavourable tensile and contracts. This contraction introduces a favourable residual stress pattern at the surface and decreases the compressive residual stress in the surface. This technique is fatigue strength. used to form tempered glass. • Inadequate quenching of steel, which produces local soft 8.11.4 Metallurgical Processes Detrimental spot having poorer fatigue strength. to Fatigue • Excessive localized surface heating during severe There are some metallurgical processes that cause a large grinding of steel, which may result in reversion of the decrease in fatigue properties. Some of these processes are: steel to austenite, and upon quenching, this austenite transforms into brittle martensite, which deteriorates fatigue performance. Hence, fatigue properties of steel may be improved considerably, if a favourable compressive residual stress field is introduced at the surface and decarburization of the steel is avoided. In fact, Harris (1965) showed that when threaded steel bolts were subjected to surface cold rolling rather than cutting and decarburization of the steel was prevented, the fatigue limit of the threaded steel bolts increased by over 400%. • Application of a soft coating to a stronger metal or alloy, 8.12 Effect of Metallurgical Variables because this reduces tensile as well as fatigue strength of the surface material. For example, when a sheet made of Some metallurgical factors that affect fatigue properties are stronger age-hardenable aluminium alloy is coated with mechanical properties, microstructures and heat treatments. soft aluminium, its fatigue strength is reduced. An overview of these effects is presented in this section. The effects of some metallurgical variables on fatigue behaviour • Decarburization of the surface of steel during a heat are usually measured by testing smooth-polished fatigue treatment, which can seriously degrade hardness, tensile specimens under conditions of purely alternating stress strength and fatigue resistance of the steel. Decarbur- cycles. Any changes in fatigue performance due to metal- ization causes loss of carbon from surface layers of steel lurgical factors that are observed by the above test are usu- and thus results in a loss of the intrinsic alloy strength at ally assumed to take place to about the same extent under the surface since carbon is the main strengthening agent more complicated fatigue conditions, such as notched in steel. Further, the higher carbon interior regions specimens loaded under conditions of combined stresses. restrain the volumetric contraction in carbon-depleted surface layers and develop unfavourable tensile residual Fatigue limit or fatigue strength of a material is often stress field at the surface, which is detrimental to fatigue related to its tensile strength. The ratio of the fatigue limit (for performance.

8.12 Effect of Metallurgical Variables 347 materials possessing a ‘knee’ in the S–N curve) or the fatigue ratio for flake-graphite cast iron is 0.42, while that for strength at 108 cycles (for materials showing a continually nodular cast iron is 0.48. The lower fatigue ratio in the decreasing S–N curve) to the tensile strength is called the flake-graphite cast iron is due to the higher notch effects of fatigue ratio. In general, the fatigue limit of cast and wrought long and sharp graphite flakes. On the other hand, if relative steels is estimated to be roughly 50% of the tensile strength of notch sensitivity between the flake-graphite cast iron and the the material, while the fatigue ratio for several non-ferrous nodular cast iron is compared (Forrest 1962), the notch metals, such as copper, nickel and magnesium alloys, is sensitivity, q, associated with an external circumferential approximately 0.35. These approximate values of fatigue ratio V-notch in the flake-graphite cast iron is 0.06 and that in the hold only for the restricted conditions where smooth-polished nodular cast iron is 0.25. The reason for the lower notch specimens are tested under zero mean stress conditions at sensitivity in the flake-graphite cast iron is that the addition room temperature. However, it has been found (Forrest 1962) of an external notch to a large number of internal flaws from rotating bending unnotched fatigue tests (with condi- already present in the form of graphite flakes creating a tions of zero mean stress) that the fatigue ratio for alloy and multitude of stress concentrations is less harmful than the carbon steels can vary between 0.35 and 0.60 and that for material having less or no internal stress concentrations wrought copper alloys can vary between 0.35 and 0.50. The arising from structural discontinuity as in the nodular cast fatigue ratio for steel in notched conditions will be lower than iron. Inclusions present in a material may also have a that in unnotched conditions and is considered to be around harmful effect on unnotched fatigue behaviour. Although 0.20–0.30. large inclusion contents in a material do not significantly alter the tensile strength, they act as potential sites for In general, finer the microstructure of a material, higher is nucleation of fatigue crack and reduce the fatigue life. So, the strength or hardness and better is the fatigue resistance of the presence of inclusions in the microstructure is not the material. So, normalized steel having fine pearlite gives desirable. If inclusions can be eliminated or reduced by better fatigue resistance than the same steel showing coarse carefully controlling melting practice, say by vacuum melt- pearlite structure obtained by annealing. By decreasing the ing, not only the fatigue life will improve but also the extent subcritical isothermal transformation temperature, the fatigue of scatter in the fatigue test results will be reduced. In fact, a limit of eutectoid steel was found (Gensamer et al. 1942) to reduction in inclusion content is not only beneficial to fati- increase in the same manner as observed for its yield strength gue performance but also to fracture toughness. On the other and tensile strength. The effect of solid-solution alloying hand, while comparing notched fatigue response between additions on the tensile properties of iron (Epremian and material with high inclusion contents and material contain- Nippes 1948) and aluminium (Riches et al. 1952) is nearly the ing no or little inclusions, the notch sensitivity of the former same as on their fatigue properties. The relation between the will be lower than the latter due to less increase in the stress fatigue limit and the tensile strength would encourage us to concentration effects in the former than in the latter. use a material with as high a tensile strength as possible to achieve the maximum resistance to fatigue, but the problem is It has been observed that the more difficult for disloca- that the environmental sensitivity increases and the fracture tions to cross slip the higher is the fatigue resistance. Dis- toughness decreases with increase in tensile strength. Note locations can easily cross slip around obstacles in materials that various strengthening mechanisms can increase yield with high stacking fault energy. This assists to develop strength of a material but the fatigue limit of the material does slip-band formation and thus the initiation and propagation not increase proportionately with its yield strength. of fatigue cracks. The fatigue strength is high in materials with low stacking fault energy because of the difficulty in The fatigue properties of metals are found to be more cross slip of dislocations. structure sensitive than their tensile properties. To verify the above, coarse pearlitic microstructure and spheroidized Generally, room temperature fatigue strength of a mate- microstructure (spheroidite) of the same tensile strength rial increases with decreasing its grain size, because the grain (Dieter et al. 1955) were produced by heat treating plain boundaries act as barriers to the propagation of a fatigue carbon eutectoid steel. When these two structural conditions crack. So, fine-grained inclusion-free alloys are usually of steel having the same tensile strength were compared, the preferred for room temperature fatigue applications. But the fatigue limit of pearlitic microstructure was found to be effect of grain size on fatigue life is found to depend on the significantly lower than that of the spheroidized stacking fault energy, i.e. slip character of material microstructure due to the higher stress concentration effects (Thompson and Backofen 1971). Tests were carried out at of long and sharp carbide lamellae in pearlite in comparison constant-stress amplitude over a life range of 104–107 cycles to spheroidite consisting of small carbide spheroids with for various grain sizes of alpha brass, copper and aluminium. large radius of curvature. Greater structure sensitivity of The grain size effect was found to be the greatest in the low fatigue properties is also seen from the fact that the fatigue stress, long-life region in which slip-band (Stage I) crack

348 8 Fatigue growth predominates. Fatigue life was found to be propor- been found experimentally (Ransom and Mehl 1952) that the tional to ðgrain diameterÞÀ1=2 in alpha brass, a low stacking fatigue limit determined in the transverse direction of steel fault energy material, in which cross-slip is difficult, whereas forgings may be only 60–70% of that in the longitudinal fatigue lives were found to be insensitive to grain size in direction. It has been confirmed (Ransom 1954b) that copper and aluminium, materials with high stacking fault non-metallic inclusions initiate almost all fatigue failures in energy in which cross-slip is easy. In high stacking fault specimens tested in the transverse direction. During work- energy materials, easy cross-slip causes dislocation cell ing, inclusions are elongated in the longitudinal direction. In structures to form readily. The cell structure masks the a transverse specimen, since the elongated inclusion stringer influence of grain boundaries so that the slip-band crack is aligned normal to the applied principal tensile stress, it can growth is not affected by change in grain sizes. On the other produce quite high stress concentration and thus results in a hand, in a low stacking fault energy material, the dislocation low transverse fatigue limit in steels containing inclusion. cell structure is absent and the crack growth is restrained When vacuum melting process is used to eliminate almost all near grain boundaries. When the grain size is increased, the inclusions, the transverse fatigue limit is increased consid- frequency with which the crack meets grain boundaries erably, but anisotropy of the fatigue limit still persists indi- decreases. Consequently, Stage I crack growth occurs more cating that factors other than inclusions may also be rapidly resulting in a decreased fatigue life with increasing responsible for the decreased transverse fatigue limit in grain size. steels. Since tensile strength and hardness are related, the fatigue 8.13 Frequency of Stress Cycling limit in a number of steel may be estimated from the hard- ness measurement, which requires relatively inexpensive Fatigue tests can be performed by applying loading cycles at machine and relatively little skill and experience in com- a great variety of frequencies. Obviously, fatigue tests at parison to use of other machines and procedures. It has been high frequencies of stress cycling are desirable to obtain test shown (Metals Handbook 1961) that fatigue limit of several data as quickly as possible. In general, as the frequency of quenched and tempered steels increases more or less linearly stress cycling is increased, the fatigue strength or fatigue life with Rockwell C hardness up to a hardness level of about RC increases, probably because of the enhanced strain rate 40, which is equivalent to a tensile strength of about associated with more rapid rate of cycling. For frequency of 1170 MPa. Above this hardness level, the determination of stress cycling over a range from about 10–200 Hz, the fatigue limit from the hardness value becomes unreliable due increase in fatigue strength is negligibly small at normal to considerable scatter in the test results. However, temperatures except for mild steel. However, in materials depending on steel, the fatigue limit of quenched and tem- with high internal friction or under stresses large enough to pered steels increases with decreasing tempering temperature produce plastic strains, considerable heating can occur at the up to hardness of RC 45 to RC 55, but at these high hardness high frequencies. Then drastic changes in the fatigue levels, the fatigue properties become extremely sensitive to strength may occur with variations in the frequency of stress surface residual stress fields, inclusion contents and surface cycling. preparation. In general, tempering of martensitic structure in steel produces the optimum structure for fatigue, as it pro- At high temperatures, the variation of fatigue strength vides maximum homogeneity. But at a hardness level above with the frequency of stress application becomes much more about RC 40, the fatigue limit of bainitic structure produced pronounced. For example, at a temperature of 650 °C, the by austempering is superior to that of the tempered fatigue strength of structural steel may be doubled when the martensitic structure with the same hardness (Borik and frequency of stress cycling is increased from 3 to 42 Hz. In Chapman 1961). The low fatigue limit observed in quenched normal fatigue tests, the reporting data consist of applied and tempered steel is due to the stress concentration effects stress and number of cycles to failure, but in fatigue tests at of needle-like or rod-shaped thin cementite plates that are high temperatures, the additional information that is to be formed by replacing epsilon carbide during the tempering of included in the report is the total time to failure. martensite usually occurring in the temperature range between 200 and 400 °C. Coffin (1971), Solomon and Coffin (1973) have analysed the effect of frequency of stress cycling on low-cycle fatigue Below a hardness level of about RC 40, the fatigue limits life at constant elevated temperature. The frequency depen- of quenched and tempered low-alloy steels of different dence of high-temperature fatigue life is illustrated chemical composition are nearly the same when the steels schematically in Fig. 8.26, drawn on a log-log plot of cycles are tempered to the same hardness level. In case of wrought to failure versus frequency of cycling. At higher frequencies products, this generalization holds good for fatigue limits of cycling, say m ! me in Fig. 8.26, the fatigue life is determined in the longitudinal direction. However, it has

8.13 Frequency of Stress CyclingCycles to failure, N, (log scale) 349 Temperature = constant Vacuum but no sudden change in fatigue properties has been indicated at temperatures below the ductile-to-brittle transition tem- Air perature. The fatigue behaviour of several FCC metals has been thoroughly investigated over the temperature range of 293–4.2 K (McCammon and Rosenberg 1957). Test shows that the fatigue ratio (ratio of fatigue strength to UTS) drops between 293 and 100 K and then rises again when the tem- perature drops below 100 K. Hence, the increase of fatigue strength becomes proportionately greater than that of tensile strength with decreasing temperature to a very low value. This indicates that the mechanism of fatigue failure at room temperature involves vacancy formation and condensation and the mechanism may change at very low temperatures. vm ve 8.14.2 High Temperature Frequency of cycling, v, (log scale) Fig. 8.26 Frequency dependence of high-temperature fatigue life at As the temperature is increased above room temperature, the constant temperature shown schematically in a plot of cycles to failure fatigue strength of metals generally decreases. A diagram of against frequency of cycling on log–log scale plastic strain amplitude, Dep=2, versus number of cycles to failure, N, at a constant frequency of cycling in air, showing independent of frequency and exhibits the maximum the effect of temperature on fatigue life, is schematically because air environment does not get sufficient time to presented in Fig. 8.27. The effect of frequency of cycling on interact and to cause fatigue damage. In such situation, the fatigue at a constant high temperature has already been mode of fracture is the usual transcrystalline fatigue failure. discussed in Sect. 8.13. Figure 8.27 shows that except at In the intermediate range of frequencies, say mm m\\me in very high strain amplitude that corresponds to low number Fig. 8.26, the fatigue life in vacuum is independent of fre- of cycles to failure, fatigue life of a material generally quency and maintains the maximum, but that in air decreases decreases with increasing temperature when frequency of with decreasing frequency of cycling because air environ- cycling is constant. Mild steel is an exception, where a ment is capable of causing local grain-boundary oxidation maximum in the fatigue limit is exhibited in the temperature that can assist significantly to initiate fatigue cracks. In this region, as the frequency of stress cycling is decreased, the Frequency of cycling, v = constant mechanism of fatigue fracture in air-tested specimens changes from transcrystalline to intercrystalline. At lower Plastic strain amplitude, ∆εp/2 frequencies, say m\\mm in Fig. 8.26, the increased fatigue damage due to pronounced interaction with the atmosphere causes the fatigue life to decrease at a faster rate with decreasing frequency of cycling, although the fatigue lives in vacuum-tested specimens will be higher than those in air-tested specimens and the mode of fatigue fracture in both cases may be intercrystalline. 8.14 Temperature Effect Low temperature 8.14.1 Low Temperature High temperature As the temperature is decreased below room temperature, the Number of cycles to failure, N S–N curve of metals shifts to higher stress levels but its shape does not alter (Allen and Forrest 1956). This means that the Fig. 8.27 Schematic graph of plastic strain amplitude, Dep=2, against fatigue strength of metals increases with decreasing temper- number of cycles to failure, N, at a constant frequency of cycling in air, ature below room temperature. At low temperatures, the showing the effect of temperature on fatigue life fatigue notch sensitivity of steel has been found to increase

350 8 Fatigue range of around 230–350 °C (503–623 K). A maximum in 8.14.3 Thermal Fatigue the rate of strain ageing in this temperature region is the cause for the existence of a maximum in the tensile strength Fatigue failure at high temperature may not only occur under as well as in the fatigue strength of mild steel. In this tem- the stresses originating from mechanical sources but also perature range, the pinning of dislocations by solute atmo- from thermal sources. In the absence of mechanical stresses, sphere, such as carbon and nitrogen atoms, would be the fluctuating thermal stresses alone can produce fatigue fail- greatest, whereas at lower temperatures, dislocations would ure. If thermal stress of a lower magnitude is repeatedly be free of solute atoms and precipitate. At higher tempera- applied to cause failure, it is called thermal fatigue. If metals, tures, the increased diffusivity of carbon and nitrogen atoms such as uranium, having high anisotropic coefficient of would lead to a greater mobility of dislocations. thermal expansion fail under repeated heating and cooling conditions, it will also be called thermal fatigue. However, if Ferrous materials which normally exhibit a well-defined a single application of thermal stress causes failure, it is fatigue limit during testing at room temperature reveal no called thermal shock. Thermal stresses develop when the fatigue limit when tests are carried out at temperatures above dimensional changes in a body resulting from a change in about 700 K (% 430 °C). Elevated temperatures cause the temperature is hindered by some sort of restriction. For fatigue notch sensitivity of material to decrease. The high example, if a member having fixed end supports is subjected local strain rate at the root of the notch results in a more to a temperature change of DT, then the thermal strain, e, pronounced strengthening effect at high temperatures. developed in the member is e ¼ aDT, and if we assume linear elasticity, the thermal stress, rT, developed in the At high temperatures, creep deformation will become member will be important, and at temperatures greater than about half of the melting point in Kelvin, failure occurs mainly due to rT ¼ Ee ¼ EaDT ð8:44Þ creep. At any given high temperature, the amount of creep which increases with increasing the static component of the where cyclic stress will be added to the effect of fatigue. During fatigue at elevated temperature, dislocations not only glide E modulus of elasticity and but also climb and form well-defined subgrains as observed a linear coefficient of thermal expansion. under creep conditions. Grain boundaries also play an important role in deformation and fracture at high tempera- High-temperature equipments often undergo failure due tures. At high temperatures and low stress levels, cavities to thermal fatigue. The susceptibility to thermal fatigue form preferentially where intense slip bands meet grain failure seems to be associated with the parameter ðrnk=EaÞ, boundaries, and at moderate temperatures and high stress where rn is the fatigue strength for N number of cycles at the levels, grain-boundary sliding leads to triple-point cracking mean temperature and k is the thermal conductivity of the in a manner similar to that occurring in creep. Hence, with material. Higher the value of this parameter, the better is the increasing temperature, there will be transition in the mode resistance to thermal fatigue and vice versa. Austenitic of failure from the usual transgranular fatigue failure to the stainless steel has a low thermal conductivity, k, and a high intergranular creep failure. At elevated temperature, local linear coefficient of thermal expansion, a, so a low value of grain-boundary oxidation can assist significantly to initiate the above parameter makes austenitic stainless steel partic- cracks (Wells 1979). In general, the fatigue strength of a ularly sensitive to the thermal fatigue phenomenon. Failure material at high temperatures will increase with increasing due to thermal fatigue in this material has been covered its creep strength and vice versa. Microstructures that give extensively in literature (Coffin 1954). better room temperature fatigue properties may not result in better creep or stress-rupture properties. For example, 8.15 Chemical Effects resistance to creep deformation in coarse-grained materials is more than that in fine-grained materials, while fine grain size The simultaneous application of cyclic stress and a corrosive results in a better fatigue strength at room temperature. medium is generally referred to as corrosion fatigue Hence, at temperatures where creep predominates, a material (McEvily and Staehle 1972; Duquette 1979). The effect of must possess coarse grain size because it provides higher corrosion fatigue is more disastrous than the individual creep strength and the corresponding better high-temperature effects of cyclic stress and corrosive medium. Chemical fatigue strength. Treatments which are beneficial to fatigue attack without fatigue loading often develops pits on the at room temperature may not be useful in high-temperature metal surfaces, which act as notches during subsequent fatigue. For example, surface compressive residual stresses fatigue loading, and fatigue strength is reduced due to notch may not be effective in reducing fatigue failure at high effects. However, reduction in the fatigue properties temperatures because they may be annealed out with increasing temperature.

