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

7.9 Concept of ECT and Elevated-Temperature Fracture 285 assumed, for example, that the work per unit area required to nucleate an intercrystalline crack is equal to the energy to create the two new fracture surfaces, then the normal stress, r; is approximately given by (Garofalo 1965) rr0 ¼ 2cs; or; r ¼ 2cs ð7:53Þ w-type r0 cracks where cs the surface energy per unit area, and Stress Mixed types r0 the interatomic distance fracture For values of cs ¼ 1:5 J mÀ2; and r0 ¼ 2:5  10À10 m; r-type the theoretical value of normal stress, r; from (7.53) is: r ¼ cavities ½ð2  1:5Þ=2:5  10À10Š N mÀ2 or Pa ¼ 12 GPa; which is found to be of the order of 100–1000 times the stresses Temperature found experimentally for the nucleation of cracks at low or high temperatures. Creep deformation, therefore, must pro- Fig. 7.23 Types of intercrystalline fracture observed at different duce stress concentrations so that it becomes possible to temperatures and stresses, schematically displayed within a band and reach the theoretical strength and initiate cracks. These high the fracture types are separated by the vertical dashed lines (McLean stress concentrations can be created at grain corners, at 1957a) particles in sliding grain boundaries and at the junction of slip bands and grain boundary particles. the relaxed boundary) to cause the formation of a microcrack • The type of intergranular fracture that will form depends under an applied tensile stress. Assuming isotropic elastic on the creep temperature and the applied stress level. In Nimonic 90 tested at different temperatures and stresses, it behaviour, the stress concentration at the grain corner can be was observed (McLean 1957a) that the type of fracture changed from ‘w-type’ to ‘r-type’ as the temperature was estimated for relaxation along the boundary mn by treating increased and the stress was decreased. Figure 7.23 (McLean 1957a) schematically shows the formation of the viscous sliding boundary as a crack. Following Inglis ‘w-type’ cracks at moderate creep temperatures under high stress levels and that of ‘r-type’ cavities at high creep (1913) analysis of stress concentration factor, which has temperatures under low stresses with the formation of mixed types of fracture in the transition zone. As high been discussed in Sect. 9.3 of Chap. 9, Zener estimated the stress favours ‘w-type’ fracture, it indicates that this type of crack propagates more rapidly than the ‘r-type’ of crack. stress concentration factor and the maximum tensile stress, A number of configurations of ‘w-type’ cracks initiated rmax; at the triple point by by grain-boundary sliding at a point of stress concentration are schematically represented in Fig. 7.24 (Chang and Grant sffiffiffiffiffiffi 1956). A mechanism for the formation of wedge-shaped cracks at grain boundary triple points was proposed by Zener rmax ¼ L Ás for L ) qt ð7:54Þ (1948). In this proposal, it was assumed that grain bound- 2qt aries behave in a viscous manner at sufficiently high tem- perature and the relative displacement of adjacent grains is where restricted near grain corners. Let mn is the grain boundary, which is not normal to the applied stress but subjected to s shear stress along grain boundary, say mn, shear stress, as shown in Fig. 7.25. According to Zener, L length of the sliding boundary and sliding along the grain-boundary mn relaxes the shear stress qt radius of curvature at the tip of the boundary along the boundary and results in a sufficiently high stress concentration at the grain corner (that is located at the end of In the above-mentioned manner, a microcrack can only nucleate if the stress concentration exceeds the cohesive strength of the grain boundary. The nucleation of a crack cannot occur if the stress concentration at a grain corner is relieved by plastic deformation in the grain ahead of the sliding grain boundary. Since grain-boundary migration relieves stress, the opening of a crack at a grain corner can

286 7 Creep and Stress Rupture by the concentrated stress at the tip of a slip band. The form A A of expression obtained is: B B rffiffiffiffiffiffiffiffiffiffiffiffi C C 12csG (a) smin ¼ pL ð7:55Þ where A A (b) smin minimum shear stress along the grain boundary, B B cs surface energy per unit area of a crack formed along a C C grain boundary, G shear modulus, and L is as defined before. BA BA Round or elliptically shaped cavities are believed to be (c) due to discontinuities lying on or in the grain boundary. The nucleation of ‘r-type’ cavities was first observed by Green- CC wood et al. (1954) who found small spherical cavities in the grain boundaries of several metals in early stage of the ter- Fig. 7.24 Schematically showing the formation of w-type cracks in tiary creep range. Chen and Machlin (1956, 1957) proposed several ways at triple points owing to grain-boundary sliding (Chang a mechanism for the formation of this type of cavities along and Grant 1956) the grain boundaries, where the grain boundary is assumed to have a pre-existing jog that obstructs the normal shear Tensile stress axis deformation along the grain boundary. Shearing along the grain boundary produces stress concentrations at the jog, as n shown in Fig. 7.26a, and as a result, a cavity develops at the jog, as indicated in Fig. 7.26b. As in the case of ‘w-type’ Sliding openings at grain boundary triple points, the formation of boundary ‘r-type’ cavities can also be prevented either by plastic deformation in the grains to relieve the stress concentration Crack or by migration of grain boundary away from the points of Region of stress stress concentration. McLean (1957) and Gifkins (1956) also proposed a mechanism for the formation of ‘r-type’ cavities m concentration by stress concentrations at jogs produced by slip in the grain boundary. The basic difference between the Chen-Machlin mechanism and McLean-Gifkins mechanism is that the former assumes that jogs are initially present along the grain boundary while the latter assumes that jogs are created by Tensile stress axis (a) Stress concentration points Fig. 7.25 Mechanism for the formation of wedge-shaped cracks at Shear deformation triple points, as proposed by Zener. Sliding along the grain boundary, Shear deformation mn, relaxes the shear stress along the boundary and results in a stress Jog in boundary concentration at the grain corner causing to form a crack under an applied tensile stress (b) also be prevented if grain boundary migrates away from the Cavity stressed region prior to the rise of stress concentration to the level where a crack can nucleate. Fig. 7.26 Mechanism for the formation of r-type cavities along the grain boundaries. a Stress concentrations produced at the jog due to Subsequently, McLean (1957a) estimated the minimum shearing along the grain boundary. b Development of a cavity at the jog shear stress needed to form a wedge-shaped crack at the head of a sliding grain boundary on the basis of the derivation by Stroh (1954) for the formation of a stable crack

7.9 Concept of ECT and Elevated-Temperature Fracture 287 slip bands penetrating the grain boundary. Further, in the cracks or cavities, once they have nucleated, can occur latter, it was assumed that the stress concentration due to mainly by the following mechanisms: dislocation pile-up at the grain boundary supplied the high stress needed to create a cavity. However, subsequently (1) Advancement of the tip of the crack or cavity due to the McLean and Gifkins have shown (Conrad 1961) that low continued operation of the shearing mechanisms which stresses generally used in creep tests are able to produce initiated the incipient fractures. cavities at jogs in the grain boundary, for which a dislocation pile-up is not required. Experimentally, it has been found (2) Condensation of vacancies into the crack or cavity. that the cavities along the grain boundary are initially about (3) A combination of both (1) and (2). 1 l (1 lm) apart, which is in agreement with the theoretical estimates. Calculations by McLean (1958) indicate that the growth of ‘w-type’ cracks occurs essentially by the above mecha- The ‘r-type’ cavities can also nucleate at fine nism (1), and that the role of vacancy supersaturation is not second-phase particles in the grain boundary when the sep- significant. This seems to be supported by experimental aration of matrix–particles interface occurs as a result of evidence. grain-boundary sliding. For example, substantial necking can occur in pure aluminium subjected to creep but the The ‘r-type’ cavities gradually increase in number and addition of a small quantity (<0.1%) of iron to form particles size with continuing creep in the tertiary stage. The density of FeAl3 at the grain boundaries results in intercrystalline and diameter of the cavities grow by the joint action of fracture without showing any localized necking. Similarly, grain-boundary sliding and vacancy diffusion and ultimately, the presence of Cr23C6 or NbC in the grain boundaries of they join together to produce a continuous crack along grain austenitic steels can remarkably reduce their creep ductility boundaries leading to total fracture. The rate at which these (Honeycombe 1984). cavities grow has been examined in silver (Oliver and Gir- ifalco 1962) by measuring the diameter of cavity perpen- McLean (1963) has shown that if the diameter of a cavity dicular to the grain boundary. These results show that the is larger than several atomic diameters it will possibly not mean radius, r; of the cavity increases linearly with time t at vanish. If the cavity becomes spherical due to surface ten- a temperature T as follows: sion, then to obtain a stable spherical cavity, the condition to be satisfied must be: r ¼ uðTÞt þ r0 ð7:57Þ r [ 2cs ð7:56Þ where in the plot of r versus t at a temperature T, r0 is the r intercept at time t ¼ 0 and uðTÞ is the slope, which is a where function of test temperature T. The rate of cavity growth, dr=dt ¼ uðTÞ; increases with increasing test temperature T. r radius of the cavity, 7.10 Presentation of Engineering Creep Data cs surface energy and r tensile stress at the grain boundary. The most important two high-temperature properties used directly in design for creep are creep strength (also called Equation (7.56) shows that as the stress is increased the creep limit), and creep-rupture strength, often referred to as size of the stable cavity becomes smaller. In the early stages stress-rupture strength, or simply rupture strength. of cavitation, ‘r-type’ cavities as small as 1 l (1 lm) or less According to ASTM (ASTM E6, sections 49, 50), these are in diameter have been observed (Gifkins 1959), which is in defined as the maximum stresses that a material can with- agreement with the values obtained from (7.56) if it is stand for a specified length of time at a given temperature assumed that cs ¼ 1 JmÀ2; and stress r [ 2 MN m−2. The without undergoing excessive strain, or rupture, respectively. number of cavities formed during creep depends on the Alternatively, creep strength can be defined as the highest purity of the material. It has been shown (Resnick and Seigle stress at a given temperature which produces a minimum 1957) that the impurities in commercial a brass increase the creep rate of a fixed amount, typically in the range of 10−8 to propensity for formation of cavities, whereas when the same 10−11 s−1. The stress referred to in the above definitions is material is remelted and purified by controlled directional usually the initial stress, which will be somewhat smaller solidification, the tendency to form cavities was consider- than the final axial stress if tests are carried out under a ably reduced. constant load. The time involved to determine the creep or After initiation of incipient fractures, their growth and coalescence are usually considered to be the cause for complete rupture of a material. The growth of the incipient

288 7 Creep and Stress Rupture rupture strength is the service life of the component, mea- the creep data are plotted in such a way that it will yield a sured in thousands of hours and sometimes in years. reasonably straight line. It is found that when the data of Creep-rupture strength is based on a rupture life of usually Fig. 7.27 is plotted as log stress versus log time, the 103, 104 or 105 h (about 11.5 years). The strain allowed for resulting curve will become a nearly straight line. Fig- measuring creep strength is the total strain—the instanta- ure 7.28 shows schematically a typical family of such curves neous strain plus the creep strain and measured in per cent. for different amounts of total strain for a given material at a Whether the amount of strain is excessive or not, that constant temperature. Hence, a curve for a given total depends on the application. For example, allowable strain allowable strain is selected in Fig. 7.28 and the stress cor- due to creep in a steam turbine blade may be 0.20% in 105 h responding to a given service life (within the test time) for a at 815 °C, but in a jet turbine, a very little strain of the order given material at the test temperature is obtained from the of 0.10% in 2000 h is allowed because of the close tolerance selected curve in Fig. 7.28 or from its extrapolation if the involved. On the other hand, 2% strain due to creep in a service life exceeds the testing time. This stress is the pressure vessel may not be excessive and strain as much as required creep strength of the given material at that tem- 4% in one year may be tolerated in creep of a plastic aerosol perature. It is to be noted that as the curves are approxi- bottle. mately linear, so the extrapolation should be restricted to short times. Extrapolation of one log cycle beyond the test 7.10.1 Prediction of Creep Strength time appears to be a short extrapolation but it actually extends the time to ten times the maximum test time. The There are several ways to determine creep strength. The slope of the nearly linear curve can sometimes change due to simplest way to obtain creep data is to test several specimens the effects of structural changes in the material and hence, of the same material at the same given temperature, but each such long extrapolation can result in serious errors. of them under a different stress level. For each specimen, the length of time required to produce the same allowable strain Another method of determining creep strength is based on is noted. In this test, the temperature and the strain experi- the creep rate given by (7.10). If (7.10) is assumed to remain enced by each specimen remain constant but the stress and valid, we can get the allowable minimum creep rate as time required for the same amount of deformation will vary for different specimens kept at a constant temperature. Now, e_s ¼ ðeÞAllowable ð7:58Þ the stress versus testing time for a given percentage of strain ðtÞService for a given material at a constant temperature can be plotted, which is shown schematically in Fig. 7.27. where Tests of this type are usually restricted to 1000 or 2000 h, ðtÞService the service life of a given material and although tests are sometimes extended up to 10,000 h. Often ðeÞAllowable the total allowable strain for that material at an these testing times are shorter than the service lives of expected operating temperature components. Then to obtain the long-time creep properties, it becomes essential to extrapolate creep data into regions A curve showing the variation of e_s with stress can be where data are not available. Obviously, it requires the obtained from engineering creep curves determined by a extension of the curve in Fig. 7.27 to the required time. series of creep-time tests at the expected operating temper- Since the extension of a curve would be easier if it is linear, ature. Each creep test is performed at a different stress level and is continued until the creep curve enters the tertiary creep stage so that the minimum creep rate can be Fig. 7.27 Stress versus testing Strain and test temperature are constant. time for a given percentage of strain for a given material at a constant temperature (schematic) Stress 0 Time, hrs 1000

7.10 Presentation of Engineering Creep Data 289 Fig. 7.28 Stress versus testing ε1 time on log-log plot for different ε3 amounts of total strain for a given material at a constant temperature ε2 (schematic) ε4 Stress (logarithmic scale) Constant temperature test; total strain, ε1 > ε2 > ε3 > ε4 0.1 1 10 100 1000 Testing time, hrs (logarithmic scale) Fig. 7.29 Family of schematic σ1 Constant test temperature; creep strain–time curves at stress, σ1 > σ2 > σ3 > σ4 > σ5; various stress levels for a given εs1 σ2 minimum creep rate, material tested at the same εs2 εs1 > εs2 > εs3 > εs4 > εs5. temperature, showing minimum creep rates σ3 Creep strain εs3 σ4 σ5 εs4 εs5 Log time determined precisely. It has been shown (Gill and Goldhoff given material is shown schematically in Fig. 7.30 using a 1970) that the value of the minimum creep rate depends on log-log plot in order to obtain linearity in the plot. the length of creep testing time. When the minimum creep rate is determined from the curve obtained by a short-time To use the plot in Fig. 7.30, first select the curve that creep test, then this value will be higher than the true value, corresponds to the expected operating temperature. Since and thus, the error will be on the conservative side. Fig- ðeÞAllowable and ðtÞService of the given material is known to us, ure 7.29 shows schematically a family of creep strain–time we can obtain the value of e_s using (7.58). Now, the stress curves at various stress levels for a given material tested at corresponding to this value of e_s can be obtained from the the same temperature. The smallest slope of each curve in selected curve in Fig. 7.30 and this stress is the required Fig. 7.29 is measured which gives the value of the minimum creep strength of the given material at that operating tem- creep rate, e_s; and this can be plotted against the corre- perature. If the time of creep testing used to construct the sponding stress ðrÞ at a constant temperature. From similar curve is shorter than the service life, extrapolation to longer tests carried out at various temperatures, plots of r versus e_s time will be required, which is based on the linearity of the at various temperatures can be made. The variation of second stage of the creep-time curve. For longer time, this minimum creep rate with stress at various temperatures for a linearity may be seriously affected due to microstructural instabilities in the material. So, such extrapolations should

290 7 Creep and Stress Rupture Fig. 7.30 Variation of minimum Stress, MPa (log scale) 10000 Temperature, T1 < T2 < T3 creep rate with stress at various 1000 temperatures for a given material (schematic) T1 T2 100 T3 10 0.0001 0.001 0.01 0.1 1 10 100 0.00001 Minimum creep rate, percent per hour (log scale) be restricted to relatively short times. However, application creep-rupture strength is reduced with a suitable factor of of (7.58) to determine e_s involves some approximation, safety. However, if the test time is prolonged, there may be a because (7.10), on which (7.58) is based, includes neither the break in the slope of stress-rupture curve at any constant temperature. This break may also appear by increasing the instantaneous strain nor the primary creep, whereas both are temperature of testing, as shown in Fig. 7.18. The reasons included in ðeÞAllowable since it represents the total strain. for such break have been mentioned in Sect. 7.8. If the time Since it is not possible to know either the instantaneous of testing used to construct the stress-rupture curve is shorter than the service life, the extrapolation of that curve becomes strain or the primary creep until the allowable stress is necessary. But such extrapolation will introduce serious errors, if there is a break in the slope of the curve somewhere obtained from the selected curve in Fig. 7.30, it is not pos- between the end of the measured data and the service life of sible to subtract them from ðeÞAllowable: If times of creep the material. testing are 1000 h or more, errors due to the above approximation are not significant. Once the allowable stress To conduct stress-rupture test is cheaper and more suit- is known from Fig. 7.30, the instantaneous strain and the able than to conduct creep test. So, it would be really helpful if creep strength could be determined from creep-rupture primary creep can be estimated from the curve of allowable strength with an accuracy that is sufficient for applications in stress in Fig. 7.29 and can be subtracted from ðeÞAllowable to design. Monkman and Grant (1956) showed empirically that obtain the steady-state creep strain, which is appropriate for when minimum creep rate, e_s; is plotted against time to rupture, tr; on a log-log scale, it results in a straight line of (7.58). This strain has to be divided by the service life to the following form: obtain the value of e_s and the corresponding allowable stress or creep strength can be obtained from Fig. 7.30. This pro- cess can be repeated for better accuracy. 7.10.2 Prediction of Creep-Rupture Strength log tr þ CMÀG log e_s ¼ KMÀG ð7:59Þ The manner of determining the creep-rupture strength, i.e. where CMÀG and KMÀG are material constants. Monkman the stress required to cause rupture after a given time at a and Grant found that for a number of aluminium-, copper-, given temperature is almost identical to the first method iron-, titanium- and nickel-based alloys, the values of con- explained to find creep strength. As already mentioned in stants are: 0:77\\CMÀG\\0:93 and 0:48\\KMÀG\\1:3: As Sect. 7.8, tests are carried out on a series of specimens up to seen from (7.59), tr is inversely proportional to e_s: If mini- the point of fracture at the same given temperature, but each mum creep rate and rupture time data are collected from tests of them under a different stress level. The rupture time for on a given material at various temperatures and then plotted, each specimen is measured and a plot is made with the the resulting curve according to Monkman–Grant relation- corresponding stress versus rupture time on a log-log scale, ship would be similar to that shown schematically in which usually shows an approximately linear variation. The Fig. 7.31. Thus, minimum creep rate required to determine creep-rupture strength for a given service life at a given the creep strength can be estimated from the rupture life temperature may be determined from such a plot by inter- obtained from the stress-rupture test if either the constants of polation. Then to obtain the working stress, the determined (7.59) are known for the given material or the experimental curve similar to that in Fig. 7.31 is available for the given

