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

238 6 Impact Loading transition temperature, as described subsequently in 6.4.1 Transition Temperature Curves Sect. 6.4.2. It may be noted that the range of brittle-to-ductile transition temperature (BDTT) is usu- For the purposes of fracture analysis, materials can be clas- ally from 0.1 Tm to 0.2 Tm in metals and from about sified depending on the values of the ratio of yield strength 0.5 Tm to 0.7 Tm in ceramics, where Tm is the absolute S0 to elastic modulus E, i.e. S0=E as follows: melting temperature of the material, expressed in K. Lower BDTT of metals makes most of them ductile at • Low-yield-strength material whose S0=E\\1=300. room temperature, while ceramics are mostly brittle at • Medium-yield-strength material whose 1=300 S0=E room temperature because of their higher BDTT. 1=150. To select a material from the angle of tendency for brittle • High-yield-strength material whose S0=E [ 1=150. fracture or notch toughness, the material to be preferred must have the lowest transition temperature. To illustrate this, the The notch toughness at a given temperature depends on fracture energy transition temperature curves for two kinds the strength level and/or crystal structure of a material. of steel, say steel ‘A’ and steel ‘B’, obtained from the Charpy Depending on the level of the notch toughness and its V-notch impact test are shown in Fig. 6.11. In this figure, the variation with temperature, the transition temperature beha- notch toughness at room temperature is given by the upper viour of a wide range of materials can be divided into three shelf energy, which is higher for steel ‘A’ than steel ‘B’, but following categories, as shown in Fig. 6.12. the transition temperature of steel ‘B’ is observed to be lower than that of steel ‘A’. Now, if steel ‘A’ is subjected to impact (1) Curve (a) in Fig. 6.12, which shows such high notch loading in service below room temperature, say at subzero toughness at all temperatures that brittle fracture does temperature, it will behave in a much more brittle fashion not take place unless there is some special reactive than steel ‘B’, and so the steel ‘B’ is preferred to the steel environment. In this case, there is a little increase in ‘A’. Thus, one must not select a material depending on the toughness with temperature. This type of curve is higher upper or lower shelf energy or the higher impact observed by low- and medium-strength face-centred- strength at only one temperature, say room temperature, cubic (FCC) metals, such as aluminium, copper, nickel, which may be misleading; rather, a lower transition tem- austenitic steel alloys, and most hexagonal closed-pack perature is the most important factor for selection of a (HCP) metals. These metals are always ductile since material. Usually, the lower the transition temperature, the there is no transition from ductile-to-brittle behaviour of greater is the notch toughness of the material in the transition materials. temperature range. (a) (b) Steel A Steel B Energy absorbed Energy absorbed (c) Room Temperature temperature Fig. 6.12 Schematically showing the effect of temperature on notch 0°C toughness for a wide range of materials. Curve (a) is exhibited by low- Temperature and medium-strength face-centred-cubic (FCC) metals and most hexagonal closed-pack (HCP) metals. Curve (b) is exhibited by low- Fig. 6.11 Schematic curves of fracture energy transition temperature and medium-strength body-centred-cubic (BCC) metals, few HCP for two steels, showing fallacy of depending on impact toughness metals and ceramics materials. Curve (c) is exhibited by high-strength values at room temperature BCC, FCC and HCP metals

6.4 Impact Properties 239 (2) Curve (b) in Fig. 6.12, which shows strong dependency capacity of the material can be obtained from the method of notch toughness on temperature, having a low used to calculate its standard strength, ignoring its fracture toughness at a low temperature and a high toughness at a properties or the stress concentration effects of flaws or high temperature. Hence, there is a transition from brittle cracks. When it is required to determine the operating tem- behaviour at low temperatures to ductile behaviour at perature by means of transition temperature curves, ascertain high temperatures. This type of transition from a temperature which must be above the transition tempera- ductile-to-brittle behaviour is observed by low- and ture of the particular preselected material so that at or above medium-strength body-centred-cubic (BCC) metals such this operating temperature, the material subjected to elastic as ferritic alloys and few HCP metals like beryllium, stress levels does not undergo brittle fracture. Obviously, the zinc and ceramic materials, because their yield strengths lower the transition temperature of the material, the greater are far more sensitive to changes in temperature and will be its notch toughness. strain rate than the yield strengths of low- and medium-strength FCC metals and most HCP metals. 6.4.2 Various Criteria of Transition Temperature With decrease in temperature and/or increase in strain rate, the yield strengths of BCC and ceramic materials There are various criteria to define the transition tempera- increase sharply, while the yield strengths of FCC metals ture, as seen from the shape of the typical fracture energy increase very slightly, showing meagre dependency on versus temperature curve (Fig. 6.8). Using a fracture energy temperature and strain rate. As pointed out in Sect. 1.10 transition temperature curve or a fracture appearance tran- of Chap. 1, this increased sensitivity in BCC and cera- sition temperature curve, obtained from the Charpy V-notch mic materials can be related to the temperature sensi- impact test, the various definitions of transition temperature tivity of their Peierls–Nabarro stresses and thereby that are illustrated in Fig. 6.13 and given below: of their yield strengths (see Table 1.5). These tempera- ture sensitivities are much larger in BCC and ceramic (1) Fracture transition plastic or FTP—It is the minimum materials than in FCC and ideal HCP metals. temperature above which the fracture is 100% fibrous or 0% cleavage, as seen from the fracture appearance (3) Curve (c) in Fig. 6.12, which shows such low notch curve, and this transition temperature corresponds to the toughness at all temperatures that brittle fracture can initiation of the upper shelf in the fracture energy curve. occur at all temperatures and strain rates under nominal FTP has been represented by the temperature T1 in stresses in the elastic range when flaws are present. Fig. 6.13. Above the FTP, the chance of occurring A very slight increase in toughness with temperature is brittle fracture is negligible, since the FTP is the tem- observed in this case. This type of curve is exhibited by perature at which there is a transition from totally ductile low-plasticity, high-strength materials of different crystal to substantially brittle fracture. FTP is the most conser- structures, such as high-strength steel (BCC), vative criterion for transition temperature and impracti- high-strength aluminium alloys (FCC) and high-strength cal in many applications. titanium alloys (HCP). Fracture occurs by brittle cleav- age at low temperatures and by low energy rupture at (2) Fracture appearance transition temperature or FATT— high temperatures. Hence, there is no transition from It corresponds to 50% cleavage and 50% shear in fracture brittle-to-ductile behaviour. For this kind of appearance curve, denoted by the temperature T2 in high-strength, low-plasticity materials, fracture Fig. 6.13. The average of upper and lower shelf energy mechanics analysis is needed, which is appropriate for values in energy curve corresponds to the transition quantitative prediction. temperature T3 in Fig. 6.13, which is slightly lower than the FATT represented by temperature T2. There is a high The main engineering application of the notched-bar probability that brittle cleavage fracture will not take place impact test is to obtain the transition temperature curves. The at or above FATT if the stress applied to the material does design philosophy using the transition temperature curves is not exceed about one-half of the yield strength (Hodjson either to select a material for a particular service temperature and Boyd 1958). This transition temperature criterion is or to determine an operating temperature for a particular less conservative than that based on FTP. preselected material. While selecting a material, it must possess sufficiently high notch toughness so that it can resist (3) Ductility transition temperature or DTT—It is based on brittle fracture when subjected to acute operating conditions. an arbitrary low value of energy absorbed, as defined by The behaviour of this material would be like a flaw-free the transition temperature T4 in the fracture energy curve material, which means a crack of any size cannot propagate of Fig. 6.13. DTT is a very common criterion to define as an unstable fracture. In such case, the load-bearing the transition temperature. Extensive tests on ship steel

240 NDT FTP 6 Impact Loading 100 Fig. 6.13 Various definitions of Fracture appearance transition temperature obtained transition temperature from a fracture energy transition temperature curve or a fracture curve appearance transition temperature curve Energy absorbed, Cv % cleavage fracture Fracture energy transition T1 = FTP temperature curve T2 = FATT T3 ∼ FATT 50 T4= DTT, (for mild steel) T5 = NDT 20 J T5 T4 T3 T2 T1 0 Temperature plates during Second World War established that brittle plastic deformation. It is a well-defined transition tem- cleavage fracture would never occur at temperatures perature criterion. greater than the transition temperature corresponding to 20-J energy absorption in V-notch Charpy test. The 6.5 Metallurgical Factors Affecting Impact energy required to fracture a V-notch Charpy specimen Properties is usually designated by CV . So, a 20 J (15 ft-lb) CV transition temperature has now become an accepted Variation in the chemical composition or microstructure of criterion for low-carbon mild steels or low-strength ship mild steel can change the transition temperature by about steels. It has been seen (McNicol 1965) that the tem- 50 °C. The transition temperature changes to the largest perature at which lateral expansion on the compression extent by varying the amount of carbon and manganese side of the notched bar is 0.38 mm (0.015 inch) rea- (Rinebolt and Harris 1951; Tetelman and McEvily 1967). sonably agrees well with the transition temperature The 20-J V-notch Charpy transition temperature (DTT) of defined by 20-J energy criterion. So, DTT of mild steels steel increases by about 14 °C for every 0.1 wt% increase in or low-strength ship steels also corresponds to a carbon content and decreases by about 5 °C for every 0.1 wt% 0.38-mm lateral expansion on the compression side of increase in manganese content. When carbon content is the notched bar, which can be obtained from ductility raised, the upper shelf energy, i.e. the maximum notch transition temperature curve, as shown in Fig. 6.10. toughness, decreases and the shape of the energy transition However, it must be noted that the energy level of 20 J temperature curve changes with a decrease in slope, as shown CV for DTT criterion is not constant, rather varies with in Fig. 6.14. If the ratio of Mn:C is raised to a higher level, it material. Gross (1970) has found that for higher strength may be possible to have a maximum decrease of about 50 °C steels with strengths in the range of 415–965 MPa, the in the transition temperature. But there are practical limita- specific energy level on which this transition tempera- tions to increases the ratio of Mn:C beyond 7:1, because a ture criterion is based would increase with increasing minimum amount of about 0.2 wt% carbon content is strength. essential to maintain the required tensile properties, while (4) Nil ductility temperature or NDT—It is the maximum manganese contents more than about 1.4 wt% may cause the temperature up to which 100% brittle cleavage fracture retention of austenite and problem associated with it. To is obtained, as seen from the fracture appearance curve. obtain satisfactory notch toughness, the ratio of Mn:C must In the fracture energy curve, NDT corresponds to the be at least 3:1. extreme point of the lower shelf energy beyond which the fracture energy begins to rise rapidly. NDT has been The effect of phosphorus in raising the transition tem- represented by the temperature T5 in Fig. 6.13. The perature is stronger than that of carbon. The 20-J CV transi- chance of occurring ductile fracture is negligible below tion temperature increases by about 7 °C for every 0.01 wt% the NDT, since the NDT is the temperature at which the increase in phosphorus content. Since nitrogen interacts initiation of fracture occurs essentially without any prior