8.15 Chemical Effects 351 resulting from corrosion fatigue is much greater than that specific number of stress cycle or fatigue life at any stress produced by corrosive attack on the surface followed by level was lower in air than in vacuum. It was found that the fatigue loading, because the rate of propagation of fatigue rate of crack initiation in air was the same as that in vacuum cracks is greatly accelerated by the corrosive attack during but the rate of crack propagation was rapid in air than in corrosion fatigue. Materials which exhibit a well-defined vacuum. It was concluded that oxygen was absorbed on the fatigue limit during testing in air at normal temperature crack surfaces, which possibly lowered the surface energy. reveal no fatigue limit when tested in a corrosive medium. Thus, the test confirmed that the fatigue life could be The effect of loading cycles within a frequency range from increased by preventing access of oxygen to the surface. about 10–200 Hz on normal fatigue tests in air at room A separate test showed that the water vapour acted as a temperature is negligible. Whereas when tests are carried out catalyst for reduction of the fatigue strength in air. This in a corrosive atmosphere at room temperature, the fatigue indicates that the relative humidity in the atmosphere may strength increases with increasing frequency of stress affect the fatigue properties of materials during testing. cycling (testing speed) because of the lesser fatigue damage However, some media are more powerful than others in due to corrosion, since corrosive attack is a time-dependent reducing fatigue properties. For example, the effect of sea phenomenon. water on the S–N curve of different materials including steel and light alloys is very large. There may be two ways to carry-out the corrosion-fatigue tests. The usual method of testing is to subject the specimen To minimize damage due to corrosion fatigue, corrosion continuously to the joint action of chemical attack and resistance materials rather than materials with higher con- fatigue loading until the specimen fractures. Tests of the ventional fatigue strengths are generally selected for appli- second type are carried out in two stages. In the first stage, cations. For example, stainless steel, beryllium copper or the testing is carried out under the combined action of cor- bronze having better corrosion resistance property would rosive attack and cyclic stress up to a certain period without probably perform better in service than heat-treated steel causing failure, and then, the first-stage test is terminated. In with higher fatigue strength. Materials with higher conven- the second stage, the specimen of first-stage test is subjected tional fatigue strengths can be used if they are protected from to fatigue loading in air until it fractures. Since the normal contact with corrosive media by providing fully effective fatigue life of the specimen in air can be obtained from its corrosion resistant metallic or non-metallic coatings, which S–N curve, the reduction in fatigue life in corrosive atmo- must not rupture under the action of cyclic stress. For sphere produced by the first-stage test can be evaluated. example, steel with cadmium or zinc coatings or alclad Tests of the second type have contributed to establish the aluminium alloys having aluminium coatings can be used mechanism of corrosion fatigue (Evans and Simnad 1947). successfully in many corrosion-fatigue applications, In corrosion fatigue, the cyclic stress causes localized split- although these coatings may reduce the fatigue strength ting of the surface oxide film and also tends to uproot or when tested in air. The formation of surface compressive remove any corrosion products whose presence might residual stresses tends to close down small pits or minute otherwise suppress the corrosive attack. Hence, the number crack and restrict easy access to the corrosive medium. of small pits produced by corrosion fatigue at the surface is Hence, nitriding is particularly useful to resist corrosion much more than that produced by only corrosive environ- fatigue and shot peening has been successfully employed ment in the absence of fatigue loading. As the tips of the pits under certain conditions. In closed systems, a corrosion are more anodic than the material beneath them, so corrosion inhibiter may be added for reduction of the corrosive attack. advances inward from the surface, which is assisted by the At last, it is very important to eliminate stress raisers by splitting of oxide film due to the action of cyclic stress. careful design when considering corrosion fatigue. When the sharpness of the pits increases to a high level, cracking will take place due to the effects of high stress Fretting is a phenomenon of surface damage related to concentration produced by sharp pits. wear and corrosion fatigue. It occurs between two surfaces in contact which undergo slight periodic movement with The influence of corrosion fatigue has been observed in respect to each other. The difference between fretting and fatigue tests performed even in air at room temperature. It wear is that the relative speed of movement of the two was demonstrated by Haigh and Jones (1930) that coating surfaces is much lower in fretting than is usually experienced the surface with oil to exclude air could produce a tenfold in wear. Fretting often occurs on the surface of a shaft with a increase in the fatigue life of lead. Gough and Sopwith press-fitted hub or bearing. A combination of mechanical (1935) showed that endurance limits of copper and brass and chemical effects causes fretting. Fretting arises from the were higher when tested in vacuum than in air. The bene- removal of metal from the surfaces either by a grinding ficial effect of excluding oxygen was also shown by subse- action or by alternate welding and tearing away of the quent work (Thompson et al. 1956) on copper tested in air asperities. The removed fragmented particles are usually and vacuum. The test showed that the fatigue strength at any associated with oxides and other corrosion products. They

352 8 Fatigue act as abrasive powder to continue the damaging process. De ¼ ðXD þ BYÞ þ DB ¼ Dee þ Dep ¼ Dr þ Dep ð8:45Þ Finally, fatigue cracks often initiate in the damaged regions. E Although surface-oxidation is not essential for fretting to occur, the severity of fatigue damage increases many times where when oxidation takes place. Dee elastic strain range ¼ XD þ BY, Fretting would not take place if all relative motion could Dep plastic strain range, which is equal to the width of the be eliminated. Increasing the force normal to the surface to stop the relative motion completely increases the fatigue loop at its centre = DB and damage, and so, this method of preventing fretting is not E Young’s modulus. satisfactory. In many cases, fretting can be eliminated by reducing the coefficient of friction between the mating parts It is important to note that fatigue damage will occur only by means of adequate lubrication. The lubricating film must when cyclic plastic strains are generated. When nominal be maintained for a long period of time and solid lubricants applied stresses are below the yield strength of material, such as MoS have been applied successfully. The other which is usually the case in stress-controlled or high-cycle approaches to reduce fretting are the reduction of surface fatigue, remember that stress concentrations in such cases welding and the elimination of oxidizing or corrosive locally raise the applied stresses above the yield strength and atmosphere from the two surfaces that are in contact with associated strains into the plastic range. each other. Since plastic deformation is not completely reversible, 8.16 Cyclic Strain-Controlled Fatigue structural modifications occurring during cyclic straining under strain-controlled and cyclic stressing under Cyclic strain-controlled fatigue occurs when the strain stress-controlled conditions can result in changes in the amplitude, consisting of some plastic strain component, is shape of the hysteresis loop. It is seen that, in both cases, the kept constant during fatigue cycles. Strain-controlled cyclic stress–strain behaviour of material changes with continued loading is observed in reversed bending between fixed dis- cycling until cyclic stability is reached. Depending on the placements or in thermal cycling, where fluctuations in the initial condition, a metal may undergo cyclic strain harden- operating temperature cause a component to expand and ing, cyclic strain softening, or remain cyclically stable. contract. Strain-controlled conditions also result where Depending on the test conditions and the initial condition of localized plastic strains are generated at a notch by sub- jecting a notched material to either cyclic stress or +σ A strain-controlled mode of loading. This plasticity near the root of notch experiences strain-controlled conditions due to ∆ε the constraining effect of the much larger surrounding mass A' of essentially elastically deformed material. ∆σ/2E σe = ∆σ/2 Figure 8.28 illustrates a stress–strain hysteresis loop under controlled constant-strain cycling in material under- –ε X D O B Y +ε going elastic and plastic deformation. During initial tensile ∆σ ∆σ/2E loading from point O to A, the stress–strain curve of the material is OA. On unloading to zero stress, the elastic strain C C' is recovered and the unloading curve follows the path AB. ∆εp When the material is loaded in compression, yielding begins ∆εe = ∆σ/E at a stress lower than that in prior tensile loading and the –σ compressive stress–strain curve is BC. Releasing the com- pressive stress from C and reapplying tensile stress, the Fig. 8.28 Stress–strain hysteresis loop under controlled stress–strain condition returns to point A along the curve constant-strain cycling in material undergoing elastic and plastic CÀDÀA. Thus, a hysteresis loop is developed, whose width deformation represents the total strain range, De ¼ XY, and height rep- resents the stress range, Dr ¼ A0C0. The hysteresis loop width will depend on the level of cyclic strain, and the area included within the loop is equal to work done or the energy loss per cycle. The total strain range, De, is given by

8.16 Cyclic Strain-Controlled Fatigue 353 the material, it is not unusual to find all three behaviours in a (a) εa 1 35 given material. Strain, ε 0 Time Cyclic hardening and cyclic softening for stress-controlled and strain-controlled conditions are shown, – εa 4 respectively, in Figs. 8.29 and 8.30. For the case of 2 stress-controlled condition, where a stress range between A0 and C0, with reference to Fig. 8.28, is used to conduct the 35 fatigue test, the plastic strain range which is the width DB of the hysteresis loop at its centre shrinks during cyclic hard- Stress, σ(b) + 1 Time ening and expands when cyclic softening occurs. The con- –0 Time dition arising from cyclic softening under stress control is particularly severe because a continually increasing strain 2Stress, σ range produced by the constant-stress range leads to an early 4 fracture. For the case of strain-controlled condition, where fatigue test is performed within a strain range limits of X and (c) + 1 Y (see Fig. 8.28), the hysteresis loop expands above A and below C during cyclic hardening and contracts below A and 3 above C during cyclic softening. The changes in the shape of 5 the hysteresis loop with successive cycles for cyclic strain hardening and softening under strain-controlled test condi- 0 tions are illustrated schematically in Fig. 8.31. Under strain-controlled conditions, cyclic hardening leads to an 4 increasing peak stress with the number of cycles causing an increase in the area of the hysteresis loop, while cyclic –2 softening results in a decreasing stress level with increasing cycles making the loop smaller. In our discussion, a Fig. 8.30 Behaviour of cycle-dependent material in a strain-controlled fatigue. a Control condition, where controlled function is strain. b Cyclic hardening, where dependent variable is stress. c Cyclic softening, where dependent variable is stress (a) 1 35 symmetrical loading with zero mean stress is assumed. If the mean stress would not be zero during a cyclic strain Stress, σσa Time experiment, then as a result of each cycle the specimen 0 Time would be found to accumulate strains, called ‘cyclic strain- induced creep’. Due to this, the specimen would be either – σa 4 elongated or shortened depending on the sense of the applied 2 mean stress. (b) + 1 Generally, the hysteresis loops stabilize after cycling for often less than 100 cycles, and the stress amplitude arrives at 3 a constant value for the imposed strain limits. The stress– 5 strain curve obtained after cyclic stabilization may then be quite different from that obtained on monotonic static load- Strain, ε 0 ing. The cyclic stress–strain curve is usually constructed by fitting a curve through the tips of stabilized hysteresis loops 4 from constant-strain amplitude fatigue tests of specimens subjected to various cycles of strain amplitudes. Such a –2 cyclic stress–strain curve for a material that cyclically hardens is shown in Fig. 8.32, which also includes mono- 35 tonic stress–strain curve for comparison purpose. Under conditions where hysteresis loops are not stabilized, the Strain, ε(c) 1 Time minimum stress amplitude for softening or the maximum stress amplitude for hardening is used to determine the cyclic + stress–strain curve. 0 In direct analogy with the Hollomon parabolic true- – stress–true-strain relation given by r ¼ K en [see (1.90a)], 2 4 Fig. 8.29 Behaviour of cycle-dependent material in a stress-controlled fatigue. a Control condition, where controlled function is stress. b Cyclic hardening, where dependent variable is strain. c Cyclic softening, where dependent variable is strain

354 (a) (b) 8 Fatigue Fig. 8.31 Schematically Stress, σ 5 Stress, σ 1 showing the changes in the shape 3 3 of the hysteresis loop with 5 successive cycles under 1 strain-controlled test conditions Strain, ε for materials that a Cyclically strain harden and b Cyclically strain soften Strain, ε 2 4 4 2 Stress, σ For most metals, n0 ranges from 0.1 to 0.2. From (8.46), it is seen that if log Dr versus log Dep is plotted, the cyclic Cyclic σ – ε curve stress–strain curve can be represented by a single straight line, but since there may be an appreciable change in the Monotonic cyclic deformation response with strain amplitude, it is not σ – ε curve unusual to get a slope in the long-life fatigue region different from that in the short one. Combining (8.45) and (8.46), we Strain, ε get the following form of the equation for the cyclic stress– strain curve: De ¼ Dr þ Dep ¼ Dr þ  Dr 1=n0 ð8:47Þ 2 2E 2 2E 2K0 Fig. 8.32 Schematic curves of monotonic and cyclic stress–strain for Several attempts have been made to determine which a material that cyclically hardens. Points on cyclic stress–strain curve represent tips of stabilized hysteresis loops obtained from materials will cyclically harden and which will soften. It has constant-strain-amplitude fatigue tests of specimens subjected to various cycles of strain amplitudes been observed (Smith et al. 1963; Manson and Hirschberg 1964) that when Su=½S0Še0¼0:002 [ 1:4, the material is it is possible to mathematically describe the cyclically sta- expected to harden cyclically, but when Su=½S0Še0¼0:002\\1:2, bilized stress–strain curve by the following power relation: cyclic softening is expected to occur, where ½S0Še0¼0:002 and Su are, respectively, 0.2% offset yield strength and ultimate Dr  n0 tensile strength of material obtained on monotonic static Dep loading. For ratios ½S0Še0¼0:002 to Su between 1.2 and 1.4, a ¼ K 0 ð8:46Þ large change in hardness is not expected. Also, if n [ 0:15, the material is likely to undergo cyclic hardening, and if 22 n\\0:15, cyclic softening is likely to occur, where n is the monotonic strain-hardening exponent. Therefore, initially where soft and low-strength materials will generally undergo Dr=2 stress amplitude; K0 cyclic strength coefficient; cyclic strain hardening and initially hard and strong mate- plastic strain amplitude; and Dep=2 cyclic strain-hardening exponent. rials will undergo cyclic strain softening. n0 The reason for softening or hardening is related to the dislocation microstructure of the material. For an initially soft material, the dislocation density is low initially, but the plastic strain cycling increases the dislocation density rapidly that contributes to significant strain hardening of the material. When a material is initially hard (for example, a highly

8.16 Cyclic Strain-Controlled Fatigue 355 work-hardened metal having high initial dislocation density), cyclically strain harden and initially hard (cold worked) subsequent cyclic strain causes dislocations to rearrange into specimens cyclically strain soften likewise, to such a degree a new configuration that reduces the stress required for plastic that the two cyclically stabilized stress–strain curves con- deformation to occur leading to strain softening of the verge regardless of the initial condition of the material, as material. The processes connected with cyclic strain softening shown in Fig. 8.33a (Feltner and Laird 1967). This material were referred to as the Bauschinger effect (Koo et al. 1967) is said to be ‘history-independent’ in a cyclic stress or strain (see Sect. 2.5 in Chap. 2) in earlier literature. The cyclic environment, because the cyclic stress–strain behaviour of strain softening tendency, i.e. the Bauschinger effect, exhib- this material in the stabilized condition is independent of ited by cold-worked metals is applied for benefit in certain prior strain history. The dislocation structure developed in a metal-forming operations. For example, when a history-independent material is also independent of initial strain-hardened work-piece is passed through roller-leveller strength and initial dislocation structure of the material. On (see Sect. 12.3.2 in Chap. 12), it bends alternatively in the other hand, a low SFE material is ‘history-dependent’ in upward and downward directions and experiences alternating a cyclic stress or strain environment, and its behaviour is tensile and compressive stresses that induce cyclic strain shown in Fig. 8.33b (Feltner and Laird 1967). In common softening and improve the ductility of material. with ‘history-independent’ materials, an initially hard material of this type usually cyclically softens and an ini- The extent and rate at which softening or hardening tially soft one cyclically hardens. In contrast to ‘history- occurs during cyclic straining or stressing depend on several independent’ materials, however, the two ‘final’ cyclically factors, out of which the most important one is the material’s stabilized stress–strain curves obtained from an initially hard stacking fault energy (SFE) that strongly affects the mobility and an initially soft condition do not converge, and an ini- of dislocations. Recall from Chap. 1 that in materials with tially hard material always remains harder than an initially high SFE, cross-slip is easier leading to greater dislocation soft one. Hence, the ‘final’ cyclically stabilized stress–strain mobility, whereas when SFE is low, cross-slip is restricted. behaviour of this material is dependent on prior strain his- As a result, the extent of cyclic hardening or softening will tory. Clearly, the restricted cross-slip in a low SFE material be more in materials having higher SFE than lower SFE. will prevent to develop a common cyclical dislocation Further, the rate of hardening or softening will be slower in structure from an initially hard and an initially soft condition, low SFE materials than in high SFE ones. For example, in a respectively. relatively high SFE metal, initially soft (annealed) specimens (a) (b) Monotonic – cold worked Monotonic – cold worked Cyclic Cyclic – cold worked Cyclic – annealed Stress Stress Monotonic – annealed Monotonic – annealed Plastic strain amplitude, ∆εp /2 Plastic strain amplitude, ∆εp /2 Fig. 8.33 Comparison of monotonic and cyclic stress–strain curves independent of prior strain history (i.e. independent of whether it is (schematic) for a high stacking fault energy (SFE) material and b low initially in annealed or cold-worked condition), whereas cyclical SFE material, in two different initial states. The cyclic stress–strain behaviour of low SFE material is history-dependent (Feltner and Laird behaviour of high SFE material in the stabilized condition is 1967)