7.10 Presentation of Engineering Creep Data 291 Minimum creep rate (log scale) In the new methods, therefore, a given material undergoes total allowable deformation or rupture within a test time Rupture time (log scale) shorter than the time for the same deformation or rupture in service, by using a test temperature higher than the service Fig. 7.31 Demonstration of Monkman–Grant relationship from tests temperature, but the stresses applied during test and service on a given material at various temperatures (schematic) periods remain the same. The test time required for a given deformation or rupture at the higher test temperature is then material. Oppositely, rupture time can also be estimated converted to the corresponding time required for the same from the value of minimum creep rate obtained from the deformation or rupture at the service temperature by means creep test according to Monkman–Grant relationship. Cor- of time-temperature parameters, as described below. This relation between the rupture time and the minimum creep new methods are usually called parameter methods. Apart rate strengthens the well-documented view that the cavities from predicting long-time creep or rupture behaviour, these form continuously throughout the process of creep. parameters are useful to compare the behaviour of materials and to assess them relatively. Finally, they can be used to 7.11 Parameter Methods to Predict extrapolate experimental data to the regions where direct Long-Time Properties evaluation is generally difficult due to limitations of the test. More than thirty time-temperature parameters have been It has already been mentioned that often long-time creep and developed (Manson and Ensign 1979; LeMay 1979). The stress-rupture data are needed for conditions for which no four popular parameter methods will be discussed below. experimental data are available. In such situations, the The first two parameter methods have been developed on extrapolation of creep and stress-rupture curves to long times theoretical grounds based on Arrhenius-type equation gov- is required, which has been described in the earlier section, erning creep process and the last two parameter methods are but this process is not generally acceptable in most cases. based on empirical grounds. Because the curves can be reliably extrapolated to long times only when it is surely known that structural changes, which Assuming that creep is a process governed by would cause a change in the slope of the curve, do not take Arrhenius-type rate equation, as expressed by (7.30), (7.30) place in the extrapolated region. Hence, the development of can be integrated to obtain the time required to reach a given new methods for prediction of long-time properties becomes creep strain or the time to rupture at a constant stress for a essential. Since 1950, several new methods have been pro- given material as follows: posed with a common central idea that raising temperature can accelerate creep. Since structural changes generally take Ze Zt ð7:60Þ place within short times by increasing temperatures, so the de ¼ A1eÀQ=RT dt curve obtained by short-time test at a temperature higher than the service temperature can be extrapolated to longer 00 times corresponding to service lives of components if no change in the slope of the test curve occurs. ) t ¼ e eQ=RT ¼ heQ=RT A1 where t is the time required to reach a given creep strain or the time to rupture, e is the corresponding given creep strain or the strain at rupture, Q, R and T have the same meanings as in (7.30), Q is assumed to be the same at all temperatures, and h is related to e and A1: Since (7.60) is same as (7.40) proposed by Dorn, so h will be termed as a temperature-compensated time parameter. Taking the common logarithm of both sides of (7.60), we get log10 t ¼ log10 h þ M Q ð7:61Þ RT where M ¼ log10 e ¼ 0:4343: If h and Q are assumed to be functions of stress only, then according to (7.61), the plot of log10 t versus 1/T will be linear for any given stress. Let us consider a set of stress-rupture data at various temperatures for a given material, which is schematically plotted as log stress versus log time to rupture as shown in

292 7 Creep and Stress Rupture Stress (logrithmic scale) T1 T2 T3 T4 Temperature, T1 < T2 < T3 < T4 < T5 T5 Rupture time (logarithmic scale) Fig. 7.32 Schematic stress-rupture curves at various test temperatures for a given material, assuming no break in the slopes of the curves Rupture time, Stress, σ1< σ2 < σ3 < σ4 < σ5 σ4 σ5 t1< t2 < t3 < t4 < t5 < t6 σ1 σ2 σ3 log10t (t in hours) 0.4343 Q =R Stress t2 t1 1 (T on absolute scale) t3 T t4 t5 Fig. 7.34 Schematic plot of rupture time on logarithmic scale against reciprocal of absolute test temperature for several different stresses, t6 showing all the lines in the plot are parallel to each other having a slope equal to ðMQÞ=R ¼ ð0:4343 QÞ=R; where Q = the activation energy for the rate-controlling process in creep and R = the universal molar gas constant. This type of plot was obtained by Orr, Sherby and Dorn Temperature these dashed lines and the constant rupture-time curves are then determined. From these data of stresses, rupture times Fig. 7.33 Constant rupture-time curves (schematic) and temperatures, a family of curves is plotted in Fig. 7.34 on coordinates of log10 t versus 1/T for several different Fig. 7.32. In this plot, it is assumed that none of the stresses for that given material, where T is the absolute test stress-rupture curves show any break in their slopes. The temperature and t is the time to rupture, expressed in hours. data from Fig. 7.32 are then replotted as constant All the curves are nearly linear. For any given straight line, rupture-time curves (schematic) on coordinates of stress (7.61) is of the form versus temperature in Fig. 7.33. To the graph of Fig. 7.33, dashed horizontal lines have been added at different stress y ¼ b þ mx ð7:62aÞ levels. Values of temperature (T) from the intersections of

7.11 Parameter Methods to Predict Long-Time Properties 293 Stress, σ1< σ2 < σ3 < σ4 < σ5 in Fig. 7.35, where all the lines converge at a common point σ1 σ4 on the log10t-axis. This evidence indicates that Q varies with σ5 stress but h does not. So, log10 h is not a function of stress σ2 and is regarded as constant. Let us take log10 h ¼ ÀCLÀM; σ3 since the point of convergence on the log10t-axis shows a negative value. Considering ðMQÞ=R ¼ ð0:4343 QÞ=R ¼ m; which is a function of stress, r; (7.61) takes the following form: log10t (t in hours)  1 log10 t þ CLÀM ¼ m T ð7:62bÞ + 20 The value of CLÀM can be determined for any material + 10 from the average intercept of the lines extrapolated to the log10t-axis as shown in a plot like Fig. 7.35. Since the slope 0 m ¼ ð0:4343 QÞ=R; is a function of stress, r; as indicated by 1 (T on absolute scale) Fig. 7.35, the Larson–Miller parameter can be formulated T from (7.62b) as – 10 – 20 CL-M = Larson – Miller constant Tðlog10 t þ CLÀMÞ ¼ m ¼ function of stress ðrÞ ð7:63aÞ where Fig. 7.35 Schematic plot of rupture time on logarithmic scale against Tðlog10 t þ CLÀMÞ ¼ Larson-Miller parameter ¼ PLÀM reciprocal of absolute test temperature for several different stresses, ð7:64aÞ showing all the lines in the plot converge at a common point on the log10t-axis as observed by Larson and Miller. This point of conver- T the absolute temperature of testing in the Kelvin gence is not a function of stress and is called Larson–Miller constant t or Rankine temperature scale, i.e. respectively in where b ¼ log10 h and m ¼ ðMQÞ=R ¼ ð0:4343 QÞ=R; CLÀM K ¼ C þ 273 or, R ¼ F þ 460; which mean that h is related to the intercept and Q to the the time required to reach a given creep strain or the slope of the line. The manner of variation of h and Q with stress will therefore determine the characteristics of such a time to rupture in h; family of straight lines shown in Fig. 7.34. There are three the Larson–Miller constant in log10 t versus 1/T possible variations: plot. For most alloys, CLÀM has been found to vary between 15 and 30 depending on material. When a 1. If h is constant while Q varies with stress, all the lines in specific value is not determined, the most com- the family will converge at a common point on the y- or monly used value of CLÀM is often assumed to be log10t-axis. A family of this type of straight line is 20, because this value has been found to be illustrated in Fig. 7.35. reasonably true for many materials. 2. If Q is constant and only h varies with stress, all the lines in the family will be parallel to each other having a slope When the creep or stress-rupture data are plotted on equal to ðMQÞ=R ¼ ð0:4343 QÞ=R; which has already been shown in Fig. 7.34. coordinates of ln t versus 1/T instead of log10 t versus 1/T, the Larson–Miller parameter will take the following form: 3. If both h and Q vary with stress, the lines may neither be parallel nor have a common intercept on the y- or log10t- Tðln t þ CLÀM1 Þ ¼ slope Q ¼ function of stress ðrÞ axis. R ð7:63bÞ where 7.11.1 Larson–Miller Parameter Tðln t þ CLÀM1 Þ ¼ Larson-Miller parameter ¼ PLÀM1 ð7:64bÞ The first possibility is the basis of the Larson–Miller CLÀM1 ¼ the Larson–Miller constant in ln t versus 1/T parameter method (Larson and Miller 1952). Larson and plot ¼ CLÀM=M ¼ CLÀM= log e ¼ CLÀM=0:4343: For most Miller analyzed large amounts of experimental stress-rupture data and plotted in accordance with (7.61). The plot is shown alloys, CLÀM1 ranges from about 35 to 69 depending on material. When input data are limited, it is often assumed

294 7 Creep and Stress Rupture Stress (logarithmic scale) correct value of the Larson–Miller constant is considered for each alloy. Since rupture time is inversely proportional to minimum creep rate, as seen from (7.59), the Larson–Miller parameter expressed in terms of minimum creep rate, e_s; instead of time to rupture will be T ðCLÀM2 À ln e_sÞ ¼ PLÀM2 ð7:65Þ Larson–Miller parameter, 7.11.2 Orr–Sherby–Dorn Parameter PL-M = T(log10t + CL-M), or, PL-M1 =T(Int + CL-M1) The second possibility in the variations of h and Q with stress mentioned above is the basis of the Orr–Sherby–Dorn Fig. 7.36 Schematic master curve for a given material based on the Larson–Miller parameter, where the most commonly used value of the parameter method (Orr et al. 1954). Experimental data Larson–Miller constant is CLÀM ¼ 20; or CLÀM1 ¼ 46: obtained by Orr, Sherby and Dorn indicated that for a given that the most commonly used value is CLÀM1 ¼ 20= 0:4343 ¼ 46: material, Q remains essentially constant while h varies with After determining the value of CLÀM or CLÀM1 from stress. This means that all the lines corresponding to various Fig. 7.35, if PLÀM or PLÀM1 is evaluated according to (7.64) for a variety of pairs of values of t and T obtained over a stresses in the plot of log10 t versus 1/T will be parallel to range of stress from Fig. 7.35 and plotted against the cor- each other having a slope equal to ðMQÞ=R ¼ responding observed stresses, a single master curve is ð0:4343 QÞ=R; as shown in Fig. 7.34. Let us take obtained for any given material. The trend of variation of ð0:4343 QÞ=R ¼ COÀSÀD; which is a time-temperature con- such a master curve is shown schematically in Fig. 7.36. At stant based on the linear relationship of log10 t versus 1/T for any given stress, PLÀM or PLÀM1 will have the same value a given material and log10 h ¼ #; which is a function of for an infinite variety of combinations of t and T, varying stress, r: Hence, (7.61) takes the following form: from the short times and high temperatures that represent test conditions to the longer times and lower temperatures that log10 t ¼ # þ COÀSÀD represent service conditions. Allowable stress for long ser- T vice can therefore be obtained by using the master curve that has been constructed from test results. This allowable stress ) log10 t À COÀSÀD ¼ intercept # ð7:66Þ is the predicted creep strength or creep-rupture strength for T long service, where t is either the time for a permissible creep strain or the time to rupture. ¼ function of stress ðrÞ Many high-temperature alloys agree well with the Lar- where son–Miller parameter. If all the master curves of different high-temperature alloys are plotted on the same graph using log10 t À COÀSÀD ¼ Sherby-Dorn parameter ¼ POÀSÀD the same scale for PLÀM or PLÀM1 ; the direct comparisons of T the alloys are possible. The same scale for PLÀM or PLÀM1 is possible to use if a single average value of the Larson–Miller ð7:67Þ constant is used for those different high temperature alloys. Often the most commonly used value of the Larson–Miller COÀSÀD ¼ ð0:4343 QÞ=R ¼ the average slope of a family constant is CLÀM ¼ 20 for PLÀM or CLÀM1 ¼ 46 for PLÀM1 : of parallel lines for various stresses in the plot of log10 t If the actual value of the Larson–Miller constant for each versus 1/T and COÀSÀD is expressed in K, since Q is alloy is close to the value used for PLÀM or PLÀM1 ; such a expressed in J mol−1 and R is expressed in J mol−1 K−1. t, graph involving the smaller amount of work may be and T have the same meanings as in (7.63a). accepted for the direct comparisons of different alloys. However, more accurate results are obtained when the Figure 7.37 schematically shows a single master curve for any given material representing (7.66). To obtain this master curve, POÀSÀD is evaluated according to (7.67) for a variety of pairs of values of t and T obtained over a range of stress taking data from Fig. 7.34 and the value of COÀSÀD is determined from the common slope of the lines in Fig. 7.34. To predict creep strength or creep-rupture strength for long service life, this master curve can also be used in a way similar to the Larson–Miller parameter method.

7.11 Parameter Methods to Predict Long-Time Properties 295 Stress, σ1< σ2 < σ3 < σ4 < σ5 (Ta , log10ta) log10ta Stress (logarithmic scale) σ1 log10t (t in hours) σ2 σ3 σ4 σ5 Ta T (absolute scale) Sherby–Dorn parameter, PO-S-D =log10t – CO-S-D Fig. 7.38 Schematic plot of rupture time on logarithmic scale against absolute test temperature for several different stresses, showing all the T lines in the plot converge at a common point, as observed by Manson and Haferd. The coordinates of the point of intersection of the extrapolated iso-stress lines are ðTa; log10 taÞ Fig. 7.37 Schematic master curve for a given material based on the where Sherby–Dorn parameter, where COÀSÀD ¼ ð0:4343 QÞ=R ¼ the aver- age slope of a family of parallel lines for various stresses in the plot of T À Ta log10 t À log10 log10t versus 1/T ta ¼ Manson-Haferd parameter ¼ PMÀH 7.11.3 Manson–Haferd Parameter ð7:69Þ An entirely empirical ground is the basis for the develop- t and T have the same meanings as in (7.63a); ment of Manson–Haferd parameter method (Manson and Ta the temperature coordinate of the converg- Haferd 1953). Manson and Haferd observed that lines ing point of lines on the graph, expressed in log10 ta the Kelvin or Rankine temperature scale; plotted on coordinates of log10 t versus 1/T as shown in the time coordinate of the converging point Fig. 7.34 or Fig. 7.35 are not perfectly linear. These inves- SMÀH ¼ 1=m of lines on the graph in log10-scale in which ta is expressed in hour; tigators therefore tried to plot test time against test temper- reciprocal of the slope of iso-stress lines and is a function of stress, r ature in alternative ways and found that when log10 t is plotted against T, the greatest linearity is obtained. Further, Following the procedure mentioned above, the Manson– Haferd master curve can be constructed by evaluating PMÀH they found that when a family of straight lines are plotted on according to (7.69) for a variety of pairs of values of t and coordinates of log10 t versus T for several different stresses T obtained over a range of stress taking data from Fig. 7.38 for a given material, these lines converge to a common point and determining the values of Ta and log10 ta from Fig. 7.38. on the graph, as shown schematically in Fig. 7.38. Let the Figure 7.39 shows schematically this master curve, which coordinates of the point of intersection of the extrapolated can be used to predict creep strength or creep-rupture iso-stress lines on the graph of log10 t versus T are Ta and strength for long service life. Since the parameter PMÀH is log10 ta and the slope of iso-stress lines is m, which is a based on a linear relation between log10 t and T for a given function of stress, r: The equation of the family of straight stress, it is also known as the linear parameter. lines in Fig. 7.38 then can be represented by It has been found (Goldhoff 1959) that consistently more log10 t À log10 ta ¼ mðT À TaÞ accurate predictions of long-time properties are obtained by using the Manson–Haferd linear parameter method than by T À Ta ta ¼ 1 ¼ SMÀH ¼ function of stress ðrÞ log10 t À log10 m ð7:68Þ

296 7 Creep and Stress Rupture Fig. 7.39 Schematic master Stress (logarithmic scale) curve for a given material based on the Manson–Haferd parameter, where Ta and log10 ta are respectively temperature (on absolute scale) and time (in hours) coordinates of the point of intersection of the extrapolated iso-stress lines in the plot of log10t versus T Manson–Haferd parameter, PM-H = T – Ta log10 t – log10 ta using the Larson–Miller or Sherby–Dorn parameter method. Stress, σ1< σ2 < σ3 < σ4 < σ5 σ1 However, the determination of the exact coordinates of the converging point of lines on the graph is a problem in the σ2 σ4 Manson–Haferd method. In this method, long extrapolations σ3 σ5 of nearly parallel lines representing log10 t versus T are required to obtain the point of convergence, which makes the log10t (t in hours) exact location of the point unreliable. This uncertainty cau- ses different values for the coordinates of the point to be 1 decided by different investigators, which in turn produces somewhat different results for the same material. However, if Tb a large enough data are available, the error due to the above 0 cause seems to be little. 1 (T on absolute scale) 7.11.4 Goldhoff–Sherby Parameter T The approach of these investigators is similar to the Man- log10tb son–Haferd method with the difference being that a family of straight lines are plotted on coordinates of log10 t versus 1/ (1/ Tb , log10tb) T for several different stresses for a given material instead of plotting log10 t versus T. It is further found that the con- Fig. 7.40 Schematic plot of rupture time on logarithmic scale against structed iso-stress lines will converge to a common point, reciprocal of absolute test temperature for several different stresses, coordinate of which is designated by ð1=Tb; log10 tbÞ: The showing all the lines in the plot converge to a common point, as plot is shown schematically in Fig. 7.40. The procedure to observed by Goldboff and Sherby. The coordinates of the point of determine Goldhoff–Sherby parameter (Goldhoff and Hahn intersection of the extrapolated iso-stress lines are ð1=Tb; log10 tbÞ 1968) is similar to that used for the Manson–Haferd parameter. The value of the Goldhoff–Sherby parameter is where the slope of lines at constant stresses on the plot of log10 t versus 1/T and this slope is a function of stress, r: Hence, the log10 t À log10 tb ¼ Goldhoff-Sherby parameter ¼ PGÀS Goldhoff–Sherby parameter is given by 1=T À 1=Tb ð7:71Þ log10 t À log10 tb ¼ slope of iso-stress lines t and T have the same meanings as in (7.63a); 1=T À 1=Tb 1=Tb the temperature-coordinate of the converging ð7:70Þ point of lines on the plot of log10 t versus 1/T, in which 1=Tb is expressed in K−1 or °R−1. ¼ function of stress ðrÞ

7.11 Parameter Methods to Predict Long-Time Properties 297 log10 tb the time-coordinate of the converging point of Consider a tensile test specimen which is subjected to a lines on the graph in log10-scale in which tb is constant total strain, et; under an initially applied stress of ri at an elevated constant temperature where creep can occur. expressed in hour The total strain et can be considered to be the summation of the following three components: 7.11.5 Limitations of Parameter Methods (1) Elastic strain, ee; (2) Time-independent plastic strain, ep; that occurs on To predict creep properties using any of the parameter methods, one must remember the fact that lines yielded by loading; any of the above methods of plotting time against temper- (3) Time-dependent creep strain, ec: ature are not perfectly linear and the complete validity of any of the parameter methods is questionable. Naturally, the The total strain after loading is therefore given as: results given by different parameter methods will be con- tradictory. However, actual errors involved in these param- et ¼ ee þ ep þ ec ¼ constant ð7:72Þ eter methods are lesser than those involved in the earlier methods described in Sect. 7.10, where extrapolations to Differentiating (7.72) with respect to time, and remem- long time would be needed. bering that the plastic strain on loading, ep; is not dependent on time and that the total strain, et ¼ constant: Another limitation of the parameter methods is the requirement of employing testing temperatures higher than 0 ¼ dee þ 0 þ dec ; or; dee ¼ À dec ð7:73Þ the service temperature of material. If the service tempera- dt dt dt dt ture is close to a transformation temperature of a given material, it is not possible to use a testing temperature that But for a linear elastic material, ee ¼ r=E; where r is the exceeds the transformation temperature, because structural instantaneous stress, which is a function of time, and E is the changes would occur. In such cases, it will not be possible to elastic modulus. Hence, obtain parameter values using higher test temperatures that would predict creep properties for long service life and for dee ¼ 1 dr ð7:74Þ this, extrapolation of the master curve itself would be dt E dt required. But such extrapolations are usually believed to be not much superior to those of the earlier methods described Substituting (7.74) into (7.73) and then integrating with in Sect. 7.10, where the lengths of time involved are the respect to time t, one obtains same. Zec Zr 7.12 Stress Relaxation decdt ¼ À 1 dr dt dt E dt ep ri ) ec À ep ¼ ri À r ð7:75Þ E Stress relaxation is the time-dependent reduction in initially where ep is the time-independent plastic strain at time t ¼ 0; applied stress in a stressed material which is constrained to a when relaxation begins, and ri is the initial stress at the same certain fixed deformation. When a stressed material is held at instant. Equation (7.75) shows that as r decreases ec a constant total strain with no dimensional change, it will increases. Creep therefore occurs under conditions of creep, if the temperature and stress are sufficiently high. decreasing stress. High temperature bolted joints and press-fitted assemblies are common practical examples where considerable stress If we assume that the tensile specimen is undergoing relaxation is often found after long periods of time as a result steady-state creep at a low stress level, then under condition of creep. The total strain of a stressed material can only of steady-state creep at the instantaneous stress r at the remain constant if the elastic strain in the material is trans- constant temperature of testing the creep rate of the tensile formed into permanent or plastic strain resulting from creep specimen according to (7.32) will be deformation, which is occurring in the material. This trans- formation causes a reduction in elastic strain, which in turn, dec ¼ e_s ¼ A02rn ð7:76Þ results in a decrease of the stress as a function of time and dt with it, the creep rate also decreases. Thus, creep at a fixed total strain results in a time-dependent stress relaxation. where A20 is independent of stress, but dependent on tem- perature. n is independent of stress and, to some extent, depends on temperature.