6.5 Metallurgical Factors Affecting Impact Properties 241 Decreasing weight% temperature lower than the rimming steel. The killed steel is of carbon in steel fully deoxidized with aluminium or silicon plus aluminium, and the 20-J transition temperature for this steel is observed Fracture energy to be around −60 °C, which is the lowest transition tem- perature among the above three varieties. In addition to the 0 0 200 beneficial effect of complete deoxidation on the transition – 200°C Temperature, °C temperature in killed steel, aluminium added for deoxidation also combines with nitrogen to form insoluble aluminium Fig. 6.14 Schematically showing the effect of carbon content on the nitride particles and thus contributes to the lowering of the fracture energy transition temperature curves for steel (Rinebolt and transition temperature, otherwise nitrogen dissolved in steel Harris 1951) would increase the transition temperature. with other elements, its role in changing the transition The transition temperature is strongly affected by grain temperature is difficult to judge. However, it is generally size of ferrite or austenite in steel. It is known that decrease considered to be harmful to notch toughness. Aluminium in grain diameter means increase in ASTM grain-size index appears to be beneficial to notch toughness because it com- number and vice versa. For mild steel, the transition tem- bines with nitrogen to form insoluble aluminium nitride and perature decreases by about 16 °C for every increase of one thus destroys the detrimental effect of nitrogen. Nickel is ASTM index number in the ferrite grain size. It has been considered to be particularly powerful in decreasing the observed (Owen et al. 1957) that the 14-J V-notch Charpy ductility transition temperature. The notch toughness gener- transition temperature decreases from about 20 to −50 °C ally increases with the amount of nickel up to 2 wt%. The when ASTM number of ferrite grain size increases from 5 to effect of molybdenum in raising the transition temperature is 10. The microstructures of highly alloyed heat-treated steels almost the same as that of carbon, while the transition tem- consist of austenite phase at room temperature. If this aus- perature is raised little by addition of chromium. Silicon tenitic grain size is decreased, the transition temperature is contents more than about 0.25 wt% seem to raise the tran- found to decrease in a similar manner. It is to be noted that sition temperature. the finer the grain size of a material, the higher is the fracture energy, i.e. the toughness. For occurrence of transcrystalline Notch toughness and transition temperature are greatly fracture, the fracture path has to pass through more affected by oxygen content and deoxidation practice. It was grain-boundary areas per unit volume in a fine-grained observed for high-purity iron (Rees et al. 1952) that oxygen material than in a coarse-grained material, and since the content greater than 0.003 wt% exhibited intergranular grain boundaries are higher energy sites than the grain fracture and corresponding low toughness. With increasing bodies, a fine-grained material will require higher energy for oxygen content from 0.001 wt% to the high value of fracture than a coarse-grained material. Normalizing treat- 0.057 wt%, the transition temperature was found to increase ment of steel produces finer grains than the annealing from −15 to 340 °C. In connection with the deoxidation treatment. So the transition temperature of normalized steel practice carried out prior to solidification of steel in ingot will be lower than that of annealed steel and normalized steel moulds, there are mainly three different types of steel ingots. will be tougher than annealed steel. The use of the lowest Depending upon the extent of deoxidation, they are the possible finishing temperature for hot working operation like killed, the semikilled and the rimming steels. No deoxidation hot rolling to obtain refined grains has also beneficial effect or small amount of deoxidation is carried out in the pro- on the transition temperature. Further, the transition tem- duction of rimming steel, and thus, it shows a higher tran- perature can decrease by about 50 °C, when the material is sition temperature, which is generally above room spray cooled from the temperature of hot working before it temperature. Semikilled steel is a balanced variety of steel, cools down (Bucher and Grozier 1965). For a given steel of which is partially deoxidized with silicon and has a transition same chemical composition and deoxidation practice, the transition temperature of a thick hot-rolled section will be considerably higher than that of a thin hot-rolled section because it is difficult to obtain small-sized grains and fine pearlite in a thick section. A new class of low-carbon ferritic steels, usually con- taining less than 0.1 wt% carbon, has been developed by the addition of small amounts, frequently less than 0.1 wt%, of niobium, titanium and/or vanadium, which are strong car- bide and nitride forming alloying elements, to take the advantage of high strength and toughness of very

242 6 Impact Loading fine-grained steels. These steels are referred to as high- The notch toughness of rolled or forged plates or bars strength low-alloy (HSLA) steels (Union Carbide Corpora- depends on the orientations of specimens cut from the tion 1977) and also described as low-carbon microalloyed products as well as on the orientations of notches in speci- steels because of the very small additions of the alloying mens. The effect of orientations of specimens ‘A’, ‘B’, and ‘C’ elements to steel having very low-carbon content (usually and that of notches in these specimens on the V-notch Charpy less than 0.1% carbon). When the steel is heated to about energy transition temperature curves are shown schemati- 1300 °C prior to the hot-rolling operation, these alloying cally in Fig. 6.15. Specimens ‘A’ and ‘B’ are oriented in the elements go into solution in austenite. Subsequently, when longitudinal direction of a rolled plate, while specimen ‘C’ is the temperature is decreased gradually during hot rolling, oriented in the transverse direction. The notch is parallel to strain-induced very fine precipitates of carbides, nitrides the plate surface in specimen ‘B’, whereas the notch lies and/or carbonitrides of these alloying elements like NbC perpendicular to the plate surface in both specimens ‘A’ and and/or Nb(CN) form from austenite because of their reduced ‘C’. This latter orientation of notch is generally preferred, solid solubility at low austenitizing temperature. These fine because it provides less resistance to the propagation of crack precipitates effectively pin down the migrating grain than the former orientation of notch and thus results in lower boundaries, which limit the austenite grain growth and even value of toughness, which is conservative. In specimen ‘B’, recrystallization during hot rolling at low finishing temper- the orientation of notch would cause the crack to propagate in atures. The amount of retardation depends on the rolling the thickness direction of the plate that would experience temperature, amount of deformation and alloy concentration. more resistance, because the plate is compressed in the This yields a fine-grained austenite in recrystallized condi- thickness direction under rolling. So, the specimen ‘B’ tion or in the extreme case, in highly deformed and elon- requires higher energy to fracture than the specimens ‘A’ and gated condition. On cooling below the upper critical and ‘C’. In the longitudinal specimen ‘A’ and transverse specimen through the lower critical temperature, ferrite forms on the ‘C’, cracks would propagate, respectively, transverse to and closely spaced grain boundaries of fine-grained austenite. As parallel to the rolling direction. The crack propagation is a result, very fine ferrite grain sizes of about 2–3 lm is easier in the longitudinal direction than in the transverse produced in the steel without a heat treatment step, which direction of rolling, and thus for fracture, the energy absorbed adds to the cost. For the same alloy concentration, in refining for the specimen ‘A’ becomes higher than that for the speci- ferrite grain size the effectiveness of Nb > that of Ti > that men ‘C’. The values of notch toughness at room temperature of V. The grain refinement and dispersion strengthening caused by the precipitate particles raise yield strengths of BA C HSLA steels that range from 350 to 550 MPa, while mild steels have yield strengths of about 210 MPa. At the same Rolling direction Longitudinal (B) time, very fine ferrite grain sizes lower the fracture appear- ance transition temperature (FATT) below −30 °C and Energy absorbed Longitudinal (A) improve the impact strength of HSLA steels. Transverse (C) When steels are cold worked, the transition temperature Room increases, but strain ageing of low-carbon steels, in which Temperature steels after cold working are heated for several hours at a 0 relatively low ageing temperature like 130 C ð%400 KÞ, 0°C exhibits a greater increase in the transition temperature, Temperature usually about 25–30 °C. In the blue brittleness region, where the cold-worked steel is heated in the temperature range of Fig. 6.15 Schematically showing the effects of orientations of spec- around 230–350 °C (503–623 K), a decreased notch imens and notches in specimens on the V-notch Charpy energy toughness is observed due to the maximum rate of strain transition temperature curves ageing. Another ageing phenomenon associated with low-carbon steels is quench ageing, a type of true precipi- tation hardening, for which plastic deformation is not nec- essary. Quench ageing that occurs on quenching low-carbon steel from the temperature of maximum solubility of carbon and nitrogen in ferrite, i.e. from around 700 °C, results in a loss of impact properties, but this loss is less than that caused by strain ageing.

6.5 Metallurgical Factors Affecting Impact Properties 243 that correspond to high levels of energy absorption differ 6.5.1 Embrittlement During Tempering quite largely for different orientations of specimens, but these differences in the values of notch toughness at low energy Steels quenched to form martensite are susceptible to two absorption levels become much less, as shown in Fig. 6.15. kinds of embrittlement during their tempering operations. Since ductility transition temperatures are evaluated in this The first type of embrittlement is known as tempered low-level energy region, DTT will not be affected greatly by martensite embrittlement (TME), which appears after tem- orientations of specimen and notch, but orientation can be an pering of hardened steels between 260 and 370 °C or 500– important variable for comparison of room temperature 700 °F, and so TME is often referred to as ‘350 °C impact properties of materials. embrittlement’ or ‘500 °F embrittlement’. The second vari- ety of embrittlement is commonly called temper embrittle- In general, tempering of martensitic structure in steel ment (TE), which develops when some hardened alloy steels produces the best combination of strength and impact containing specific impurity elements are tempered isother- toughness, but in the hardness range between Rockwell C 40 mally in the temperature range of 400–600 °C or slowly and 60, the impact strength of austempered product in steels cooled through that critical temperature range after initial is superior to that of the tempered martensitic structure with tempering at a high temperature of 600–700 °C. The names the same hardness. Austempering is a hardening treatment in suggested by McMahon are ‘one-step embrittlement’ for which austenite is isothermally transformed to bainite, TME and ‘two-step embrittlement’ for TE, because a single without formation of the high-temperature transformation tempering treatment is sufficient to induce TME, while two product, by quenching the austenitized steel in a molten lead tempering treatments or a heating step or a cooling step is or salt bath held at a temperature below the region of fine sometimes required for TE to occur (McMahon 1975). The pearlite formation and above the martensite start tempera- heat treatments involved in one-step and two-step embrit- ture. Comparison of impact toughness of austempered and tlements are shown in Fig. 6.16 (Briant and Banerji 1978). conventionally quenched-and-tempered carbon steel as a function of hardness showed that austempering is clearly Both TME and TE cause a decrease in the notched-bar most beneficial in the hardness range between RC 50 and 55 impact toughness and an increase in the impact transition (Grossmann and Bain 1964). For example, at a hardness of temperature. In spite of these similarities between the two RC 50, 0.74% carbon steel austempered at 305 °C showed types of embrittlement, TME and TE are separated into two much better impact toughness and ductility than the same different phenomena from the practical point of view, steel quenched from the same austenitizing temperature of because TME is a much more rapid process than TE and 790 °C and tempered at 315 °C. The impact toughness was they take place in the two separate temperature ranges. TME 47.9 J for austempered steel against 3.9 J for occurs during tempering for short times, normally within one quenched-and-tempered steel, and the ductility measured by hour time period, in the critical temperature range of reduction of area was, respectively, 34.5% against 0.7%. The embrittlement, whereas it requires many hours for TE to low impact toughness observed in quenched-and-tempered occur. Hence, thicker sections, like large shafts and steam steel was most probably due to tempered martensite turbine rotors for power generating equipment, are suscep- embrittlement, which has been subsequently discussed in tible to TE, because even after tempering at high tempera- Sect. 6.5.1. tures of 600–700 °C, thicker sections cool very slowly over Fig. 6.16 Heat treatments of (a) (b) temper embrittled steels: γ a one-step embrittlement γ (TME) and b two-step embrittlement (TE) (Briant and Banerji 1978) Temperature 600°C Temperature 350°C 400°C 200°C Time Time