356 8 Fatigue Fig. 8.34 Schematic Plastic strain amplitude, ∆εp / 2 (log scale) 1 Fatigue ductility coefficient, ε′f ≈ εf representation of low-cycle fatigue test results by plotting the 10–1 ∆εp plastic strain amplitude, Dep=2, 2 against the number of strain = ε′f (2 N)c reversals to failure, 2N, on log-log coordinates 10–2 Fatigue ductility exponent, c = slope of this line 10–3 10–4 101 102 103 104 105 106 1 Number of strain reversals to failure, 2N (log scale) 8.16.1 Low-Cycle Fatigue Dep ¼ ef0 ð2NÞc ð8:48Þ 2 During high-cycle fatigue, where the number of cycles to failure is very large, usually more than 104 cycles, the where applied stress at the macroscopic level is such that the material grossly undergoes only elastic deformation, Dep plastic strain range and although fatigue damage involves generation of plastic Dep=2 deformation at the microscopic level, irrespective of the 2N plastic strain amplitude. natures of macroscopic strain. Conversely, in the low-cycle fatigue range, the material is typically subjected to plastic e0f total number of strain reversals necessary to cause strain at microscopic as well as macroscopic level, and fatigue failure; note that two strain reversals make failure occurs at relatively high stress and low number of c one strain cycle, i.e. N = number of strain cycles to cycles to failure, usually less than 103 cycles. Hence, failure. application of cyclic plastic strain amplitude is responsible fatigue ductility coefficient, defined by the strain for low-cycle fatigue failure. Low-cycle fatigue conditions intercept at one strain reversal, i.e. at 2N ¼ 1. For are often produced in places where thermal expansion or many metals, e0f can be approximately equated to contraction of material creates repeated thermal stresses. In the true fracture strain ef . this case, cyclic strain rather than cyclic stress causes fatigue fatigue ductility exponent, a material property to occur. Examples of low-cycle fatigue failure include which varies from −0.5 to −0.7. steam turbines, nuclear pressure vessels and various types of power machinery. Since the relationship given by (8.48) was first discovered by Coffin (1954) and Manson (1954), it is usually referred to The results of low-cycle fatigue test are usually presented as the Coffin–Manson relation. Fatigue life is expected to by plotting the plastic strain amplitude, Dep=2, against the increase with an increasing fatigue ductility coefficient ef0 number of strain reversals to failure, 2N, on log-log coor- dinates. The plot of results will be linear as shown in and a decreasing fatigue ductility exponent c. Morrow Fig. 8.34. In comparison to the S–N diagram shown in (1965) has shown that c ¼ À½1=ð1 þ 5n0ފ, from which it can Fig. 8.9 displaying test results of stress-controlled or high-cycle fatigue, one noticeable difference is that the be said that a material with a higher value of cyclic plastic strain amplitude Dep=2 rather than the stress S is strain-hardening exponent n0 will result in an increased plotted logarithmically rather than linearly in this figure displaying strain-controlled or low-cycle fatigue test results. fatigue life. An empirical relationship between Dep=2 and 2N for low-cycle fatigue is best described by Coffin–Manson rela- 8.16.2 Strain–Life Equation and Curve tion (Tavernelli and Coffin 1959; Manson and Hirschberg 1964): Let us now consider how the life of a component or a specimen experiencing strain-controlled loading is affected by cyclically stabilized properties of material. The relation- ship between plastic strain amplitude and fatigue life has been described by (8.48) in the region of low-cycle

8.16 Cyclic Strain-Controlled Fatigue 357 high-strain fatigue. For the region of high-cycle low-strain materials give the best performance for high-cycle low-strain fatigue, where the gross strain at the macroscopic level is fatigue applications, because nucleation of fatigue cracks is elastic, the stress amplitude and the elastic strain range are more difficult in strong materials and absorbs much of correlated to the number of stress cycles necessary to cause high-cycle fatigue life. On the other hand, ductile but not fatigue fracture through the following equation: strong materials give the best performance for low-cycle high-strain fatigue applications, because propagation of ra ¼ Dr ¼ Dee E ¼ r0f ð2NÞb ð8:49Þ cracks is more difficult in ductile materials and most of the 2 2 low-cycle fatigue life is occupied by Stage II crack growth. A tough material having both reasonable strength and duc- where tility is a good compromise when resistance to both high-cycle and low-cycle fatigue is necessary. The value of ra stress amplitude in an alternating stress cycle and fatigue life, Nt, at which the transition from high cyclic strain Dr stress range. condition to low one occurs, can be obtained from Dee ¼ Dee elastic strain range and Dep as follows: Dee=2 elastic strain amplitude. E Young’s modulus. rf0 ð2Nt Þb ¼ ef0 ð2Nt Þc ; or; ð2Nt ÞbÀc ¼ e0f E N number of stress cycles necessary to cause fatigue E r0f fracture. 2N number of stress reversals necessary to cause ) 2Nt ¼ e0f E!1=ðbÀcÞ ð8:51Þ fatigue failure. r0f r0f fatigue strength coefficient, defined by the stress intercept at one stress reversal, i.e. at 2N ¼ 1. r0f This transition typically takes place at around 103 cycles b can be approximately equated to the uniaxial true (2  103 reversals), i.e. at Nt ffi 103 cycles, which corre- fracture stress rf . sponds to a total strain range of about 0.01 (Landgraf 1970). fatigue strength exponent, which varies from −0.05 to −0.12 for most metals. Fatigue life is expected to improve with an increasing Strain amplitude, ∆ε/2 (log scale) ε′f ∆εe + ∆εp fatigue strength coefficient rf0 and a decreasing fatigue Slope = –c 22 strength exponent b. Morrow (1965) has shown that b ¼ À½n0=ð1 þ 5n0ފ, from which it can be said that a σ′f Elastic material with a lower value of cyclic strain-hardening E ∆εe exponent n0 will result in a longer fatigue life for fatigue controlled by (8.49). 2 Since the relationship between the strain amplitude and the number of cycles to failure at the ‘extremes’ of fatigue behaviour, i.e. in the regions of low-cycle and high-cycle fatigue, is represented by (8.48) and (8.49), respectively, an equation for the total strain amplitude over the entire range of fatigue lives can be obtained by combining those (8.48) and (8.49). Since from (8.45), De ¼ Dee þ Dep ∆εp Slope = –b 22 2 2 Plastic ) De ¼ rf0 ð2NÞb þ ef0 ð2NÞc ð8:50Þ 1 2Nt Х 2 × 103 2 E Stress or strain reversals to failure, 2N (log scale) The relationship given by (8.50) is schematically plotted Fig. 8.35 Fatigue strain–life curve (schematic) approaching towards in Fig. 8.35, which shows that the total fatigue strain–life the plastic strain–life curve at large total strain amplitudes and the curve approaches towards the plastic strain–life curve at elastic strain–life curve at low total strain amplitudes. At low N values large total strain amplitudes and the elastic strain–life curve where De ffi Dep, the logarithmic slope is –c and the intercept of the at low total strain amplitudes. Figure 8.36 (Landgraf 1970) plastic line is e0f . At high N values where De ffi Dee, the logarithmic shows a comparison of total fatigue strain–life plots for slope is −b (with b < c) and the intercept of the extrapolated elastic strong, tough and ductile materials. Strong but not ductile portion of the curve is rf0 =E

358 8 Fatigue Fig. 8.36 Schematic plots showing comparison of total fatigue strain–life for strong, tough and ductile materials (Landgraf 1970) log ∆ε 2 ∆ε ~ 0.01 2Nt Х 2 × 103 Strong Tough Ductile log 2N Further, n0 and K0, which are defined in (8.46), can be conditions at the same time. This needs to be discussed elaborately because several high-temperature structures, such correlated with the coefficient and exponent parameters of as turbine blades and materials operating in nuclear reactors, are subjected to fatigue environments and the elevated tem- (8.48) and (8.49) by substituting the expression for Dr=2 perature damage of material is due to an interaction between both creep and fatigue processes. Creep enhanced by the from (8.49) and the expression for Dep=2 from (8.48) into fatigue environment or vice versa depends on several factors, such as cyclic stress/strain amplitude, applied frequency of (8.46) as follows: loading, temperature. The deformation may be considered as creep accelerated by fatigue, and fracture surfaces exhibit a r0f ð2NÞb ¼ K 0 h ð2N Þc in0 ¼ K 0  n0 ð2N Þcn0 : tendency towards intergranular fracture when ef0 ef0 ) r0f ¼ K 0  n0 ; and ð2NÞb ¼ ð2NÞcn0 ef0 b ¼ cn0; or Hence, • Cyclic stress/strain amplitude is small compared to the mean stress/strain, or n0 ¼ b=c ð8:52aÞ • Operating temperature is high or/and applied cyclic fre- And quency is low. K0 ¼ rf0  n0 ¼ rf0  b=c ð8:52bÞ On the other hand, the deformation may be considered as = ef0 = e0f fatigue accelerated by creep and fracture surfaces are man- ifested by fatigue striations and regions of transgranular Equation (8.49) is applicable when the mean stress is fracture when zero, i.e. for purely alternating stress cycle. But when the • Cyclic stress/strain amplitude is large compared to the mean stress/strain, or mean stress is present, i.e. is not zero, then (8.49) is cor- • Operating temperature is low or/and applied cyclic fre- rected by introducing the mean stress term in it: quency is high.  ð8:53Þ Out of several empirical creep–fatigue correlations so far ra ¼ r0f À rm ð2NÞb developed, the important approaches suggested for design under conditions of creep–fatigue interaction are as follows: However, after a little number of cycles in strain-controlled fatigue tests, there will be usually relax- ation of mean stresses (Lorenzo and Laird 1984). 8.17 Creep–Fatigue Interaction (1) Cumulative Damage Rule In Sect. 8.14.2, it has already been mentioned that a member In this approach, the damage due to fatigue and creep is subjected to fatigue loading at elevated temperature also considered separately. These independent damages are then undergoes creep deformation under cyclic stress/strain added to develop the fracture criterion. To account for

8.17 Creep–Fatigue Interaction 359 fatigue damage, Palmgren–Miner cumulative damage rule, 1 Cyclically work- given by (8.25), is considered. One such cumulative damage hardening materials rule for creep was proposed by Robinson in 1952 (Courtney 1990). This rule estimates the time required for creep frac- Linear model ture under varying stress conditions. If the time spent at one stress rl is tl and the fracture time at that stress is ðtf Þl, then creep fracture will occur when Xl¼m  t  l=m l¼1 tf l ∑ t ¼ 1 ð8:54Þ tf l l=1 An estimate for creep–fatigue interaction can be obtained Cyclically work- by adding these fatigue and creep life fraction laws given, softening materials respectively, by (8.25) and (8.54). Hence, the fracture cri- terion is expressed as: 0 j=k n 1 ∑ Xj¼k  n  Xl¼m  t  j=1 N j j¼1 N j l¼1 tf l þ ¼ 1 ð8:55Þ where Fig. 8.37 Combined effects of fatigue and creep showing three types of interaction. If the combined life fraction rule given by (8.55) is n number of fatigue cycles applied at stress/strain obeyed, the interaction will be plotted as a straight line labelled ‘linear amplitude condition j; model’. Cyclically softening materials have lives less than that predicted by linear model (8.55) and lives of cyclically hardening N number of fatigue cycles to failure at stress/strain materials are greater than that predicted by linear model (8.55) amplitude condition j; with some modification can be considered. The modification t time spent under the applied creep load condition l; will be to replace ultimate uniaxial tensile strength, Su, by ðtf Þl time needed for creep fracture at the applied creep either creep strength, rcr, for a given allowable creep strain or creep-rupture strength, rr, in the Goodman relation given load condition l. by (8.21). Thus, we get A plot of the combined effects of fatigue and creep showing three types of interaction is shown in Fig. 8.37. If Goodman relation for cre!ep-fatigue interaction: (8.55) is obeyed, creep–fatigue interaction will be plotted as ra ¼ rn 1 À rm ð8:56Þ ðor; a straight line labelled ‘linear model’ as shown in Fig. 8.37. rcr rr Þ Other interactions are shown for cyclically softening and where hardening materials. Materials that cyclically soften have lives less than that predicted by linear model (8.55), and rcr creep strength for a given allowable creep strain and rr creep-rupture strength; hence (8.55) is an ‘unsafe’ design criterion for them. On the rn limiting fatigue strength at some design designated other hand, (8.55) is a conservative design rule for cyclically number of cycles. hardening materials, because their lives are greater than that predicted by linear model (8.55). The matter of interest is Other terms are defined in connection with (8.21). The plot of (8.56) is shown in Fig. 8.38. The relationship that creep lives of cyclically hardening materials can be given by (8.56) seems to be conservative design criterion. The area bounded by the straight line predicted by (8.56) and benefitted by a minor fatigue interaction, which is evident the two axes is expected to be a safe region as far as creep– fatigue interaction is concerned. from Fig. 8.37 by the positive slope of the fracture criterion Pj¼k for those materials at small values of ðn=NÞj. j¼1 (2) Modification of Goodman Law (3) Frequency-Modified Fatigue Relation In this approach, it is assumed that at any given high tem- In Sect. 8.13, the effects of frequency of stress cycling and perature, fatigue is caused by the stress amplitude, ra, and environment on fatigue life at constant elevated temperature the amount of creep increases with increasing the static have been discussed and illustrated in Fig. 8.26. It has been component, rm, of the cyclic stress. Then an approach seen that the effect of environment like air on similar to the construction of Goodman line (see Sect. 8.7)

360 8 Fatigue Stress amplitude, σa, causing fatigue σn Temperature = constant where k is an empirical constant that depends on temperature σa + σm = 1 and is also sensitive to environment. The value of k for a σn σcr (or, σr) vacuum environment is taken as unity, i.e. fatigue life in vacuum becomes essentially independent of frequency while in air, k is less than 1, usually about 0.8, and hence, the number of cycles to failure is reduced at lower frequencies. According to (8.58), a log-log plot of Dep versus NmkÀ1 must be a straight line at a given temperature in a given envi- ronment, which is often to be the case for low-cycle elevated temperature fatigue, as shown in Fig. 8.39. This figure shows that for low-cycle fatigue, temperature has no or little effect on fatigue life (see also Fig. 8.27), whereas when the plastic strain range is low, the increase in temperature reduces the fatigue life in the same environment. σcr or σr (4) Strain-range Partitioning Method (Manson et al. Mean stress component, σm, causing creep 1971a, b) Fig. 8.38 Schematic diagram showing fatigue–creep interaction at a Any purely alternating inelastic strain cycle can be disinte- constant temperature grated into four separate strain-range components, in which two kinds of deformation are taken into account. One is high-temperature fatigue life is detrimental, the degree of time-dependent deformation or creep and the other is which depends on the frequency, m, of the applied fatigue time-independent plastic deformation. The former will be cycle. Coffin (1974) introduced a term, known as designated by the subscript c and the latter by the subscript frequency-modified fatigue life, which is: p. The strain-range components are: N mkÀ1 ð8:57Þ • Purely alternating plastic deformation, Depp; • Plastic deformation in tension followed by creep in Similar to Coffin–Manson relationship given by (8.48), an empirical relationship between frequency-modified fati- compression, Depc; gue life given by (8.57) and plastic strain range, Dep, has • Creep in tension followed by plastic deformation in been developed, which is: compression, Decp; and DepÀNmkÀ1ÁÀc¼ constant ð8:58Þ • Purely alternating creep, Decc. In the above notation, the type of deformation is desig- nated by the two-letter subscripts, in which the first one refers to the tensile portion and the second one to the 1 Temperature, T1 < T2 < T3; Environment is same (air). 0.1 ∆εp 0.01 T3 T2 T1 0.001 1 10 102 103 104 105 0.1 Frequency-modified fatigue life, N vk–1 Fig. 8.39 Schematic log–log plot of plastic strain range, Dep, versus frequency-modified fatigue life, NmkÀ1, for elevated temperature fatigue in a given environment, showing the reduction in fatigue life with increasing temperature in the same environment when the plastic strain range is low