298 7 Creep and Stress Rupture Substitution of (7.74) and (7.76) into (7.73) gives in stress. Further, the rate at which the stress decreases is initially rapid but the rate of drop diminishes continually and 1 dr ¼ ÀA02rn ð7:77Þ ultimately, the curve flattens because the creep rate decreases E dt with decreasing stress level. The time required to relieve residual stresses by thermal treatment can also be estimated Equation (7.77) is the differential equation for the ideal- by using these types of curves. Equation (7.80) is derived basically to point out the principles on which the theory of ized case of stress relaxation where steady-state creep at a stress relaxation is based. For accurate predictions a more exhaustive method, in which primary creep and the defor- low stress level and a fixed total strain, et; are assumed. mations of the testing machine and the connected parts are Integrating (7.77), we get considered, must be applied. Such a procedure includes solution by step-by-step numerical integration. ZZ dr However, it has been shown (Trouton and Rankine 1904) rn ¼ ÀA02E dt for creep of lead under a tensile force at 25 °C that the tensile stress, r; relaxes logarithmically with time, which is 1 ð7:78Þ given by 1ÞrnÀ1 À ðn À ¼ ÀA20 Et þ C where C is the integration constant. At the start of testing, i.e. at time t ¼ 0; the initial stress r ¼ ri: From this condi- tion, C can be evaluated from (7.78): 1 r ¼ ri À cr lnð1 þ mrtÞ ð7:81Þ 1Þrni À1 C ¼ À ðn À ð7:79Þ where ri is the initial stress, and cr and mr are time-independent constants. This relation has been found to Substitution of the value of C from (7.79) into (7.78) gives the relation between stress and time in stress relaxation apply quite well on a-iron, copper, and a-brasses in the as follows: temperature range of 77–358 K (Feltham 1961) and in magnesium and an Al-9.7% Cu alloy at 298 K (Laurent and 1 ¼ 1 þ A20 Eðn À 1Þt ð7:80Þ rnÀ1 rinÀ1 Eudier 1950). The parameter cr is not very sensitive to the initial stress ri; but it decreases with increasing temperature, The stress-relaxation curve, i.e. stress r versus time t, for and is very sensitive to composition in the case of brass. On a given material can be plotted from (7.80) if values of A02; E, the other hand, the frequency factor, mr; is sensitive to ri and and n for that material at a given temperature and the initial temperature. mr is found to vary between 10 and 30 min−1 in stress ri are known. Stress–relaxation curves for a given the work on a-brasses and copper. Logarithmic creep material at different temperatures for the same initial stress behaviour appears to be valid during stress relaxation, since are shown schematically in Fig. 7.41. This figure shows that elastic strain is transformed into plastic creep strain and the the stress relaxation is also a function of temperature and that amount of creep strain involved is small. Substitution of the higher the temperature the higher is the rate of decrease (7.75) into (7.81) gives the time dependence of creep strain as follows: ec ¼ ep þ cr lnð1 þ mrtÞ ð7:82Þ E Initial stress Data given by Feltham (1961) for 65/35 a-brass show at t = 0 that cr is related to c of (7.6), which has also been found to be insensitive to the initial stress and this relation is Temperature, T1 < T2 < T3 cr=E % c: Stress Tests have been designed to study the stress relaxation behaviour. A hard testing machine should be used to avoid T1 the deformation of the machine, but even then the effect of machine stiffness must be considered (Guiu and Pratt 1964). T2 Depending on the application, the initial stress applied during loading to deform the specimen to a fixed strain, T3 which is then held constant, may be below, at or above the 0 Time, t yield stress of material. The specimen may or may not experience permanent plastic strain but will always undergo Fig. 7.41 Schematic stress-relaxation curves for a given material at elastic strain on loading. various temperatures for the same initial stress, showing the rate of decrease of stress increases with increasing temperature

7.12 Stress Relaxation 299 7.12.1 Step-Down Creep Test within Æ3 C or Æ0:5%; whichever is greater. The stability of the furnace temperature is very important because tem- Step-down creep test at constant extension is a perature variation can cause thermal expansion or contrac- stress-relaxation test performed in tension. Quantitative tion that will affect the strain. This strain and temperature measurement of the amount of relaxation in very long times control must be maintained over the whole period of test from a given stress is the objective of this test. The test is duration, which may be often several years to relate to actual often used as a convenient means for evaluating materials for service times. Moreover, the loading rate can significantly high-temperature bolting service. Since the applied initial affect the stress relaxation. stress is often below the yield stress, great precision and stability of the testing equipment are required. A mo- In this step-down test, the strain is normally maintained tor-driven lever arm-type creep machine is commonly used between limits. The common procedure to approximate a (Loveday and King 1982), which consists of a means of constant extension is illustrated schematically in Fig. 7.43a periodically adjusting the weight to maintain constant (Kuhn and Medlin 2000). When the strain reaches the upper specimen constraint. A typical setup of stress-relaxation test limit, the stress is reduced by a small amount. This instan- equipped for step-down tension testing is shown schemati- taneously leads to a proportionate elastic contraction. The cally in Fig. 7.42. This mainly consists of a furnace, a specimen then creeps under the reduced stress until the temperature-compensated extensometer system, an elec- limiting strain is again reached, at which time the stress is tronic control module and a data recorder. Its structure has a again reduced by the same amount, and so on. By successive precisely balanced lever arm supported on knife edges. reductions of stress, the relation between stress and time is A stress versus time relaxation curve is constructed directly obtained for the specimen under conditions in which the by using the output from the load cell. total strain fluctuates between the stipulated upper and lower limits and this stress–time relation is shown schematically in According to ASTM Standard (ASTM E328 2013), it is Fig. 7.43b (Kuhn and Medlin 2000). required to control strain within Æ0:0025% and temperature 7.13 Materials for High-Temperature Use Lever arm Knife edge As mentioned before, many engineering processes are support required to be operated at elevated temperatures where creep Load train predominates. For example, high temperatures are needed in Electronic cracking stills used in chemical and petroleum industries to Specimen control accelerate the reaction rates. Thus, machine parts and Furnace, module structural components operated at elevated temperatures oven, or must be made of creep-resistant materials. For example, environmental creep-resistant materials must be used for turbine blades chamber otherwise creep deformation of the blades during service may result in seizing of the blades with the turbine casing. Extensometer In general, the higher the melting point of materials, the Load cell greater is the creep resistance because creep occurs to a great extent at a homologous temperature greater than about 0.5, Motor and at a given temperature, the diffusion coefficient on which drive creep rate depends is lower in materials with high melting point. Since refractory oxides such as MgO and Al2O3 Fig. 7.42 Schematic setup of stress-relaxation test equipped for possess high melting points, so they are very suitable for step-down tension testing high-temperature applications. At the same time, since they are brittle, their uses are limited to applications where only compressive stresses are experienced. On the other hand, metals and alloys can be used under more flexible service conditions. Most creep-resistant alloys contain a base metal of higher melting point. But apart from the high melting point, to select a metal for high-temperature use one must also consider that the metal should have ability to be fabri- cated into the required shape of the component and its cost of production and density should be low. For example,

300 7 Creep and Stress Rupture Fig. 7.43 Schematic (a) (b) stress-relaxation curve derived from step-down test. AC E G ∆σ = ∆P a Approximation of a constant A0 extension by step-down test. σ3 σ4 Derived stress b Stress–time relation (Kuhn and σ1 σ2 H (etc.) relaxation curve Medlin 2000) Extension σi σ1 StressBDF σ2 (not to scale) σ3 Time 0 σ4 0 (etc.) O Initial Time loading time tungsten with a high melting point exceeding 3000 °C has Thus, a material possessing a high melting point and not been widely used because of fabrication difficulty, higher modulus of elasticity and low diffusion coefficient is suitable density and production cost. Iron-based, nickel-based and for high-temperature applications. Further, such materials cobalt-based heat-resistant alloys (superalloys), in which the must possess a combination of superior creep strength, melting points of all the three base metals are moderately thermal fatigue resistance, and oxidation and hot corrosion high of the order of around 1500 °C, are commonly used at resistance. An alloy for creep resistance must be designed to temperatures exceeding 540 °C. These superalloys exhibit provide a structure possessing the desired properties, which the best combination of high strength, superior resistance to involves the proper balance of composition and treatment. It creep and fatigue, good corrosion resistance and thermal may not be always possible for the same structure to yield stability, i.e. the ability to operate at high temperatures for the maximum strength, ductility and stability under the prolonged periods of time. No other metallic material can service conditions. For those situations where fabricability is provide such combination of elevated temperature strength most important or substantial ductility is needed at the and resistance to surface degradation. Further, fibre- expense of strength, the most appropriate strengthening reinforced superalloys are also being used as structural method is solid-solution alloying. The best strength will be materials at elevated temperatures. It has been seen that the achieved for a completely saturated alloy and by additions of service temperatures of fibre-reinforced superalloys may solutes possessing higher valency which produces a strong increase by 175 °C over that of unreinforced superalloys. decrease in stacking fault energy. Metals with lower stacking Tungsten fibres have been found suitable for reinforcement fault energy have greater creep resistance because of having in superalloys and tungsten-reinforced superalloys exhibit wider extended partial dislocations which cannot easily superior high-temperature strength and creep resistance overcome obstacles by cross-slipping or climbing. (Petrasek et al. 1986). Fabricability, rupture life and ductility at fracture of superalloys are considerably improved by Out of aforementioned superalloys, nickel-based super- vacuum melting, probably because of the decrease in the alloys have the best creep resistance property. They are used number and size of inclusions. as turbine blades in engines operating at temperatures as high as 1290 °C, which is equivalent to a homologous Metals with lower diffusivities creep to a lesser extent and temperature of 0.9. Tungsten (W), molybdenum (Mo) and have higher creep resistance. Since FCC metals because of titanium (Ti), the constituent elements of nickel-based closed-pack structure have lower diffusivities than BCC superalloys, act as very effective solid-solution strengthen- metals, creep resistance of FCC structure is usually superior ers. Further, W and Mo assist to reduce the diffusivity of the to BCC structure for metals with the same melting point. An alloy. Although increase in solid-solution strengthening is increase in modulus of elasticity decreases the strain rate and small with increase of chromium (Cr) content in nickel improves a material’s resistance to dislocation creep. As far (Ni) superalloys, but the overall solid solution strengthening as improved creep resistance is concerned, the role played by effect of Cr is large since Cr can be dissolved in the Ni ‘diffusivity hardening’ is primary and that by ‘modulus matrix to a large extent. In many nickel-based superalloys, hardening’ is secondary. Ceramics, which have higher submicron-size Ni3ðAl; XÞðc0Þ precipitates form within the melting points and moduli of elasticity and lower diffusivity nickel solid solution ðcÞ; where X within the c0 phase rep- than metal system, have been used to develop components of resents the constituent elements Ti, niobium (Nb) or tanta- gas turbine engine, but their low ductility and brittle beha- lum (Ta). If cobalt (Co) is present in the alloy, it assists to viour in tension limit their use in such applications. improve the stability of the precipitate, though it gives

7.13 Materials for High-Temperature Use 301 comparatively small solid-solution strengthening. In these growth of coarser particles requires the dissolution of finer alloys, dislocation motion through the ordered c0 precipitates particles. For example, in thoria-dispersed nickel is difficult, which increases creep strength of the alloys at (TD-nickel) which is a dispersion-hardened alloy, the elevated temperature. It may be particularly noted that c0 interparticle spacing of fine particles of thoria is small phase exhibits threefold to sixfold increase in strength when enough to hinder effectively the motion of dislocation in the the temperature is increased from ambient to 700 °C nickel matrix and maintains the creep strength up to a (Thornton et al. 1970; Stoloff 1971; Jensen and Tien 1981). homologous temperature of 0.9. Further, the fine dispersions in dispersion-hardened alloys, such as sintered aluminium The constituent elements that are introduced as surface powder (SAP), prevent large-scale grain-boundary migration stabilizers in nickel-based superalloys include Cr, aluminium and the associated decrease in strength, because the migra- (Al), Zirconium (Zr), boron (B) and hafnium (Hf). Cr present tion of grain boundary usually reduces the strain hardening in solid solution allows forming Cr2O3, which lowers the in the crystal leading to marked softening and loss of rate of oxidation and hot corrosion. The presence of Al adds strength. The driving force for grain-boundary migration is to improved oxidation resistance and resistance to oxide the energy of strain hardening present in one of the grains spalling. Finally, improved hot strength, hot ductility and situated next to the grain boundary before recovery. If the rupture life are imparted by addition of B, Zr and Hf (Decker migration of grain boundary cannot occur, the creep rate is and Freeman 1961). much slower. The dispersion of carbide particles along grain boundaries It is to be noted that cold working (plastic deformation) in polycrystalline alloys prevents grain-boundary sliding and cannot be used as a strengthening process to improve creep migration and thus provides creep resistance. Carbide resistance. Because recrystallization will occur quite readily formers such as Cr, W, Mo, vanadium (V), Ti, Nb and Ta at a homologous temperature greater than about 0.5, where lead to the formation of carbide particles, such as MC, M2C, creep predominates and the cold-worked strength will be lost M6C, M7C3, M23C6. Out of these, MC carbides, e.g. TiC, are on recrystallization, as shown in Fig. 10.11 in Chap. 10. most stable and Cr23C6 precipitates are formed when Cr contents are relatively high. Cobalt-based superalloys A material with a fine grain size is desirable for better acquire their strength from a combination of solid-solution strength, hardness and toughness in a low-temperature strengthening and dispersion strengthening caused by car- application, where creep is insignificant. On the other bide precipitates. In creep-resistant steels, the carbides can hand, coarse-grain materials must be used to have better be precipitated by heat treatment prior to the creep defor- creep resistance for high-temperature applications, where mation or may be precipitated preferentially at dislocations diffusional flow-controlled creep dominates. For diffusional during the creep deformation. The thermal stability of these creep, increasing the grain size reduces the creep rate (see precipitates, which form in a finely dispersed fashion, is Sect. 7.6.2). As discussed in Sect. 7.5.1, vacancy diffusion is essential to maintain high strength at elevated temperature relatively slow in a coarse-grain material, which results in a and is thus useful to improve creep resistance. But finely lower creep rate and higher resistance to creep. Further, as dispersed small-sized precipitates, which give the greatest grain-boundary sliding can add to the creep deformation and strengthening, are the most unstable. reduce the creep resistance, so fine-grain materials are to be avoided because a fine-grained material has greater In many nickel-based superalloys, small amounts of Ti grain-boundary areas per unit volume available for sliding and/or Al are added that combine with Ni to form fine than a coarse-grained material. Since the nucleation sites for precipitates of intermetallic compounds, such as Ni3Ti, fracture at high temperatures are grain boundaries, so control Ni3Al, or Ni3(Al, Ti). In this alloy, the coarsening of the fine or removal of grain boundaries will restrain fracture and precipitates at elevated temperatures is prevented by main- improve rupture life. Directional solidification (Ver Snyder taining coherency between the precipitate particles and the and Shank 1970) can be used to control the alignment of matrix. For example, an interface of a very low energy of the grains in such a way that the grain boundaries are oriented order of about 0.005 J m−2 (5 erg/cm2) is formed between predominantly parallel to the tensile stress axis. Directional the precipitate particles of Ni3ðAl; TiÞ and the matrix. As the solidification of a turbine blade (i.e. solidifying the material driving force for coarsening is the reduction in the total sequentially from the bottom to the top of the blade) pro- surface energy, the driving force available here is very little, duces columnar grains having boundaries oriented predom- and thus, the precipitates do not grow and remain stable for inantly parallel to the longitudinal axis of the blade so that longer time. the bending stresses on the grain boundaries are low. This minimizes grain-boundary sliding and significantly reduces In dispersion-hardened alloys, the strengthening phases formation of cavities at grain junctions leading to delayed are finely dispersed in a metallic matrix as small-sized par- fracture and appreciably greater elongation at fracture. ticles which are thermally stable because they are practically Grain-boundary sliding is not a problem in a single crystal insoluble in the matrix. This negligible solubility stabilizes fine particles by preventing their coarsening because the

302 7 Creep and Stress Rupture because grain boundaries have been entirely eliminated. to additions of such solutes also tends to be more pronounced Even greater improvements in rupture time and ductility can than that due to additions of solutes of lower valency. be obtained by using cast single crystal titanium turbine blades although the cost is an inhibiting factor here. When 3. Use Solid-Solution Hardened Alloy grain sizes in a turbine blade made of a nickel-based superalloy are increased from 100 lm to 10 mm, which is of Solid-solution hardening can increase the strength of the the order of the thickness of the turbine blade, the creep rate alloy and thus contribute to higher creep strength. Higher is reduced by approximately 6 orders of magnitude, where hardening effect could be achieved if atomic size and the creep mechanism is predominantly Coble creep that valency of solutes differ considerably from the parent metal, often occurs under typical blade operating conditions. but unfortunately, these factors are not favourable for extensive solid solubility and thus limit the formation of 7.13.1 Rules to Develop Creep Resistance solid solution. From the discussions in the previous section, it is possible to 4. Use Structure with Long-Range Order state a series of general rules, which should be followed in order to obtain increased creep resistance at elevated tem- A further contribution to the creep strength of solid solutions peratures. Some of these rules are: is provided by long-range ordering in solid solutions, because the superlattice dislocations group together in pairs 1. Use Metals with a High Melting Point to preserve order across the slip plane. The disorder created by the first dislocation of a pair is eliminated by the passage The rate of recovery by climbing of edge dislocation is of second dislocation, and thus, extra energy is needed to proportional to the rate of self-diffusion. At a given tem- move the pair of dislocations if order is to be preserved perature, since self-diffusion is usually slower for a metal of across the slip plane. Hence in long-range ordered structure higher melting point than for a metal of lower melting point, (superlattice), the pairs of dislocations with domain bound- so climbing process at a given temperature becomes more aries between them behave in a fashion similar to the difficult in metals and alloys of high melting point leading to extended dislocations in metals of low stacking fault energy. higher resistance to creep. Recovery by climbing cannot occur significantly at a homologous temperature below 5. Use Thermally Stable Precipitates about 0.5. Formation of thermally stable precipitates is needed to 2. Use FCC Metals with Low Stacking Fault Energy improve further the creep strength of a solid solution at elevated temperature. Precipitates block the glide planes and Metals with low stacking fault energy have greater creep prevent dislocations from excessive gliding. It is theoreti- resistance, because the perfect dislocations are dissociated cally estimated that the most effective dispersion of precip- into partial dislocations, for which it is more difficult to itates for optimum strength should have an interparticle overcome obstacles by cutting through, cross-slipping, or spacing of about 10−5 mm, so that it can prevent dislocations climbing. The lower the stacking fault energy the higher will bending around and bypassing the particles. Unfortunately, be the separation between partial dislocations, i.e. the width such closely dispersed fine precipitates usually coarsen at of dissociated dislocations and the more will be the difficulty higher temperatures resulting in a wider interparticle spac- in overcoming the obstacles. Such dissociated dislocations ing, which is not an effective barrier to dislocation motion exist in FCC metals. Further, FCC metals having and results in a decrease in strength properties. The insta- close-packed structures than BCC metals result in lower bility of fine dispersions at elevated temperatures occurs due diffusivities that limit creep deformation. So, creep resis- to diffusion of elements from the precipitates to the matrix tance of FCC metals is usually superior to BCC metals at and this can be minimized in the following ways: equivalent homologous temperatures. • Choice of precipitating elements which have a slow Solute additions to a pure metal can lower the stacking diffusion rate and form the most stable precipitating fault energy. For this purpose, solutes possessing higher compounds. valency are better, because they increase the electron to atom ratio to a greater extent causing a higher decrease in the • Use of a dispersed phase which is nearly insoluble in the stacking fault energy. Fortunately, the rise of flow stress due matrix, so that the resolution of fine precipitates and the growth of coarser precipitates are slow.