244 6 Impact Loading a long period of time involving several hours through the 1977), the plates of cementite lead to brittle transgranular critical temperature range for embrittlement and cause TE. fracture. These cementite platelets have no effect on the On the other hand, TME is independent of section size tensile ductility measured by reduction of area, but they and/or cooling rate after tempering, because it takes shorter severely reduce the room temperature notch toughness of duration to develop. steel. The common mode of fracture associated with TME is the intergranular fracture along prior austenite grain 6.5.1.1 Tempered Martensite Embrittlement boundaries of steels. It occurs when impurity elements such The fracture energy in the notched-bar impact test at room as phosphorus (P), sulphur (S), nitrogen (N), antimony (Sb), temperature generally increases with increasing tempering arsenic (As) and/or tin (Sn) are segregated to prior austenite temperature of a hardened steel, except showing a minimum grain boundaries during austenitizing. Figure 6.18 (Mater- in the curve in the critical tempering temperature region of kowski and Krauss 1979) shows schematically the room 260–370 °C due to the development of TME, but hardness temperature Charpy V-notch impact energy versus temper- of steel decreases continuously with increase in tempering ing temperature for two steels of the similar composition, but temperature, as shown schematically in Fig. 6.17 (Briant and one with higher phosphorus content of 0.03 wt% and Banerji 1978). Studies have shown that TME develops due another with lower phosphorus content of 0.003 wt%. Both to the stress concentration effects of needle-like or of them are quenched to form martensite after austenitizing rod-shaped thin cementite plates that are formed by replac- at the same temperature and then tempered for 1 h at dif- ing epsilon carbides during the tempering of martensite in ferent temperatures up to 500 °C. Both steels show a trough the above temperature range. TME can occur at temperatures in their corresponding curves between tempering tempera- as low as 200 °C and as high as 400 °C, depending on the ture of 250 to about 400 °C, and the room temperature notch time allowed for tempering. In commercial practice, hard- toughness of the steel containing 0.03% P is found to be ened steels are not generally tempered between 200 and inferior to that containing 0.003% P at all tempering tem- 400 °C to avoid TME. Additions of about 2–2.25 wt% sil- peratures. Moreover, when steel containing 0.03% P is icon to steels increase the temperature of formation of broken at room temperature under impact loading after cementite platelets by stabilizing the epsilon carbide, and tempering in the temperature zone of embrittlement, it shows thus, the embrittling reaction shifts to a higher temperature intergranular fracture (Materkowski and Krauss 1979). of about 400 °C. This allows tempering of steels between Indeed, the impact energy reaches to its minimum value 200 and 400 °C, without severe embrittlement, and steels when the extent of intergranular fracture is observed to be with strength levels above 1400 MPa can be obtained. the maximum (Bandyopadhyay and McMahon 1983). Although phosphorus that is segregated to prior austenite TME may or may not be associated with segregation of grain boundaries during austenitizing remains present in the embrittling impurity elements along prior austenite grain as-quenched martensite and after tempering up to the tem- boundaries of steels. For steels containing low concentration perature of embrittlement, TME does not fully develop of impurity elements and/or where the embrittling effects of unless cementite plate forms in the tempered martensite. It impurities have been minimized by the scavenging action of may be concluded from the latter observation that an inter- an alloying element, for example the interaction of molyb- action between the impurity element and cementite is denum with impurity like phosphorus (McMahon et al. Fig. 6.17 Schematically Hardness showing the effect of tempering temperature on hardness and Hardness and notch toughness room temperature notched-bar fracture energy for hardened steel (Briant and Banerji 1978) Notched-bar 600 fracture energy 0 200 400 Tempering temperature, °C

6.5 Metallurgical Factors Affecting Impact Properties 50 245 Fig. 6.18 Schematic curves of Charpy impact energy, Joules Steel containing room temperature Charpy 0.003 wt% P V-notch impact energy versus tempering temperature for steel containing different amounts of phosphorus (Materkowski and Krauss 1979) Steel containing 0.03 wt% P TIME zone 0 0 100 200 300 400 500 Tempering temperature, °C required to cause the intergranular mode of fracture in TME. The composition analysis of atomic layers adjacent to the On the other hand, when steel containing 0.003% P is intergranular fracture surface by AES has contributed to the tempered at 350 °C followed by breaking at room temper- understanding of cause of TE. AES shows that the alloying ature under impact loading, transgranular fracture with flat element such as Ni stimulates segregation of embrittling cleavage facets is observed and no intergranular fracture is impurity elements at prior austenite grain boundaries visible (Materkowski and Krauss 1979). Further, the trans- (McMahon et al. 1976). It has been argued (Banerji et al. granular fracture mode associated with TME is of two types 1978; McMahon and Vitek 1979; Kameda and McMahon —translath cleavage and interlath cleavage that may be 1980; McMahon et al. 1981) that the cohesive energy of the related to thickness of carbide remaining between laths of grain boundaries is lowered by the segregation of embrittling martensite. The thicker carbides promote translath cleavage, impurities such as Sb and in turn the local stress required to while the thinner ones promote interlath cleavage. The produce an accelerating microcrack is reduced that leads to translath cleavage, where the cleavage facets are oriented intergranular fracture and lower toughness. To reduce the across the martensite laths of a packet, initiates due to susceptibility to TE, the best procedure is to reduce the cracking of relatively thick cementite plates (King et al. embrittling impurities through control of raw materials and 1977), whereas the interlath cleavage is induced by cracking melting practice. parallel to thinner plate of cementite formed (Schupmann 1978). Compositional factors required to induce TE have certain characteristics that are as follows: 6.5.1.2 Temper Embrittlement • In plain carbon steels with manganese content less than The heat treatment conditions and compositional factors that 0.5 wt%, TE does not occur. cause temper embrittlement in steels are well known from many review articles (Hollomon 1946; Woodfine 1953; • Specific impurities must be present in steel to cause TE. McMahon 1968; Capus 1968; Olefjord 1978), but the exact From the work of Steven and Balajiva (1959), it has been mechanisms remain unclear. Heat treatment factors that known that the potent embrittling impurity elements are cause TE, as mentioned earlier, involve isothermal temper- antimony, phosphorus, tin, arsenic (in decreasing order ing in or slow cooling through the temperature range of 400– of their embrittling effects) and high-purity steels do not 600 °C. experience TE. These results have also been verified by others (Low et al. 1968). TE is found to occur even in the It is generally believed that TE results from segregation of presence of very small quantities of these detrimental embrittling impurity elements at prior austenite grain impurities on the order of 100 ppm (0.01 wt%) or less boundaries as a result of exposure to the critical temperature and increase in severity with the concentration of these range for embrittlement. By using Auger electron spec- embrittling impurities. These offending impurities seg- troscopy (AES), a surface analysis technique, this has been regate to the prior austenite grain boundaries and cause verified (Marcus and Palmberg 1969; Stein et al. 1969; TE. Marcus et al. 1972) and the extent of segregation of embrittling impurity elements has been determined quanti- • TE is found to occur in certain alloy steels of commercial tatively (Palmberg and Marcus 1969; Mulford et al. 1976). purity, but comparable alloys of high purity are not susceptible to TE, as demonstrated by Balajiva et al.