8.17 Creep–Fatigue Interaction 361 (a) σ (b) Depc. Hence the total inelastic strain range, Dei, is given by: Dei ¼ Depp þ Decc þ Depc. Plastic σ Each of the strain range components follows a Coffin– Creep Manson-type relation, and so, a straight-line relationship is ε ε obtained for each of them, when logarithmically plotted ∆εpp ∆εcp against number of cycles to failure, N, as shown in Fig. 8.42. Plastic Plastic Each of the strain range components constitutes a certain fraction of a given value of total inelastic strain range, Dei. These fractions are: Fpp ¼ Depp ; Fpc ¼ Depc ; Fcc ¼ Decc ; Fcp ¼ Decp : Dei Dei Dei Dei (c) σ (d) Hence, for the combined creep and fatigue strain cycle, σ according to the creep–fatigue interaction damage rule the Plastic predicted life, Npred, is given by Creep 1 ¼ Fpp þ Fpc þ Fcc þ Fcp ð8:59Þ Npred Npp Npc Ncc Ncp ε ε 8.18 Increasing Amplitude Tests ∆εpc ∆εcc Testing procedures at increasing stress amplitudes are developed to save both specimens and time of testing. These Creep Creep test methods are used for rapid estimation of the fatigue limit of materials using a less number of specimens in comparison Fig. 8.40 Four kinds of separated strain-range cycles to conventional means of testing. In increasing amplitude tests, each specimen is subjected to stress cycles at compressive portion of the cycle. For example, all four kinds increasing levels of stress amplitude until fracture occurs. Hence, all specimens contribute to the test data and undergo of separated strain-range cycles are shown in Fig. 8.40. fracture. Since each specimen is subjected to stress cycles at more than one stress level, the specimen may experience Let us consider a purely alternating strain hysteresis cycle Stress Creep as shown in Fig. 8.41, in which plastic deformation and Plastic flow creep portions of the cycle are labelled. In this figure, the total inelastic strain range in the tensile part of the cycle is A BC D Strain Dei ¼ AD, which is equal to the sum of a plastic strain AC and a creep strain CD. Similarly, the addition of a plastic Creep Plastic flow strain DB and a creep strain BA constitutes the inelastic Fig. 8.41 Purely alternating strain hysteresis cycle (schematic), in which plastic flow and creep portions of the cycle are labelled strain range DA in the compression part of the cycle. Out of the two plastic strain components AC and DB, the smaller one is the purely alternating segment of plastic strain range, Depp, i.e. Depp ¼ DB. Similarly, out of the two creep strain components CD and BA, the smaller one is the purely alternating segment of creep strain range, Decc, i.e. Decc ¼ CD. For Fig. 8.41, AC þ CD ¼ DB þ BA, or AC À DB ¼ BA À CD, i.e. the difference between the two plastic strain components is equal to that between the two creep strain components, and this difference is either Depc or Decp. In any hysteresis cycle, only one and not both of Depc and Decp will exist. If plastic strain component in the tensile part is greater than that in the compressive part of the cycle, Depc will exist, while for greater creep strain component in the tensile part than in the compressive part of the cycle Decp will exist. In Fig. 8.41, the aforementioned difference is

362 8 Fatigue Total inelastic strain range, ∆εi, (log scale) ∆εpp this new stress level. If the specimen does not break, the stress is again increased, and in this way, the testing con- ∆εcc ∆εpc tinues with increasing stress level and maintaining the same number of stress cycles, N, until the specimen fractures at ∆εcp stress cycles N. The fatigue strength of the specimen, for N stress cycles, rn, is taken as the stress halfway between the Ncp Ncc Npc Npp stress amplitude at fracture and the highest stress amplitude Number of cycles to failure, N, (log scale) of survival of specimen preceding the fracture stress. Fig. 8.42 Coffin–Manson-type straight-line relationship for each of The procedure of ‘step’ testing of a single specimen is the strain range components illustrated graphically in Fig. 8.43. Several specimens are tested in the manner described above to yield several values of rn. The median fatigue strength for N cycles can then be obtained from these above results using statistical procedure. The advantage of step test is that only a few specimens are required for tests. Their disadvantage is that each specimen is usually subjected to a specified large number of stress cycles for several times. This process involves long time which may make the test unpopular. coaxing effects and cumulative fatigue damage. Since the 8.18.2 Prot Test manner of loading in these tests is such as to cause coaxing effects, these tests are usually restricted to materials which In Prot test, developed by Marcel Prot, the stress amplitude are not sensitive to coaxing. The values of fatigue limit is increased steadily rather than by steps, as used in the step produced by these tests may not be in agreement with those test. Prot test is faster than step test and thus, less time is obtained from testing at constant stress for materials which consumed in Prot test. Equation (8.34) developed from are strengthened by coaxing at stresses below the fatigue cumulative fatigue damage theory discussed in the preceding limit. Since cumulative fatigue damage theory is based on Sect. 8.8.2 is repeated below for convenience. the application of series of overstress cycles, it can be assumed that no fatigue damage will take place until the ) rR ¼ re þ k0ai ð8:34Þ applied stress level reaches the fatigue limit. So, when materials that possess definite fatigue limits are tested and where k0 and i are unknown material constants; rR is the stress cycling is started at a stress lower than the fatigue rupture stress of test specimen, whose value will be available limit, cumulative fatigue damage is minimized. For materials from the test; a is the rate of increase of stress amplitude possessing no definite fatigue limits, these tests are occa- whose value has to be fixed prior to the start of test; and re is sionally performed to determine long-life fatigue strengths the fatigue limit of test material, which is to be determined (ASTM STP 1958). There are mainly two types of test—one by the test. So, a and rR are the variables in the test. Several is known as the step test and the other is the Prot test— which have been discussed in the following sections. 8.18.1 Step Test Applied stress amplitude Failure Estimated fatigue strength of specimen In step test, each specimen is subjected to purely alternating × tested stress cycles at a series of stress levels, in which each applied level of stress amplitude is higher than the preceding one. In Successfully the test, stress cycling is started at a stress level below the completed fatigue limit, at about 70% of the expected fatigue limit. At consecutive the starting stress, the specimen is subjected to a specified stress cycles large number of stress cycles, say N, which is usually taken without break as 107 or 108 cycles. If the specimen remains unbroken, the subsequent level of stress amplitude to be applied is Starting stress Stress increment increased by a certain small amount and the specimen is again subjected to the same number of stress cycles, N, at 103 104 105 106 107 108 Number of stress cycles, N Fig. 8.43 Schematic representation of ‘step’ testing of a single specimen

8.18 Increasing Amplitude Tests 363 different values of each of these variables are required to (√α3, )σR3 compute the fatigue limit of test material, re. So, a number of specimens are tested at several different values of stress amplitude increase per cycle. Since the data of all fatigue Average rupture stress, σR tests shows considerable scatter, so it is recommended (√α1, )σR1 (ASTM STP 1958) that a total of 20 specimens must be (√α1, )σR2 tested using at least three rate of increase of stress ampli- σe (Endurance limit) tude, i.e. 3 values of a, say a1; a2; and a3. Any standard fatigue testing machine, such as the R.R. Moore rotating-beam fatigue testing machine, can be used for the Prot test, if the deadweights used in this machine are replaced with some arrangements to increase the load at a prefixed constant rate. For example, the uniform increase in 0 Rate of increase of stress amplitude,√α the rate of loading can be obtained by replacing deadweights Fig. 8.44 Prot method showing average rupture stresses, with a container into which small metal balls or a stream of rR1 ; rR2 ; and rR3 , at three rate of increase of stress amplitude, a1; a2; and a3p, ffiwffi here the straight line averaging these three points is water can be poured at a uniform rate. extended to a ¼ 0 to determine the endurance limit, re According to the cumulative fatigue damage theory, an initial stress amplitude equal in magnitude to the fatigue considering the value of i to be unknown, although the limit, re, of the test material should be applied at start of the solution of (8.34) then becomes more difficult. ASTM rec- test, but since re is not known, the standard procedure is to ommended a trial-and-error method to solve (8.34), which apply the starting stress amplitude that equals to about can be written as 60–70% of expected value of re. Since it is assumed that the material is not affected by the stress cycles that are applied at logðrR À reÞ ¼ log k0 þ i log a ð8:60Þ stresses below the fatigue limit, so re taken as the lower limit of the integral of (8.33) holds good. From the experimental data, (8.34) could be easily solved Equation (8.60) shows that if logðrR À reÞ is plotted against log a, a straight line should be obtained. Since the for re if the value of i was known. For simplicity, let us consider Prot’s assumption that i ¼ 1=2, and Prot equation, slope, i, of this linear line is unknown, the procedure is to select trial values of re and make a series of trial plots until such a (8.35), as shown in the preceding Sect. 8.8.2, is repeated value of the fatigue limit, re, is selected that the resulting data points lie on a straight line in the plot. As before, the arithmetic below for convenience. average values of the observed rupture stresses, rR1 ; rR2 and rR3 , obtained, respectively, at loading rates of a1; a2; and a3 rR ¼ re þ k0pffiaffi ð8:35Þ are used. The first trial value selected for re should be close to but slightly lower than the observed lowest average rupture Equaptiffioffi n (8.35) represents a straight line in a plot of rR stress. Using this value of re; ½log a1; logðrR1 À reފ;, versus a. The intercept of this straight line on the rR-axis ½log a2; logðrR2 À reފ and ½log a3; logðrR3 À reފ are plotted, gives the value of the fatigue limit re. The procedure of Prot respectively, as points A, B and C as in Fig. 8.45. The selected method is illustrated in Fig. 8.44. The initially applied stress value of re is substituted in (8.60) to yield three equations amplitude is increased at a constant rate per cycle for each corresponding to points A, B and C. The respective slopes i1; i2 specimen until the specimen fractures. Let, the rates of and i3 of the lines AB, BC and CA are computed by solving increase of stress amplitude are a1; a2; and a3. For each of those three equations and then compared. If the maximum the three rates of loading, several specimens are tested until fracture occurs and the rupture stress for each specimen at difference between any two of the three slopes exceeds one or each rate of loading is noted. In Fig. 8.44, the arithmetic two per cent, a new trial value of re must be selected and the average values of the observed rupture stresses obtained for procedure is repeated. The selected value of re that causes the each of the three rates of loading have been plotted. Points slopes of the lines joining the data points not to differ by more rR1 ; rR2 and rR3 , shown in Fig. 8.44, are the average rupture stresses obtained, respectively, at loading rates of than one or two per cent from each other is considered as the a1; a2; and a3. A straight line is fitted to these points. The value of fatigueplffiiffimit, re, is obtained by extrapolating this required fatigue limit. straight line to a ¼ 0. The above test method for estimation of fatigue limit is Since the experimental data are better accommodated by the more general equation (8.34) than by the Prot equation limited to materials that seem to possess a fatigue limit and (8.35), it is better to analyse the experimental data by are not sensitive to coaxing. The rate of increase of stress amplitude and the frequency of stress cycling must both be

364 8 Fatigue rmax ¼ Pmax ¼ 0:3 MPa; and A A B rmin ¼ Pmin ¼ À 0:1 MPa: C A A A log (σR – σe) The mean and alternating stresses of the stress cycle will be, respectively: rm ¼ rmax þ rmin ¼ ð0:3=AÞ þ ðÀ0:1=AÞ ¼ 0:1 MPa; and 2 2 A ra ¼ rmax À rmin ¼ ð0:3=AÞ À ðÀ0:1=AÞ ¼ 0:2 MPa: 2 2 A The conservative design criterion is Goodman line given by (8.21), which on substitution of the values of log α ra; r106 ; rm; and Su gives Fig. 8.45 Trial plot of logðrR À reÞ against log a, using a selected  0:1 trial value of re. Points A, B and C are average results of several tests 0:2 1 À 0:1=A 6A 1:3 conducted, respectively, at each of three rate of increase of stress A ¼ 25 150 ¼ 25 À ; or; 6A ¼ 25; amplitude, a1; a2; and a3. Since points A, B and C are not close to the same straight line, a new trial value of re must be selected ) A ¼ 1:3 m2 ¼ 8:67  10À3 m2 ¼ 8666:67 mm2 ¼ pD2 ; 6  25 4 precisely controlled, because little variations in either may where D is the rreqffiffiuffiffiiffired rdiaffiffimffiffiffiffieffiffitffieffiffirffiffiffiffioffiffifffiffiffitffihffiffiffie bar. produce significant scatter in the results. Several precautions 4A 4  8666:67 and limitations are specified by ASTM STP (1958). Hence, D ¼ p ¼ p mm ¼ 105 mm. 8.19 Solved Problems 8.19.2. A cantilever cylindrical metallic beam with length 150 mm and diameter 25 mm is subjected to a loading cycle 8.19.1. A metallic round bar of uniform cross-section is of PðNÞ ¼ P0½1 þ 2 sin 100tŠ at its free end, where t is the subjected to a fluctuating tension–compression axial loading, time and P0 ¼ P at t ¼ 0. Determine the maximum per- in which the load varies from a maximum of 300 kN in missible value of P and P0 so that the beam tension to a minimum of 100 kN in compression. Consid- ering a safety factor of 2, compute the bar diameter that (a) does not undergo yielding and gives a fatigue life of 106 number of stress cycles based on (b) has an infinite fatigue life on the basis of Gerber the conservative design criterion. The mechanical properties of the material are: parabola. UTS ¼ 150 MPa; Yield strength ¼ 100 MPa; and The mechanical properties of the metal are: Fatigue strength at 106 number of stress cycles ¼ 50 MPa UTS = 450 MPa; Yield strength = 300 MPa; Fatigue limit = 150 MPa. Solution Solution Given that the maximum load is Pmax ¼ 0:3 MN, and the minimum load is Pmin ¼ À0:1 MN. The UTS of the material Given that the length of beam is L = 0.15 m, and the is Su ¼ 150 MPa, and the fatigue strength at 106 number of diameter is D = 0.025 m. The fixed end of the cantilever stress cycles is rn ¼ r106 ¼ 50 MPa. Considering a safety beam is subjected to a maximum bending moment, which is factor of 2, r106 ¼ 50=2 ¼ 25 MPa. MB ¼ PL ¼ P0½1 þ 2 sin 100tŠ  0:15 N m, if the units of P and L are, respectively, N and m. The stress cycle acting Let us assume the cross-sectional area of the round bar is on the surface of the beam at the fixed end is obtained from A m2. So, the maximum and minimum stresses are, (4.14), which is: r ¼ MB=Z, where for a circular beam, the respectively: section modulus is (4.16c): Z ¼ pD3 ¼ pð0:025Þ3 m3: 32 32

8.19 Solved Problems 365 ) r ¼ 0:15P0½1 þ 2 sin 100tŠ  32 Or, pð0:025Þ3 ðCP0Þ2 þ À  109 Á À 2:025  1017 ¼ 0: ¼ ð97784:797ÞP0½1 þ 2 sin 100tŠ 2:7 CP0 ¼ C Á P0½1 þ 2 sin 100tŠ N=m2; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) CP0 ¼ À2:7  109 Æ ð2:7  109Þ2À4ðÀ2:025  1017Þ where C ¼ 97784:797 mÀ2, and P0 is in N. 2 Since the maximum and minimum values of sin 100t are, ¼ 73:025  106 N=m2; ðneglecting negative valueÞ: respectively, +1 and −1, therefore the maximum and mini- Hence, mum values of stress in the stress cycle are, respectively: rmax ¼ CP0½1 þ 2Š ¼ 3CP0 N=m2; and P0 ¼ 1  73:025  106 ¼ 73:025  106 N ¼ 746:79 N: rmin ¼ CP0½1 À 2Š ¼ ÀCP0 N=m2: C 97784:797 ) Pmax ¼ 3P0 ¼ 3  746:79 N ¼ 2240:37 N ¼ 2:24 kN: Hence, the mean or static stress is: rm ¼ rmax þ rmin ¼ 3C À C P0 ¼ CP0 N=m2: 8.19.3. The loading spectrum on a material at each stress level 2 2 is defined by the number of cycles occurring in 105 number of stress cycles, as tabulated below. The median fatigue life for And the alternating stress or stress amplitude is: 50% survival at each stress is also provided in the table. Determine the expected total median fatigue life of the ra ¼ rmax À rmin ¼ 3C þ C P0 ¼ 2CP0 N=m2: material on the basis of the cumulative fatigue damage theory. 2 2 (a) If yielding is not allowed, then the Soderberg relation Stress Frequency of occurrence in 105 Median fatigue (MPa) number of stress cycles life, N50, cycles given by (8.24) is applicable. Given that the fatigue limit 4.0  104 65 0.4  104 3.2  105 is re ¼ 150  106 N=m2, and the yield strength is 60 1.6  104 1.5  106 S0 ¼ 300  106 N=m2. Hence, substituting the values of 55 3.0  104 2.5  107 ra; re; rm and S0 in (8.24), we get 50 5.0  104  CP0 ¼ 150  106 À CP0 ; 2CP0 ¼ 150  106 1 À 300  106 2 Or, Solution CP0ð2 þ 0:5Þ ¼ 150  106; Let N is the expected total median fatigue life of the material on the basis of the cumulative fatigue damage theory. Sup- ) P0 ¼ 1  150  106 ¼ 1  150  106 N pose, n1; n2; n3 and n4 are, respectively, the numbers of C 2:5 97784:797 2:5 stress cycles endured at stress levels of 65, 60, 55 and ¼ 613:59 N: 50 MPa. These numbers of stress cycles can be obtained from the given data in terms of fraction of the total median life N as follows: Hence, 0:4  104 105 Pmax ¼ 3P0 ¼ 3  613:59 N ¼ 1840:77 N ¼ 1:84 kN: n1 ¼ N ¼ 0:04 N cycles; (b) Given that UTS, Su ¼ 450  106 N=m2. For infinite n2 ¼ 1:6  104 N ¼ 0:16 N cycles; fatigue life, considering Gerber parabola given by 105 (8.23), we can write n3 ¼ 3:0  104 N ¼ 0:3 N cycles; 105 \"  CP0 2# 2CP0 ¼ 150  106 1 À 450  106 n4 ¼ 5:0  104 N ¼ 0:5 N cycles: 105 ¼ 150  106 À ðCP0Þ2 : The given virgin fatigue life at each stress level is 1350  106 respectively