7.13 Materials for High-Temperature Use 303 • Selection of such a precipitate that there is a close crys- • Considering creep deformation to be viscous flow, we can tallographic matching, or coherency between the lattices assume principal shear-strain rates are proportional to of the precipitate and the matrix. principal shear stresses. This gives that the principal true strain rates are proportional to the applied principal true 6. Use Also Elements Which form Precipitates in Associa- stresses, i.e. e_1 / r1; e_2 / r2; and e_3 / r3: tion with Crystal Defects During Service Time If C is the constant of proportionality, we can write from If precipitates form on dislocations or nucleates in associa- the second assumption: tion with stacking faults, it becomes an important source of strengthening both at low and high temperatures. It is indi- e_1 À e_2 ¼ Cðr1 À r2Þ; ð7:83Þ cated by some experiments that precipitates which form during creep deformation, particularly if they nucleate on e_2 À e_3 ¼ Cðr2 À r3Þ; ð7:84Þ dislocations, are more effective in reducing the creep rate than precipitates which form prior to creep deformation. For e_3 À e_1 ¼ Cðr3 À r1Þ; ð7:85Þ this, the explanation given (Schoeck 1961) is that atmo- spheres of solute atoms around dislocations may not sig- Subtracting (7.85) from (7.83), (7.83) from (7.84) and nificantly influence their ability to climb but if precipitates (7.84) from (7.85), we get respectively form on dislocations, it may be possible to prevent climbing of dislocation. If precipitation occurs at grain boundaries, it 2e_1 À ðe_2 þ e_3Þ ¼ C½2r1 À ðr2 þ r3ފ ð7:86Þ reduces grain-boundary sliding and thus produces high creep strength but often at the expense of creep ductility, because 2e_2 À ðe_3 þ e_1Þ ¼ C½2r2 À ðr3 þ r1ފ ð7:87Þ in many cases, the precipitates cause early cavity formation and premature intercrystalline failure. If grain-boundary 2e_3 À ðe_1 þ e_2Þ ¼ C½2r3 À ðr1 þ r2ފ ð7:88Þ precipitates form an interface with the matrix of a very low energy, the probability of intergranular cracking is likely to Now from the first assumption, substituting e_2 þ e_3 ¼ decrease. Àe_1 into (7.86), e_3 þ e_1 ¼ Àe_2 into (7.87), and e_1 þ e_2 ¼ Àe_3 7. Use Materials with Large Grain Size into (7.88), we obtain respectively Creep resistance at elevated temperature can be improved by 1 ! increasing grain size or developing an elongated structure 2 through directional solidification technique so that virtually 3e_1 ¼ 2C r1 À ðr2 þ r3Þ or; all grain boundaries perpendicular and inclined to the tensile stress axis are eliminated. For example, directionally solid- 2 1 ! ð7:89Þ ified superalloys are used as creep-resistant materials. 3 2 e_1 ¼ C r1 À ðr2 þ r3Þ 7.14 Creep Under Multiaxial Stresses 1 ! In the absence of metallurgical changes, the following basic 2 simplifying assumptions of plasticity theory that hold rea- 3e_2 ¼ 2C r2 À ðr3 þ r1Þ or; sonably well for multiaxial stress conditions during steady-state creep are as follows: 2 1 ! ð7:90Þ 3 2 • Considering the constant volume deformation process for e_2 ¼ C r2 À ðr3 þ r1Þ a metal, which is essentially incompressible, it has been shown that the summation of true strains in three prin- 1 ! cipal directions is zero, i.e. e1 þ e2 þ e3 ¼ 0; [see (1.21)]. 2 This leads to the assumption that the summation of true 3e_3 ¼ 2C r3 À ðr1 þ r2Þ or; strain rates in three principal directions will also be zero, i.e. e_1 þ e_2 þ e_3 ¼ 0: 2 1 ! ð7:91Þ 3 2 e_3 ¼ C r3 À ðr1 þ r2Þ For engineering applications, let us consider the stress dependence of steady-state creep rate expressed by (7.32), in which for combined stress conditions e_s and r must be substituted by the significant strain rate e_ and the significant stress r: Thus, (7.32) can be written as e_ ¼ A02rn ð7:92Þ By substituting e_1; e_2; e_3; from (7.89), (7.90) and (7.91) in the significant strain rate relation obtained from (1.77a), the constant C, can be evaluated as follows:

304 7 Creep and Stress Rupture e_ ¼ pffiffi h À e_2Þ2 þ ðe_2 À e_3Þ2 þ ðe_3 À e_1Þ2i1=2 7.15 Indentation Creep ¼ 2 ðe_1 In Chap. 3, it has been assumed that the deformation p3ffiffi \"  À 3r22 þ 4  À 3r32 response of the material subjected to indentation is instan- 2 4 C2 3r1 C2 3r2 taneous, or nearly so, which is the fact for most metals and ceramics when they are indented at room temperature. In 39 2 2 9 22 general, however, the deformation due to indentation can depend on time at or above a homologous temperature of þ 4  À 3r12#1=2 about 0.5, where time-dependent creep is an important C2 3r3 phenomenon in metals and ceramics and the amount of the 9 22 time-dependent deformation due to indentation at constant load or stress increases with increasing temperature. Inden- ¼ pffiffi h À r2Þ2 þ C2ðr2 À r3Þ2 tation creep can be defined (Mahmudi and Rezaee-Bazzaz 2 C2ðr1 2005) as the time-dependent penetration of a hard indenter 3 into the material under stress at a constant temperature. The possibility to obtain information on creep behaviour using þ C2ðr3 À r1Þ2i1=2 long-time hardness or indentation tests have been reported by several authors (Sundar et al. 2000; Fujiwara and Otsuka ¼ pffiffi C p2ffiffi &pffiffi h À r2Þ2 þ ðr2 À r3Þ2 2001; Dorner et al. 2003; Mahmudi and Rezaee-Bazzaz 2 2 2 ðr1 i1=2) 2005; Bhakhri and Klassen 2006). The most common 3 þ 2 method of measuring creep by the indentation technique is to maintain the applied load at a constant maximum value and ðr3 À r1Þ2 measure the change in depth of the indentation as a function of time. When an indenter is pressed into the test surface of a ¼ 2 C r: ½with the help of(1.76)Š hot material under a constant load at a constant temperature, 3 the indenter displacement increases gradually with time due to creep, following the instantaneous deformation that ð7:93Þ occurs just after the application of the load. The time-dependent indenter displacement is recorded and the Hence combining (7.92) and (7.93), we can write indenter displacement versus indentation time is plotted, which shows a typical indentation creep curve obtained at a C ¼ 3 e_ ¼ 3 A02rn ¼ 3 A02rnÀ1 ð7:94Þ constant temperature under a constant load. For example, in 2 r 2 r 2 one study (Fujiwara and Otsuka 2001) a conical diamond tip was pressed into the test surface of hot material in an argon Substituting for C from (7.94) into (7.89), (7.90) and gas atmosphere at different temperatures above a homolo- (7.91), we get the true significant strain rates, respectively, in gous temperature of 0.6 under a constant load using an three principal directions: indentation creep tester, and the time-dependent indenter displacements were recorded in real time on a PC. The 2 3 1 ! indenter displacement versus indentation time obtained at a 3 2 2 constant load was plotted at each temperature to generate e_1 ¼ A20 rnÀ1 r1 À ðr2 þ r3Þ indentation creep curves at various temperatures. ! 1 ð7:95Þ There are various ways to perform the indentation creep 2 tests. One of the ways to characterize and quantify important ¼ A02rnÀ1 r1 À ðr2 þ r3Þ creep parameters is to use instrumented indentation testing (IIT) method. The indentation creep test is conducted by 2 3 1 ! applying a constant load to the indenter and monitoring its 3 2 2 displacement as a function of time. Figure 7.44a, b shows e_2 ¼ A20 rnÀ1 r2 À ðr3 þ r1Þ typical micro-indentation curves for creep obtained by ! applying a low load and using a depth-sensing indentation 1 ð7:96Þ 2 ¼ A20 rnÀ1 r2 À ðr3 þ r1Þ 2 3 1 ! 3 2 2 e_3 ¼ A20 rnÀ1 r3 À ðr1 þ r2Þ ! 1 ð7:97Þ 2 ¼ A02rnÀ1 r3 À ðr1 þ r2Þ The significant strain rate and the significant stress are useful parameters for correlating steady-state creep data. When they are plotted on log-log coordinates, a linear relationship is obtained.

7.15 Indentation Creep (a) 305 Fig. 7.44 a Load–depth and, Pmax (b) b depth–time plots for indentation creep h2 h1 Load, P Depth, h h1 h2 t1 t2 Depth, h Time, t technique. Figure 7.44 shows that the test load is applied pm ¼ P ð7:100Þ over a time period of 0 to say, t1; when the depth of Apc indentation is say, h1 because of yielding. The test load is then kept constant from the time t1 to say, t2; during which where the indentation depth increases from h1 to say, h2 by creeping. The change in the indentation depth from h1 to h2 P the applied load, and is measured and the relative change of the penetration depth Apc the projected area of contact under load. is referred to as the creep of the test material. The creep For conventional creep test conducted in uniaxial tension value of the indentation test, denoted by CIT; is represented under low stress level, the temperature and stress depen- as a percentage and is computed from: dence of steady-state creep rate, e_s; are often described by (7.35). Noting that the mean contact pressure pm in an CIT ¼ h2 À h1 Â 100 ð7:98Þ indentation test is equivalent to the stress r of (7.35) and the h1 indentation strain rate e_i is equivalent to the steady-state creep rate e_s of (7.35), the analogue of (7.35) for an inden- where h1 is the depth of penetration at time t1; at which the tation creep test is: test load is reached, and h2 is the depth of penetration at a later time t2; up to which the test load is held constant. Note e_i ¼ Ai0pmn eÀQ=RT ð7:101Þ that displacement per unit time is not used to express CIT: The reported creep value includes the relative change of the where A0i is a material constant [which has replaced the constant A′ in (7.35)], n is the stress exponent for creep, Q is penetration depth CIT expressed as percentage, together with the activation energy for creep, R is the universal molar gas the test conditions. For example, CIT0:3=15=300 ¼ 3% constant, and T is the test temperature on absolute scale. means a 3% creep determined by applying a test load of It has been found (Atkins et al. 1966; Mayo and Nix 0.3 N, which was reached after a time of 15 s and thereafter 1988; Raman and Berriche 1992; Stone and Yoder 1994; Poisl et al. 1995; Lucas and Oliver 1999) that creep beha- held constant for a dwell time of 300 s. viour of some but not all materials are adequately described The indentation strain rate, e_i; i.e. the normalized rate of by (7.101). When (7.101) is followed, a log-log plot of the indentation strain rate e_i against the mean contact pressure indentation displacement (Mayo and Nix 1988; Lucas and pm produces a straight line with a slope that gives the stress Oliver 1999) can be defined as exponent, n. Interestingly, the construction of such a plot can often be made from data produced in a single indentation e_i ¼ dh=h ¼ dh=dt ¼ h_ ð7:99Þ test. For example, indium, a low melting point metal that dt h h creeps at room temperature was loaded with a Berkovich indenter at a constant rate of loading to a peak load (Lucas where h is the displacement of indenter. The above definition is appropriate if cone or pyramid indenter (Pollock et al. 1986; Atkins et al. 1966) is used. The mean contact pressure, pm; in an indentation test is given by

306 7 Creep and Stress Rupture and Oliver 1999). The data were obtained by holding the creep time and the large pointer on the Rockwell dial will maximum load for an extended period of time and moni- rotate gradually with increase of creep time in an anticlock- toring the displacement of indenter as a function of time. As wise direction. The number of divisions rotated by the large the indenter penetrates due to creep deformation during the pointer, which is related to the increment of indentation depth dwell time, the contact area increases resulting in a reduction under the application of the major load on the test piece, is of contact pressure which in turn causes a corresponding noted as a function of creep time. The rate of increase in decrease in the rate of displacement. In this test, creep data indentation depth gradually decreases during the initial per- over several orders of magnitude in e_i were obtained. From iod of creep and reaches an almost constant value after a the test data, an approximate linear plot of indentation strain certain length of time, say a dwell time of tn; at which the rate e_i against mean contact pressure pm was made. The creep deformation is halted by withdrawal of the major load. stress exponent deduced from the slope of this linear plot The dial readings are noted just before and after the with- was found to be n ¼ 6; which is very close to the value drawal of the major load at the end of the creep period, tn; derived using conventional creep testing techniques. while still keeping the minor load applied. The conventional hardness tester (Roumina et al. 2004; Now without removing the minor load, the major load is Deming et al. 2007; Bhaduri 2007) can also be used to withdrawn after a total creep time of tn; causing the large produce a constant-load creep curve of a metal at a constant pointer to rotate instantaneously in a clockwise direction from temperature, from which the steady-state creep rate can be the final division mark on the dial to a new division mark due determined. to elastic recovery of metal, because the indentation will recover elastically by a certain amount depending on the 7.15.1 Method to Obtain Creep Curve Using material and the applied load. The number of divisions moved Rockwell Hardness Tester by the large pointer from its final position to its new position is noted, which is used for calculation of the elastic recovery The aim of using the hardness tester for indentation creep is experienced by the metal. The amount of elastic recovery not to determine the hardness value of the test piece but to occurring after a total dwell time of tn is also applicable to find the increment of indentation depth as a function of time each of the smaller dwell times that increases from 0 to tnÀ1; under load with a purpose of constructing creep curve. The because the applied major load is constant throughout the test method of generating creep curve described below (Bhaduri and a single sample of the same metal is used for the test. 2007) is based on the use of the conventional Rockwell hardness tester, which has been discussed in Sect. 3.8 of Next, the minor load is withdrawn. Elastic recovery of Chap. 3. metal, due to release of minor load being very small, is neglected. After a total creep time of tn; when the total For indentation creep test, a steel ball indenter of prefer- (minor plus major) load is released an elastically recovered ably larger size is attached to a Rockwell hardness tester to permanent spherical indentation is formed on the surface of minimize personal error in measuring the indentation size and the test piece. For this indentation, the diameters at right a test piece is maintained at a temperature where creep pre- angles are measured by a graduated optical microscope, like dominates, for example, lead specimen can be used at room a Brinell microscope, and an arithmetic average of the above temperature. After application of the minor load on the test two readings of the measured diameter is noted. piece with the Rockwell hardness tester, when a suitable major load is applied the large pointer rotates instantaneously To generate creep data from the above indentation test, in an anticlockwise direction from its initial set point to a the following steps are required to be followed, preceding certain division mark on the dial. The number of divisions which certain terms related to different lengths of creep times moved by the large pointer from the set point gives a measure from 0 to tn; are defined below: of the precise depth of additional penetration of the indenter into the surface of the test piece on instantaneous application N.D. the number of divisions moved by the large pointer of major load. The total depth at this instant corresponds to from the set point in anticlockwise direction on the the instantaneous strain due to yielding of metal. On holding Rockwell dial prior to release of the major load; the total load at the same level, metal starts to deform by creeping from that instant which is taken as creep time t ¼ 0: hm the depth of indentation due to application of minor Hence, the penetration depth of the indenter into the surface load; of the specimen will gradually increase at the test temperature with holding time of the applied constant load that is, with hr the depth recovered elastically due to release of the major load, which is assumed to be the same for each creep time and applicable to total load, since elastic recovery due to release of minor load being very small is neglected;

7.15 Indentation Creep 307 ht the total depth of indentation prior to elastic in mm corresponding to N.D. for each creep time can recovery upon application of the major and minor be calculated from ð0:002  N.D.Þ: loads combined; (7) For creep times increasing from 0 to tnÀ1; the value of h for each creep time is calculated by means of (7.102). h the depth of permanent indentation (after elastic Values of Dht; hm and hr can be obtained, respectively, recovery) upon withdrawal of the applied major load from steps (6), (5) and (3). with the minor load still applied ¼ ht À hr; (8) For creep times increasing from 0 to tnÀ1; the value of d for each creep time is calculated from the corre- Dht the indentation depth due to major load = the sponding value of h, founrd hffiiffinffiffiffiffisffiffitffieffiffipffiffiffiffi(ffiffi7ffiffi)ffiffi,ffiffiffibffiffiyffiffiffiffitffihffiffiffieffiffiffiffirffieiffiffilation difference between the total depth of indentation obtained from (3.4): d ¼ D2 À ðD À 2  hÞ2 : prior to elastic recovery due to the total load and the depth of indentation due to the minor (9) Since dn is obtained from the measurement of average load ¼ ht À hm; diameter of the permanent indentation after the creep time of tn; and once other values of d is known for other Dh the depth difference between indentation upon creep times increasing from 0 to tnÀ1; from the step (8), withdrawal of the applied major load with the minor the true strains in creep, e; can easily be calculated for load still applied and indentation made by the minor the corresponding creep times using Tabor relation load ¼ h À hm ¼ ðht À hrÞ À hm ¼ ðht À hmÞ À hr given by (3.58) which is e ¼ 0:2ðd=DÞ: ¼ Dht À hr; The indentation creep data of the test piece under the d the diameter of the elastically recovered permanent applied total load obtained from the above steps are pre- indentation made by the total load. sented in Table 7.1 for different lengths of creep time, from which the creep curve for the test metal at the test temper- Hence from the above, we can write ature under the applied load can be constructed. Generally, the indentation creep curve will show the first two stages h ¼ ht À hr ¼ Dht þ hm À hr ð7:102Þ similar to the creep curve obtained from the conventional The required calculation steps are: creep test. The first stage of the indentation creep curve will show an increase in the true strain with time at a decreasing (1) After a creep time of tn; if the measured average rate, followed by a steady-state region where the true strain diameter of the permanent indentation due to the total increase almost linearly at a minimum rate with time. The load is dn; then the total depth of permanent indenta- slope of the linear part of the creep curve will give the tion,h hn;piffiffisffiffiffiffiffifficffiffiaffiffilffifficffiffiuilated by using (3.4), which is steady-state creep rate. As fracture of a specimen does not h ¼ D À D2 À d2 =2; where d = the chordal diameter usually occur in a hardness test, it is not possible to produce the third stage of the creep curve as normally observed in of indentation after unloading, and D = the diameter of constant-load creep test. spherical indenter. (2) At a creep time of tn; if N.D. = N.D.n as found from the The indentation creep test can be particularly advanta- Rockwell dial reading, then ðDhtÞn¼ geous when the material is only available as small test ½0:002  N.D.nŠ mm; since each division represents pieces and/or there are difficulties in shaping or machining 0.002 mm vertical motion of the indenter. of specimens from materials such as lead alloys. To obtain (3) Upon the withdrawal of the major load after a creep creep characteristics of materials using indentation creep time of tn; if the number of divisions moved by the tests, not only small amounts of material are needed, but large pointer from the final position to a new position is specimen preparation is also simple, since only a flat N.D.r; then hr ¼ ð0:002  N.D.rÞ mm: specimen surface is required. On the other hand, specimens (4) Then at creep time tn; from the values of ðDhtÞn in step for standard creep test must have shapes and dimensions (2) and hr in step (3), we get corresponding to standards specified by some organization such as ASTM. Further, preparation of a large number of Dhn ¼ ðDhtÞnÀ hr mm creep specimens may be expensive. Besides, a hardness ¼ ½0:002  N.D.nŠ À ð0:002  N.D.rÞ mm; tester like Rockwell is relatively inexpensive and its operation requires relatively little skill and experience in (5) From steps (1) and (4), hn and Dhn are known, so at a comparison to use of a creep tester. Thus, the indentation creep time of tn; hm can be calculated from hm ¼ creep test provides a simpler method of investigating creep hn À Dhn; ðsince Dhn ¼ hn À hmÞ: This value of hm is behaviour of a metal. However, a hardness tester cannot applicable to each creep time as the applied minor load on the test piece is constant. (6) Now from the readings of N.D. noted during test for creep times increasing from 0 to tnÀ1; the value of Dht

308 7 Creep and Stress Rupture Table 7.1 Presentation of indentation creep data using Rockwell hardness tester Creep time, Number of divisions moved by Depth difference Total depth of Permanent indentation True strain after total large pointer from set point in Dht ¼ ht À hm; diamqeffitffieffiffirffiffi,ffiffiffidffiffiffiffiiffinffiffiffiffimffiffiffiffimffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! in creep, e load is anticlockwise direction on dial, in mm, permanent ½¼0:2ðd=Dފ applied N.D. ð¼0:002  N.D.Þ ¼ D2 À ðD À 2  hÞ2 indentation, h, e0 d0 e1 in mm,  d1 e2 ¼ht À hr ¼ d2 e3 d3 ... Dht þ hm À hr ... ... ... ... (7.102) ... enÀ1 dn−1 en 0 N.D.0 Dht0 h0 dn t1 N.D.1 (measured data) t2 N.D.2 Dht1 h1 t3 N.D.3 ... ... Dht2 h2 ... ... ... ... Dht3 h3 ... ... ... ... ... ... tn−1 N.D.n−1 DhtnÀ1 hn−1 tn N.D.n Dhtn hn [taken from step (1)] completely substitute a creep tester; it may provide an e_s ¼ A expðnarÞ estimate about creep properties of homogeneous and iso- ¼ 2n tropic metals, where there is a dearth of conventional creep   103sÀ1 machine. 1:67  10À8 exp 3:5  1:5  10À6  36 A0  7.16 Solved Problems 23:5  ¼ 1:476  10À9exp 0:189 sÀ1: A0 7.16.1. Garofalo empirical relation between steady-state Since the allowable steady-state creep rate, creep rate ðe_sÞ and stress ðrÞ at constant temperature is given by e_s ¼ Aðsinh arÞn and it reduces to exponential depen- e_s ¼ 10À2 hÀ1 ¼ 10À2=3600 sÀ1 ¼ 2:78  10À6 sÀ1: dence of stress at high stress level (for values ar [ 1:2)  which is observed by a rod subjected to an axial load of ) 1:476  10À9exp 0:189 sÀ1 ¼ 2:78  10À6 sÀ1; or; 36 kN at a constant temperature of 816 °C. Compute the A0 initial cross-sectional area of the rod based on an allowable e_s of 10−2 per hour. Given that, n ¼ 3:5; a ¼ 1:5  0:189 ¼ ln 2:78  10À6 ¼ 7:541: 10À6 m2=N; and A ¼ 1:67  10À8 sÀ1: A0 1:476  10À9 Solution Hence, the initial cross-sectional area of the rod, Given that, n ¼ 3:5; a ¼ 1:5  10À6 m2=N; and A ¼ 1:67  A0 ¼ 0:189 m2 ¼ 0:025063m2 10À8 sÀ1: Axial load applied to a rod P ¼ 36 kN ¼ 36  7:541 103 N: Let the initial cross-sectional area of the rod ¼ A0 m2: So, the applied stress will be r ¼ P=A0 ¼ ¼ 25063 mm2: ½ð36  103Þ=A0Š N=m2: 7.16.2. A steel bolt clamping two rigid plates together is kept Garofalo empirical relation at constant temperature, e_s ¼ Aðsinh arÞn; at high stress level (for values ar [ 1:2) over a period of 5 years at a constant temperature of 650 °C. reduces to exponential dependence of stress as shown by (7.34), which is It is found that the stress ðr in MPaÞ dependence of steady-state creep rate ðe_s in sÀ1Þ for this steel at 650 °C is given by e_s ¼ constant ðrÞ5: Test of the bolt steel at this temperature indicates that e_s ¼ 7  10À9 sÀ1 at a stress of 41 MPa. If Young’s modulus of the steel at 650 °C is