246 6 Impact Loading (1956), Steven and Balajiva (1959). The additions of Energy absorbed Unembrittle alloying elements to steels that enhance the susceptibility steel to embrittlement are Cr, Mn, Ni and Si. For a given impurity level, steels alloyed with either Ni or Cr are Temper embrittled embrittled less than Ni–Cr alloy steels (Low et al. 1968). steel Thus, the severity of embrittlement depends not only on the concentration of the embrittling impurity elements 0 Room temperature present, but also on the overall composition of the steel. – 200 0 • There are certain alloying elements that suppress grain-boundary segregation of the embrittling species and Testing temperature, °C thus inhibit the embrittlement. The onset of embrittle- ment is delayed by addition of a small amount of W, Mo, Fig. 6.19 Shift in fracture energy transition temperature curve, Ti or Zr to steel. The advantageous effects of these obtained from notched-bar impact test, to higher temperature as a alloying elements on preventing embrittlement are due to result of temper embrittlement (TE) produced in steel by isothermal their scavenging actions on harmful impurity elements. holding and furnace cooling through the critical temperature range However, the actions of these alloying elements are only to make the rate of embrittlement slow because they temperature embrittlement (TTE) curve, as shown in slowly react with carbon to form stable carbides, Fig. 6.20, similar to TTT (time–temperature transforma- releasing the impurity atoms that segregate to grain tion) diagram for steel. The TTE curve shows isoem- boundaries and give rise to TE. In other investigation brittlement lines as a function of tempering time and (Garcia et al. 1985), it was reported that the tendency for temperature with a nose or the shortest hold time at about TE in alloy steels was reduced through additions of 550 °C. One investigation (Carr et al. 1953) has shown lanthanide metals, which acted as scavenger elements to that it requires an isothermal hold time of about one hour form thermodynamically stable harmless compounds in at 550 °C to notice the first increase in transition tem- the matrix by combining with embrittling impurity ele- perature and several hundred hours at 375 °C for the first ments, such as Sb, P, Sn and As. As a result, the seg- sign of embrittlement. In TE developed on isothermal regation of impurity elements to the prior austenite grain tempering of alloy steel, the embrittlement as well as the boundaries was prevented that in turn reduced the sus- fracture appearance transition temperature (FATT) is ceptibility to TE. found to increase with tempering time (King and Wig- more 1976). TE is demonstrated in the following three ways: Tempering temperature, °C ~ 600 (1) The embrittlement that results is primarily detected by a remarkable increase in the impact-notch transition TE temperature, as shown schematically in Fig. 6.19. In 1 Cr–Ni steel with 0.3 wt% C, the presence of 0.04% Sb can raise the transition temperature from −40 °C in the TE unembrittled condition to 600 °C in the embrittled 2 condition. It is to be noted that the hardness and tensile properties are not affected by TE. ~ 400 (2) TE is always associated with intercrystalline fracture Temper embrittlement, TE2 > TE1 that occurs along prior austenite grain boundaries. Hold time (3) A grain-boundary etching effect is observed after the Fig. 6.20 Schematically showing isoembrittlement lines (fixed shift in occurrence of TE. This can be used for the measure- ductile–brittle transition temperature) as function of tempering temper- ment of austenite grain size. ature and hold time Other important special characteristics of TE are as follows: • The embrittling kinetics that are quantified by measuring the transition temperature as a function of tempering temperature and hold time exhibit a C-shaped time–

6.5 Metallurgical Factors Affecting Impact Properties 247 • TE is a reversible process unlike TME. The temper Zt Zt Zt embrittled steels can be de-embrittled; i.e., their original E1 ¼ load  displacement ¼ PðV0 Á dtÞ ¼ V0 Pdt toughness in the unembrittled condition can be restored. This de-embrittlement can be obtained by reheating the 0 00 embrittled steel above 600 °C, holding at that tempera- ture for only few minutes and then cooling rapidly to ð6:8Þ below about 300 °C, with a rate faster than or at least equal to the cooling rate given by the tangent to the nose where of TTE curve. If unembrittled steel is cooled slowly from above 600 °C so that the cooling rate intersects the TTE E1 total fracture energy when the pendulum velocity is curve, then TE will develop in that steel. assumed to be constant; t time; V0 initial pendulum velocity; P instantaneous load. 6.6 Instrumented Charpy Impact Test In fact, the assumption of a constant pendulum velocity V is not valid. Rather, V decreases in proportion to the The total energy absorbed in fracturing the test specimen is instantaneous load on the test specimen. From the work of measured by the ordinary Charpy impact test. But it is Augland (1962), it is found that possible to obtain additional information by instrumenting the Charpy impact tester so as to provide a load–time history ET ¼ E1ð1 À aÞ ð6:9Þ of the test specimen during the test (Turner 1970; Wullaert 1970). Figure 6.21 shows a typical load–time curve obtained where from such an instrumented Charpy impact test. A curve of ET total fracture energy and this kind provides information on the general yield load, the maximum load and the fracture load. The energy required to a ¼ E1 ; initiate fracture as well as that required to propagate fracture 4 E0 can also be determined from this type of record. where E0 is the initial energy of the pendulum. However, the fracture energy of the test specimen can be The fracture energy values based on the final position of computed by integrating a load–displacement record. If the velocity V of the striking pendulum is known and assumed the pendulum are conventionally determined by direct to be constant throughout the test, then the fracture energy read-out from the dial of an impact tester. When total frac- E1 is calculated from a load–time curve as follows: ture energy values are computed from (6.9) and compared with the conventionally determined results, good correlation Yield load, Py Maximum load, Pmax is obtained. Because of such good agreement, the initiation Fracture load, Pf and propagation energies required for fracture at any given test temperature can be calculated separately by using (6.9). Load Propagation Further, the determination of the data related to the yield energy, Ep load, the maximum load and the fracture load has facilitated us to identify more clearly the various stages in the fracture Initiation energy, EI process. Moreover, for comparison of material properties and selection of material, the instrumented Charpy impact Time for brittle fracture test is relatively inexpensive. Time The notch root of a fatigue precracked specimen used in Fig. 6.21 Schematic load–time history from an instrumented Charpy fracture mechanics test (described in Chap. 9) is sharper than test that of a Charpy specimen. So to determine dynamic fracture toughness KID by using the instrumented Charpy impact test, the standard Charpy specimen is precracked by introducing a fatigue crack at the tip of the V-notch. Hence, it is to be concluded that the instrumented Charpy impact test can be used to establish the existence of a temperature-induced transition in dynamic notch toughness response.

248 6 Impact Loading 6.7 Additional Large-Scale Fracture Test sections. To solve this problem, Pellini (1971) and his Methods co-workers developed tests to be performed on specimens at least 25 mm thick and their rational method of analysis. It was found that small laboratory specimens like Charpy Different large-scale tests are described below. specimen made of ship steel did not fracture at service temperature at stresses below the gross yield stress, while 6.7.1 Explosion-Crack-Starter Test brittle fractures were observed in ship structures at the same temperature at elastic stress levels. Because the degree of Explosion-crack-starter test was the first developed (Puzak triaxial constraint in a thinner section is likely to be less than et al. 1954) large-scale test. In this test, a flat steel plate of that in a thicker section. Further, considerable scattered data 350 Â 350 Â 25 mm thick is considered as a specimen. are obtained in a thin section than in a thick one. Thus, A short brittle weld bead is deposited on one surface of the Charpy impact test carried out with a standard 10-mm-thick plate. A small natural crack, similar to a weld-defect crack, is specimen shows a higher toughness at a given service tem- introduced by the deposited weld bead in the test plate. The perature and a lower transition temperature than that plate is placed over a circular die, weld bead face down and exhibited by a thicker section of the same material, as shown dynamically loaded with an explosive charge on the face in Fig. 6.22. This situation necessitates the development of opposite to weld bead after it has reached a desired test other large-scale tests to be performed on much thicker temperature. The same test is repeated at various tempera- tures, and the various transition temperatures are determined Charpy from the appearance of the fracture, as shown in Fig. 6.23. specimen From this test, nil ductility temperature (NDT) is determined by the highest temperature up to which a flat elastic fracture Fracture energy Thicker that propagates completely to the edges of the test plate is section developed. With increase in temperature above NDT, a plastic bulge is formed at the centre of the plate, but still the Service temperature flat elastic fracture runs completely to the plate edge. At Temperature certain higher temperature, the elastic fracture remains confined within the bulged region instead of propagating to Fig. 6.22 Schematically showing the effect of section thickness on the plate edge. The temperature at which the elastic fracture fracture energy transition temperature curves no longer spreads to the plate edge is called the fracture transition elastic (FTE). FTE is defined as the maximum temperature above which purely elastic stresses cannot propagate a crack. With further increase of the test temper- ature, plasticity of the plate increases that gives rise to for- mation of a helmet-type bulge and complete ductile tearing. The minimum temperature above which this occurs is the fracture transition plastic (FTP). Fig. 6.23 Fracture appearance NDT FTE FTP as a function of temperature in explosion-crack-starter test and determination of various transition temperatures Flat Bulge Bulge Helmet-type Bulge fracture & & & fracture partial fracture ductile tearing Temperature

6.7 Additional Large-Scale Fracture Test Methods 249 6.7.2 Drop Weight Test (DWT) The length, width and thickness dimensions of specimens are as follows: Drop weight test (DWT) was developed (Puzak and Pellini 1962; ASTM E208 2012) particularly to measure NDT of • 360 mm  90 mm  25 mm, with weld bead length of 15.9-mm-thick or more thick structural materials with an 63.5 mm. accuracy of ±5 °C and is quite reproducible. This test is not recommended for steels less than 15.9 mm thick. For this • 130 mm  50 mm  19 mm, with weld bead length of test, there are three standard flat plate-shaped specimens 44.5 mm. according to ASTM E208–06 (2012). A centrally located weld bead, approximately 50 mm long and 12.7 mm wide, • 130 mm  50 mm  16 mm, with weld bead length of is deposited on one surface of the plate specimen. At the 44.5 mm. centre of the length of the weld bead, a minute notch is introduced for initiation of crack. Care must be taken to The length of the weld bead is not critical, provided the ensure that only the weld deposit is notched without cutting crack-starter notch is at the centre of specimen and the weld the specimen surface. The plate is placed with weld bead bead does not contact the anvil support in fully bent con- face down, as a simple beam in a holder having an anvil stop dition of the specimen. just below the weld bead and heated in a constant temper- ature bath to a desired test temperature. Then, the specimen The same test, as mentioned above, is repeated by sub- is impacted with a falling weight on the face opposite to jecting each of a series (generally four to eight) of specimens weld bead as shown in Fig. 6.24, and as a result, the of a given material to a single impact load sequentially at crack-starter brittle weld bead deposited on the tensile face various temperatures to determine the maximum temperature of the specimen is fractured at near yield stress levels. The at which a specimen breaks. When the test specimen is impact load is provided by a guided, free-falling weight dynamically loaded in three-point bending, the anvil stops whose energy varies from 340 to 1630 J depending on the restrain the bending of the specimen, which in turn does not yield strength of the specimen material. The placement of allow the stress on the tension face of the specimen to anvil stop will be such as to prevent the specimen from exceed the yield strength. Since very little energy is required deflecting more than a few tenths of an inch. Note that when for a crack to start propagation at the base of notched bead of the specimen is fully bent or deflected under load, the weld brittle weld metal, the critical factor is whether the base test bead must not contact the anvil support. plate can resist the propagating crack or not, i.e. whether the plate does not break or breaks. If a crack spreads to one or more edges on the tension face, the ‘break’ condition of the specimen is established. For the specimen to be considered Fig. 6.24 Schematic diagram of Impact loading drop weight test (DWT) from falling weight 360 mm 90 mm Specimen 25 mm Creacked weld bead Anvil stop