366 8 Fatigue N1 ¼ 4:0 Â 104 cycles; N2 ¼ 3:2 Â 105 cycles; ÂÈ É È ÉÃ N3 ¼ 1:5 Â 106 cycles; N4 ¼ 2:5 Â 107 cycles: log 103 Â ð639:2 À 480Þ = 103 Â ð555:7 À 480Þ i1 ¼ logð1:43=0:41Þ From (8.25), we can say fatigue failure will occur when ¼ logð159:2=75:7Þ ¼ 0:595: logð1:43=0:41Þ n1 þ n2 þ n3 þ n4 ¼ 1; i.e.; The slope of the line joining the data points of test N1 N2 N3 N4 number 2 and 3 is obtained by subtracting (8.63) from 0:04N 0:16N 0:3N 0:5N (8.62), which is denoted by, say, i2. 4:0 Â 104 3:2 Â 105 1:5 Â 106 2:5 Â 107 þ þ þ ¼ 1; ÂÈ É È ÉÃ log 103 Â ð555:7 À 480Þ = 103 Â ð495:1 À 480Þ Or, i2 ¼ logð0:41=0:05Þ Nð1:0 þ 0:5 þ 0:2 þ 0:02Þ Â 10À6 ¼ 1; ¼ logð75:7=15:1Þ ¼ 0:766: logð0:41=0:05Þ 106 ) N ¼ 1:72 cycles ¼ 581; 395 cycles: The slope of the line joining the data points of test 8.19.4. Determine analytically the fatigue limit by trial-and- number 3 and 1 is obtained by subtracting (8.63) from error from the following data obtained in a Prot fatigue test: (8.61), which is denoted by, say, i3. Test Mean rate of increase of Mean rupture stress of ÂÈ É È ÉÃ number stress amplitude, kPa/cycle test specimen, MPa log 103 Â ð639:2 À 480Þ = 103 Â ð495:1 À 480Þ i3 ¼ logð1:43=0:05Þ 1 1.43 639.2 2 0.41 555.7 ¼ logð159:2=15:1Þ ¼ 0:7024: 3 0.05 495.1 logð1:43=0:05Þ The difference between the maximum and the minimum values of i, i.e. between i2 and i1, is: Solution i2 À i1 0:766 À 0:595 i1 0:595 As a rule, the first value selected for the fatigue limit, re, Â 100 ¼ Â 100 ¼ 28:7%: should be close to but slightly lower than the observed lowest Hence, a new trial is necessary. Skipping intermediate average rupture stress value, i.e. 495.1 MPa. Let us assume trials, let us go directly to the final value of re, which will that re ¼ 480 MPa ¼ 480 Â 103 kPa. Substitution of the result in a maximum difference of i less than 1%. given three values of mean rate of increase of stress ampli- tude, a, in units of kPa=cycle, given three values of mean Final Trial: Let us assume that re ¼ 464:7 MPa ¼ rupture stress, rR, in units of kPa and the assumed value of re 464:7 Â 103 kPa and repeat the above process. (in units of kPa) into (8.60) gives three simultaneous equa- À Â 103 À 464:7 Â 103Á ¼ log k0 þ i logð1:43Þ tions as follows: log 639:2 À Â 103 À 480 Â 103Á ¼ log k0 þ i logð1:43Þ ð8:64Þ log 639:2 ð8:61Þ À Â 103 À 464:7 Â 103Á ¼ log k0 þ i logð0:41Þ log 555:7 À 103Á log 555:7 Â 103 À 480 Â ¼ log k0 þ i logð0:41Þ ð8:65Þ ð8:62Þ À Â 103 À 464:7 Â 103Á ¼ log k0 þ i logð0:05Þ log 495:1 À 103Á log 495:1 Â 103 À 480 Â ¼ log k0 þ i logð0:05Þ ð8:66Þ ð8:63Þ Subtracting (8.65) from (8.64), where i is the slope of the plot of logðrR À reÞ versus log a, ÂÈ É È ÉÃ and k0 is the intercept of the plot on the logðrR À reÞ-axis. log 103 Â ð639:2 À 464:7Þ = 103 Â ð555:7 À 464:7Þ The slope of the line joining the data points of test number 1 i1 ¼ logð1:43=0:41Þ and 2 is obtained by subtracting (8.62) from (8.61), which is ¼ logð174:5=91Þ ¼ 0:521155: logð1:43=0:41Þ denoted by, say, i1.

8.19 Solved Problems 367 Subtracting (8.66) from (8.65), (a) Fatigue strength reduction factor and fatigue notch sen- sitivity index. ÂÈ É È ÉÃ (b) The effective maximum and minimum stresses of the log 103 Â ð555:7 À 464:7Þ = 103 Â ð495:1 À 464:7Þ imposed stress cycle. i2 ¼ logð0:41=0:05Þ (c) Maximum allowed stress amplitude based on the most preferred design criterion. Comment on whether the above ¼ logð91=30:4Þ ¼ 0:521077: shaft will survive for an infinite fatigue life or not. logð0:41=0:05Þ (d) If the above shaft fails to survive for an infinite fatigue life, determine the minimum UTS of such steel which can Subtracting (8.66) from (8.64), survive for an infinite fatigue life. All other aforesaid con- ditions remain the same. ÂÈ É È ÉÃ log 103 Â ð639:2 À 464:7Þ = 103 Â ð495:1 À 464:7Þ Solution i3 ¼ logð1:43=0:05Þ ¼ logð174:5=30:4Þ ¼ 0:5211065: logð1:43=0:05Þ The difference between the maximum and the minimum Given that the elastic stress concentration factor is values of i, i.e. between i1 and i2, is: Kt ¼ 2:55, UTS of the given steel is Su ¼ 550 Â 106 N=m2, i1 À i2 Â 100 ¼ 0:521155 À 0:521077 Â 100 ¼ 0:015%: the Neuber’s constant of the steel is qN0 ¼ 0:15 Â 10À3 m, i2 0:521077 the diameter of the shaft is D ¼ 0:015 m and the radius of the hole existing in the shaft is qt ¼ 0:75 Â 10À3 m. Since Hence, the values of i differing by a maximum of 0.015% the fatigue ratio, which is the ratio of fatigue limit to UTS, is indicate that the assumed value of re in this trial is the assumed to be 0.5, so the fatigue limit of the steel shaft correct one. without any stress concentration is: Thus, the fatigue limit is re ¼ 464:7 MPa. The value of the intercept, k0, can be obtained by using re ¼ 0:5 Â Su ¼ 275 Â 106 N=m2: (a) From (8.38a), the fatigue strength reduction factor is: the average of the three computed values of slope i, and substituting the average i value, i.e. i, into the (8.64), (8.65) Kpt Àffiffiffiffi1ffiffiffiffiffiffiffiffi 1 þ qN0 =qt or (8.66): Kf ¼ 1þ i ¼ i1 þ i2 þ i3 ¼ 0:521155 þ 0:521077 þ 0:5211065 ¼ 1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffiffiffi:ffi5ffiffiffi5ffiffiffiÀffiffiffiffiffi1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2:071: 33 1 þ ð0:15 Â 10À3Þ=ð0:75 Â 10À3Þ ¼ 0:52111: k0 639:2 Â 103 À 464:7 Â 103 From (8.37a), the fatigue notch sensitivity index is: ð1:43Þ0:52111 ) ¼ or 555:7 Â 103 À 464:7 Â 103 q ¼ Kf À 1 ¼ 2:071 À 1 ¼ 0:691: Kt À 1 2:55 À 1 or ð0:41Þ0:52111 (b) Given that the axial static load is Pm ¼ 15 Â 103 N. 495:1 Â 103 À 464:7 Â 103 Hence, the static or mean stress is: ð0:05Þ0:52111 Pm 4 Â 15 Â 103 ðpD2Þ=4 pð0:015Þ2 ¼ 144:8 Â 103: rm ¼ ¼ N=m2 ¼ 84:88 MN=m2: 8.19.5. A cylindrical steel shaft with diameter of 15 mm is Given that the completely reversed bending moment is subjected to a completely reversed bending moment of ± 30 N m and an axial static load of 15 kN. The steel shaft MB ¼ Æ30 N m. The stress amplitude, ra, of the imposed has a transverse hole with diameter of 1.5 mm that produces stress cycle is obtained from the applied bending moment an elastic stress concentration factor of 2.55. The Neuber’s constant of the steel having UTS of 550 MPa is found to be MB using (4.14), in which the section modulus for a circular 0.15 mm. Assume that the fatigue ratio is 0.5 for the shaft cross-section from (4.16c) is: having no geometrical discontinuity and determine the following: Z ¼ ÀpD3Á=32 ¼ À Â 0:0153Á=32 m3: p

368 8 Fatigue Since steel is a ductile metal, so for the notched condition, Exercise the factor Kf should be applied only to the stress amplitude component as shown below: 8.Ex.1. A cylindrical shaft of 12 mm diameter shows an infinite fatigue life when it is subjected to completely ra ¼ Kf MB ¼ 2:071  30  32 N=m2 reversed bending moment of ±30 N m, beyond which the Z p 0:0153 shaft fails to survive for an infinite fatigue life. If fatigue ¼ 187:5 MN=m2: ratio of the shaft is 0.47, what will be the UTS of the shaft-material? Hence, the effective maximum and minimum stresses of the imposed stress cycle are, respectively, given by (8.6) and 8.Ex.2. It is seen from fatigue tests of 10 specimens of the (8.7) as shown below: same material at a stress level of 215 MPa that they fail after stress cycles of 19,200, 17,700, 17,600, 17,100, 16,400, rmax ¼ rm þ ra ¼ 84:88 þ 187:5 ¼ 272:38 MPa ðtensileÞ: 16,300, 16,100, 16,000, 15,900 and 15,400. For this stress rmin ¼ rm À ra ¼ 84:88 À 187:5 ¼ À102:62 MPa ðcompressiveÞ: level, compute the cyclic life that provides 95% confidence ðcÞ that 99.9% of the components would survive (p). Values (c) For the steel shaft having geometrical discontinuity, its of q for S–N data assuming normal distribution can be obtained from Table 8.1, where n denotes the number of unnotched fatigue limit, re, will be replaced by its notched specimen. fatigue limit, re0 , according to (8.36b): 8.Ex.3. The unnotched fatigue strength of an aluminium re0 ¼ re ¼ 275 MPa: alloy at 106 cycles in a purely alternating stress cycle of Ær Kf 2:071 is 200 MPa. A circular shaft of this alloy has a transverse groove with radius of 2 mm that produces a theoretical stress The most preferred design criterion is Goodman relation concentration factor of 2.2. If the Neuber’s constant of the alloy is found to be 0.4 mm, calculate the working stress for given by (8.21). Obviously, re in Goodman relation will be the shaft for a fatigue life of 106 cycles under the same stress replaced by re0 for the steel shaft having geometrical dis- cycle of Ær, taking a safety factor of 3. continuity, and the modified relation can be used to obtain 8.Ex.4. Neuber’s constant of an aluminium alloy is 0.6 mm and the fatigue ratio for the aluminium alloy without any the value of the maximum allowed stress amplitude crack is given by the typical fatigue ratio value of several non-ferrous metals tested in smooth-polished condition at ðraÞallowed, which will give an infinite fatigue life for the room temperature for a purely alternating stress cycle. What given steel shaft subjected to a static stress of will be the fatigue ratio for an infinitely wide plate of the same aluminium alloy having a central through thickness rm ¼ 84:88 MPa. circular hole of 2 mm diameter (the plane of hole is oriented normal to the applied stress axis)?    rm 275 1 À 84:88 MPa 8.Ex.5. An infinitely wide steel plate has a through thickness ðraÞallowed¼ re0 1 À Su ¼ 2:071 550 elliptical internal crack with a length of 4 mm and a tip radius of 0.5 mm, which is oriented normal to the stress axis ¼ 112:29 MPa: applied in the longitudinal direction. If the fatigue ratio of the steel with the above crack geometry is 0.21 and that in Since the actual stress amplitude, ra, of the imposed smooth condition, i.e. without crack, is 0.49, then calculate stress cycle is greater than the maximum allowed stress the following: amplitude, ðraÞallowed, for an infinite fatigue life, the shaft (a) Fatigue notch sensitivity index. fails to survive for an infinite fatigue life. (b) Peterson constant. (c) UTS of the steel empirically from Peterson constant. (d) The minimum UTS of such steel which can survive for 8.Ex.6. Assume that the variation of alternating component an infinite fatigue life can be determined by substituting the with the static component of the stress cycle is represented by a parabola whose vertex is at the point of UTS on the axis actual stress amplitude ra of the imposed stress cycle in of static stress and passing through the point of endurance Goodman relation modified by the replacement of re by r0e in (8.21). Hence, substitution of ra ¼ 187:5 MPa gives 0:5  Su  2:071 84:88 187:5 ¼ 1 À Su ; or, Su À 84:88 ¼ 187:5  2:071 ; 0:5 ) Su ¼ 187:5  2:071 þ 84:88 MPa ¼ 861:505 MPa: 0:5

8.19 Solved Problems 369 limit on the axis of alternating stress. A material is subjected Stress Fraction of expected total median Median fatigue to a stress cycle of rðkPaÞ ¼ 25½1 þ n sin 200tŠ, where t is (MPa) fatigue life of the material life, N50, cycles the time and n is the multiplication factor. Compute the 3  106 maximum tolerable value of r for an infinite fatigue life 180 0.30 4  107 based on the above design criterion. Compare this computed value with those derived based on the relations due to 160 0.55 Goodman and Gerber and comment. 8.Ex.11. Suppose, there is a sharp elastic through-thickness The mechanical properties of the metal are: surface crack with depth 2 mm from one surface in an infi- nitely wide plate of an alloy, where Y ¼ 1:1. The plate is UTS ¼ 175 kPa; Yield strength ¼ 125 kPa; subjected to uniaxial fatigue stresses of constant amplitude, Endurance limit ¼ 75 kPa: in which the stress varies from 150 MPa in tension to 50 MPa pinffiffifficffi ompression. Fracture toughness of the alloy is 8.Ex.7. A steel bar is subjected to a tensile–tensile fluctu- 30 MPa m, and tensile strength is 520 MPa. Neglecting the ating stress of r to 2r MPa. What will be the highest per- small influence of mean stress on the crack growth, estimate missible value of r for an infinite fatigue life based on the the number of cycles required for fracture of the plate if the Gerber parabola? The mechanical properties of steel are: fatigue crack growth rate relation obeyed by the alloy is: Tensile strength ¼ 900 MPa; Yield strength ¼ 600 MPa; da ðm=cycleÞ ¼ 2  10À37ðDKÞ4ÀPapmffiffiffiffiÁ4: Fatigue limit ¼ 420 MPa: dN 8.Ex.8. For a completely reversed stress cycle, the following 8.Ex.12. Indicate the correct or most appropriate answer data were obtained in a Prot fatigue test of a certain metal: from the following multiple choices: Mean rate of increase of stress Mean rupture stress of test (a) State which of the following respective combinations of amplitude, kPa/cycle specimen, MPa microstructure and UTS result in the highest fatigue limit for eutectoid steel: 1.7 730 (A) Spheroidite of 500 MPa; 0.007 490 (B) Spheroidite of 650 MPa; (C) Pearlite of 500 MPa; Using Prot equation, determine the fracture stress of that (D) Pearlite of 650 MPa metal when the rate of increase of stress amplitude is 0:22 kPa=cycle. (b) Fatigue strength of steel is reduced when surface treat- ment is: 8.Ex.9. A metallic member is subjected to cyclic strain-controlled fatigue with a total strain range Æ0:005 and (A) Flame hardening; (B) Chromium plating; stress range of Æ100 MPa. The metal shows a true fracture strain of 40% and an elastic modulus of 200 GPa. Assuming (C) Nitriding; (D) Shot peening. fatigue ductility exponent to be −0.5, calculate the (c) To make rubber tyre treads less susceptible to fatigue, the (a) Elastic and plastic strain range. treatment used is: (b) Number of strain cycles to failure. (A) Shot peening; 8.Ex.10. The loading spectrum on a material at each stress (B) Surface cold rolling; level is described by saying that a certain fraction of the (C) Thermal contraction; expected total median fatigue life of the material is loaded (D) Heat softening. at each stress level, as tabulated below. The median fatigue life for 50% survival at each stress is also provided in the (d) Out of the followings, the best resistance to thermal table. Determine the expected total median fatigue life of the fatigue is indicated if the linear thermal coefficient of material on the basis of the cumulative fatigue damage expansion and the thermal conductivity of an alloy are theory. (A) both high; (B) respectively high and low; (C) both low; (D) respectively low and high. Stress Fraction of expected total median Median fatigue (e) The fatigue resistance of a non-ferrous metal is improved (MPa) fatigue life of the material life, N50, cycles by the following technique: 7  104 310 0.05 6  105 (A) Carburizing; (B) Nitriding; (C) Anodizing; (D) Shot peening. 225 0.10 (continued)