7.16 Solved Problems 309 124 GPa and the stress in the bolt must not drop below (a) Larson–Miller parameter, where the value of Larson– 3 MPa during the 5 years, determine the initial stress to Miller constant is 17. which the bolt must be tightened. (b) Sherby–Dorn parameter, where the value of activation energy for creep is 350 kJ/mol. Solution (c) Manson–Haferd parameter, where the coordinates of the intersection point of the extrapolated lines on the graph of Given that e_s ¼ constant ðrÞ5; at 650 °C. Further, it is indi- log10t versus T are Ta ¼ 172 K; and log10 ta ¼ 20: cated that when the stress is r ¼ 41 MPa; the steady-state creep rate is e_s ¼ 7 Â 10À9 sÀ1; therefore Solution Constant, say; A02 ¼ e_s ¼ 7 Â 10À9 MPaÀ5 sÀ1 Given that the temperature of stress-rupture testing is TT ¼ r5 ð41Þ5 650 C ¼ ð650 þ 273Þ K ¼ 923 K; the time of rupture dur- ing test is tT ¼ 103 h; and the service temperature is TS ¼ ¼ 6:04197 Â 10À17 MPaÀ5 sÀ1 560 C ¼ ð560 þ 273ÞK ¼ 833 K: Let the rupture life at service is tS h; which is to be determined. The stress level for Since it is given that the dependence of steady-state creep the test and the service conditions remains constant at rate ðe_sÞ on stress ðrÞ is governed by power relation, so the 250 MPa. relation between stress and time in stress relaxation will be given by (7.80), which is: 1 ¼ 1 þ A20 Eðn À 1Þt (a) From (7.63a), the Larson–Miller parameter is: PLÀM ¼ rnÀ1 rinÀ1 Tðlog10 t þ CLÀMÞ ¼ constant at constant stress level, in which CLÀM ¼ Larson–Miller constant ¼ 17; (given); where where T = temperature in K; and t = time in h. Due to the same r the stress remaining after 5 years ¼ 3 MPa; stress level, we can equate the test and the service conditions n the power index of stress ðrÞ in the relation between by the Larson–Miller parameter as follows: steady-state creep rate ðe_sÞ and stress ðrÞ ¼ 5; TTðlog10 tT þ 17Þ ¼ TSðlog10 tS þ 17Þ; ri the initial stress in MPa, which is to be determined; A20 the constant in the relation between steady-state creep Hence; À 103 þ Á ¼ 833ðlog10 tS þ 17Þ; or; 923 log10 17 rate ðe_sÞ and stress ðrÞ ¼ 6:04197 Â 10À17 MPaÀ5 sÀ1; E Young’s modulus of the steel at 650 °C 833ðlog10 tS þ 17Þ ¼ 18460; ¼ 124 Â 103 MPa; Or, log10 tS ¼ 18;460 À 17 ¼ 5:1608; 833 t the time in seconds ) tS ¼ 144:8 Â 103 h: ¼ ð5 Â 365:25 Â 24 Â 3600Þ s ¼ 157:788 Â 106 s; Hence, substituting the above values into (7.80), we get 1 ¼ 1 À À Â 10À17 Á Â À Â 103Á (b) Given that the value of activation energy for creep is r5i À1 ð3Þ5À1 6:04197 124 Q ¼ 350 Â 103 J=mol: Â ð5 À 1Þ Â À Â 106 Á From (7.66), the Sherby–Dorn parameter is: POÀSÀD ¼ 157:788 log10 t À ðCOÀSÀD=TÞ ¼ constant at constant stress level, where COÀSÀD ¼ 0:4343ðQ=RÞ; in which R = molar gas ¼ 7:617 Â 10À3 MPaÀ4: constant ¼ 8:314 J molÀ1 KÀ1: Therefore, the value of Or; r4i ¼ 7:617 1 10À3 ¼ 131:285 MPa4; COÀSÀD will be: Â ) ri ¼ ð131:285Þ14MPa ¼ 3:385 MPa: COÀSÀD ¼ 0:4343 Â 350 Â 103 ¼ 18:283 Â 103 K: 8:314 7.16.3. A ferrous alloy shows a rupture life of 103 h when Due to the same stress level, the test and the service tested in a stress-rupture test at a temperature of 650 °C conditions can be equated by the Sherby–Dorn parameter as under a stress level of 250 MPa. Calculate the time to rup- follows: ture of the alloy subjected to the same stress level during service at an operating temperature of 560 °C according to log10 tT À COÀSÀD ¼ log10 tS À COÀSÀD ; each of the following time–temperature parameter methods: TT TS

310 7 Creep and Stress Rupture Hence; log10 103 À 18;283 ¼ log10 tS À 18;283 ; or; Creep time, under application Number of divisions moved by 923 833 of total load of 30 kg (min) large pointer, from set point in anticlockwise direction on dial, log10 tS À 21:948 ¼ À16:808; 5 N.D. 7 Or, log10 tS ¼ À16:808 þ 21:948 ¼ 5:14; 11 174 ) tS ¼ 138  103 h: 15 200 20 240 (c) From (7.68), the Manson–Haferd parameter is: PMÀH ¼ 25 260 35 279 ðT À TaÞ=log10 t À log10 ta ¼ constant at constant stress 45 297 level, in which it is given that Ta ¼ 172 K; and log10 ta ¼ 20: 332 368 Due to the same stress level, we can equate the test and the service conditions by the Manson–Haferd parameter as Solution follows: log10 tT À log10 ta ¼ log10 tS À log10 ta ; TT À Ta TS À Ta From (3.4), the total depth of permanent indentation after 45 min of creep time is: Hence; log10 103 À 20 ¼ log10 tS À 20 ; h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 923 À 172 833 À 172 DÀ D2 À d425 ¼ 5 À 52 À 3:552 or; log10 tS À 20 ¼ À22:636  10À3; h45 ¼ 2 mm 661 2 Or, log10 tS ¼ À22:636  10À3  661 þ 20 ¼ 5:038; ¼ 0:7395 mm ) tS ¼ 109  103 h: ð7:103Þ 7.16.4. A Rockwell superficial hardness tester attached with At 45 min of creep time, from the given N.D.45 ¼ 368; a spherical steel ball indenter (a Brinell indenter) of diameter we get the difference between the total depth of indentation D ¼ 5 mm is used to indent a lead specimen at 30 °C by prior to elastic recovery due to the total load and the depth of applying a total load of 30 kg. After a dwell time of 45 min, indentation due to the minor load: when the major load of 27 kg is withdrawn the number of divisions moved by the large pointer in the clockwise Dht45 ¼ ht45 À hm ¼ N.D.45  0:002 direction is 15, and after the release of total load of 30 kg, ¼ 368  0:002 mm the diameter of permanent indentation formed on lead is ¼ 0:736 mm: measured to be 3.55 mm by means of an optical microscope. Neglecting very small elastic recovery upon withdrawal of (This is the indentation depth due to major load only.) the minor load, compute the creep strains (only plastic part Elastically recovered depth of indentation on release of of creep strains) for the corresponding creep times, using Tabor relation given by (3.58), and draw the creep curve for major load, lead. Also draw steady-state creep curve and determine the steady-state creep rate from this curve. The following hr ¼ N.D.r  0:002 ð7:104Þ experimental data are provided: ¼ ð15  0:002Þ mm ¼ 0:03 mm Creep time, under application Number of divisions moved by At 45 min of creep time, the depth difference between of total load of 30 kg (min) large pointer, from set point in indentation upon withdrawal of the applied major load with anticlockwise direction on dial, the minor load still applied and indentation made by the 0 N.D. minor load is: 1 2 85 Dh45 ¼ Dht45 À hr 3 ¼ ð0:736 À 0:03Þ mm 112 ¼ 0:706 mm: 126 Indentation depth due to minor load (neglecting very small elastic recovery upon withdrawal of the minor load) is: 145 (continued)

7.16 Solved Problems 311 Table 7.2 Indentation creep data Creep time, Number of divisions moved by Depth difference Total depth of Permanent indentation True strain after total large pointer from set point in Dht ¼ ht À hm; permanent diamqeffitffieffiffirffiffi,ffiffiffidffiffi,ffiffiffiiffinffiffiffiffimffiffiffiffimffiffiffiffiffiffiffiffiffiffiffiffiffiffi! in creep, e, load is anticlockwise direction on dial, in mm, indentation, h, in ½¼0:2ðd=Dފ applied (min) N.D. ð¼0:002  N:D:Þ mm,  ¼ D2 À ðD À 2  hÞ2 0.07321 0 85 0.17 ¼ht À hr ¼ 1.8302 0.08336 1 112 0.224 Dht þ hm À hr 2.084 0.08808 2 126 0.252 [hr from (7.104) 2.202 0.09402 3 145 0.29 and hm from 2.3506 0.10226 5 174 0.348 (7.105)] 2.5565 0.10895 7 200 0.4 2.7237 0.11822 11 240 0.48 0.1735 2.9555 0.12247 15 260 0.52 3.0617 0.12629 20 279 0.558 0.2275 3.1574 0.12975 25 297 0.594 3.2438 0.13605 35 332 0.664 0.2555 3.4011 0.142 45 368 0.736 3.55 (measured data) 0.2935 0.3515 0.4035 0.4835 0.5235 0.5615 0.5975 0.6675 0.7395 [from (7.103)] hm ¼ h45 À Dh45 ð7:105Þ 7.Ex.2. A round bar is found to creep at a fixed steady-state ¼ ð0:7395 À 0:706Þ mm creep rate of 10À5 hÀ1 over a period of 104 h under a con- ¼ 0:0335 mm stant load of 2 kN. The bar has an initial length of 400 mm and initial diameter of 20 mm. Assuming that the volume The indentation creep data of lead at 30 °C under a total remains constant, compute the following after a steady-state load of 30 kg are presented in Table 7.2 for different lengths creep period of 103 h: of creep time. (a) The final length of the bar. Figure 7.45 shows the creep curve for lead at 30 °C (b) The engineering and true strains. under a load of 30 kg, for which the data have been taken (c) The engineering and true stresses. from Table 7.2. 7.Ex.3. Determine the activation energy for creep in J=mol Creep time and true strain data in the region of from the following data, which were obtained in creep test of steady-state creep of lead metal are taken from Table 7.2 steel at a constant stress of 110 MPa: and presented separately in Table 7.3. Temperature Steady-state creep rate From the data of Table 7.3, creep strain versus creep time 480 °C 7:22  10À9 sÀ1 in the region of steady-state creep is plotted in Fig. 7.46. It is 650 °C 2:36  10À4 sÀ1 found from Fig. 7.46, that steady-state creep rate ¼ 6:4596  10À4 minÀ1 ¼ 1:0766  10À5 sÀ1. Exercise 7.Ex.4. Derive the relation between stress and time in stress relaxation, when the exponential relation, as proposed by 7.Ex.1. The initial clearance between the ends of 200 mm Garofalo and given by (7.34), governs the stress dependence long blades and the housing of a steam turbine is 0.08 mm. of steady-state creep rate. During operation, the blades extend elastically which is calculated to be 0.02 mm. If it is desired to maintain a final 7.Ex.5. A rod, whose creep properties are given below, is minimum clearance of 0.03 mm, what is the maximum per subjected to an axial load of 35 kN at 800 °C. Determine the cent creep that can be allowed in the blades?

312 7 Creep and Stress Rupture 0.16 Fig. 7.45 Creep curve for lead at 303 K under a constant load of 30 kg = (30  9.81) N = 294.3 N 0.14 0.12 True Strain in Creep 0.10 0.08 Steady State Creep Primary Creep 0.06 0.04 5 10 15 20 25 30 35 40 45 0 Time, in Minute Table 7.3 Steady-state creep Steady-state creep time after total load of 30 kg is applied, in min True strain in steady-state creep, e, data taken from Table 7.2 ½¼0:2ðd=Dފ 15 0.12247 20 0.12629 25 0.12975 35 0.13605 45 0.142 initial cross-sectional area of the rod based on an allowable Dwell time Number of divisions moved by the large pointer creep rate of 1% per 104 h. Assume that the dependence of (min) minimum creep rate ðe_sÞ on stress ðrÞ is governed by 30 50 in anticlockwise direction under the application of total load of 60 kg exponential relation. 60 70 in anticlockwise direction under the application of total load of 60 kg r (MPa) e_s at 800 °C (s−1) 60 20 in clockwise direction, when the major load of 70 2  10À10 50 kg is withdrawn 100 7  10À9 7.Ex.6. A specimen of an isotropic metal is loaded in a After the dwell time of 60 min when the total load of Rockwell hardness tester using 1/8 in. diameter steel ball 60 kg is released, the diameter of permanent indentation indenter at a homologous temperature greater than 0.5. formed on that metal is measured to be 1.2 mm by means of Compute the creep strain (only plastic part of creep strain) an optical microscope. Neglect very small elastic recovery for a dwell time of 30 min, using Tabor relation given by that occurs upon withdrawal of the minor load. (3.58), from the following data:

7.16 Solved Problems 313 Fig. 7.46 Steady-state creep True Strain in Steady State Creep 0.145 Y=m*X+C curve of lead at 303 K under a 0.140 Parameter Value constant load of 30 kg, showing 0.135 linear relationship between true m 6.45964E-4 strain and creep time C 0.11322 0.130 0.125 0.120 15 20 25 30 35 40 45 Steady State Creep Time, in Minute 7.Ex.7. A steel bolt clamping two rigid plates at 537 °C is maximum service temperature below which the alloy must be held according to each of the following time–temperature initially tightened to a stress of 69 MPa and Àte_hs einshtrÀe1sÁs parameter methods: ðr in MPaÞ dependence of steady-state creep rate (a) Larson–Miller parameter, where the value of Larson– for that steel at 537 °C is given by e_s ¼ ð1:33 Â 10À12Þ Á r3: Miller constant is 22. (b) Sherby–Dorn parameter, where the value of activation If Young’s modulus of the steel at 537 °C is 134 GPa, what energy for creep is 460 kJ=mol: (c) Manson–Haferd parameter, where the coordinates of the will be the stress remaining after one year has elapsed? intersection point of the extrapolated lines on the graph of log10 t versus T are Ta ¼ 311 K; and log10 ta ¼ 18: 7.Ex.8. The rupture time of a material is found to be 104 h at a service temperature of 540 °C and the corresponding 7.Ex.11. A metal with melting point of 1100 °C creeps Larson–Miller test condition on the same material shows a above a homologous temperature of 0.5. Assume that it rupture life of 13 h at a test temperature of 650 °C, where undergoes climb–glide creep exhibiting power law relation both service and test conditions are performed under the with the most common value of stress exponent. The acti- same stress level of 150 MPa. Compute the activation vation energy for lattice self-diffusion is 200 kJ=mol; which energy for creep in kJ=mol; corresponding to that stress is assumed to be independent of stress and temperature level. above the homologous temperature of 0.5. Steady-state creep rate is found to be 10À8sÀ1 at a homologous temperature of 7.Ex.9. If stress-rupture test at 650 °C under a stress of 0.6, when the applied stress is 10 MPa. Determine the 200 MPa shows a rupture life of 15 h for an alloy, what will steady-state creep rate for the following: be the time to rupture of the same alloy subjected to the same stress level at a service temperature of 500 °C, based on Larson–Miller (L–M) parameter, assuming the L–M parameter constant to be 42.5? 7.Ex.10. The stress-rupture test on Cr–Mo–V-based steel at (a) When the stress level is increased to 20 MPa at the same 650 °C under a stress level of 100 MPa shows a rupture life homologous temperature of 0.6. of 20 h. If it is desired to have a service life of 104 h for the (b) When the homologous temperature is increased to 0.7 at the same stress level of 10 MPa. alloy subjected to the same stress level, calculate the

314 7 Creep and Stress Rupture 7.Ex.12. Indicate the correct or most appropriate answer(s) Answer to Exercise Problems from the following multiple choices: 7.Ex.1. 0.015%. (a) In Nabarro-Herring creep, the main deformation mech- 7.Ex.2. (a) 404.02 mm; (b) 1.005% and 1%; (c) 6.366 and anism is: 6.43 MPa. 7.Ex.3. 353.33 kJ mol−1. (A) Cross-slip of dislocation; 7.Ex.4. expðÀbrÞ ¼ expðÀbriÞ þ A200bEt. (B) Dislocation climb; 7.Ex.5. 480.8 mm2. (C) Stress-assisted lattice diffusion; 7.Ex.6. 6.17%. (D) Stress-assisted grain-boundary diffusion. 7.Ex.7. 17.323 MPa. 7.Ex.8. 376.9 kJ mol−1. (b) To obtain increased creep resistance above half of 7.Ex.9. 96822.5 h. homologous temperature, the stacking fault energy of a FCC 7.Ex.10. (a) 554 °C; (b) 563 °C; (c) 551 °C. metal and its grain size should be respectively 7.Ex.11. (a) 3:2  10À7sÀ1; (b) 6:48  10À7sÀ1: 7.Ex.12. (a) (C) Stress-assisted lattice diffusion. (b) (B) low (A) high and coarse; (B) low and coarse; and coarse. (c) (C) d−1. (d) (D) stress-assisted grain- (C) high and fine; (D) low and fine. boundary diffusion. (e) (A) high and rapid. (f) (A) moder- ate and higher. (g) (D) balance between strain hardening and (c) In many alloys, at high temperature, the steady-state recovery. (h) (A) diffusional creep; (B) climb; (D) diffusion creep rate is found to vary as by vacancy mechanism. (A) d; (B) d2; (C) d−1; (D) d−2; where d is the grain size of the alloys. (d) In Coble creep, the main deformation mechanism is References (A) cross-slip of dislocation; (B) dislocation climb; (C) stress-assisted lattice diffusion; (D) stress-assisted grain- Ashby, M.F.: A first report on deformation mechanism maps. Acta Metall. 20, 887–897 (1972) boundary diffusion. ASTM E328: Standard Test Methods for Stress Relaxation for (e) Transgranular fracture in stress-rupture test at a given Materials and Structures. Designation: E328–13, ASTM Interna- temperature is most favoured when applied stress and strain tional, West Conshohocken, PA. doi:https://doi.org/10.1520/E0328- rate are respectively 13 (2013) (published in 2014) (A) high and rapid; (B) low and rapid; (C) high and slow; (D) low and slow. Atkins, A.G., Silverio, A., Tabor, D.: Indentation creep. J. Inst. Metals. 94, 369–378 (1966) (f) During tertiary stage of creep deformation intergranular cracks of w-type usually form, if creep temperature and Bhaduri, A.: Use of hardness tester for the measurement of different applied stress level are respectively mechanical properties of metals. J. Mater. Ed. 29(3–4), 269–288 (2007) (A) moderate and higher; (B) moderate and lower; Bhakhri, V., Klassen, R.J.: Scripta Mater. 55, 395–398 (2006) (C) higher and lower; (D) higher and higher. Chang, H.C., Grant, N.J.: Trans. AIME 197, 1175 (1953) Chang, H.C., Grant, N.J.: Trans. AIME 206, 544 (1956) (g) The secondary creep deformation is characterized by Chen, C.W., Machlin, E.S.: Acta Metall. 4, 655 (1956) Chen, C.W., Machlin, E.S.: Trans. AIME 209, 829 (1957) (A) pronounced strain hardening; Coble, R.L.: J. Appl. Phys. 34, 1679 (1963) (B) balance between residual stress and deformation; Conrad, H.: The role of grain boundaries in creep and stress rupture. In: (C) pronounced void formation; (D) balance between strain hardening and recovery. Dorn, J.E. (ed.) Mechanical Behavior of Materials at Elevated (h) Some of the processes which have the same activation Temperatures, p. 264. McGraw-Hill Book Company Inc, New York energy in a given material are: (1961) (A) diffusional creep; (B) climb; (c) cross-slip; Cottrell, A.H.: The time laws of creep. J. Mech. Phys. Solids 1, 53–63 (D) diffusion by vacancy mechanism. (1952) Courtney, T.H.: Mechanical Behaviour of Materials, International edn., p. 504. McGraw-Hill Publishing Company, New York (1990) da Andrade, E.N.C.: Proc. Roy. Soc. London, Ser. A, 84, 1 (1910) da Andrade, E.N.C.: The flow in metals under large constant stresses. Proc. Roy. Soc. London, Ser. A, 90, 329–342 (1914) da Andrade, E.N.C., Chalmers, B.: Proc. Roy. Soc. London, Ser. A, 138, 348 (1932) Decker, R.F., Freeman, J.W.: Trans. AIME 218, 277 (1961) Deming, H., Yungui, C., Yongbai, T., Hongmei, L., Gao, N.: Mater. Lett. 61, 1015–1019 (2007)