250 6 Impact Loading as ‘broken’, cracking on the compression face of the speci- of applied stress, the material and the thickness of specimen. men is not required. According to ASTM Standard E208, the From crack-arrest tests, it has been seen that CAT of mild nil ductility temperature (NDT) is defined as the maximum steel below the NDT is independent of temperature, but the temperature up to which the plate ‘breaks’. Above NDT, the stress level for crack arrest is 35–55 MPa, above which plate does not break and so NDT reflects a break, no-break brittle fracture will occur. Since the allowed stress level is condition of the specimen. too low for practical applications, mild steels cannot be used in service below the NDT. Although crack-arrest test pro- 6.7.3 Robertson Crack-Arrest Test vides quantitative measurements, but it is not used widely, because large specimens and large testing machines are Robertson crack-arrest test (Robertson 1951) provides the required for this test. relationship between the magnitude of applied stress and the temperature at which the material is capable of arresting a 6.7.4 Dynamic Tear (DT) Test rapidly propagating crack. In this test, at one side of a 150-mm-wide plate specimen, there is a saw cut that acts as Dynamic tear (DT) test is a large Charpy test carried out on a starter crack. The specimen is subjected to a thermal gra- specimens of typically 455 Â 120 Â 15–25 mm thick, as dient across the plate width by applying heat at one side and shown in Fig. 6.26, but the thickness of specimens may be using liquid nitrogen coolant at the other side such that the as high as 300 mm. The energy capacity of a single-blow starter crack is at the lowest temperature, as shown in swinging pendulum-type tester used for DT test is 14 kJ in Fig. 6.25. A uniform elastic tensile stress is applied normal contrast to 330 J for a standard Charpy tester. An electron to the plane of the starter crack. When the plate is impacted beam narrow weld that acts as a notch is produced on one at the starter crack on the cold side of the specimen, an side of the specimen. This weld is embrittled by addition of unstable crack starts to grow from the root of the starter alloying element, for example addition of Ti produces a crack. The crack propagates across the plate width towards brittle Fe–Ti alloy. The specimen is supported as a simple the hot side until it experiences such a high plate temperature beam with embrittled weld-face down and is impacted with that provides enough ductility to blunt the crack tip and the pendulum on the face opposite to weld after it has resists further crack growth. This temperature is called the reached a desired test temperature. When the test specimen is crack-arrest temperature (CAT). Hence, CAT is defined as dynamically loaded in three-point bending, the embrittled the highest temperature up to which unstable propagation of narrow weld fractures easily and provides a reproducible crack can occur at any stress level in a given material. sharp crack that causes the fracture of the specimen. The Crack-arrest tests are carried out over a range of stress levels. above DT test is repeated over a range of temperature in a Such tests have shown that CAT depends on the magnitude pendulum-type machine (Puzak and Lange 1969; Lange and Fig. 6.25 Schematically showing Load Robertson crack-arrest test, in which specimen is uniformly Saw cut loaded and subjected to thermal Impact gradient. After impact loading, crack propagates from starter notch (saw cut) and arrested at TCAT Weld Heat applied TCAT Liquid N2 Cold Warm coolant Load

6.7 Additional Large-Scale Fracture Test Methods 251 Impact load FTE FTP 455 mm DT energy Slant fracture Shear lip 15 to 25 mm 120 mm Embrittled NDT electron-beam Temperature narrow weld Fig. 6.26 Specimen details and method of loading in dynamic tear test Fig. 6.27 Schematic curve of DT energy versus temperature from (DT) dynamic tear test Loss 1970) from which the energy absorbed in fracturing 6.8 Fracture Analysis Diagram (FAD) each specimen at every temperature is measured and the resulting test data are used to construct a curve of DT energy Pellini and Puzak (1963) collected data from large-scale versus temperature as shown in Fig. 6.27. From this plot of fracture tests like Robertson crack-arrest test and DWT and DT test and the appearance of fracture surface of specimen, prepared the information in the form of fracture analysis various transition temperatures can be defined. NDT is the diagram (FAD). The ratio of fracture strength to yield maximum temperature up to which fractures are fully brittle strength versus temperature for a steel containing flaws of with flat surfaces and without formation of any shear lips. various initial sizes is presented as FAD, as shown in In DT energy versus temperature curve, NDT corresponds to Fig. 6.28. In this figure, the curve marked a5 corresponding the extreme point of the lower shelf energy beyond which to a large-sized initial flaw is a CAT curve determined from the DT energy begins to rise sharply. At temperatures above the Robertson crack-arrest test carried out over a range of the NDT, the DT energy required for fracture sharply stress levels. The CAT curve is a limiting curve of fracture increases and shear lips begin to appear on the fracture strength because brittle fracture will not take place for any surfaces. point to the right of this curve. Decreasing initial flaw sizes progressively moves the fracture strength curves upward and With rise in temperature to the FTE, the development of to the left as evident from Fig. 6.28, which in turn increases shear lips becomes progressively more noticeable. The the allowable stress levels for a given minimum service average of upper and lower shelf DT energy values corre- temperature. Thus, FAD provides allowable stresses as a sponds roughly to the FTE, as shown in Fig. 6.27. function of flaw size and operating temperature for Above FTE, as temperature is increased to FTP the fracture low-strength ferritic steels of the type used for construction mechanism becomes void coalescence type, resulting in a of ships and has been applied in the design of structures. For ductile fracture, and the fracture surface appears as a fibrous mild steel below the NDT, the CAT curve is parallel to the slant fracture. Further, FTP corresponds to the initiation of temperature axis because CAT is independent of temperature the upper shelf in the DT energy curve. The DT test pos- as mentioned earlier and the safe stress level is 35–55 MPa, sesses high flexibility because it is equally applicable to above which brittle fracture will occur irrespective of the low-strength ductile materials that have a high value of original flaw size. Above the NDT, the CAT curve rises upper shelf energy and high-strength low-toughness mate- sharply with increasing temperature causing to increase the rials that exhibit a low value of upper shelf energy. The stress required for the unstable propagation of a large-sized large-sized specimen used in DT test produces a high degree initial flaw. Apart from NDT, some other transition of triaxial constraint and reduces the scattering of data to a minimum.

252 Fracture strength, σf / yield strength, σ0 Initial flaw size, a1 < a2 < a3 < a4 < a5 6 Impact Loading UTS Fig. 6.28 Fracture analysis FTP diagram showing allowable a1 stresses as a function of initial 1.00 Plastic flaw size and operating 0.75 a2 FTE temperature. Curve a5 corresponds to crack-arrest Elastic temperature (CAT) as obtained from Robertson test a3 CAT 0.50 a4 0.25 a5 CAT 35 to 55 MPa NDT NDT NDT 0 +33°C +67°C NDT +17°C Temperature temperatures can be defined with respect to CAT curve. 4. When r UTS, Tmin ! NDT þ 67 C ð120 FÞ or; Hence, FTE corresponds to the point on the CAT curve, Tmin ! FTP. Because above FTP, the behaviour of a where allowable stress is equal to the yield strength and FTP material will be like a flaw-free material that means any is the point at which allowable stress is equal to the UTS. It crack whatever be its size cannot propagate as an has been seen that for various structural steels, CAT curve is unstable fracture and only ductile failure will occur when related to NDT in the following way: the CAT curve is the applied stress is equal to the UTS of the material. estimated by • NDT þ 17 C ð30 FÞ 6.8.1 Design Philosophy Using FAD at maximum allowable stress ¼ 0:5 Â yield strength. There are three variables with respect to use of FAD in • NDT þ 33 C ð60 FÞ ¼ FTE design: (a) selection of steel, (b) service temperature and at maximum allowable stress ¼ yield strength. (c) applied stress level. Once any two variables out of the above three variables are decided by the designer, the third • NDT þ 67 C ð120 FÞ ¼ FTP factor will automatically be fixed by the FAD, so that the at maximum allowable stress ¼ UTS. brittle fracture can be avoided. But the designer must first decide the most important factor required for the design of Thus, once the NDT for structural steels has been mea- the structure so that it not can only safely operate but can sured, the entire CAT curve can be well established and also be economical. The design philosophy is illustrated with applied for design in service, although the above relation the following examples: may be a function of the thickness of specimen and the value of temperature that is added to NDT may increase for very Example I thick sections on the order of 150–300 mm. Based on the above relation, the permissible minimum operating temper- Suppose the minimum expected operating temperature, Tmin, ature, Tmin, for structural steels containing sharp long flaws is known or fixed by the designer. For application, the or cracks depending on the levels of applied stress can be designer selects such steel whose FTE is lower than or equal given (Pellini 1969) as follows: to the Tmin. Hence, from FAD, one can easily visualize that as long as the applied stress does not exceed the yield 1. When applied stress r 35–55 MPa, Tmin ! NDT. strength, the worst expected flaw will not propagate and the 2. When r 0:5 Â yield strength, Tmin ! NDT þ 17 C component will work safely, but this design criterion against brittle fracture is over conservative and not at all economical. ð30 FÞ. 3. When r yield strength, Tmin ! NDT þ 33 C ð60 FÞ or; Tmin ! FTE.