370 8 Fatigue (f) The fracture surface morphology of a fatigue fracture Evans, U.R., Simnad, M.T.: Proc. R. Soc. London 188A, 372 (1947) shows: Feltner, C.E., Laird, C.: Cyclic stress-strain response of FCC metals and (A) Dimples; (B) Cleavage; (C) Veins; (D) Striations. alloys I. Acta Metall. 15, 1621 (1967) Findley, W.N.: An explanation of size effect in fatigue of metals. Answer to Exercise Problems J. Mech. Eng. Sci. 14(6), 424–428 (1972) 8.Ex.1. 376 MPa. Forman, R.G., Kearney, V.E., Engle, R.M.: Trans. ASME J. Basic Eng. 8.Ex.2. 10,862 stress cycles. 8.Ex.3. 36.4 MPa. 89, 459 (1967) 8.Ex.4. 0.1645. Forrest, P.G.: Fatigue of Metals. Addison-Wesley, Reading (1962) 8.Ex.5. (a) 0.443; (b) 0.629 mm; (c) 348 MPa. Gensamer, M., Pearsall, E.B., Pellini, W.S., Low, J.R.: Trans. ASM. 8.Ex.6. 94.44 kPa; Goodman: 89.28 kPa; Gerber: 98.47 kPa. The given design criterion is more conservative 30, 983–1020 (1942) than Gerber parabola but less conservative than Goodman Goodman, J.: Mechanics Applied to Engineering. Longmans, Green & line. 8.Ex.7. 422.8 MPa. Co., Ltd., London (1899) 8.Ex.8. 565.8 MPa. Gough, H.J.: Proc. ASTM 33(2), 3–114 (1933) 8.Ex.9. (a) Elastic strain range = 10−3; Plastic strain Gough, H.J., Sopwith, D.G.: Some further experiments on atmospheric range = 9 Â 10−3; (b) 3950 cycles. 8.Ex.10. 1,005,325 cycles. action on fatigue. J. Inst. Met. 56, 55–89 (1935) 8.Ex.11. 276,652 cycles. Grosskreutz, J.C.: Fatigue mechanism in the sub-creep range. In: 8.Ex.12. (a) (B) Spheroidite of 650 MPa. (b) (B) Chromium plating. Manson, S.S. (ed.) Metal Fatigue Damage-Mechanism, Detection, (c) (D) Heat softening. (d) (D) respectively low and high. Avoidance, and Repair, ASTM STP 495, p. 32. American Society (e) (D) Shot peening. (f) (D) Striations. for Testing and Materials, Philadelphia (1971) Haigh, B.P., Jones, B.: J. Inst. Metals 43, 271 (1930) References Halford, G.R.: Cumulative fatigue damage modelling—crack nucle- ation and early growth. Int. J. Fatigue 19(1), S253–S260 (1997) Allen, N.P., Forrest, P.G.: Inst. Mech. Eng. ASME International Harris, W.J.: The influence of decarburization on the fatigue behavior Conference on the Fatigue of Metals, p. 327 (1956) of steel bolts. S&T Memo 15/65, Ministry of Aviation, U. S. Govt. Report 473394, Aug 1965 (1965) ASTM STP: Manual on Fatigue Testing. ASTM STP No. 91, American Koo, G.P., Riddell, M.N., O’Toole, J.L.: Polym. Eng. Sci. 7, 182 Society for Testing Materials, Philadelphia, Pa, pp. 6–65 (1949) (1967) Laird, C.: The influence of metallurgical structure on the mechanisms ASTM STP: A Tentative Guide for Fatigue Testing and the Statistical of fatigue crack propagation. Fatigue Crack Propagation, Analysis of Fatigue Data (Supplement to Manual on Fatigue ASTM STP No. 415, American Society for Testing and Materials, Testing, STP No. 91). 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NASA Technical Notes, National Advisory Committee for strength of a high carbon steel. Trans. ASM 53, 447–465 (1961) Aeronautics, Washington, D.C., p. 2933 (1954) Coffin, L.F.: Proc. I. Mech. E. 188, 109 (1974) Manson, S.S., Hirschberg, M.H.: Fatigue: An Interdisciplinary Coffin Jr., L.F.: Metall. Trans. 2, 3105–3113 (1971) Approach, p. 133. Syracuse University Press, Syracuse (1964) Coffin Jr., L.F.: A study of cyclic thermal stresses in a ductile metal. Manson, S.S., Halford, G.R., Hirschberg, M.H.: Symposium on Design for Elevated Temperature Analysis, American Society of Mechan- Trans. ASME Am. Soc. Mech. Eng. 76, 931–950 (1954) ical Engineers, New York, pp. 12–23 (1971a) Corten, H.T., Dimoff, T., Dolan, T.J.: An appraisal of the prot method Manson, S.S., Halford, G.R., Hirschberg, M.H.: Creep-Fatigue Anal- ysis by Strain-Range Partitioning. NASA TM X-67838. Technical of fatigue testing. Proc. ASTM 54, 875–894 (1954) Paper proposed for presentation at the First National Pressure Courtney, T.H.: Mechanical Behavior of Materials, p. 540. Vessel and Piping Conference sponsored by the American Society of Mechanical Engineers San Francisco, 10–12 May 1971 (1971b) McGraw-Hill Publishing Company, New York (1990) McCammon, R.D., Rosenberg, K.M.: A discussion on work hardening Dieter, G.E., Horne, G.T., Mehl, R.F.: NACA Tech. Note 3211 (1954) and fatigue. Proc. R. Soc. A242, 203 (1957) Dieter, G.E., Mehl, R.F., Horne, G.T.: Trans. ASM 47, 423–439 (1955) McEvily, A.J., Staehle, R.W. (eds.): Corrosion Fatigue. Nat. Assoc. Duquette, D.J.: Fatigue and Microstructure, pp. 335–363. American Corrosion Eng., Houston (1972) Miner, M.A.: Cumulative damage in fatigue. J. Appl. Mech. 12(3), Society for Metals, Metals Park, Ohio (1979) A159–A164 (1945) Epremian, E., Nippes, E.F.: Trans. ASM 40, 870–896 (1948) Morrow, J.D.: Cyclic plastic strain energy and fatigue of metals. 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References 371 Palmgren, A.: Bertschrift des Vereines Ingenieure 58, 339 (1924) Soderberg, C.R.: Factor of Safety and Working Stress. Trans. ASME Paris, P.C., Erdogan, F.: A critical analysis of crack propagation laws. 52(pt. APM-52-2), 13–28 (1930) Trans. ASME Series D J. Basic Eng. Am. Soc. Mech. Eng. 85, 528– Solomon, H.D., Coffin Jr., L.F.: Fatigue at Elevated Temperatures. 534 (1963) ASTM STP No. 520, ASTM, Philadelphia, Pa., pp. 112–122 (1973) Peterson, R.E.: Stress-Concentration Design Factors. Wiley, New York (1974) Tavernelli, J.F., Coffin Jr., L.F.: Trans. ASM 51, 438 (1959) Phillips, C.E., Heywood, R.B.: Proc. Inst. Mech. En. (London) 165, Thompson, A.W., Backofen, W.A.: The effect of grain size on fatigue. 113–124 (1951) Prot, E.M.: Fatigue testing under progressive loading; a new technique Acta Metall. 19(7), 597–606 (1971) for testing materials. Rev. de Metall. XLV(12), 481 (1948) Thompson, N., Wadsworth, N.J., Louat, N.: The origin of fatigue Ransom, J.T.: Proc. ASTM 54, 847–848 (1954a) Ransom, J.T.: Trans. ASM. 46, 1254–1269 (1954b) fracture in copper. Phil. Mag. 1, 113–126 (1956) Ransom, J.T., Mehl, R.F.: Proc. ASTM 52, 779–790 (1952) Weibull, W.: Statistical Representation of Fatigue Failures in Solids. Richards, C.W.: Engineering Materials Science, p. 386. Wadsworth Publishing Company Inc., Belmont (1961) Trans. Royal Inst. of Tech., No. 27, p. 49 (1949) Riches, J.W., Sherby, O.D., Dorn, J.E.: Trans. ASM 44, 852–895 Wells, C.H.: Fatigue and Microstructure, pp. 307–333. American (1952) Ritchie, R.O.: Near threshold fatigue crack propagation in steels. Int. Society for Metals, Metals Park, Ohio (1979) Met. Rev. 24, 205–230 (1979) Wöhler, A.: Zeitschrift für Bauwesen 10 (1860). Cited in: Hertzberg, R. Sherrett, F.: The Influence of Shot-Peening and Similar Surface Treatments on the Fatigue Properties of Metals. Part I, S&T Memo W.: Deformation and Fracture Mechanics of Engineering Materials, 1/66, Ministry of Aviation, U. S. Govt. Report 487487, Feb 1966 3rd edn., p. 463. Wiley, New York (1989) (1966) Wood, W.A.: Bull. Inst. Met. 3, 5–6 (1955) Sinclair, G.M.: Proc. ASTM 52, 743–758 (1952) Wood, W.A.: Some basic studies of fatigue in metals. In: Fracture. Sines, G., Waisman, J.L. (eds.): Metal Fatigue. McGraw-Hill Book Wiley, New York (1959) Company Inc., New York (1959) Wulpi, D.J.: How Components Fail. American Society for Metals, Smith, R.W., Hirschberg, M.H., Manson, S.S.: Fatigue Behavior of Metals Park, Ohio (1966) Materials Under Strain Cycling in Low and Intermediate Life Range. NASA Technical Note D-1574, National Aeronautics and Web Site Space Administration, Washington, D.C., Apr 1963 (1963) Chapter 7—Notches and Their Effects, University of Toledo, p. 36. https://www.efatigue.com/training/Chapter_7.pdf. Accessed 02 Nov 2016

Fracture 9 Chapter Objectives • Variation of interatomic bonding force (cohesive force) with interatomic spacing. Evaluation of ideal fracture strength (cohesive strength in an ideally perfect crystal). • Relation of geometrical discontinuity (flaw) in a body with theoretical (or, elastic) stress concentration factor and material’s fracture strength. • Effects of notch on material’s fracture behaviour: ‘notch strengthening’ and ‘notch weakening’. Distributions of elastic stresses ahead of a notch and of elastic/plastic stresses during local yielding in the vicinity of a notch in plane stress and plane strain conditions. • Characteristic features of fracture process. Fractography describing dimpled fracture (different shapes of the dimple depending on the state of stress), cleavage fracture, quasi-cleavage fracture and intergranular fracture. • Griffith theory of brittle fracture and its applicability. • Modification of Griffith theory by Orowan relation for brittle metals. Modification by Irwin approach introducing ‘elastic strain energy release rate’, its significance and experimental measurement. • Stress intensity factor and its expressions depending on the types of loading and the geometry of crack and specimen configurations. • Different modes of crack surface displacement. Relationship between energy release rate and stress intensity factor. • Plastic-zone size at crack tip and effective stress intensity factor due to crack-tip plasticity in plane stress and strain conditions. • Fracture toughness: plane stress versus plane strain. Test to determine plane-strain fracture toughness and design philosophy using it. • Problems and solutions. © Springer Nature Singapore Pte Ltd. 2018 373 A. Bhaduri, Mechanical Properties and Working of Metals and Alloys, Springer Series in Materials Science 264, https://doi.org/10.1007/978-981-10-7209-3_9

374 9 Fracture 9.1 Introduction If they continue to approach further making the separation distance less than r0; the repulsive force predominates and Fracture refers to breaking or disintegration of a sold sub- increases more rapidly than the attractive force for dimin- stance into two or more pieces under application of stress. If ishing values of r and tends to push them back to their a material becomes unusable for application in service due to equilibrium spacing. It will be easy to visualize the equilib- either fracture or excessive distortion or any other reason, it rium spacing if one assumes that the two atoms or ions are is called failure. So failure of a material may or may not connected by a spring whose neutral length is the same as that involve fracture of the material. Fracture can occur under all of the equilibrium spacing. service conditions. Fracture due to mechanical causes is the subject of our concern and that caused by chemical factors As the attractive forces in interatomic bonds are largely like corrosive environment or any other reasons are beyond electrostatic, the form of the attractive force ðA=rMÞ is usually the scope of the text. considered to be the same as that for the force between electric charges, which is inversely proportional to the square of the The fracture strength of the material is due to the inter- spacing as per Coulomb’s law. Therefore, the value of M is atomic bonding force or cohesive force between two adja- usually taken as 2, but values of N vary from 7 to 10 for cent atoms or ions. So, let us consider the variation of the metallic bonds and 10 to 12 for ionic and covalent bonds. The interatomic bonding force or cohesive force, FðrÞ; between equilibrium spacing r0 normally varies with the different two adjacent atoms or ions with their centre-to-centre bonds from 1 to 4 Å. The equilibrium spacing r0 can be spacing, r; which is shown schematically in Fig. 9.1. The expressed in terms of the constants in (9.1) as follows: curve for the cohesive force between atoms or ions is the resultant of the curves for the repulsion and attraction forces At r ¼ r0; FðrÞ ¼ 0: between atoms or ions. The equilibrium distance of atoms or ions separation, r0; corresponding to zero value of the Therefore, A ¼ B ; or, r0NÀM ¼ B ; bonding force, is associated with a balance of the forces of r0M r0N A repulsion and attraction between two adjacent atoms or ions. The relationship between F(r) and r may be expressed  1 approximately by the following equation: B NÀM ) r0 ¼ A ð9:1aÞ FðrÞ ¼ A À B ðN [ MÞ ð9:1Þ If the atoms or ions are pulled apart from r0; i.e. the rM rN interatomic spacing in the unstrained condition, the repulsive force diminishes much more rapidly than the attractive force where A=rM and B=rN represent, respectively, the forces of with increase of atomic or ionic separation. As a result, a net attraction and repulsion, and A, B, M and N are constants that force between atoms or ions is built up that just balances the depend on the form of bond. As two atoms or ions approach, applied force, and when the applied force is released, atoms or they are drawn together by the attractive force until they reach ions will return to equilibrium spacing at r0: When the sep- equilibrium spacing (spacing in unstrained condition) r0: aration of the atoms or ions is increased further by application of higher and higher pulling force, the resultant cohesive force Inter-atomic bonding force, F(r) Attractive force curve will arrive at its peak point because the repulsive force becomes negligible, while the decrease of the attractive force F(r) = A B is not so great. To separate the atoms or ions completely, a rM rN force equal to the maximum ordinate of the resultant curve, Fmax; must be applied. This force, then, corresponds to the r cohesive strength, rc; of the material. Generally, the higher the cohesive strength of the material, the higher will be the 0 values of its elastic constant and melting point and the smaller will be the coefficient of its thermal expansion. (Equilibrium atomic spacing) Since the problem of brittle fracture is considered to be 0 Inter-atomic spacing, r important in the field of engineering applications and most of the research works have been focussed on this topic, the Fmax engineering aspects of brittle fracture will be mainly consid- ered in this chapter. Since brittle fractures in tensile loading Repulsive force involve little or no plasticity, so the change in cross-sectional area of a member undergoing brittle fracture is negligible; i.e., Fig. 9.1 Schematic variation of the interatomic bonding force or the cross-sectional area is assumed to remain practically cohesive force, FðrÞ; between two adjacent atoms or ions with their constant. Hence, the engineering stress, S %, the true stress, r; centre-to-centre spacing, r:

9.1 Introduction 375 and the stress in this chapter will be designated by the term r, In case of a brittle elastic solid, where Hook’s law is although initial cross-sectional areas of the members are applicable till the point of its fracture, we can write mostly used to determine the stresses. r ¼ Ee ¼ E x r0 ð9:4Þ 9.2 Theoretical Cohesive Strength where E = modulus of elasticity. Now, we can eliminate x after equating (9.3) with (9.4) The magnitude of theoretical cohesive strength (cohesive strength in an ideally perfect crystal) is many times greater and get a solution for rc as follows: than the values of fracture strengths commonly observed in engineering materials. So it is appropriate to evaluate the rc 2px ¼ E x ; or; rc ¼ k E ð9:5Þ theoretical cohesive strength, rc; and discuss the reason for k r0 2p r0 the above discrepancy with the real fracture strength values. For this, let us take the cohesive stress versus atomic dis- If we make a reasonably accurate assumption that placement curve as shown in Fig. 9.2. The theoretical r0 % k=2, then from (9.5) cohesive strength can be well approximated if it is assumed that the shape of the cohesive stress curve in Fig. 9.2 is rc % E ð9:6Þ represented by a sine curve with a half period of k=2; where p k is the wave length of the curve. Hence, the shape of the curve is approximated by Equation (9.6) shows that the value of theoretical cohe- sive strength is likely to be extremely high, which indicates r ¼ rc sin 2px ð9:2Þ the probability of having extremely high fracture strength for k materials. As an example, the fracture strength of very fine diameter silica fibres % E=4: where r ¼ the tensile stress applied to pull the atoms apart; r0 ¼ the equilibrium interatomic spacing, i.e. the interatomic The value of k can be determined in terms of surface spacing in the unstrained condition; and x ¼ r À r0 ¼ the energy cs by considering the energetics of the fracture pro- atomic displacement in a lattice, in which r is the interatomic cess. All the work expended during fracture of a brittle solid spacing. When x ¼ k=4; we can see from (9.2) as well from is utilized to create two new fracture surfaces. The work Fig. 9.2 that r ¼ rc: For small displacements of atom, done per unit area of surface during fracture is the area under sin x % x and (9.2) reduces to the cohesive stress versus atomic displacement curve in Fig. 9.2, which is given as follows: Zk=2 Zk=2 2px k 2px Fracture work per unit area ¼ r dx ¼ rc sin dx k r ¼ rc ð9:3Þ 00 ¼ rc À cosð2px=kÞ!x¼k=2 ¼ rck Cohesive stress, σ 2p=k x¼0 2px!x¼0 cos 2p k x¼k=2 r0 σc ¼ rck ½1 À ðÀ1ފ 0 2p λ x=0 2 r ¼ rck (Inter-atomic spacing) p ð9:7Þ Atomic displacement, x = r – r0 But all this work given by (9.7) is equal to the energy required to form the two new fracture surfaces in a unit area. If cs ¼ surface energy per unit area of fracture surface, then Fig. 9.2 Schematic curve of cohesive stress versus atomic rck ¼ 2cs; or, k ¼ 2pcs ð9:8Þ displacement p rc