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Fatigue 8 Chapter Objectives • Fatigue fracture and its characteristic appearance. • Various types of fluctuating stress cycle and its components. • Standard fatigue test and S–N diagram. Fatigue properties, and reason for existence of fatigue limit. • Statistical nature of fatigue and estimation of anticipated fatigue life by means of standard statistical procedure. • Fatigue crack initiation through persistent slip bands, such as slip-band extrusions and slip-band intrusions. • Fatigue crack growth: Stage I and Stage II. Crack growth rate and ‘Paris’ law. • Cumulative fatigue damage with definitions of overstressing, understressing and coaxing. • Methods of presenting S–N data, where mean stress is not zero. Design criteria for mean stress effects: Gerber relation, Goodman relation or Goodman diagram to prevent fatigue failure, and Soderberg relation to prevent yielding. • Effects of stress concentration, specimen size, metallurgical variables, frequency of stress cycling, corrosive environment, low and high temperature on fatigue and thermal fatigue. • Effects of specimen surface, such as its roughness, properties and residual stress on fatigue. Surface treatments beneficial to fatigue and metallurgical processes detri- mental to fatigue. • Cyclic strain-controlled fatigue, describing cyclic strain hardening and cyclic strain softening, and their dependency on material’s stacking fault energy. • Low-cycle fatigue and Coffin–Manson relation. Total fatigue strain–life equation, and its plot approaching towards the plastic strain–life curve at large total strain amplitudes and the elastic strain–life curve at low total strain amplitudes. • Creep–fatigue interaction, where suggested important design approaches are: cumulative damage rule, modification of Goodman law, frequency-modified fatigue relation and strain-range partitioning method. • Increasing amplitude tests, such as step test and Prot test. • Problems and solutions. © Springer Nature Singapore Pte Ltd. 2018 317 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_8

318 8 Fatigue 8.1 Fatigue Failure manner creating a rough dull fibrous region. The growth of the crack captures a large part of the total life of the speci- Most structural assemblies are subjected to fluctuating or men prior to its failure due to fatigue. It is to be noted that repetitive stresses of sufficient magnitude for sufficient the critical (limiting) crack length at which fracture occurs number of times, although the maximum value of this increases with decreasing stress levels. On a macroscopic applied fluctuating stress may be considerably less than the scale, the fracture surface consisting of a rough dull fibrous static tensile strength of the material. Such condition of region and a flat smooth region lies usually at right angles to dynamic loading produces a permanent damage to the the applied principal tensile stress. material that leads to failure after a considerable period of service. This progressive failure of the material at a stress In many cases, particularly when the member fails after a much lower than that required to cause fracture on a single long period of time, the visible examination of smooth application of load is called a fatigue failure. Hence, fatigue region of the fracture surface will often reveal a series of can be defined as the phenomenon leading to a progressive concentric ring of non-uniform colour, which is known in fracture under fluctuating or repeated stresses having a the literature as ‘beach markings’ and/or ‘clamshell mark- maximum value less than the static tensile strength of the ings’. The centre of curvature of these curved markings is material. However, a fatigue failure can take place even the point of initiation of the fracture from which theses when the maximum stress in a fluctuating stress cycle is well marks progress inward. The distances between these mark- below the static tensile yield strength of the material. Fatigue ings correspond to different periods of crack growth and do is encountered in a large number of components, such as not represent individual load excursions, i.e. they are not a aircraft wings, springs, rubber tires, rotating and recipro- measure of crack extension per fatigue cycle. Rather grow- cating parts in vehicles, turbines, pumps, compressors and ing crack advances only microscopic distances on each cycle all other machines, which are subjected to vibration or and the spacing between consecutive ‘beach’ marks corre- repeated loading during service. It has been estimated that sponds to thousands or even tens of thousands of fatigue 80–90% of all mechanical failures in service are due to cycles. For example, crack extension occurs during the fatigue. It is to be noted that for fatigue failure to occur, the working period of an aircraft or a machine and crack growth maximum stress of a time-varying applied stress cycle must stops during the non-operational period of that aircraft or be a tensile stress, i.e. if the maximum stress of the applied machine, and during these alternate crack growth and dor- stress cycle is compressive in nature, fatigue failure would mant periods, areas on the fracture surface are corroded not occur. and/or oxidized by varying amounts. This results in the formation of a series of concentric ring of non-uniform A fatigue fracture occurs all of a sudden without an colour (beach marks) on the fracture surface. The flat smooth appreciable amount of gross plastic deformation during region with characteristic ‘beach’ markings corresponds to service life and appears to be brittle. A fatigue fracture can the slow fatigue crack growth while the rough dull fibrous usually be identified by the characteristic appearance of section is the fast-fracture region. A rough measure of the distinct fracture surface markings. A fatigue failure is initi- magnitude of material’s tensile strength and/or the maximum ated from a minute crack. Most cracks originate at the point applied cyclic stress may be obtained from the relative areas of stress concentration caused by geometrical discontinu- of the slow- and fast-fracture regions. For example, the ities, such as notches, sharp corners, fillets, holes, screw fast-fracture region increases with the maximum applied threads and keyways, or at the point of stress concentration cyclic stress at a given tensile strength, and the region cor- arising from surface roughness and metallurgical stress responding to the slow crack growth increases with the raisers like blowholes or porosity, inclusions and decarbur- material’s tensile strength for a constant maximum applied ization. In a fluctuating stress cycle, when the applied cyclic cyclic stress. stress reaches the maximum value in tension this minute crack opens up, and when the applied stress diminishes to Often microcracks form on separate planes, and subse- the minimum value this crack tends to close down. This quently, they link to create a vertical step on the fracture creates a rubbing action on the surface along which the crack surface. This junction surface between adjacent crack origins propagates through the section up to a limiting distance. is called ratchet line. So, in addition to the set of horizontal Hence, this rubbing region of the fracture surface becomes beach markings, a second set of vertical ratchet line mark- smooth and flat indicating the absence of gross plastic ings may be observed on the fracture surface. Once the deformation and resembling macroscopically to brittle frac- initial cracks originated at adjacent regions have joined ture. When the crack becomes sufficiently long so that the together, the ratchet line disappears. Hence, the neighbour- remaining cross-section of the member is no longer capable ing areas where cracks had originated separately are con- of carrying the applied load, the member fails in a ductile nected by the ratchet lines. In general, the greater the applied stress and/or the severity of a design-imposed stress

8.1 Fatigue Failure 319 In fatigue, a fluctuating stress cycle is composed of two components, a steady or mean stress, rm, and an alternating or variable stress or stress amplitude, ra. Prior to proceeding for discussion on the general types of fluctuating stresses that can cause fatigue, let us first define the terms and symbols associated with a fluctuating stress cycle: Low applied stress and High applied stress and • Maximum stress, rmax, is the highest algebraic value of low stress concentration low stress concentration stress in a cycle. • Minimum stress, rmin, is the lowest algebraic value of stress in a cycle. • Mean or steady stress, rm, is the algebraic mean of the maximum and minimum stresses in a cycle. The oscil- lating stress is superimposed on it. rm ¼ rmax þ rmin ð8:1Þ 2 Low applied stress and High applied stress and • Stress amplitude or alternating stress, ra, is the ampli- high stress concentration high stress concentration tude of superimposed oscillating stress and given by one-half of the algebraic difference between the maxi- Fig. 8.1 Diagram (Wulpi 1966) showing several patterns of typical mum and minimum stresses in a cycle. fatigue fracture surface appearance for low and high levels of applied stress and stress concentration rmax À rmin 2 ra ¼ ð8:2Þ concentration, as in the case of a fillet of smaller radius, the • Range of stress, rr, is the algebraic difference between more will be the number of crack nucleation sites and the maximum and minimum stresses in a cycle. associated ratchet lines. Diagram in Fig. 8.1 (Wulpi 1966) shows several patterns of typical fracture surface appearance rr ¼ rmax À rmin ¼ 2ra ð8:3Þ in fatigue for low and high levels of applied stress and stress concentration. Any one of these patterns may be exhibited • Stress ratio, R, is the ratio of the minimum to the max- on the fracture surface depending on the magnitude of imum stress in a cycle. applied stress and the number of crack nucleation sites. If possible, the stress concentration must be avoided and the R ¼ rmin ð8:4Þ level of stress applied on the component should be main- rmax tained to a minimum. In fact, only one crack nucleation site finally causes total fatigue failure in most services. The size • Amplitude ratio, A, is the ratio of the stress amplitude to of this fatigue crack at the point of final failure is related to the mean stress in a cycle. the fracture toughness of the material and the applied stress level (which becomes fracture stress at the point of fracture) A ¼ ra ¼ rmax À rmin ¼ 1ÀR ð8:5Þ through (9.65), given in Chap. 9. rm rmax þ rmin 1þR 8.2 Stress Cycles Hence, the maximum and minimum stress can be expressed in terms of the mean stress and the alternating Since majority of the fatigue failures occurs without an stress as follows: appreciable amount of gross plastic deformation during service life, so the difference between the original and the rmax ¼ rm þ ra ð8:6Þ instantaneous cross-sectional area is insignificant and the engineering stress, S %, the true stress, r. Hence, fatigue rmin ¼ rm À ra ð8:7Þ stresses will be described by the term r subsequently in this chapter, although initial cross-sectional areas of the members Typical time-varying fluctuating stress cycles in fatigue are mostly used to determine the stresses. are illustrated in Fig. 8.2. It may be noted that the tensile stress will be considered as positive and the compressive stress as negative. The simplest type of repeated stress cycle

320 8 Fatigue Fig. 8.2 Typical time-varying (a) Stress ratio, R = – σmin =–1 (b) Range of stress ratio, R : – 1 < – σmin <0 fluctuating stress cycles in σmax σmax fatigue. a Completely reversed σmax stress cycle, where jrmaxj ¼ 0Stress σa σm σmax σr σa jrminj ¼ jraj and rm ¼ 0. b Partly σr Time reversed tension–compression σmin Stress σm type of stress cycle, where 0 jrmaxj [ jrminj and rm [ 0. Time c Pulsating stress between rmax σmin and rmin ¼ 0, where rm ¼ ra ¼ rmax=2. d Tension– (c) Stress ratio, R = – σmin = 0 =0 (d) Range of stress ratio, R : 0 < σmin < +1 tension type pulsating stress σmax σmax σmax cycle, where all stresses are positive. e Irregular or random stress cycle σmax σa σm σmax σa σmin Stress σr σm σr σmin Stress 0 Time 0 Time (e) Stress Tension + 0 Time – Compression is a completely reversed stress cycle or a purely alternating top fibre of the beam with overload will be like that shown in stress cycle of sinusoidal form like that shown in Fig. 8.2a, Fig. 8.2c. Here, the mean stress is no longer zero but has a where the mean stress rm ¼ 0; jrmaxj ¼ jrminj ¼ jraj, and positive value resulting from the constant load of additional R ¼ rmin=rmax ¼ Àrmax=rmax ¼ À1. This is an ideal con- mass. This mean stress is given by rm ¼ ra ¼ rmax=2. Fig- dition which is produced by an R.R. Moore rotating-beam ure 8.2c shows a repeated stress cycle varying from a max- fatigue testing machine. In service, a rotating axle operating imum tensile stress rmax to a minimum stress, rmin ¼ 0, at constant speed without overloads approaches this condi- having R ¼ 0. Figure 8.2d shows a tension–tension type of tion where the bending stresses vary in a completely reversed repeated stress cycle, where the maximum, the minimum, the stress cycle pattern. Fatigue testing machines, specimens and mean and the alternating—all stresses—are positive, i.e. their preparation, test procedure and technique are described tensile, and R lies between 0 and 1. Figure 8.2b shows a from page 6 to 65 in the reference (ASTM STP 1949). partly reversed tension–compression type of stress cycle, where the maximum stress is tensile and the minimum stress If a cantilever flexure beam, as shown in Fig. 8.3a, is compressive but jrmaxj [ jrminj. This gives a mean stress undergoes same amount of deformation alternatively on rm [ 0, i.e. tensile with R lying between −1 and 0. either side of its equilibrium configuration, the variation of flexural stress will be like that shown in Fig. 8.2a. If, how- The above-mentioned stress variations of sinusoidal form ever, the same beam carries an additional load or mass, say at are found to occur in many rotating and reciprocating its free end, as shown Fig. 8.3b, and is deformed alternatively machine parts. However, there are many examples where from its equilibrium configuration without overload to fatigue loading may involve complicated stress cycle in downward direction, then the variation of flexural stress in the which the variation of stress is far from regular. Figure 8.2e

8.2 Stress Cycles 321 (a) Cantilever beam (b) Crank Mass Cantilever beam Crank Fig. 8.3 a A cantilever beam oscillating with equal displacement oscillating alternatively between its equilibrium configuration without alternatively on either side of its equilibrium configuration. b A overload and downward bent configuration cantilever beam carrying an additional load or mass at its free end is shows this kind of an irregular or random stress cycle which In addition to the above factors, fatigue failure is influ- may be encountered in an aircraft wing subjected to periodic enced by many other variables, such as stress concentration, unpredictable overloads due to storm. The random stress residual stresses, overstressing, temperature, corrosion and variation may also occur in machines operating intermittently metallurgical structure. The relationship between each of due to occurrence of natural vibrations of variable amplitudes these factors and fatigue will be discussed subsequently. during starting and stopping. This kind of irregular stress cycle must be reduced to a simpler form in order to use in 8.3 Standard Fatigue Test design for fatigue. The usual procedure is to split the actual stress cycle into many sets of simple sinusoidal variations. In Many different kinds of fatigue testing machines developed each set, the number of oscillations should be the same as that so far (the ASTM Manual on Fatigue Testing refers to more of actual oscillations with nearly the same stress amplitude than thirty) may be classified with respect to the type of and mean stress. In this manner, the actual complicated applied load and the way of its application. The basic types loading spectrum is transformed into an equivalent simplified of loading used in the laboratory fatigue tests are rotating spectrum of sinusoidal form which can be used in analysis. bending, reversed-flexure bending, tension–tension or ten- sion–compression type of axial loading (push–pull type of The nature of stress variations is usually designated by loading), torsion and combinations of them. Specimens are either of the following two ways: loaded by applying either a constant maximum load or moment or a constant maximum displacement or strain. In (1) A statement of the numerical value of the maximum constant-load machines, although the specimen is subjected stress, rmax, together with the stress ratio, R. to a fixed cycle of loading throughout the experiment, its deflection usually increases as it becomes weaker. In (2) A statement of the numerical value of the mean stress, constant-displacement machines, a fixed alternating deflec- rm, together with the stress amplitude, ra. tion is imposed on the specimen and the resulting stress may change as fatigue progresses. When fatigue tests are con- The kind of stresses in a fatigue stress cycle may be ducted with a fixed cycle of load or stress limits, it is called a tension, compression or shear, which may be caused by stress-controlled fatigue. It is a high-cycle fatigue (often axial, flexural, shearing or torsional loading or by combi- simply termed as fatigue) because fatigue failure takes place nations of them. So for complete definition of a stress con- at high numbers of stress cycles, usually more than 104 dition, the kind of stress must also be stated in addition to cycles. When fatigue tests are conducted with a fixed cycle designating the degree of stress variation. of elastic plus plastic strain limits, it is called a strain-controlled fatigue or a low-cycle fatigue because Three basic factors that are related to the stress cycle are fatigue failure takes place when the number of cycles nec- essential for fatigue failure to occur. These are: essary to cause fatigue failure, N\\103 cycles. Since majority of the fatigue failures in service occurs at N [ 104 (1) A sufficiently high value of the maximum tensile stress, cycles, the fatigue in the high-cycle region (stress-controlled rmax, in the applied stress cycle, fatigue) has received an engineering importance and our (2) A high stress amplitude, ra, in the applied stress cycle, and (3) A sufficiently large number of oscillations in the applied stress cycle.

322 8 Fatigue present discussion is restricted to this. The cyclic axis. Equal loads on these two bearings are applied by means strain-controlled or low-cycle fatigue will be considered of deadweights so that the specimen between the loaded subsequently in Sect. 8.16. bearings is subjected to a uniform bending moment and bending takes place in a vertical plane. To apply cycles of One of the earliest investigations of stress-controlled stress, a motor directly connected to one of the shaft exten- cyclic loading effects on fatigue life was conducted by sions rotates the specimen at speeds varying from 3000 to Wöhler (1860), who studied the fatigue failure of the wheel 10,000 rpm and the rotation continues until the specimen axles of the railway cars. Wöhler designed the first ‘rotating fractures. Since the topmost fibres of the specimen are always bending’ fatigue tester to determine the fatigue strength of in compression while the bottom fibres are in tension, it is the railway axles under alternating stress condition. obvious that a complete cycle of reversed stress of sinusoidal form with zero mean stress is produced at each point on the Among different types of fatigue testing machines/ surface of the specimen during each revolution. A counter is methods used to analyse the fatigue properties of materials, provided with this machine for recording the number of rotating bending is the most widely used standard method of revolution, i.e., stress cycles applied. There is a switch or testing, which is used exclusively for applying constant-stress some device that disengages the counter and stops the testing amplitude with condition of zero mean stress. The most machine automatically when the specimen breaks. This popular standard rotating bending machine is the R.R. Moore machine loaded with a simply supported beam specimen can rotating-beam fatigue testing machine. Its popularity is due be used to produce either unnotched or notched test data. For to its simplicity of operation, inexpensiveness and the fact unnotched test data, the specimen has to be smooth, whereas that it produces a commonly observed condition of stress. for notched test data, a circumferential notch is introduced in Schematic diagram of this machine is presented in Fig. 8.4, the gage section of specimen to create stress concentration. and a diagram of rotating-beam fatigue test specimen in the However, the limitations of the rotating bending machine are form of a simple beam used for this machine is shown in that the test data cannot be used in applications where mean Fig. 8.5 (Richards 1961). The cross-section of specimen used stresses are not zero and the necessity to use a specimen of for this machine must always be circular so that its section circular cross-section. modulus remains constant as it rotates. Fatigue specimens must have very smooth surfaces and be carefully prepared to Instead of a simply supported beam specimen as descri- avoid stress concentrations and tensile residual stresses. bed above, a cantilever round specimen, for which the Between both grip ends of the specimen, a reduced section is loading condition is shown in Fig. 8.6, can also be used by formed by using a constant curvature from end to end without one variant of the rotating bending machine, which has been the necessity of using any fillet so that undesirable stress shown schematically in Fig. 8.6b. A gravity load is applied concentrations are avoided. The diameter of the specimen at to the free end of the cantilever specimen while it is rotating. the centre of the reduced section will be the minimum at In this case, bending moment is not uniform rather increases which the specimen is supposed to fracture. Both ends of the linearly with increasing distance from the point of applied specimen are rigidly held between the ends of two shafts, and load along the gage length of the specimen. If the gage thus, the specimen becomes part of a long beam which is length of the specimen is of uniform cross-section as shown subjected to pure bending. The specimen is loaded through in Fig. 8.6, then the bending moment and thereby the two ball bearings on the shaft extensions (equidistant from bending stress will be the maximum at the base of the fillet at the centre of the span) so that it can rotate freely around its Fig. 8.4 Schematic diagram of Specimen Shutoff switch R.R. Moore rotating-beam fatigue Flexible coupling testing machine Ball bearing Shaft Shaft Motor Revolution counter Roller support Roller support W