6.8 Fracture Analysis Diagram (FAD) 253 Example II ðb2 À b3Þ=10 ¼ ð160:1 À 156:5Þ=10 ¼ 0:36: A slightly less conservative and more practical design cri- Thus, during either a downward or an upward swing of terion against brittle fracture is to base on an allowable the pendulum through an angle of 160.1°, the energy lost in air drag and the bearing friction of the pendulum is repre- applied stress level that does not exceed half of the yield sented by an angle of strength. From FAD, it can be seen that no crack will 160:1 Â 0:36 ¼ 0:182: propagate so long as the service temperature does not fall 316:6 below NDT þ 17 C. If minimum operating temperature, Tmin, is not expected to fall below −7 °C, the selection of Hence, according to (6.5), the corrected angle of fall, steel with the help of FAD would be such that its NDT is acorrect, is as follows: −7° −17° = −24 °C. 6.9 Solved Problems acorrect ¼ 160:1 þ 0:182 ¼ 160:282: 6.9.1. The weight of the hammer of a pendulum impact During an upward swing of the pendulum through an machine is 20 kg. The distance from the centre of gravity of angle of 160.1°, the total energy lost in friction involved to the pendulum to the axis of rotation is 800 mm, with ref- move the indicator and in air drag and the bearing friction of erence to Sect. 6.3.1, given that b1 ¼ 160; b2 ¼ 160:1; the pendulum is represented by an angle of b3 ¼ 156:5. The angle of rise of the hammer after breaking the specimen is observed to be 100°. Neglect the loss of 0:1 þ 0:182 ¼ 0:282: energy required to move the broken test piece. Applying necessary correction for the loss of energy due to air drag on Hence, this observed angle b needs correction, and the the pendulum, for the energy absorbed through friction in the corrected angle of rise according to (6.6) is as follows: machine bearing and by the movement of the indicator arm, determine accurately bcorrect ¼ 100 þ 0:282 Â 100 ¼ 100 þ 0:176 160:1 ¼ 100:176: (a) The potential energy of the impact machine. (a) Replacing a by acorrect in (6.1), we can get (b) The striking velocity of the hammer. (c) The impact strength of the material of the specimen. The potential energy of the impact machine Solution ¼ WRð1 À cos acorrectÞ ¼ 196:14 Â 0:8ð1 À cos 160:282Þ N m or J In problem, it is given that the weight of the hammer is: ¼ 304:6 J: W ¼ 20 Â 9:807 ¼ 196:14 N; the distance from the centre of gravity of the pendulum to the axis of its rotation is: (b) Replacing a by acorrect in (6.2), we can write R ¼ 0:8 m; and the observed angle of rise of the pendulum after break of the specimen is: b ¼ 100. WVF2all ¼ WRð1 À cos acorrectÞ; where VFall is the striking 2g For the movement of the indicator through an angle of b2 ¼ 160:1, the energy required is represented by an angle of velocity of the hammer. b2 À b1 ¼ 160:1 À 160 ¼ 0:1: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) VFall ¼ pffi2ffiffigffiffiRffiffiffiðffiffi1ffiffiffiffiÀffiffiffifficffiffioffiffiffisffiffiaffiffifficffioffiffirffirffieffifficffitffiÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The average angle of rise of the pendulum between readings b2 and b3 is as follows: ¼ 2 Â 9:807 Â 0:8ð1 À cos 160:282Þm=s ¼ 5:52 m sÀ1: ðb2 þ b3Þ=2 ¼ ð160:1 þ 156:5Þ=2 ¼ 158:3: (c) Replacing a by acorrect and b by bcorrect in (6.4), we get the Hence, the angle of a complete swing including fall and corrected impact strength of the material of the specimen rise of the pendulum is approximately which is equal to 158:3 Â 2 ¼ 316:6: Energy to break the specimen During one average forward swing, the energy lost in air drag and the bearing friction of the pendulum is represented ¼ WRðcos bcorrect À cos acorrectÞ by an angle of ¼ 196:14 Â 0:8ðcos 100:176 À cos 160:282Þ N m or J ¼ 119:989 J % 120 J:

254 6 Impact Loading 6.9.2. Suppose the temperature dependence of uniaxial ten- (a) The potential energy of the impact machine. sile yield strength ðr0Þ and fracture strength ðrf Þ of a (b) The striking velocity of the hammer. material is given by r0ðMPaÞ ¼ 1500 À 2:5 T and rfðMPaÞ ¼ 1000 À 0:1 T, respectively, where T is the tem- 6.Ex.2. The weight of the hammer of a pendulum impact perature in Kelvin. Calculate the ductile-to-brittle transition machine is 30 kg. The starting height of fall of the hammer is temperature and comment on the behaviour of material at 1.1 m, and after breaking the specimen, the height of rise of room temperature for: the hammer is 0.9 m. Neglect all losses and compute the following: (a) Simple tension and (b) Notched specimen, where plastic constraint factor is 2.5. (a) The impact value of the specimen. (b) The striking velocity of the hammer. Solution 6.Ex.3. The standard specimen for Charpy impact test is (a) The temperature at which r0 ¼ rf is the transition tem- 10  10  55 mm in size. But it is required to perform perature of unnotched specimen subjected to uniaxial ten- Charpy tests on 8-mm thick and 12-mm thick specimens of sion. Therefore, the same metal having identical microstructures to evaluate their toughness. What percentage difference in the values of 1500 À 2:5 T ¼ 1000 À 0:1 T; impact energy, if any, would you expect if results from both Or; 2:4  T ¼ 1500 À 1000 ¼ 500; specimens are compared with each other? ) Ductile-to-brittle transition temperature; 6.Ex.4. The yield strength and UTS of a structural steel containing worst flaws are, respectively, 210 and 320 MPa. T ¼ 500 K ¼ 208:3 K ¼ À 64:7 C: Assuming that the thickness of the steel is 25 mm and the 2:4 NDT of the steel is at −10 °C, Comment (a) Draw the CAT curve for the steel. (b) Find the highest allowable stress against brittle fracture The smooth specimen of the material is ductile at room of the steel at temperature ð300 K ¼ 27 CÞ. (i) NDT; (ii) NDT þ 17 C; (iii) FTE; and (iv) FTP. (b) In problem, it is given that plastic constraint factor is: q ¼ 2:5. The notch transition temperature will be the tem- (c) When the maximum applied stress does not exceed the perature at which rf = yield strength of the notched yield strength of the steel, what is the minimum allowable specimen = q  r0. service temperature for the steel? 1000 À 0:1 T ¼ 2:5  ð1500 À 2:5 TÞ 6.Ex.5. Indicate the correct or most appropriate answer(s) ¼ 3750 À 6:25 T; or; from the following multiple choices: 6:15 T ¼ 3750 À 1000 ¼ 2750; Hence, ductile-to-brittle notch transition temperature, (a) Ductility transition temperature of mild steel is the tem- perature at which lateral expansion on the face opposite to T ¼ 2750 K ¼ 447 K ¼ 174 C: V-notch of a Charpy specimen, after fracture in Charpy 6:15 impact test, will show a value of Comment (A) 0.14 mm; (B) 0.26 mm; (C) 0.38 mm; (D) 0.50 mm; (E) none of the above. The notched specimen of the material is brittle at room temperature ð300 K ¼ 27 CÞ. (b) Out of the followings, the best impact toughness is achieved when eutectoid steel is Exercise (A) Annealed; 6.Ex.1. The hammer of a pendulum impact machine weighs (B) Normalised; 25 kg. The distance from the centre of gravity of the pen- (C) Hardened and tempered at 300 °C; dulum to the axis of rotation is 1.25 m. If the pendulum arm (D) Austempered at 300 °C. is horizontal before striking the specimen, compute assum- ing the ideal situation

6.9 Solved Problems 255 (c) Among the following choices, lowest transition temper- (l) In selecting a material from the standpoint of notch ature results in case of toughness, the most important factor is: (A) Annealing of killed steel; (A) Transition temperature; (B) Annealing of rimmed steel; (B) Impact value at service temperature; (C) Normalizing of killed steel; (C) Lower self energy; (D) Normalizing of rimmed steel. (D) Upper shelf energy. (d) State, which of the following elements, if added, lowers (m) Regardless of the size of initial flaw, the maximum safe the 20-J ductility transition temperature of mild steel: stress level against brittle fracture below NDT for mild steel (A) Nickel; (B) Chromium; is (C) Molybdenum; (D) Phosphorus. (A) 5–25 MPa; (B) 35–55 MPa; (C) 65–85 MPa; (D) 95–115 MPa. (e) State, which of the following conditions, shows the highest probability of brittle cleavage fracture: (n) Ductile–brittle transition temperature for steels depends significantly on (A) High strain rate and plane strain; (B) High strain rate and plane stress; (A) grain size; (B) shear modulus; (C) Low strain rate and plane strain; (D) Low strain rate and plane stress. (C) tensile strength; (D) strain rate. (f) Unless there is some special chemical reactive environ- (o) Unless there is some special chemical reactive environ- ment, brittle fracture is not observed for ment, brittle fracture is not observed for one of the following materials. Select it. (A) Aluminium alloy with S0 [ E=150; (A) Nickel; (B) Zinc; (C) Mild Steel; (D) Glass. (B) Aluminium alloy with S0\\E=300; (C) Steel with S0 [ E=150; (p) For improvement of notch toughness in mild steel, there (D) Steel with S0\\E=300. are practical limitations to increase the ratio of Mn:C beyond where S0 is the yield strength, and E is the Young’s (A) 5:1; (B) 7:1; (C) 9:1; (D) 11:1. modulus. Answer to Exercise Problems (g) To obtain satisfactory notch toughness, the ratio of Mn:C in mild steel must be at least 6.Ex.1. (a) 306.4 J; (b) 4.95 m s−1. 6.Ex.2. (a) 58.8 J; (b) 4.64 m s−1. (A) 1:2; (B) 2:1; (C) 1:3; (D) 3:1. 6.Ex.3. Energy absorbed by 12-mm-thick specimen will be 50% more than that by 8-mm-thick specimen. (h) Suggest one of the following materials for use as a 6.Ex.4. (b) (i) 35–55 MPa; (ii) 105 MPa; (iii) 210 MPa; container for liquid oxygen: (iv) 320 MPa. (c) 23 °C. 6.Ex.5. (a) (C) 0.38 mm. (b) (D) Austempered at 300 °C. (A) Glass; (B) Mild Steel; (c) (C) Normalizing of killed steel. (d) (A) Nickel. (e) (A) (C) Copper; (D) 18/8 Austenitic stainless steel. High strain rate and plane strain. (f) (B) Aluminium alloy with S0\\E=300. (g) (D) 3:1. (h) (C) Copper. (i) (B) FATT. (i) Out of the following transition temperatures, the one (j) (C) 50. (k) (B) 8 and 0.01 wt%. (l) (A) Transition tem- which cannot be determined by explosion-crack-starter test is perature. (m) (B) 35–55 MPa. (n) (A) grain size; and (D) strain rate. (o) (A) Nickel. (p) (B) 7:1. (A) NDT; (B) FATT; (C) FTE; (D) FTP. References (j) FATT is the temperature at which % of brittle cleavage fracture becomes ASTM E208: Standard test method for conducting drop-weight test to determine nil-ductility transition temperature of ferritic steels. (A) 0; (B) 25; (C) 50; (D) 75 (E) 100. Designation: E208–06 (reapproved 2012), ASTM International, West Conshohocken, Pa (2012). doi:https://doi.org/10.1520/E0208- (k) Out of the following respective values of ASTM 06R12 grain-size number and oxygen content of mild steel, the lowest transition temperature will be observed for Augland, B.: Brit. Weld. J. 9(7), 434 (1962) Balajiva, K., Cook, R.M., Worn, D.K.: Nat. Lond. 178, 433 (1956) (A) 4 and 0.01 wt%; (B) 8 and 0.01 wt%; (C) 8 and 0.03 wt%; (D) 4 and 0.03 wt%.