376 9 Fracture Substituting for k from (9.8) into (9.5), we get near the discontinuity will be higher than the average stress in the body, which will be found to exist at a distance away from rc ¼ 2pcs E ; or, r2c ¼ csE ; the discontinuity. At the tip of the discontinuity, a stress rc 2pr0 r0 concentration is observed that is indicated by the crowding of the elastic lines of force at the crack tip, as shown in Fig. 9.3. rffiffiffiffiffiffiffi Thus, a geometrical discontinuity in a body acts as a stress Ecs raiser. Stress concentration can also arise from structural ) rc ¼ r0 ð9:9Þ irregularities or metallurgical stress raisers such as inclusions, graphite flakes in cast iron, blowholes or porosity, and We have approximated and expressed the shape of the decarburization. The stress concentration is usually expressed cohesive stress–displacement curve by a simple sine wave, but by a factor known as theoretical stress concentration factor use of more complicated expressions for the shape of the curve or elastic stress concentration factor, which is normally shows that rc varies from E=4 to E=15: A convenient choice denoted by Kt; and defined as follows: for the value of theoretical cohesive strength is to take E=10: For the purpose of illustration, the following example is cited. Example: For silica fibre, if E = 97.1 GPa, cs ¼ 1 J/m2 and Kt ¼ maximum stress at the tip of discontinuity nominal stress based on the net cross-sectional r0 ¼ 1.6 Å, then the theoretical cohesive strength or ideal applied area fracturreffiffiffiffisffiffiffitffirffiffieffiffinffiffiffigffiffiffitffihffiffiffiffiffioffiffiffifffiffiffiffisilica fibre from (9.9) will be given by ¼ rmax 97:1 Â 109 Â 1 E E ra rc ¼ 1:6 Â 10À10 Pa ¼ 24:635 GPa ¼ 3:94 % 4 : ð9:10Þ Very fine diameter silica fibre is probably the only material which shows the ideal fracture strength % E=4: This applied nominal stress (or engineering stress) ra is usually calculated on the basis of the net cross-sectional area Some materials like tiny defect-free metallic whiskers of a specimen without taking into account the effect of geometric discontinuities such as cracks, notches, holes, approach the theoretical value of fracture strength, for grooves, fillets and normally determined from the applied load divided by the initial cross-sectional area at the notch, example the approximate ratio of modulus of elasticity to although some workers calculate this stress based on the gross cross-sectional area of a specimen, i.e. the entire area fracture strength for iron whisker is 23, for silicon whisker is 26 and for alumina whisker is 33. Patenting of 1 wt% C steel with a heavy deformation above 95% to form thin wire can produce the highest value of real UTS obtainable in indus- trial product which is about 4.5 GPa. This strength is roughly E=44:5; where E for steel is 200.1 GPa, but fracture strengths of high-strength steels in exceptional cases exceed Stress, σ 2 GPa. The observed fracture strengths of engineering materials are normally 10–1000 times lower than the values of their theoretical cohesive strengths, i.e. ideal fracture strengths. From the above findings, it is concluded that the flaws or cracks present in engineering materials make their fracture strengths many times lower than the theoretical values. The above defects may be introduced during the manufacturing process of the material, which may include shrinkage cavities, porosities, such as pinholes and blow- holes. Welding and heat treatment may also introduce cracks 2a or microcracks, e.g. formation of quench cracks. Cracks can Surface crack of depth a further form from the microstructural constituents like inclusions, brittle grain-boundary films and second-phase particles on application of stress above certain level. 9.3 Inglis Analysis of Stress Concentration (Thitckness) Factor Stress, σ Any geometrical discontinuity or defect such as a crack or a notch or a hole or any flaw in a body causes a non-uniform Fig. 9.3 Crowding of the elastic lines of force at the tips of internal distribution of stress in the vicinity of the defect. The stress crack of length 2a and surface crack of length a indicates the presence of stress concentration

9.3 Inglis Analysis of Stress Concentration Factor 377 of cross-section in a region where there is no stress con- Putting the value of b from (9.12) in (9.11), we get centrator. Values of Kt for simple shapes can be computed from the theory of elasticity. For complicated geometries, Kt    rffiffiffiffi can be determined from photoelastic measurements. For 2a a various standard geometrical discontinuities like holes, not- rmax ¼ ra 1 þ pqffiffiffitffiaffiffi ¼ ra 1 þ 2 qt ð9:13Þ ches or fillets, values of Kt are available in the literature (Peterson 1974). Since in most cases, a ) qt; i.e. rffiaffiffiffi ) 1; neglecting 1 qt From the idea of stress concentration, Inglis (1913) was able to show how the presence of cracks in a body can result from (9.13), we get in a reduction of the fracture strength. Figure 9.4 shows a two-dimensional view of a thin and small elliptical hole rffiffiffiffi resembling a crack in an infinitely wide plate. Assume that a the major length of the crack-like elliptical hole is 2a; its rmax ¼ 2ra qt ð9:14aÞ minor length is 2b; and the radius of curvature at its tip is qt: Suppose a tensile stress is applied to the plate in a direction rmax rffiffiffiffi normal to the major length of the elliptical hole. By ana- ra a lysing the above plate, Inglis showed the equation for the Or; Kt ¼ ¼ 2 qt ð9:14bÞ theoretical stress concentration factor, Kt, as follows: pffiffiffiffiffiffiffiffiffi So, the term 2 a=qt expresses the effect of crack geometry on the local rise of the stress level at the tip of the crack and can be used to define the theoretical stress con- Kt ¼ rmax ¼ 1þ 2a ð9:11Þ centration factor, Kt: ra b When the applied average tensile stress ra is increased, the maximum stress at the tip of the crack, rmax; also increases and arrives locally at the value of the theoretical cohesive where strength, rc, at certain level of the applied stress ra; although ra remains at a much lower value than rc: It is proposed that rmax maximum stress at the ends of the major axis of the when rmax ¼ rc; the material containing cracks or defects elliptical crack; will fracture and the applied average tensile stress ra becomes ra average tensile stress based on the net cross-sectional the fracture strength rf of the material, i.e. ra ¼ rf : Hence, area applied to the plate in a direction normal to the substituting rmax ¼ rc; and ra ¼ rf ; in (9.14a), we get major axis of the elliptical crack; rffiffiffiffi a half of the major length of the elliptical crack; a b half of the minor length of the elliptical crack. rc ¼ 2rf qt ð9:15Þ It is known that the radius of curvature, qt; at the tip of Hence, equating (9.9) and (9.15), we can solve for the the ellipse is given by fracture strength rf : b2 b ¼ pqffiffiffitffiaffiffi rffiffiffiffiffiffiffiffiffiffiffi ¼ a ; or; ð9:12Þ Ecsqt qt rf ¼ 4r0a ð9:16Þ Equation (9.16) shows the effect of crack geometry on Applied tensile stress, σa the fracture strength of the material. The sharper is the crack, Maximum stress, Maximum stress, i.e. the smaller is the crack-tip radius, qt, and the longer is σmax σmax the crack, i.e. the higher is the value of a; the lower will be the value of fracture strength, rf , of the material. For the sharpest possible crack, where qt ¼ r0 (equilibrium inter- atomic spacing), (9.16) reduces to σa σa rffiffiffiffiffiffiffi Crack-tip radius, ρ b Ecs rf ¼ 4a ð9:17Þ t 2a Applied tensile stress, σa As an example, consider a brittle material having the Fig. 9.4 Stress distribution due to elliptical crack in an infinitely wide sharpest possible crack with length in the micron level. As the plate equilibrium spacing r0 normally varies from 1 to 4 Å, assuming half crack length a ¼ 104 Â r0; the relation between fracture strength of that material and its theoretical cohesive strength can be obtained from (9.9) and (9.17) as follows:

378 9 Fracture rf ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffi tensile test. Upon yielding, this applied stress ra becomes the rc Ecs r0 ; ¼ rc : yield strength for the notched specimen and will be denoted 4ð104 Â r0 Þ Â Ecs or, rf 200 by r0N: The presence of notch raises the flow curve by the amount which can be expressed by a plastic constraint factor, Note that the fracture stress is found to be 200 times usually denoted by q; and it is defined as follows: lower than the theoretical cohesive strength. Thus, we see that the presence of a very small crack can cause a very large Plastic constraint factor; decrease in the stress for fracture. q ¼ flow stress of notched component flow stress of smooth component ð9:18Þ 9.4 Effects of Notch ¼ r0N r0 The effects of notch introduced in a specimen are very similar to those of crack in the specimen. Three conse- The verification of, r0N [ r0; has been given below. quences result from the presence of a notch: From the axial symmetry of the specimen of round (1) Stress concentration, i.e. a rise of stress to a high value cross-section, it can be written that incremental transverse at the notch root; true tensile strain ¼ deh ¼ der ¼ incremental radial true tensile strain. Applying the Levy–Mises equations (1.84b) (2) Set-up of a steep stress gradient from the tip of the notch towards the centre of specimen; and (1.84c), it is shown in Sect. 1.6.2.7 of Chap. 1 by (1.99) that rh ¼ rr; for specimen having axial symmetry. With (3) Creation of a triaxial state of stress field. reference to Von Mises’ yielding criterion given by (1.64), we can write r1 ¼ r0N ; and r3 ¼ rh ¼ rr ¼ r2; in the pre- The notch introduced in a material has an important sent context. influence on the fracture behaviour of the material. The presence of notch can strengthen or weaken a material With the help of Von Mises’ yielding criterion given by depending on whether the notched body is capable of (1.64), the fact that r0N is higher than r0; has been proved, as undergoing appreciable amount of plastic deformation or shown by the following (9.19). not. Let us consider a round bar of a ductile material sub- jected to a tensile load in the elastic level, where a circum- r0 ¼ p1ffiffi h À rr Þ2 þ ðrr À rr Þ2 þ ðrr À r0N Þ2i1=2 ferential notch is introduced in the specimen. The smaller 2 ðr0N cross-sectional area in the notched section causes to develop a higher true stress in that section than in the smooth region ¼ p1ffiffi h ðr0N À rr Þ2 i1=2 ; of the specimen under a given applied load. With increase of 2 2 applied tensile load, the true stress in the notched section reaches the yield strength level of the ductile material, while or; r0 ¼ r0N À rr; ð9:19Þ the true stress in the smooth region still remains in the elastic or; r0N ¼ r0 þ rr; i:e:; r0N [ r0 level. The more highly stressed material in the reduced section tends to extend plastically in the direction parallel to Equation (9.19) shows that the yield strength found in a the axis of loading, and at the same time, the material at the notched-bar tensile test of a substantially ductile material is notch root seeks to contract laterally because conservation of greater than the yield strength of a smooth specimen of the volume must be maintained during the plastic deformation same material. Further, the deeper the notch, the higher is the process. But the unnecked regions of the specimen still constraining tensile stresses rr and rh; and the more is the experiencing an elastic stress level constrain such lateral rise of the yield strength of a notched specimen, r0N ; over that contractions, and in doing so, they induce radial, rr; and of a smooth specimen, r0; of the same material, provided the transverse, rh; true tensile stresses in the vicinity of the material is capable of undergoing appreciable amount of notch tip. Thus, these induced constraining tensile stresses plastic deformation. Figure 9.5a shows the elevation of flow along with the true tensile stress applied at the notch root, curve in a ductile material produced by the plastic constraint say ra; develop a triaxial state of tensile stress, which acts to resulting from a triaxial tensile stress field developed at the constrain the material from plastically deforming in the notch root. On the other hand, in materials with a little notched region. Consequently, the development of the intrinsic plastic deformation capacity, the stress concentra- plastic constraint in the notched section raises the applied tion at the root of the notch causes a local rise of stress to such axial tensile stress ra required to initiate plastic deformation a high level that it reaches the critical value for fracture before above the uniaxial flow stress, r0; found in a smooth-bar the material would undergo general plastic yielding, and thus, notches weaken these brittle materials. Figure 9.5b shows that the introduction of a sharp notch induces premature brittle failure for a material with a limited intrinsic plastic

9.4 Effects of Notch (a) (b) 379 Fig. 9.5 Schematic diagram Deep notch Smooth bar showing effect of notch on flow Shallow notch Notched bar curve. a Elevation of flow curve Smooth bar due to the presence of notch in ductile materials. b Premature failure due to introduction of notch in brittle materials Stress, σ Stress, σ Strain, ε ~ 0.2 Strain, ε ~ 0.05 deformation capacity. Further, with the increase of notch The distribution of stresses near a notch, which is a stress depth in a brittle material, the magnitude of elastic stress concentrator, will now be discussed. The presence of a notch concentration factor Kt increases, which causes to decrease in a plate can change the state of stress from plane stress to the applied stress required for the fracture of material. But in plane strain condition with increase of thickness of the plate; a ductile material, plastic deformation at the root of the notch i.e., a notch will produce a biaxial state of stress in a thin blunts the notch tip causing to increase the notch-tip radius plate in contrast to a triaxial state of stress in a thick plate. and thereby decreases the magnitude of stress concentration Suppose a plate with a through-thickness edge notch is factor Kt to a low level. Thus, the triaxial stress field produced loaded with a tensile stress less than the elastic limit acting in in the presence of a notch results in ‘notch strengthening’ in a the longitudinal direction of the plate, say y-direction. The material with appreciable ductility, while ‘notch weakening’ longitudinally developed stress in the plate will be denoted occurs in a material prone to brittle fracture. The above by ry: In analogy with the earlier discussion on the round bar effects of notch can be observed by performing a series of in this section, there will be development of the constraining notch tensile tests on brittle and ductile metals. For example, tensile stresses at the notch root acting in the two lateral the brittle metal may be a high-strength steel hardened to directions of the plate—one in the width direction of the form martensite in an untempered condition, which will plate, say x-direction, and another in the thickness direction notch weaken, and this weakening effect will increase with of the plate, say z-direction. Suppose the laterally induced increasing the depth of notch in the specimen. On the other stress in the x-direction (along width of the plate) is rx; and hand, a low-carbon steel or aluminium alloy may be con- that in the z-direction (along thickness of the plate) is rz; sidered as an example of the ductile metal, which will not which replace, respectively, the radial, rr; and the trans- only notch strengthen, but this strengthening effect will also verse, rh; stresses considered for the round bar. Since no increase with increasing the depth of notch in the specimen. stress can exist normal to a free surface, the value of through-thickness stress rz must be zero at both free surfaces Two factors must be remembered in relation to the notch of the plate but may rise rapidly with movement inside the strengthening effect, which are as follows: plate from both free surfaces along the notch root and reach a comparatively high value at the mid-thickness plane. If the (1) The load required for fracture of a notched component plate is thin, the stress cannot rise significantly in the showing notch strengthening is lower than that of an thickness direction and so the through-thickness stress rz can unnotched component, when the gross cross-sectional be ignored, which can be expressed as area is considered. rz % 0 ð9:20Þ (2) Whatever is the sharpness and depth of the notch, there is a limit to the magnitude of the plastic constraint Equation (9.20) shows that a biaxial stress condition factor q that has been defined by (9.18). Orowan (1945) consisting of stresses ry and rx; exists at the tip of notch in a has shown that the value of the plastic constraint factor thin plate; i.e., a plane stress condition is developed if the exhibited by a material cannot exceed 2.57, in contrast thickness of a notched or cracked plate is small relative to to the elastic stress concentration factor Kt that can the notch or crack depth. On the other hand, in a notched or exceed a value of 10 with increasing the depth and cracked thick plate the stress rz can increase substantially, sharpness of the notch.

380 (a) y (b) 9 Fracture x Fig. 9.6 Schematic distribution v (σx + σy) of through-thickness stress rz z (along thickness direction z) at the σz notch root, where x ¼ 0; in (a) a thin sheet under plane-stress condition and (b) a thick plate under plane-strain condition σz σz σz which along with stresses ry and rx; develop a triaxial state elements were free from restraints in the x-direction, it can be shown that in plane stress, ex ¼ Àm ey [from (1.35)], and in of tensile stress at the root of notch or crack and severely plane strain, ex ¼ À½m=ð1 À mފ ey [from (1.39)]. Since eyðaÞ [ eyðbÞ [ eyðcÞ; . . .; etc., due to rapid decrease in the limit straining in the z-direction so that ez ¼ 0: Thus, for a longitudinal tensile stress ry in going away from the notch tip, it would then follow that exðaÞ [ exðbÞ [ exðcÞ; . . .; etc., and plate of thickness large enough in comparison with the notch the a=b; b=c; . . .; etc., interfaces would pull apart. Hence, in order to maintain the continuity of elements ‘a’ and ‘b’, ‘b’ or crack depth, a condition of plane strain will be created, and ‘c’, etc., a tensile stress rx must be present across each interface of the tensile elements. Since at the free surface of where the through-thickness stress according to (1.85) is the notch ðx ¼ 0Þ lateral contraction of the tensile element ‘a’ can occur without any restraint from the empty notch, so given by rx ¼ 0 at x ¼ 0: Since the value of ry is large near the notch root, the lateral stress rx; required to maintain continuity, rises ÀÁ ð9:21Þ steeply with increasing distance x from the notch tip, attains a rz ¼ m ry þ rx maximum value and then decreases slowly to a low value at where m ¼ Poisson’s ratio. The peak value of rz strongly depends on the thickness of plate, and the thicker the plate, large values of x, where the longitudinal strains acting on the higher is the peak value of rz: But the values of ry and rx adjacent elements (e.g. ‘p’ and ‘q’) become nearly identical, are almost independent of the thickness of plate; they because the distribution of ry flattens out at distances remote Àfrom theÁnotch. The distributions of biaxial elastic stresses decrease by less than 10% as the plate thickness decreases. ry þ rx in a thin plaÀte, i.e. in plaÁne stress condition and The distribution of the stress rz through the plate thickness, triaxial elastic stresses ry þ rx þ rz in a thick plate, i.e. in i.e. with z at the notch root, where x ¼ 0; is shown in plane strain condition from the notch root at x ¼ 0; with Fig. 9.6, for both conditions of plane stress and plane strain. increasing distance in the x-direction, i.e. along the width of It is to be noted that the longitudinal elastic tensile stress ry the plate, are shown, respectively, in Fig. 9.7a, b. will reach to a peak value at the root of the notch due to the Let us now consider plastic deformation at stress con- effect of stress concentration. This peak value will gradually centrators. In an elastically stressed notched piece, high decrease with increasing distance in the x-direction and will stresses generated near the notch may locally exceed the arrive at the value of the average applied body stress at a yield strength of material to form a small plastic zone or distance away from the notch. The development of a lateral ‘enclave’. The distribution of stresses within this plastic zone elastic tensile stress rx due to the application of the longitu- is very much dependent on whether the deformation is tak- dinal elastic tensile stress ry in a notched plate can be understood physically by imagining a series of small tensile ing place in plane strain or in plane stress. elements: a; b; c; . . .; p; q; . . .; etc., starting from the notch tip and arranged along the x-axis as shown in Fig. 9.7a. The In plane stress (obtained by loading a thin notched average longitudinal stress ryðaÞ acting on element a is specimen), the through-thickness principal tensile stress rz is appreciably larger than ryðbÞ acting on element b, so the the smallest stress and equal to zero. Here, yielding takes longitudinal tensile strain eyðaÞ would be higher than eyðbÞ; if each tensile element were able to deform freely. However, in free deformation, there would also be lateral contractions due to Poisson’s ratio, such as exðaÞ; exðbÞ; . . ., associated with longitudinal tensile strain eyðaÞ; eyðbÞ; . . .; etc. If the tensile