8.3 Standard Fatigue Test 323 Fig. 8.5 Specimen for R.R. 90 mm Moore rotating-beam fatigue test (Richards 1961) 0.25 mm 19 12 mm mm M6 x 1 Tap D 4.75 mm 9.5 4.75 mm R mm Taper – 52 mm/m Diameter, D – 5 to 10 mm, selected on the basis of ultimate strength of material. Radius, R – 90 to 250 mm. Fig. 8.6 a Rotating bending of a (a) (b) cantilever round specimen loaded at its free end. b Cantilever-type Main bearing Test piece specimen loaded in rotating bending fatigue testing machine Motor Flexible coupling Load bearing W W the end of the gage section where fatigue failure will take superimposed on both tensile mean stress acting on the place. In fact, this produces a notched fatigue test since the lower wing skin and compressive mean stress acting on the test results will strongly depend on the geometry of fillet. For upper wing skin. The specimen in the axial loading (push– an unnotched test data, the specimen is tapered to obtain the pull) type fatigue tester is held at two ends and subjected to condition of constant bending stress along the gage length of pure axial loading of tension–tension or tension–compres- the specimen. Rotating bending fatigue test with cantilever sion type in which the load varies cyclically between two specimen also represents conditions of zero mean stress just extreme (maximum and minimum) values. The configura- like test with simple beam. tion of such axial loading is shown in Fig. 8.8. Reversed bending fatigue testing machine with fixed 8.4 The S–N Diagram and Fatigue Properties displacement can be used to overcome the above-mentioned limitations of rotating bending machine. A schematic dia- The engineering (stress-controlled) fatigue data are pre- gram of reversed bending fatigue testing machine is shown sented by means of S–N diagram, also known as ‘Wöhler’ in Fig. 8.7. This machine can test either a cantilever or a diagram, which is a plot of stress S against the number of simply supported beam specimen with a variety of shapes stress cycles necessary to cause fatigue failure, N, for high instead of only circular section as required by rotating numbers of cycles, usually more than 104 cycles. In S–N bending machine. This machine can test flat and square bars plot, all parts of the range, except for high values of N, show and sheet metals. In this machine, an arm is attached to the considerable curvature because with decrease of applied specimen, which is mounted as a stationary beam. A crank stress S, N increases slowly at first and then more and more or eccentric drives the arm and produces a constant alter- rapidly. But if a log scale is used for N, the first part of the nating deflection. A variety of alternating and static loads curve often becomes nearly linear and so, N is almost always can be provided by adjusting the arm length. plotted on a logarithmic scale. The data for S–N diagram can be obtained from the standard rotating bending fatigue tests The above modes of loading may be suitable when a at constant-stress amplitude, described in Sect. 8.3, and component is subjected to rotating or reversed loads. How- typical S–N curves for ferrous and non-ferrous metals ever, it is often more appropriate to use the axially loaded obtained from such tests are shown schematically in specimen for applications involving direct loading, where steady stress is not zero, rather an important variable. For example, in aircraft wing, fluctuating stresses are

324 8 Fatigue Crank Motor To determine an S–N curve, a group of fatigue specimens is tested at different stress levels and at each of the several Built-in end stress levels the loaded specimen is rotated until it fractures. After completion of each test at the point of failure, both the Connecting rod applied stress level, S, and the number of cycles (revolu- tions) necessary to cause fatigue fracture, N, are noted and Specimen this (N, S) represents a point on the S–N curve. Several tests at different stress levels from high stress values to very low Fig. 8.7 Schematic diagram of a reversed bending fatigue testing ones with corresponding cycles to fracture will create several machine with constant displacement test points for the construction of an S–N curve. Usually, the first fatigue specimen is tested at a high stress level of the Fig. 8.8 Configuration of tension–tension or tension–compression order of about two-thirds the static tensile strength of the type axial loading material, where failure is expected to take place at a rela- tively low value of N. But the applied stress levels should Fig. 8.9. In these tests, since the stress cycles are completely not be so high that fatigue failures occur at N\\103 cycles, reversed, i.e. the mean stress is zero, so the value of stress because application of higher stresses produces gross plastic S that is plotted can be rmax; ra, or rmin, because deformation that makes the interpretation difficult in terms of jrmaxj ¼ jraj ¼ jrminj. The S–N diagram for cases where the stress. When the number of cycles to failure, N [ 104 mean stress rm 6¼ 0 will be considered in Sect. 8.7. cycles, the stresses, on a macroscopic scale, are usually elastic, but plastic deformation of metals occurs in a highly localized way. However, the test stress is progressively lowered from high value for each successive specimen until the number of cycles applied to the specimen reaches at least 107 cycles. For certain metals and alloys, such as iron and titanium alloys, the specimen does not fracture at or after the specified 107 number of cycles and in such cases, the S–N curve becomes asymptotic to the horizontal line, as shown in Fig. 8.9. The stress corresponding to this horizontal asymptote is called the fatigue limit or endurance limit of a metal, designated as re, and at stresses lower than re, the specimen can endure an infinite number of cycles without failure. Hence, fatigue limit or endurance limit can be Fig. 8.9 Typical schematic S–N curves for ferrous and non-ferrous metals Stress, S (= σmax or σa) Iron and titanium alloys Fatigue limit Copper, aluminium, and magnesium alloys 105 106 107 108 Number of cycles to failure, N (logarithmic scale)

8.4 The S–N Diagram and Fatigue Properties 325 defined as the maximum stress below which a material can endurance limit, re, are determined. Fatigue strength is presumably endure an infinite number of stress cycles defined as the maximum stress which a material can with- without failure. This indicates that the number of cycles is no stand repeatedly for a specified number of cycles longer a factor at stresses below re, where no fatigue dam- ðsay; N cyclesÞ without failure. It is very useful in design. age is expected to take place. Usually, the fatigue limit is far A highly desirable design criterion is the fatigue limit, which below the yield strength of the material. is a special case of fatigue strength, because it may be defined as the limiting value of fatigue strength as the Most non-ferrous metals and alloys, such as copper, number of cycles, N, becomes very large (ASTM STP aluminium and magnesium alloys, often do not have a true 1958). Obviously, fatigue strength and fatigue limit are not fatigue limit, because the S–N curve of such materials never directly measurable from experimental observation, because becomes horizontal, as shown in Fig. 8.9. Rather the S–N the applied stress must always be decided prior to the start of curve slopes gradually downward as the applied stress is the test and it would be impossible to preselect a stress for decreased causing the number of cycles to increase. So, the each specimen such that it would either fail at just specified material showing a continually decreasing S–N curve has no number of cycles ðsay; N cyclesÞ, or have an infinite fatigue apparent lower stress limit below which the material can be life. As a result, the fatigue strength and the fatigue limit considered to be completely ‘safe’. For such materials, the have to be determined by interpolation from the S–N curve. test is continued beyond 107 cycles and usually terminated However, when either the fatigue strength or the fatigue limit when N reaches about 108 or 5 Â 108 cycles, which requires is mentioned for a material, it will be assumed that the stress at least 5 weeks of test running time at the usual cycling is completely reversed within each cycle unless otherwise rates. For such materials, the endurance limit is defined as stated. If the stress cycle is not completely reversed, the the maximum stress which the material can withstand value of the minimum stress, mean stress or the stress ratio repeatedly for 108 or 5 Â 108 number of stress cycles must be specified. without failure. For example, the maximum stress sustained without failure corresponding to 5 Â 108 number of stress 8.4.1 Reason for Existence of Fatigue Limit cycles is the endurance limits for aluminium and magnesium alloys, while that corresponding to 108 number of stress Some materials, especially iron and titanium alloys, possess cycles is the endurance limits for copper- and nickel-based a horizontal ‘knee’ in the S–N curve, the stress corre- alloys (ASTM STP 1958). sponding to which is the fatigue limit. It has been shown (Levy and Sinclair 1955; Lipsitt and Horne 1957; Lipsitt and The S–N curve for a given material depends on the type Wang 1961; Levy and Kanitkar 1961) that the existence of a of test used to determine it. According to experimental fatigue limit depends on the presence of interstitial elements evidence, the S–N curve for a given material produced by the in those materials. Figure 8.10 schematically illustrates the test data is: steps in the development of a fatigue limit. Curve A is the continuously decreasing S–N curve for a pure metal where • The highest for the reversed-flexure bending test. N increases with decreasing stress. When a solute element is • The next highest for the rotating bending test. added to form solid solution, the yield strength is raised and • The lowest for the push–pull type test. the initiation of a slip band becomes more difficult. This solid solution strengthening results in the shift of the S–N Hence, test data of the push–pull type produce the most curve of the solid solution to curve B, which is located above conservative fatigue design and that of the rotating bending and to the right side of curve A. If suitable amount of test gives the next conservative fatigue design. So, while interstitial elements is present in the solid solution alloy, using S–N curve to design the machine parts, one must be there will be additional strengthening of the alloy due to aware of the type of test data that is used to construct the strain ageing from interstitials. Since strain ageing is not curve. strongly dependent on the applied stress, the fatigue damage caused by the applied stress and the localized strengthening Fatigue life, N, is the number of cycles required to cause due to strain ageing will balance each other at a certain fatigue failure of a material for a stated condition, e.g. given limiting stress which results in the fatigue limit. The fatigue rmax and rm, or given rm and ra. It is the basic fatigue limit is shown in curve C, which is located above and to the property since it is the only one which can be measured right side of curve B. Either raised temperature during fati- directly from experimental observation. We shall see shortly gue or increased amount of interstitial elements in the alloy that this experimentally measured fatigue life for a given will enhance the phenomenon of strain ageing. This condition is a property of the individual specimen and may not be representative of the property of the material from which the specimen is made. On the S–N curve, fatigue properties like fatigue strength, rn, and fatigue limit or

326 8 Fatigue Stress D of the variation of fatigue life with stress and other factors. Hence, the fatigue data should be expressed as a C three-dimensional surface to represent a relationship between stress (S), number of cycles to failure (N) and B probability of survival (P). The method of presenting this in A a two-dimensional plot is shown in Fig. 8.11. This figure shows a diagram of stress versus number of cycles to failure log N with schematic illustration of a distribution of fatigue life at constant stress, based on which the curves of constant Fig. 8.10 Schematically illustrating the mechanism in the develop- probability of survival are drawn on the same plot. A family ment of a fatigue limit. Curve A is for a pure metal. Curve B is due to of curves of this type is known as a P–S–N diagram. In this solid solution strengthening effect by addition of solute element to diagram, the solid line is the median S–N curve (50% sur- A. Curve C shows fatigue limit that arises due to balance between vival) and the dashed lines are S–N curves for p per cent fatigue damage and strengthening due to strain ageing from interstitials. survival. For example, at stress r1, 99% of the specimens Curve D shows raised fatigue limit due to enhanced strain ageing would be expected to survive at N1 cycles, 50% at N2 cycles and 1% at N3 cycles. Figure 8.11 indicates that the scatter in enhanced strain ageing is shown by the curve D, where the fatigue life decreases with increasing stress level. It is fatigue limit is elevated and the horizontal ‘knee’ starts at a believed that the period of crack initiation prior to its lower number of stress cycles compared to curve C. A propagation is much shorter at higher stress levels resulting well-defined fatigue limit exists in quenched and tempered in a smaller scatter. The scatter in the test data may originate steels, which generally do not show strain ageing in the from various sources. These include alignment of the test tension test. This fatigue limit results probably from local- machine, preparation of the specimen surface, variations in ized strain ageing at the root of the fatigue crack. testing environments and a number of metallurgical vari- ables. With respect to the testing machine, rotating bending 8.5 Statistical Nature of Fatigue machines produce the least amount of scatter since misalignment is less critical than axially loaded machines. Since S–N curves represent the fatigue behaviour of only laboratory specimens, the fatigue properties of a part in At a given stress level, the anticipated fatigue life for a service may vary considerably from the laboratory results. desired per cent of survival under completely reversed stress The existence of scatter in data is very common in fatigue cycle may be estimated by means of standard statistical tests. When identical several specimens are tested in fatigue procedure. For a given alternating stress-cyclic life data at the same stress level, their fatigue lives are generally not population, if N is the mean value of the range of fatigue the same, but may vary or scatter to a great extent, often as lives of the specimens tested at a given stress level, and s is much as one log cycle between the minimum and maximum the value of the associated standard deviation, then using values. Hence, the fatigue life for p per cent survival, Np, is standard statistical procedure it can be written considered for design purpose, where p refers to the proba- bility factor and Np represents the fatigue life up to which N ¼ Pi¼n Ni ð8:8Þ p per cent of the specimens tested at the same stress level is expected to survive; for example, N90 is the fatigue life up to i¼1 which the probability of survival of the specimens is 90% and the probability of failure is 10%. The fatigue life for n 50% survival, N50, is known as the median fatigue life. It is estimated by the middle value of the observed fatigue lives \"Pi¼n À À N Á2 #1 of the specimens tested at the same stress level when they are Ni 1 2 arranged in order of magnitude. If there is an even number of i¼1 the observed fatigue lives of the specimens, the average of s ¼ ð8:9Þ the two middle values is considered as the median. It is often nÀ convenient to deal with the median fatigue life in the study where Ni the value of fatigue life of one specimen (the ith specimen) tested at a given stress level; and n the number of specimens tested at the same stress level With the above statistical parameters, the confidence limits of the probability of survival may be determined. If the specimens are subjected to purely alternating stress cycles, the fatigue life anticipated with a desired confidence level of c% that at least p% of the specimens will survive at any given stress level may be given by (Forrest 1962):

8.5 Statistical Nature of Fatigue 327 Fig. 8.11 Schematic P–S–N Median S – N curve curves to represent fatigue data on a probability basis σ1 Stress P = 0.01 P = 0.10 P = 0.01 P = 0.90 P = 0.50 P = 0.99 Median fatigue limit P = 0.99 N1 N2 N3 Number of cycles to failure, N (log scale) Anticipated cyclic life ðc; pÞ ¼ N À qs ð8:10Þ fatigue lives, because the anticipated fatigue lives decrease with increasing values of q. where q ¼ f (confidence level, c%; probability of survival, p%; and the number of specimens used to determine N Similar to fatigue life, fatigue strength and fatigue limit and s). can also be defined by a distribution, and like fatigue life, the fatigue strength or fatigue limit is a statistical quantity. If Assuming normal distribution, values of q for S–N data fatigue strength and fatigue limit are determined from the are provided in Table 8.1 (ASTM STP 1963). The value of median S–N curve, they are, respectively, called the median q increases with any one of the followings: fatigue strength at N cycles and the median fatigue limit. The median fatigue strength at N cycles is the stress at which • An increase in the confidence level; 50% of the specimens would survive N cycles, and the • An increase in the probability of survival; median fatigue limit is the stress at which 50% of the • A decrease in n, i.e. the number of specimens tested at specimens would have an infinite fatigue life. It will often be more desirable to use the fatigue strength for p per cent the same stress level. survival at N cycles or the fatigue limit for p per cent sur- vival, which may be found by interpolation from the S–N With the aid of (8.10) and Table 8.1, the P–S–N diagram curve for p per cent survival. consisting of a family of curves representing the probability of survival is possible to develop, which may be used for To establish the necessary distribution of fatigue lives, decision-making in engineering design. Generally, a higher each group of fatigue specimens tested at each stress level confidence level of survival is selected for more important for construction of an S–N curve should consist of at least 10 components. This increases the value of q. In such cases, the specimens. In general, at least six to eight stress levels are design or operating stresses must be reduced, especially if required to determine the S–N curve including the fatigue few test data are available for determining the anticipated limit. Since the scatter in the test results increases with Table 8.1 Values of q for S–N data assuming normal distribution (ASTM STP 1963) c 0.75 0.90 0.95 0.75 0.90 0.95 0.75 0.90 0.95 0.75 0.90 0.95 p ¼ 99:9% n p ¼ 90% p ¼ 95% p ¼ 99% 5.062 4.273 5.556 6.612 3.981 3.858 4.629 5.203 6 1.860 2.494 3.006 2.336 3.091 3.707 3.243 4.242 3.520 3.661 4.215 4.607 3.295 3.561 4.009 4.319 10 1.671 2.065 2.355 2.103 2.568 2.911 2.927 3.532 3.158 3.497 3.882 4.143 15 1.577 1.866 2.068 1.991 2.329 2.566 2.776 3.212 20 1.528 1.765 1.926 1.933 2.208 2.396 2.697 3.052 25 1.496 1.702 1.838 1.895 2.132 2.292 2.647 2.952

328 8 Fatigue decreasing stress level, more specimens are required to be Crack Initiation tested at the low stress levels and fewer at the high stress levels to maintain an approximate equal degree of precision. It is well established that the formation of a fatigue crack can The total number of specimens required for determination of occur prior to consumption of 10% of the total fatigue life. the S–N curve varies from 60 to 80 or more. Fatigue cracks usually initiate at free external surfaces, commonly at geometrical notches or discontinuities, but Several attempts have been made to express S–N curves rarely at internal surfaces if the metal contains an interface, by mathematical equations. One of the most useful equations such as case–core interface created during surface-hardening formulated on the basis of statistical theory was suggested by treatments like carburizing and nitriding, or defects such as Weibull (1949), in which it was assumed that the S–N curve voids and cracked second-phase particles. is a hyperbola. This equation is: Gough (1933) has shown that a metal subjected to fatigue ðr À reÞmN ¼ k ðfor r [ reÞ ð8:11Þ deforms by slip on the same atomic planes and in the same crystallographic directions as in unidirectional loading. But where m and k are material constants, r and N are the slip in unidirectional deformation is usually widespread variables and re is the fatigue limit. If we take logarithms of throughout all the grains, whereas in fatigue, slip lines are both sides of (8.11), it gives a linear relation between exhibited in some grains and there is no evidence of slip in logðr À reÞ and log N, as indicated by the following form: other grains. In fatigue, the first few thousands of stress cycles generally form slip lines and a systematic build-up of fine slip m logðr À reÞ ¼ log k À log N ð8:12Þ lines, which correspond to the movements of the order of 1 nm instead of steps of 100–1000 nm as observed for static 8.6 Fatigue Crack Nucleation and Growth slip bands, results in cyclic slip bands. Additional slip bands are produced by successive stress cycle, but the number of Fatigue process starting from crack nucleation to fracture can slip bands produced is not directly proportional to the number be divided into the following stages: of stress cycles. In a number of metals, the increase in visible slip is found to reach a saturation limit in a short time. If the • Crack initiation, which occurs at the early stage of fati- deformed surface of specimen is electropolished by inter- gue. Crack initiates at heterogeneous nucleation sites rupting the fatigue test, some of these cyclic slip bands will which are pre-existing flaws or generated during the persist and remain visible while the rest will be removed by cyclic straining process. polishing. These persistent slip bands (PSBs) have been observed to form after the specimen has undergone only 5% • Stage I crack growth, also called slip-band crack growth, of the total fatigue life. The extent of plastic strain within the where the initial crack grows along slip planes, i.e. planes PSB can reach up to 100 times of that in the surrounding of high shear stress. material. PSBs are sources of fatigue cracks, because the application of small tensile strains opens them up into wide • Stage II crack growth, also known as crack growth on cracks. Once fatigue cracks are formed, they tend to grow planes of high tensile stress, where a well-defined crack initially along slip planes, but subsequently propagate in a propagates in a direction normal to the maximum applied direction normal to the maximum applied tensile stress. The tensile stress. propagation of fatigue crack is normally transcrystalline. • Ultimate ductile failure, which takes place when the The fine structure of a slip band as obtainable by an crack becomes sufficiently long so that the remaining electron microscope is shown schematically in Fig. 8.12, cross-section can no longer sustain the applied load. which illustrates the mechanism of formation of fatigue crack based on Wood’s concept (Wood 1955). Under similarly The fraction of the total fatigue life that will be shared by applied stress conditions, the slip in a slip band is analogous each of the above stages depends on the material and the test to the movement of playing cards in a pack. Figure 8.12a conditions. Generally, a larger fraction of total fatigue life is shows that slip caused by static deformation would produce involved with the Stage II crack growth in high-stress a contour similar to a staircase at the surface of a metal. In low-cycle fatigue than in low-stress high-cycle fatigue, while contrast, the back-and-forth movements of fine slip in a slip the largest part of long-life fatigue consists of Stage I crack band of fatigue deformation lead to the formation of ridges, growth. Stage I crack growth may not occur at all in case of called slip-band extrusions (see Fig. 8.12b), and grooves or high tensile stress, for example, in case of fatigue of sharply notches, called slip-band intrusions (see Fig. 8.12c), at the notched materials.