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Creep and Stress Rupture 7 Chapter Objectives • Creep phenomenon. • Creep curve and its different stages. • Strain–time relations to describe the basic shapes of creep curves mathematically. A general empirical equation for time laws of creep. • Creep rate–stress–temperature relations, showing influence of stress and temperature on steady-state creep rate. • Effect of grain size on steady-state creep rate. • Activation energy for creep, its determination and relation with activation energy for self-diffusion. • Creep deformation mechanisms: dislocation glide, dislocation creep or climb–glide creep, diffusional creep (Nabarro–Herring creep and Coble creep), and grain boundary sliding. • Deformation mechanism map. • Stress-rupture test and its difference with the creep test. • Concept of equicohesive temperature (ECT) and deformation features at ECT. • Fracture at elevated temperature. Creep cavitation: wedge-shaped cracks and round or elliptically shaped cavities. • Presentation of engineering creep data, and prediction of creep strength and creep-rupture strength. • Prediction of long-time properties by means of parameter methods, such as Larson– Miller parameter, Orr–Sherby–Dorn parameter, Manson–Haferd parameter, Gold- hoff–Sherby parameter and limitations of parameter methods. • Stress-relaxation and step-down creep test. • Creep-resistant materials for high-temperature applications and rules to develop increased creep resistance at elevated temperatures. • Creep under multiaxial stresses. • Indentation creep and method to obtain creep curve using Rockwell hardness tester. • Problems and solutions. © Springer Nature Singapore Pte Ltd. 2018 257 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_7

258 7 Creep and Stress Rupture 7.1 Long-Time Loading at High Temperature been observed at the edge of roof due to occurrence of creep of lead under its own weight. Plastics also undergo signifi- In Chap. 1, it has been mentioned that recoverable defor- cant creep at room temperature, whereas creep may be mation may be instantaneous (time independent), known as observed in tar and asphalt at temperatures far below room elastic deformation, or time dependent, known as anelastic temperature. deformation. Similarly, permanent deformation may be instantaneous (time independent), or time dependent. So far Since many materials at high temperatures behave our discussions have been limited to the instantaneous per- somewhat like a very viscous liquid, so a stress often much manent deformation response to the application of a load. less than that required to cause failure in a short time (in a Our previous discussions have considered permanent few minutes) may cause failure if sufficient time is allowed deformation under short-time loading, as in conventional at high temperatures (where creep predominates). Creep may tension test, in which deformation increases simultaneously often cause undesirable structural distortion over a period of with the load or stress, as can be observed in static stress– time prior to reaching the stage of rupture in material. Such strain diagram. If, under any conditions, deformation excessive distortion seriously impairs the dimensional sta- extends over a period of time when the load or stress is kept bility and thus the usefulness of the material. Examples of constant, this time-dependent permanent deformation under such effects are the undesirable changes in clearance constant load or stress is called creep and is observed in both between the ends of the blades and housing of a steam tur- crystalline and amorphous materials. Initially creep defor- bine and the loosening of flanged joints caused by creep in mation may be small, but over the lifetime of the structure the connecting bolts. deformation can grow large and even result in final fracture without any increase in load. However, when both recov- Our discussions to creep will be restricted to metallic erable and permanent deformations occur together and materials only. It is essential to know the various changes dependent on time, it is called viscoelastic deformation, that take place in metallic materials with increasing tem- which is commonly observed in non-crystalline organic perature. Some of these are mentioned below: polymers. • Diffusion-controlled mechanisms become active due to Creep deformation is possible at all temperatures above rapid increase in the mobility of the atoms. absolute zero and has been observed down to the tempera- ture of liquid helium. However, since it depends on thermal • Equilibrium concentration of vacancies increases. activation, the strain rate at a given stress level is extremely • The presence of a large number vacancies aids climbing sensitive to temperature. As a result, the higher the tem- perature, the more pronounced is the creep phenomenon. of edge dislocation, which in turn results in greater Since temperature is a relative quantity for any material, i.e. mobility of dislocations. a high temperature for one material may not be high for • New slip systems may be introduced. another material, it is normal practice to express temperature • Deformation at grain boundaries may occur. as a homologous temperature, which is the ratio of the • Metallurgical stability of metals and alloys may be dis- operating temperature to the melting temperature on an turbed resulting in loss of strength, for example, loss of absolute temperature scale. In crystalline solids, such as strength caused by recrystallization and grain coarsening metals and ceramics, creep occurs appreciably at a homol- of cold-worked metals and overaging of age-hardened ogous temperature ! 0.5, although some crystalline mate- alloys. rials exhibit measurable creep deformation at a homologous • Oxidation and intergranular penetration of oxide may temperature as low as 0.25. In amorphous materials, such as occur. glasses and polymers, creep occurs at measurable rates above the glass transition temperature. Approximate tem- Thus, for successful applications at high temperatures, peratures for the onset of creep deformation are 540 °C for materials require better high-temperature strength and oxi- austenitic iron-based heat resisting alloys, 650 °C for nickel dation resistance. Many types of engineering machines and and cobalt-based heat resisting alloys, and 980–1540 °C for structures are required to sustain steady loads for a long period refractory metals and alloys. On the other hand, low melting of time at temperatures as high as 650 °C or higher, e.g. point metals, such as lead and tin, exhibit significant creep at internal combustion and jet engines, high-pressure boilers and room temperature because room temperature is equivalent to steam turbines and cracking stills used in chemical and pet- a homologous temperature of about 0.5 for lead and greater roleum industries. The operating temperature of turbine than 0.5 for tin. In old time roofs, a thick rim of lead has bucket of gas turbine engine is around 800 °C and that for ballistic-missile cone and rocket engine is much higher. Hence for elevated temperature applications, one of the important factors is the selection of proper creep-resistant materials.

7.1 Long-Time Loading at High Temperature 259 At lower temperatures, where creep is negligible, per- under load, i.e. undergo a time-dependent increase in length. manent deformation of a structure can be prevented if the Now, the extension over the gage length of the specimen is applied stress does not exceed the yield strength of the measured as a function of time, t, and the true strain is material and yield strength is the limiting design factor. For determined from the natural logarithm of the ratio of the example, the design of bridges, ships and other large struc- instantaneous length found from extension to the initially tures made of steel which are to be used at ambient tem- measured original gage length of tensile specimen. Hence, peratures is based primarily on elasticity theory, because the creep curve is constructed by plotting the determined true creep is usually not considered to be important for such strain, e; against log time ðlog tÞ: It is important to note that applications. On the other hand, at higher temperatures, the creep testing time may extend from several months to as where creep is predominant, permanent deformation can take high as more than 10 years. Since it is not generally feasible place over a period of time at stresses well below the to extend creep tests for periods more than 10 years, so creep material’s yield strength. Because high thermal energy tests are frequently conducted for periods of 2,000 h and makes plastic deformation possible under lower stresses if often to 10,000 h and the results of such short tests are sufficient time is provided. Ultimately, whether creep or extrapolated to long time for getting the information about time-independent flow (yielding) would be considered in the performance of a long-lasting component, which has engineering design will be decided by which type of been discussed later in this chapter. However, as the defor- deformation dominates at different operating temperature mation continues the length of specimen increases but the and stress combinations. cross-sectional area decreases resulting in an increase in the axial stress, since the applied load is held constant Most often two types of time-dependent tests at elevated throughout the engineering creep test. Hence, the reported temperatures are conducted to determine the strength and value of stress is usually the initially applied stress based on lifetime of materials for applications in different kinds of the original cross-sectional area of specimen. long-time high-temperature services. These tests are creep test and stress-rupture test. The former one is a long-time The idealized shape of a constant-load creep curve at a test that may extend from several months to more than constant temperature is shown by curve A in Fig. 7.2. The 10 years and generally performed to know useful informa- creep rate, de=dt; or e_, is determined by the slope of this tion about the performance of a long-lasting component, curve. Two dominant external variables that affect the shape such as a steam pipeline which is used in elevated- of creep-time curve are test temperature and stress. Another temperature service for more than 10 years. On the other external variable is certainly environment because oxidation hand, stress-rupture test is usually carried out for relatively and corrosion are important in the fracture process. Unfor- shorter time, usually between one to two months, with an tunately, consideration of this variable is beyond the scope aim to get the information about the high-temperature per- of the text. formance of a short-lived item, such as a missile nose cone or a rocket engine. Other test includes stress-relaxation test The plot of creep rate versus total creep strain, as shown that involves imposing a constant displacement under load at in Fig. 7.3, illustrates the large variation in the creep rate that elevated temperatures. occurs during the creep test. Since the test temperature and the applied load remain constant, the creep rate varies due to 7.2 The Creep Curve the changes in the internal structure of the material that occurs with creep strain and time during creep process. The ASTM [ASTM E6] defines creep as ‘the time-dependent part overall creep rate at a given stress and temperature is of the strain resulting from stress’. To carry out creep testing, strongly affected by a continually changing interaction a constant load or a constant stress is applied to a tensile between strain hardening and recovery or softening pro- specimen maintained at a sufficiently high constant homol- cesses. It is believed that strain hardening at elevated tem- ogous temperature, where measurable creep deformation can perature involves formation of subgrain associated with occur. For engineering purposes, the creep curve is deter- rearrangement of dislocations (Mukherjee et al. 1969), while mined under constant load because in actual service condi- two dominant recovery processes are thermally activated tions, usually the load but not the axial stress is maintained climb of edge dislocations and cross-slip of screw constant, whereas constant stress testing is carried out to dislocations. study the mechanisms of creep deformation. When the load is first applied, normally a small strain, e0; Typical loading apparatus for constant-load creep test is occurs very rapidly, which is termed as instantaneous strain. shown in Fig. 7.1. To obtain an engineering creep curve, a Even if the applied initial stress is below the yield strength, tensile specimen is subjected to a constant tensile load at a the instantaneous strain, e0; may not be fully elastic, rather it constant elevated temperature. The specimen will creep may be partly elastic, partly anelastic and partly plastic and the extent of each part depends upon the material, the tem- perature and the stress. Thermal expansion may also be

260 7 Creep and Stress Rupture Upper pull rod Lower pull rod Loading lever Furnace Specimen Weights Fig. 7.1 Loading apparatus for constant-load creep testing (Moore and Moore 1953) Primary creep Secondary or Tertiary creep Stage I steady state creep Stage III Rupture Stage II AB Strain, ε Stage I Stage II Stage III Creep rate, ε= dε dε = minimum dtdtcreep rate, εs Instantaneous strain, ε0 Log time (log t) Rupture time, tr Total strain, ε Fig. 7.2 Constant-temperature typical creep curve showing the three Fig. 7.3 Creep rate as a function of total strain distinct stages of creep. Curve ‘A’ for constant-load test and curve ‘B’ for constant-stress test included in this component if it is significant. The instanta- structural members. The creep strain is obtained by sub- neous strain, e0; is not considered as creep, but it is important tracting the instantaneous strain, e0; from the total strain that with regard to the design of the component because it forms the creep specimen undergoes. Excluding the initial instan- taneous strain, e0; the constant-load ideal creep curve can be an appreciable part of the allowable total strain in the