9.4 Effects of Notch 381 Fig. 9.7 Schematic distribution (a) Stress (b) Stress of elastic stresses ahead of a notch (a) in a thin plate (plane stress y σy (a) y y condition) and (b) in a thick plate σy (b) σy z (plane strain condition) σ y (c) σy σz σx x σx x 0 x 0 abc pq place on planes at 45° to the y- and z-axes. From Tresca rises steeply just ahead of the notch. Within the plastic criterion of yielding given by (1.68a), we know that enclave at x [ 0; the rate of rise of the lateral stress, rx; is r1 À r2 ¼ r0; where r1 ¼ ry (longitudinal principal tensile higher than that for the elastic situation. This may be stress) and r2 ¼ rz (through-thickness principal tensile stress) in the present context. Hence, the longitudinal tensile understood from the following consideration of the defor- stress throughout the plastic zone, according to Tresca cri- mation of tensile elements lying along x-axis as in Fig. 9.7a. terion, is ry À rzð¼ 0Þ ¼ ry ¼ r0; where r0 is the uniaxial Once yielding initiates, the first tensile element ‘a’ at the yield stress. Thus, the maximum stress in the plastic zone is equal to the uniaxial yield strength of material, as shown in notch tip [see Fig. 9.7a] deforms plastically at constant vol- ume with the plastic value of Poisson’s ratio m ¼ 1=2 instead Fig. 9.8a. of the most common elastic value of m ¼ 1=3: Therefore, the lateral strain ex becomes greater for the plastic deformation In plane strain (obtained by loading a thick notched than for the elastic deformation. For example, in elastic specimen), the distribution of stresses changes remarkably. deformation for m ¼ 1=3; ex ¼ À½m=ð1 À mފ ey ¼ Àey=2, and The smallest lateral principal tensile stress is now r2 ¼ rx; in plastic deformation, ex ¼ Àey; since m ¼ 1=2: Therefore, a and yielding occurs in the yx plane. Within the plastically greater value of rx has to be applied for maintaining cohesion yielded zone, the value of ry; according to Tresca criterion, at interfaces of the tensile elements. Thus, the steep rise of rx is given by ry ÀÀ rx ¼ rÁ0; or ry ¼ r0 þ rx; and that of rz is within the plastic zone in turn also increases ry and rz. The given by rz ¼ ry þ rx =2; which is obtained from (9.21) peak value of rx is now found at the elastic–plastic interface. by substituting m ¼ 1=2 for the constant volume plastic It is important to note that ry and rz have their maximum deformation process. values at the notch root at x ¼ 0 for the elastic deformation, but on yielding, the peak values of ry and rz occur at some Yielding in plane strain starts at the notch root, i.e. at distance beneath the notch, i.e. at the elastic–plastic interface, x ¼ 0; where ry ¼ r0; because rx ¼ 0, at the free surface of where the value of rx is the maximum. notch, i.e. at x = 0. During local yielding, ry drops from its high value in the elastic condition [see Fig. 9.7b] to the Figure 9.8b shows a sketch of the distribution of all three value of uniaxial flow stress r0 at the notch root. Therefore, the onset of plastic deformation at the notch root relieves the principal stresses in plane strain condition from the notch high elastic longitudinal stress and restricts the peak longi- root at x ¼ 0; with increasing distance in the x-direction, i.e. along the width of the plate during local yielding in the tudinal stress to the yield strength of the material. According vicinity of the notch. At x [ 0; all the three stresses increase to (9.21), the value of the through-thickness stress rz at the sharply, reach the maximum values at the interface of the notch root, i.e. at x ¼ 0; is rz ¼ mry ¼ ry=2 ¼ r0=2; since ry ¼ r0 and rx ¼ 0 at x ¼ 0; and m ¼ 1=2: However, just plastic zone ahead of the notch tip and the rest elastic region below the notch in the x-direction (along width of the plate), ry required for yielding is higher than the smooth-bar uni- of the plate and then fall gradually with increasing the dis- axial tensile flow stress r0; because ry ¼ r0 þ rx; and rx tance x. With increasing stress, the movement of the plastic zone occurs inwards until at some point, the entire zone beneath the notch deforms plastically.

382 9 Fracture (a) y intercrystalline). When a crack travels through the grains, it z is called a transgranular fracture, whereas if a crack propa- y gates along the grain boundaries, it will produce an inter- σy granular fracture. A brief comparison between a ductile and σ0 Plastic zone x a brittle fracture is given below. x=0 (1) Tearing of a ductile material occurs slowly with a slow rate of crack propagation, and the crack does not usually ry x extend unless the applied stress is increased. Conversely, (b) y σy in a brittle material, once the crack is initiated, it con- tinues to grow very rapidly without warning and without 3 σ0 σz the requirement of increase in stress; i.e., the length of an σx initiated crack increases in magnitude leading to brittle 2 σ0 fracture at constant stress. σ0 (2) Often, an appreciable amount of energy is expended to cause ductile fracture because of a large amount of plastic deformation involved in the fracture process, while the amount of energy required for a brittle fracture is very low due to the absence of or very less amount of plastic deformation. (3) Ductile fracture is usually a transgranular fracture, whereas brittle fracture shows either transgranular or intergranular fracture that will depend upon whether the grain boundaries are stronger or weaker than the grains. (4) The appearance of fracture surface in a ductile fracture is generally fibrous, grey and dull, whereas that in a brittle fracture is normally flat, granular and bright. x=0 x Brittle fractures are normally noticed in BCC and HCP metals usually at low temperatures, ionic and covalently Plastic zone bonded ceramic materials, glass, ice, but not in FCC metals ry unless they are attacked by some special reactive chemical environment or exposed to factors causing grain-boundary Fig. 9.8 Schematic distribution of elastic/plastic stresses during local embrittlement. At times materials exhibit fractures, which yielding in the vicinity of a notch in (a) plane stress condition (where are partially ductile and partially brittle. The demarcation rz ¼ 0) and (b) plane strain condition (where ez ¼ 0) between a brittle and a ductile fracture is arbitrary and depends on the case being examined. For example, nodular 9.5 Characteristic Features of Fracture cast iron is considered as brittle in comparison with mild Process steel, but it is ductile in comparison with grey iron having graphite flakes. Fractures of engineering materials are broadly classified as either ductile or brittle depending on the amount of plastic Many terms have been used from different disciplines to deformation that a material can undergo prior to its fracture. characterize the fracture process. From the metallurgical A ductile fracture is one that shows a large amount of plastic point of view, it is convenient to discuss the fracture process deformation, while a brittle fracture exhibits little or no of an engineering material in terms of the three following plastic deformation. The fracture process consists of crack characteristics: (1) energy to fracture, (2) macroscopic mode initiation and crack propagation. Based on the path of crack of fracture and (3) microscopic mechanism of fracture that is propagation in polycrystalline materials, fractures are clas- studied by fractography. sified as either transgranular, (also known as transcrys- talline or intragranular) or intergranular (also called 9.5.1 Energy to Fracture As discussed in Sect. 1.9.5 of Chap. 1, the tensile toughness of a material is defined by the mechanical energy absorbed

9.5 Characteristic Features of Fracture Process 383 during deformation till the point of its fracture. It is mea- Stress Stress sured by the total area under the engineering tensile stress– strain curve or more accurately the true tensile stress–strain curve up to the point of fracture. The equations for engi- neering and true toughness are repeated, respectively, below for convenience of reference. Zef ð1:139aÞ Energy=volume ¼ Sde 0 Zef ð1:139bÞ Energy=volume ¼ rde Brittle Tough 0 where S and r are engineering and true stresses, respectively, Stress Stress and ef and ef are engineering and true strains, respectively. Fig. 9.9 Extent of plastic zone developed (shaded region) at fracture If the energy to break a material calculated by (1.139) is for brittle and tough materials high, the material is described as tough or possesses high fracture toughness, and conversely, if the fracture energy is absorbed to break the material is very low, it is brittle, while found to be low from (1.139), the material is said to be the high energy absorption reflects that the material is tough. brittle. It is important to mention here that materials are broadly divided into two categories: ductile and brittle based Whether a material is relatively tough or brittle may also on the extent of strain prior to fracture. But depending on the be judged by noting the amount of plastic zone developed amount of energy absorbed in fracturing a material, it is surrounding the crack tip. When the stress concentration better to classify the material as tough and brittle rather than ahead of a crack tip raises the applied stress at or above the ductile and brittle, because even a highly ductile material value of the yield strength of a material, the material sur- with poor strength may not be sometimes suitable for rounding the crack tip will plastically deform and a zone of application as an engineering component due to lower plasticity will be created there. This plastic zone will be toughness possessed by this material. This is explained with embedded within an elastically deformed mass of the reference to the engineering tensile stress–strain curve in material. The higher the volume of this plastic zone, the Fig. 1.47. This figure shows that the material C having greater is the toughness of the material, because the energy optimum level of strength and ductility absorbs high energy, expended during plastic deformation is much more than that so it is considered to be a tough material and its fracture during elastic deformation. As shown in Fig. 9.9, if the size toughness is high. On the contrary, material A is classified as of the plastic zone just prior to fracture is little – maximum brittle although its strength is high and the toughness of 2% of the crack length [as shown in (9.59c) in Sect. 9.11] – material B possessing high ductility is poor. the material will have low toughness and will be called brittle. On the other side, if the plastic zone extends con- The toughness of a notched specimen can also be mea- siderably from the crack tip to cover the unbroken ligament sured by the notch strength ratio, NSR, evaluated by the of the material (Fig. 9.9), the fracture energy will be high notch tensile test, as discussed in Sect. 1.13 of Chap. 1. NSR and the material will be classified as tough. is defined as the tensile strength ratio of notched to unnot- ched specimen, as given in (1.165). If NSR < 1, then the 9.5.2 Macroscopic Mode of Fracture notch sensitivity of the material is considerably high and the material is said to be notch brittle. When the material The macroscopic fracture path may also inform us about the exhibits a high value of NSR – say, more than 2 – the toughness level possessed by a component before its fracture. material possesses high toughness and its notch sensitivity Macroscopic view of the fracture surface may reveal fully decreases to a minimum. slant or completely flat or partly slant and partly flat fracture. Further, the notch toughness (also called impact tough- ness) of a material can be determined from the notched-bar impact test by noting the energy absorbed in fracturing a notched bar under a rapid rate of loading over a wide range of temperature, as discussed in Chap. 6. If the energy

384 9 Fracture Particularly in sheet- or plate-type component, the toughness powder sintering may pre-exist in a material, but these level can be assessed by the relative proportions of slant and must not be confused with these mechanically induced flat fracture. A given material in thin size usually exhibits a microvoids. Either the interfacial failure between an full slant fracture and possesses a higher value of fracture inclusion or second-phase particle and the surrounding toughness, whereas with increasing thickness, the same matrix or the cracking of particle is mainly responsible material generally results in a complete flat fracture that for the nucleation of microvoids. Consequently, the corresponds to a lower value of fracture toughness. Similarly, spacing between neighbouring microvoids is intimately a mixed mode of fracture, i.e. part slant and part flat fracture, associated with the interparticle distance. Different sized indicates an intermediate level of toughness, which increases microvoids are generally observed in a material con- with increasing the amounts of slant fracture. taining more than one type of inclusion associated with more than one size distribution. The criteria for void 9.5.3 Microscopic Mode of Fracture nucleation depend on a number of factors that include or Fractography stress and strain levels, inclusion size, local deformation modes and alloy purity (Van Stone et al. 1985). The examination of the facture surface using microscope to (2) The second stage is the growth of the microvoids. It has obtain useful information about the nature of fracture is been seen (Hertzberg 1989) that the most of the energy normally called fractography. The fractographic work is required for ductile fracture associated with MVC is conducted recently using the scanning electron microscope expended during this growth process. The growth (SEM) while that had been conducted on the transmission mechanisms involve (a) plastic deformation of the electron microscope (TEM) in the past. The advantages of matrix surrounding the site of nucleation and (b) plastic SEM over TEM are that the actual fracture surface can be deformation enhanced by separation of tiny particles in examined in SEM, while the preparation of a replica of the the matrix. fracture surface is required in TEM due to the limited pen- (3) The final stage of MVC that produces the ultimate etrating power of the electrons. The replica technique used ductile fracture involves the coalescence of numerous for TEM produces electron images that are reversed with microvoids into big cracks. Coalescence may occur by respect to the actual fracture surface morphology, whereas joining together big microvoids with several smaller such reversal is absent in SEM images. Further, SEM has a microvoids. Frequently, the process of coalescence takes large depth of focus, which is useful to examine rough place when ligaments of the material situated between fracture surfaces. adjacent microvoids neck down and eventually fracture, thereby leading to the linking of adjacent microvoids. The commonly observed four classical microscopic modes of fracture are (a) microvoid coalescence or MVC, Depending on the state of stress (Beachem 1975), the producing dimpled fracture, (b) cleavage, (c) quasi-cleavage shape of the dimple may be equiaxed or elongated. Under and (d) intergranular. However, there are other fracture uniaxial tensile loading conditions, the microvoids grow out modes such as fatigue and creep rupture that have been in a plane usually normal to the loading axis, which takes discussed in their respective chapters. place in the fibrous zone located in central region of the cup-and-cone fracture, as shown in Fig. 1.62 of Chap. 1. 9.5.3.1 Dimpled Fracture The above results in the formation of micron-sized spherical When a microvoid on the fracture surface appears as a shaped ‘equiaxed dimples’, also known as ‘spherical dim- micron-sized cup-like depression under SEM, it is called a ples’, which are found in the fibrous zone of the fracture dimple. This type of fracture surface represents a ductile surface. Since the process of plastic deformation is involved fracture occurring in a transgranular manner. The micro- during the growth and coalescence of these microvoids, the scopic mechanism observed in dimpled fracture of most total energy required for fracture, i.e. toughness of the metallic alloys and many engineering plastics is microvoid material, is expected to depend on the size of these dimples. coalescence or shortly MVC, which consists of the follow- It has been experimentally observed that the energy to ing three stages: fracture increases as the width and depth of the observed dimples increase (Birkle et al. 1966; Passoja and Hill 1974). (1) The initial stage is the nucleation of microvoids that takes place in a material subjected to stress. It has been When shear stresses control the fracture process, the experimentally confirmed that plastic deformation can nucleated microvoids grow and finally coalesce along planes produce microvoids or microcracks in metals. Some- of maximum shear stress oriented at 45° to the loading axis. times, microporosities produced during casting or As a result, those voids are elongated and form parabolic depressions on the fracture surface, which are called

9.5 Characteristic Features of Fracture Process 385 ‘elongated dimples’ or ‘parabolic dimples’. If two matching (a) fracture surfaces are compared, it will be observed that these parabolic dimples are elongated in the direction of the σ1 applied shear stress and thus point in opposite directions on σ2≅ σ3 both halves of the fracture surface. Such dimples are observed in the slant shear lip located in outer zone of the (b) cup-and-cone fracture. σ1 When the state of stress is one of combined tension and bending, the material undergoes tearing associated with this σ3 non-uniform stress. The resulting tearing process produces ‘elongated dimples’, also known as ‘tear dimples’. In con- (c) σ 1 trast to the parabolic dimples, the tear dimples point in the same direction on the two matching fracture surfaces. It is to σ2≠ σ3 be noted that the tear dimples point back towards the origin of crack and help us to view the crack origin. Figure 9.10 Fig. 9.10 Diagrams illustrating the effect of the state of stress on the presents schematically to illustrate the effect of the state of morphology of microvoids. a Equiaxed microvoids known as ‘spherical stress on the morphology of microvoids. dimples’ produced by tensile stresses. b Pure shear stresses produce microvoids, called ‘parabolic dimples’, which are elongated in the 9.5.3.2 Cleavage Fracture shearing direction; i.e., these voids point in opposite directions on both Cleavage fracture occurs in a transcrystalline manner along halves of the fracture surface. c Tearing process associated with specific low-index (Reed-Hill 1973) crystallographic planes non-uniform stress produces elongated dimples (known as ‘tear and is usually associated with low fracture energy and thus dimples’) that point back towards crack origin (Beachem 1965), i.e. represents brittle fracture. However, brittle fracture may not point in the same direction on the two matching fracture surfaces be always accompanied by cleavage mode of fracture. For example, brittle fracture in high-strength aluminium alloys produced, which will cause high stresses, and as a result, can occur without cleavage. As a further example, a deeply microcracks will be initiated easily. Cleavage cracks may notched tensile specimen exhibits little gross plastic defor- also be initiated at mechanical twins (Hull 1961, 1963). The mation, i.e. a brittle behaviour, yet the fracture can occur by preferred nucleation sites of crack are those where the twins a shear mode, and not by a cleavage mode. It is to be intersect with each other and with grain boundaries. This remembered that a fracture is said to be brittle if the fracture may be observed in BCC metals subjected to low tempera- energy is low or the crack tip plasticity is limited, while tures and high strain rates, where twinning is the preferred cleavage describes a microscopic fracture mechanism. mechanism of plastic deformation. Low-strength ferritic steel is a ductile material, but when it is subjected to some combinations of low temperature, high The characteristic feature of transgranular cleavage frac- strain rate and/or a high triaxial tensile stress field, as created ture surface is that at high magnification, it usually shows in the presence of a notch, the steel will produce a brittle typical flat cleavage steps or facets, whose size is approxi- cleavage fracture. In this steel, cleavage is so closely related mately equal to that of the ferrite grain in steel. Usually, to the brittle behaviour that ‘cleavage’ and ‘brittle’ are often these cleavage steps exhibit a ‘river pattern’ or ‘river used synonymously in the literature of fracture. marking’ of branching cracks, wherein fine steps are observed to combine progressively into bigger ones. The The process of cleavage fracture is made up of three ‘flow’ direction of the ‘river pattern’ is normally believed to steps: (1) plastic deformation for the development of dislo- cation pile-ups, (2) initiation of crack and (3) propagation of crack. It has been experimentally demonstrated that cracks responsible for brittle cleavage fracture do not initially exist in the material but are created by the process of deformation. The nucleation of microcracks occurs commonly by the cracking of second-phase particles during deformation, but the nature of these particles can largely influence the process of crack initiation. The resistance to cracking increases if particles are strongly bonded with the matrix or spherical and fine having radius less than 1 lm. If the dispersed second-phase particles are easily cut by the dislocations, then comparatively large pile-up of dislocations will be