8.6 Fatigue Crack Nucleation and Growth 329 (a) (b) (c) Fig. 8.12 Mechanism of formation of fatigue crack at the surface of a formation of slip-band extrusion (ridge at surface). c Cyclic deforma- metal, based on Wood’s concept. a Slip in static deformation produces tion leads to the formation of surface groove or notch called slip-band a contour similar to a staircase. b Cyclic deformation leads to the intrusion surface. The notch, thus produced, will act as a stress raiser image of fatigue striations obtained by examining the fatigue with a notch root radius of atomic dimension and may ini- fracture surface with a scanning electron microscope. tiate a fatigue crack. The formation of slip-band extrusions Although the presence of striations indicates that the failure and intrusions seems to be unique to fatigue deformation. is due to fatigue, their absence does not mean that the fatigue Metallographically, it has been shown (Wood 1959) that the failure has not occurred. initiation of fatigue cracks occurs at extrusions and intru- sions. Hence, the above mechanism for a fatigue crack ini- Stage II fatigue crack propagation proceeds by an unin- tiation agrees well with the facts that fatigue cracks start at terrupted process of crack sharpening followed by plastic surfaces and that they have been found to initiate at blunting, as illustrated schematically in Fig. 8.14 (Laird slip-band extrusions and intrusions. 1967). Figure 8.14a shows a sharp-tip crack with charac- teristic striation spacing, x, at the start of stress cycle when Stage I Crack Growth stress is zero. As the tensile stress is applied and increased, the crack is opened up and the stress concentration at the After initiation, fatigue crack propagates initially along slip double notch of the crack tip causes plastic deformation at planes, i.e. parallel to the persistent slip bands at approxi- 45° to the plane of the crack (Fig. 8.14b). With increasing mately 45° to the applied principal tensile stress axis. The tensile stress (Fig. 8.14c), the crack grows in length by rate of Stage I crack growth is normally very low, on the plastic shearing preceded by its widening to the maximum order of nm per cycle. A practically featureless fracture extent. At the same time, plastic deformation blunts the surface is produced during Stage I. In a polycrystalline crack tip. On reversing the stress to compression metal, the Stage I crack may spread over a few grain diameters before the crack enters Stage II. Stage II Crack Growth When the Stage I crack attains a critical length that is suf- 6.98 mm from notch tip ficient for the stress field at the crack tip to become domi- nant, the crack propagation changes from Stage I to Stage II. Fig. 8.13 Fatigue striations in Fe-3 wt% Si alloy (Courtesy Prof. The critical size of Stage I crack is determined by mechan- R. Mitra, IIT Kharagpur). Crack propagation direction is shown with an ical properties of material and the applied stress level and arrow state. A Stage II crack propagates in a direction normal to the applied principal tensile stress direction, and the crack-propagation rate is related closely to the range of stress, rr, in each cycle; the greater the rr, the greater is the rate of crack extension. The crack-propagation rate in Stage II is on the order of microns per cycle, i.e. much higher than Stage I crack growth rate of nm per cycle. Further in contrast to Stage I crack growth, crack propagation in Stage II often produces a pattern of ripples or fatigue stri- ations. The spacing between these striations represents the crack growth per stress/strain cycle. These striations marks are not visible to the naked eye, and Fig. 8.13 shows an

330 8 Fatigue (a) x (d) (b) (e) (c) (f) Fig. 8.14 Schematic mechanism of growth of Stage II fatigue crack crack-tip by plastic deformation. d Removal of blunting and by a process of crack sharpening followed by plastic blunting. re-sharpening of the crack tip on reversal of stress. e and f Repetition a Sharp-tip fatigue crack with characteristic striation spacing, x. b Open- of the process (Laird 1967) ing up of the crack with increasing tensile stress. c Blunting of the (Fig. 8.14d), the slip direction at the crack tip is reversed. stress concentration, or those which are subjected to The compressive stress forces the crack to close, removes low-cycle fatigue. blunting and resharpens the crack tip. Note that the stress cycle produces an increase in the crack length, which is A precracked specimen of the type used for plane-strain equal to the characteristic distance x. The above process is fracture toughness testing (see Chap. 9) is sufficient for the then repeated as shown in Fig. 8.14e, f. It is clear from this measurement of Stage II crack growth rate, which is com- model that for the advancement of crack, stress reversal is monly performed in laboratory. A fixed stress (or sometimes, necessary because it removes the blunting and resharpens the strain) amplitude at a specified stress ratio or mean stress is crack tip by altering its shape. It is clear, too, that the usually applied to the specimen, and the crack length incre- characteristic striation spacing, x, will increase with the ment is measured in a variety of ways, such as directly with range of alternating stress. an optical microscope or by electrical resistivity measure- ment across the cracked specimen, accompanied by a suitable However, ultimately failure occurs in a ductile manner calibration procedure. The measured crack length is recorded when the fatigue crack becomes sufficiently long so that the and plotted as a function of the number of cycles. Figure 8.15 remaining cross-section is no longer capable of carrying the shows such a plot of crack length, a, versus the number of applied load. fatigue cycles, N, where the crack grows from an initial size, ai, to a critical size, ac, corresponding to the number of 8.6.1 Fatigue Crack Growth Rate fatigue cycles to failure, Nf . The crack growth rate, da=dN, which is determined from the slope of the a À N curve, is If Stage II crack growth rates are measured, then the mea- found to increase continuously with the number of cycles, as sured values can be used for engineering design. For seen from the figure. In other words, we can say that the example, if we know the Stage II crack growth rate and the crack growth rate is initially slow, but increases with fracture toughness (see Chap. 9) of the material, the number increasing crack length. It is to be noted that for a fixed stress of Stage II cycles prior to catastrophic fatigue fracture, i.e. range, the crack growth rate increases with increase in the the approximate service life under specific loading condi- stress ratio, R. tions and service environment, can be estimated. Thus for materials, for which the Stage II crack growth occupies a The linear elastic fracture mechanics (LEFM) approach major portion of their fatigue lives, the number of fatigue involving the elastic stress intensity factor, K, described in cycles endured by them prior to their failures, i.e. their Chap. 9, can be applied to fatigue crack growth even in service lives, can be approximated. For example, such low-strength, high-ductility material, because fatigue crack materials are those which contain pre-existing surface cracks growth needs very low values of K and the sizes of plastic or flaws, where the necessity of nucleating a fatigue crack is zone at the crack tip are quite small. It is possible to estimate eliminated and the applied tensile stress is raised due to the safe lifetime in fatigue from the correlation existing between the elastic stress intensity factor, K, and the crack growth rate or the amount of crack extension per stress

8.6 Fatigue Crack Nucleation and Growth Region I Region II 331 Region III Fatigue life ac ac Crack length, aai da Crack growth rate da / dN (log scale)dN No propagation of fatigue cracks Stable crack growth Rapid or unstable crack growth ai p 0 Nf Number of fatigue cycles, N Fig. 8.15 Schematic plot of crack length, a, versus the number of da = A (∆K)p dN fatigue cycles, N, showing the increase of crack growth rate, da=dN, for linear portion with increasing N from 0 to the number of cycles to failure, Nf , or with increasing a from an initial crack length, ai, to a critical crack size, ac cycle, da=dN. Figure 8.16 shows an idealized curve of ∆Kth da=dN versus DK on log-log scale, where DK is the range of Stress intensity factor range ∆K (log scale) stress intensity factor. DK can be defined and expressed in terms of the stress range, Dr, with the help of (9.48) as Fig. 8.16 Schematic curve of fatigue crack propagation in follows: non-aggressive surrounding pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi DK ¼ Kmax À Kmin ¼ Yrmax pa À Yrmin pa ¼ Yrr pa ð8:13Þ where the parameter Y depends on the types of loading and temperature, frequency of cycling and environment. Usually the geometry of crack and specimen configuration (see Chap. 9). Since the crack is closed during compression for metals, p is in the range of 2–6 depending on the mag- loading, so if rmin is a compressive stress, the stress intensity factor in compression, Kmin, will be taken as zero. nitude of DK (Grosskreutz 1971) and a compromise value of The curve in Fig. 8.16 having a sigmoidal shape can be p over the entire range of da=dN, suggested by Paris, is divided distinctly into three regions. In Region I, crack p ¼ 4. The material parameters A and p must be determined growth rates are very low of the order of 0.25 nm per cycle experimentally from crack growth rate measurements. The or less corresponding to low DK values. Region I is bounded by a threshold value DKth, which is at the lower end of DK value of A can be obtained by extenpdiffinffiffigffi the linear part of the range, where the crack growth rate approaches zero. Below curve in Fig. 8.16 to DK ¼ 1 MPa m, and p can be found DKth, there is no observable fatigue crack growth (Ritchie 1979). DKth typically lies between 5 and 10% of critical from the slope of the curve in Fig. 8.16. stress intensities. When DK is very high, Kmax approaches the fracture In Region II, the crack growth rate is stable and there is a linear relationship between logðda=dNÞ and logðDKÞ, which toughness of the material and we get a region of accelerated can be represented by the following power law equation: crack growth, which is designated as Region III in Fig. 8.16. Substitution of DK from (8.13) into the Paris law given by (8.14), applicable during Stage II or linear crack growth region, results in: da ¼ À rr ppffiffiffiaffiffiÁp¼ AY p ðrr Þp ðpaÞp=2 ð8:15Þ dN AY da ¼ AðDKÞp ð8:14Þ When fracture toughness of a material is known under dN relevant loading conditions, the fatigue crack growth life of Equation (8.14) is often referred to as ‘Paris law’ (Paris and Erdogan 1963), in which A and p are constant param- the material can be determined by rearranging and inte- eters and are related to material variables, stress ratio, grating (8.15) between the limits of initial crack size ai and critical crack size, ac, at which fracture occurs. The value of

332 8 Fatigue critical size, ac, can be obtained by substituting where a ¼ ac; ra ¼ rmax, which is the applied maximum tensile stress, and K ¼ Kc, the fracture toughness of the material, in Kc the fracture toughness of the material that depends on specimen thickness; (9.48), R stress ratio ¼ rmin=rmax ¼ Kmin=Kmax  Kc 2 1 Note that even if a structural member contains a crack ac ¼ ð8:16Þ which can be detected using several non-destructive tech- p Yrmax niques, its use can be continued in service, provided that the crack is periodically inspected. The damage-tolerance design If Nf is the fatigue life of material, i.e. the number of approach is based on this philosophy. fatigue cycles required to cause fatigue fracture, then we get from (8.15), ZNf dN ¼ Zac da 8.7 Effect of Mean Stress AY p ðrr Þp ðpaÞp=2 0 ai Our discussions on fatigue have been so far limited to the conditions of purely alternating stress cycles, where the ) Nf ¼ 1 Zac da ð8:17Þ mean stress rm ¼ 0. However, the stress condition in many AY pðrrÞppp=2 ap=2 cases in practice often comprises an alternating stress and a ai superimposed tensile or compressive mean stress, which will be now discussed. where it has been assumed that Y is independent of crack An S–N diagram can be constructed in several ways for a length, which may be true for a few geometries. For other situation where the mean stress rm 6¼ 0. The two most common methods of presenting S–N data for a given mate- geometries, a suitable average value of Y, say Y, may be rial are: used to replace Y in (8.17). It has also been assumed that rr, i.e. ðrmax À rminÞ; is constant and taken outside the integral. (1) To plot the maximum stress, rmax, versus log N (log of If p 6¼ 2, integration of (8.17) yields cycles to failure) for a constant value of the stress ratio R ¼ rmin=rmax. This gives a single S–N curve, each Nf ¼ ai1Àðp=2Þ À ac1Àðp=2Þ ð8:18Þ point of which must have the same value of R. To obtain ½ðp=2Þ À 1ŠAYpðrrÞppp=2 different points on the same S–N curve, a series of stress cycles with different maximum stresses is applied, and Equation (8.18) is the appropriate solution of the Paris for each case of change in the maximum stress, the equation when p 6¼ 2, and Y ¼6 f ðaÞ. Considering the more minimum stress is adjusted so that it becomes a constant general case where Y ¼ f ðaÞ, (8.17) must be written as: fraction of the maximum stress. In the similar fashion, several such S–N curves for the same material at dif- Nf ¼ 1 Zac da ð8:19Þ ferent values of R can be determined, which are AðrrÞppp=2 Y pap=2 schematically plotted in Fig. 8.17a. The S–N curve for ai completely reversed stress cycle is the curve plotted at R ¼ À1. Note that as R becomes more positive, which This integration is generally solved using numerical corresponds to the increase in the mean stress, the S–N techniques. As mentioned earlier, increasing the stress ratio, curve for the same material shows greater allowable R, increases the crack growth rate in all segments of the maximum stress at a specified number of cycles or sigmoidal curve in Fig. 8.16, but the effect of increasing R is greater fatigue limit or shows greater fatigue life for a more pronounced in Regions I and III, i.e. at low and high given rmax level. values of DK, than in Region II. Since the Parris equation does not include the influence of R, so the expression (2) To plot the alternating stress, ra against log N (log of developed by Foreman and his associates (Forman et al. cycles to failure) at a constant value of the mean stress 1967), in which the sensitivity of the Stage II crack growth rm, which gives a single S–N curve having the same rate to the stress ratio has been considered, is given below: value of mean stress at each point. Several such S–N curves for the same material at different values of rm da ¼ ð1 AðDKÞp DK ð8:20Þ have been schematically plotted in Fig. 8.17b. Note that dN À RÞKc À

8.7 Effect of Mean Stress 333 Fig. 8.17 Two most common (a) (b) methods of presenting S–N data for a given material, when the σm1 > σm > σm3> σm4 mean stress rm ¼6 0. a Schematic S–N curves for different values of σm4 2 stress ratio, R. b Schematic S–N σm3 curves for different values of Maximum stress, σmax R4 ( 0 < R4< + 1) Alternating stress, σa σm2 mean stress, rm, showing the σm1 effect of stress amplitude, ra, and R3 = 0 mean stress, rm, on fatigue life, N R2 ( –1< R2< 0) R1 = –1 105 106 107 105 106 107 Number of cycles to failure, N (log scale) Number of cycles to failure, N (log scale) Fig. 8.18 Haig-Soderberg Invalid region of σe or σn diagram (Soderberg 1930) Gerber parabola showing combined effect of alternating and mean stress Gerber parabola components on fatigue by means of Gerber parabola, Goodman and Alternating Goodman line Soderberg lines stress, σa Experimental findings Soderberg line 0 Yield stress, S0 Su (UTS) Mean stress, σm Compressive region Tensile region as the algebraic value of rm increases, the allowable diagram in Fig. 8.18 shows Goodman line, Gerber parabola alternating stress at a specified number of cycles or the and Soderberg line, which are discussed below. fatigue limit decreases or the fatigue life decreases for a given ra level. Goodman suggested a straight line starting from the ordinate at the point of the fatigue strength at a specified number of cycles, Other ways of presenting S–N data are to plot rmax say N cycles, or fatigue limit for the condition of mean stress, against log N at constant rm and rmax against log N at con- rm ¼ 0, and ending at the point of the UTS, Su, on the abscissa. stant minimum stress, rmin. This line is referred to as Goodman line. If a point corre- sponding to the value of applied mean stress and alternating To account for the effect of mean stress on fatigue life, stress, i.e. ðrm; raÞ; lies within the safe region bounded by the empirical relations showing the variation of mean stress with Goodman line and the two axes, then fatigue failure would not alternating stress have been developed for design purposes, be expected to occur according to Goodman law. Let, the graphical representation of which is displayed schemat- ically in Fig. 8.18. This figure is often called the ra fatigue strength at a specified number of cycles, say Haig-Soderberg diagram (Soderberg 1930). In this diagram, N cycles, or fatigue limit for infinite life, in terms of the alternating stress, ra, is plotted against the mean stress, stress amplitude, where rm 6¼ 0; rm. This figure is obtained for alternating bending or axial stresses with tensile or compressive mean stress or alter- rm mean stress; nating torsion with tensile mean stress. On the abscissa, i.e. rn fatigue strength at a specified number of cycles, say on the rm-axis, the ultimate tensile strength, Su, of the material is the boundary of the plot, i.e. considered to be the N cycles, in terms of stress amplitude, where rm ¼ 0; or end of the curve, because the material is expected to break at re fatigue limit for infinite life, in terms of stress rm ¼ Su, when there is no alternating stress, i.e. ra ¼ 0. The amplitude, where rm ¼ 0; and Su ultimate tensile strength, obtained from uniaxial ten- sion test.

334 8 Fatigue Now, the equation of the Goodman line joining the ) Gerber relation: ra ¼ rnðor; \" À rm2# ð8:23Þ coordinate points of ð0; rnÞ or ð0; reÞ and ðSu; 0Þ is called reÞ 1 Su Goodman relation, which is given by ra À rnðor; reÞ rm À 0 To define completely both Goodman and Gerber relations, rnðor; reÞ À 0 0 À Su ¼ ; or; two mechanical properties of material are required—the reÞ ra À rnðor; reÞ ¼ Àrnðor; rm ; ultimate tensile strength, Su, and the fatigue strength, rn, at a Su specified number of cycles or fatigue limit, re, for a com- pletely reversed stress cycle. Since most of the experimental  rm data lie between the Goodman line and Gerber parabola, the ) Goodman relation : ra ¼ rnðor; reÞ 1 À Su ð8:21Þ Goodman relation represents a more conservative design Another empirical relation proposed by Gerber, known as criterion for mean stress effects. Although the Gerber relation Gerber relation, represents graphically a parabolic curve whose vertex is at the coordinate point of ð0; rnÞ or ð0; reÞ on is economical, the Goodman relation is generally preferred in the ordinate and passing through the coordinate point of ðSu; 0Þ on the abscissa. If a point corresponding to the value engineering design because fatigue test data shows a lot of of applied mean stress and alternating stress, i.e. ðrm; raÞ; lies within the safe region bounded by the Gerber parabola and scatter, as discussed earlier, and the test data for notched the two axes, then fatigue failure would not occur according to Gerber law. Experimental data for ductile metals generally specimens lie closer to the Goodman line. lie closer to the Gerber parabola. One flaw of the Gerber parabola is that it is not valid on the left side of the ordinate, So far, we are discussing the prevention of fatigue failure. i.e. the ra-axis, where the mean stress is compressive (neg- ative) because the drop in parabola reduces the allowable Sometimes, it is required to prevent yielding of the material total value of alternating stress and mean stress with increase in the compressive mean stress, but it has been found in in the cyclic loading condition where the mean stress is not practice that a compressive mean stress has a beneficial effect on fatigue life. This beneficial effect can be easily visualized zero, and in such cases, the point corresponding to the value by extending the Goodman line to the left side of the ordinate, which shows an increase in the allowable total value of of applied mean stress and alternating stress, i.e. ðrm; raÞ; alternating stress and mean stress with increasing the com- must lie within the safe region bounded by the Soderberg pressive mean stress. The equation of Gerber parabola gives the Gerber relation, which is derived below: line and the two axes. Since the Soderberg line is based on the yield strength, S0, rather than the ultimate tensile strength, Su, we get the Soderberg relation by substituting S0 for Su in (8.21), derived from Goodman line, as follows:  1 À rm ) Soderberg relation: ra ¼ rnðor; reÞ S0 ð8:24Þ ðrm À 0Þ2¼ À4a½ra À rnðor; reފ ð8:22Þ where where a represents the distance between the vertex and the S0 yield strength. focus of the parabola and negative sign is given to indicate that it is a down facing parabola. Since the parabola passes Soderberg relation is completely defined when two through the point ðSu; 0Þ, so by substituting it for ðrm; raÞ mechanical properties of material, S0 and rn ðor; reÞ, are into (8.22), a can be evaluated as follows: known. Soderberg line or relation is also termed as ‘peak stress’ criterion. Su2 ¼ À4a½0 À rnðor; reފ; ) 4a ¼ Su2 reÞ ; rnðor; An alternative graphical representation showing the mean stress effects on fatigue life is the Goodman diagram Substitution for 4a into (8.22) yields (Goodman 1899), as shown in Fig. 8.19, which is often used in engineering design. The maximum stress range, rm2 ¼ À S2u reÞ ½ra À rnðor; reފ; or; rmax À rmin, which can be withstood without failure is rnðor; plotted against the mean stress, rm, in the Goodman dia- gram. Hence, the Goodman diagram is a plot showing the ra À rnðor; reÞ ¼ Àrnðor; reÞ r2m ; variation of the maximum allowable stress range, Su2 rmax À rmin, with the mean stress, rm. Figure 8.19 shows that the maximum allowable stress range is different for each value of the mean stress and decreases with increasing the tensile mean stress. Ultimately, the maximum allowable stress range reduces to zero when the mean stress reaches the value of the ultimate tensile strength, Su, of the material. On the other hand, the available data (Ransom 1954a) for