7.2 The Creep Curve 261 divided into three distinct stages which may be observed at Microstructural softening due to recovery process in the only certain combinations of stress and temperature. The low-temperature region includes cross-slip of screw dislo- relative strain extent of these three stages depends strongly cations while that at a homologous temperature above 0.5 on the testing temperature and the applied stress. includes rearrangement and annihilation of dislocations and climbing of edge dislocations. Further, at constant-load Primary Creep creep test, the reduction in the cross-sectional area of the specimen with increasing strain causes a rise in the axial After the initial instantaneous strain, e0; the creep rate stress, which increases the creep rate, because the creep rate decreases rapidly with time from a large initial value, as of a metal is a sensitive function of stress. So, geometrical exhibited in Fig. 7.2. This is the Stage I or the first stage of softening due to the increase in stress is also required to be creep, known as primary creep, which is a period of pre- balanced by the strain hardening. Recovery process is slow dominantly transient creep. At low stresses and creep at low temperature while that is rapid at elevated temperature strains, primary creep may exhibit logarithmic behaviour, because high thermal energy makes the diffusion-controlled known as logarithmic creep. The strain hardening caused mechanisms to become operative. Higher the rate of recov- primarily by the changes in the number, type and arrange- ery the more is the creep deformation. Thus, the creep ment of dislocations is more effective than the recovery or deformation is significant only at elevated temperatures. softening processes in the Stage I, leading to decrease in the strain rate. Transmission microscopic examination shows the The average value of the creep rate during steady-state increase of dislocation density, and in many materials, for- creep is called the minimum creep rate. The minimum creep mation of subgrain structure with a cell size decreasing with rate, e_s; which can be derived from the second stage of creep increasing strain. These structural changes are indicative of curve, is usually the most important design parameter. This decreasing creep rate with increasing time in primary creep design parameter is used for components in which the region. The elastic after-effect, which always forms a part of fracture is not at all tolerated but a low permanent strain transient creep, is practically insignificant in crystalline without disturbing the structural integrity may be allowed. materials. Commonly used two standard criteria are: Secondary Creep (1) The stress to produce a minimum creep rate of 1% in 104 h or 0.0001% per hour. This is applied to those Following the primary creep, the creep rate reaches essen- components which are used for hundreds of hours at tially a steady state, in which the creep rate changes little high temperatures, e.g. a jet engine turbine blade. with time. This region of approximately constant creep rate is the Stage II or the second stage of creep, known as sec- (2) The stress to produce a minimum creep rate of 1% in ondary creep, or steady-state creep, or sometimes called 105 h (about 11.5 years) or 0.00001% per hour. This is viscous creep. During this stage, the steady state is achieved applied to those components which are used for many because of an approximate balance between two opposing years at high temperatures without replacement, e.g. factors: the strain hardening that tends to reduce the creep boiler tubing and blades in a steam turbine. rate and the softening or recovery process that tends to increase it. Thus, softening reactions tends to oppose or Tertiary Creep nullify the strain hardening. In this stage, the dislocation density does not increase or a subgrain structure, if present, Subsequent to the second stage creep, the Stage III or the does not become progressively finer with increasing strain. third stage of creep, known as tertiary creep, or accelerated This invariant microstructure indicates that the intrinsic creep is observed mainly in the constant-load creep test at strain-hardening capacity of the material is balanced by high stresses or/and at high temperatures. During this stage, softening or recovery effects. If the rate of strain hardening is the creep rate is greater than that during Stage II and h ¼ @r=@e; and the rate of recovery is r ¼ À@r=@t; then for increases continuously till the material undergoes fracture, as a steady-state creep rate e_s, the flow stress r must remain seen from Fig. 7.2. Tertiary creep occurs when the effective constant, i.e. dr ¼ 0. This gives cross-sectional area of the specimen is reduced remarkably either due to localized necking or internal void formation. In dr ¼ @r @ t þ @r @e ¼ 0; or; @r @e ¼ À @r @ t; a constant-load creep test, this causes a rapid increase in the @t @e @e @t axial stress which overcomes the strain hardening and accelerates the creep rate during tertiary creep. Further, ) e_s ¼ de ¼ À @r=@t ¼ r ð7:1Þ metallurgical changes that often occur during tertiary stage dt @r=@e h are recrystallization of strain-hardened grains and overaging in a precipitation-hardened alloy. These changes lead to

262 7 Creep and Stress Rupture softening that supersede the strain-hardening effects and thus undergoes and this is an important characteristic of most tend to increase the creep rate. Fracture in tertiary creep is creep phenomena. also related to several other factors which include a number of weakening metallurgical instabilities, such as grain At small strains, the creep curve is essentially the same in boundary shearing leading to intercrystalline fracture, pre- both constant-load and constant-stress creep tests. In cipitation of brittle second-phase particles, resolution of fine constant-stress creep test, the transition from Stage II to second-phase particles that originally strengthened the alloy, Stage III creep is greatly delayed. At a constant temperature, and corrosion. In most cases, the creep test is terminated as the shape of a constant-stress creep curve excluding the soon as the creep curve enters the third stage, because the tertiary creep stage is shown by the curve B (dashed line) in main aim of the creep test is to establish the stage of Fig. 7.2. However, tertiary creep can also be observed in steady-state creep and to measure the minimum creep rate, constant-stress tests in which its onset is related to recrys- which will be used for design purpose in components tallization, coarsening of precipitate particles and/or internal without allowing them to fracture. However, the duration of void formation. To maintain a constant stress during creep tertiary creep is also important because the time to fracture test, it is usually necessary to decrease the load during the provides the information about the safety margin before a test in order to compensate for the decrease in the catastrophic fracture of a component takes place. cross-sectional area of the specimen as it deforms. Since creep essentially involves plastic strain, so constancy in The variations in the shape of creep curve obtained from volume is applicable to the creep process. Hence, the constant-load tests carried out at a constant temperature cross-sectional area varies inversely with length, as indicated under various initial stress levels or carried out at a constant in (1.26). In a constant-stress test, the instantaneous applied initial stress for various temperatures are shown in Fig. 7.4, load P is: because both variables, temperature and initially applied stress, tend to alter the shape of the creep curve in a similar P ¼ r A ¼ r A0L0 manner. The family of creep curves in this figure clearly L shows that high stresses or high temperatures reduce the extent of primary creep and practically eliminate the As long as the creep strain is uniform, in order to maintain steady-state creep, with the result that the creep rate accel- a constant stress, r; the applied load P at each moment has to erates almost from the start of the test. It can be seen from vary inversely as the instantaneous length L (which is this figure that the greater the stress or the higher the tem- increasing during the test), since the initial cross-sectional perature, the greater will be the creep rate. At intermediate area A0 and length L0 remain constant. Methods to carry out stresses or intermediate temperatures, the primary and sec- the constant-stress creep test have been developed (Andrade ondary stages become more prominent. Low levels of stress and Chalmers 1932; Fullman et al. 1953), which involve or low temperatures show long, well-defined regions of deadweights with adjustable lever arms and screws controlled secondary creep. Therefore, conditions that favour a very by electronic devices. The creep stress, which is usually slow creep rate make the second stage of creep most pro- determined from the initial cross-sectional area of a member nounced. Further, this figure shows that the longer the life of in a constant-load test or held constant in a constant-stress a creep test, the smaller the total extension that the specimen test, will be designated by the term r in this chapter. Fig. 7.4 Schematic (a) Temperature, T = constant (b) Stress, σ = constant representation of creep curves σ6 > σ5 > σ4 > σ3 > σ2 > σ1 T6 > T5 > T4 > T3 > T2 > T1 a under various initial stresses ðfrom r1 to r6Þ at one σ6 T6 temperature, T, and b at various σ5 T5 temperatures ðfrom T1 to T6Þ σ4 σ3 under the same initial stress, r Creep strain Creep strain σ2 T4 σ1 T3 T2 T1 Log time Log time

7.3 Strain–Time Relations 263 7.3 Strain–Time Relations graphite below 1500 °C exhibit logarithmic creep, where creep strain, e; varies linearly with the logarithm of time, t, as Numerous attempts have been made over the years to given below: describe the basic shapes of creep curves mathematically and e ¼ c ln t þ C ð7:4Þ many equations have been proposed with varying degree of success. One of the earlier relationships provided by where c and C are time-independent constants. At stresses and temperatures, where creep strains are greater, the vari- Andrade (1910) in terms of specimen length was: ation of e with ln t deviates from linearity. Most experiments showing logarithmic creep behaviour have run for a short  ð7:2Þ duration, involving only a few days, when creep rate is l ¼ l0 1 þ bt13 expðjtÞ found to decrease continually with time as shown by (7.4), which indicates the absence of secondary creep at low where l0 is the instantaneous length of specimen observed temperatures. But at low temperatures, secondary creep is upon loading, i.e. at time t ¼ 0; and ðl À l0Þ is the creep observed at sufficiently high stresses or in very long time extension of specimen after a time t. b and j are constants. tests at low stress levels. Thus, (7.4) describes primary creep at low creep strains and stresses. Equation (7.4) is valid only After converting to true strain, (7.2) becomes: when time t [ 0. Because (7.4) shows that at any stress level at time t ¼ 0; the initial creep rate is infinitely high which ln l l0  bt13  ! cannot be accepted because no known creep mechanism can 1 expðjtÞ ; lead to infinitely high rates. On the other hand, initial creep ¼ ln þ rates are finite under finite stresses. So, to eliminate the li li objection associated with (7.4), let us put C ¼ c ln m in (7.4), which then becomes where li is the initial gage length of specimen. ð7:3Þ  e ¼ e0 þ ln 1 þ bt31 þ jt where e0 is the instantaneous true strain on the application of e ¼ c ln mt ð7:5Þ load and e in the total true strain in time t. Andrade per- formed the experiment under constant stress and considered Since C ! 0 and c ! 0; m ! 1: Hence, without introducing that constant-stress creep curve, excluding e0; consists of much error, (7.5) can be replaced by the following relation: two creep components. The first one is the transient creep having a creep rate decreasing with time and describes the e ¼ c lnð1 þ mtÞ ð7:6Þ primary creep. This is also known as b flow (the second term in the above equation), which occurs when j ¼ 0: The From (7.6) we get, the initial creep rate at t ¼ 0 as second one is the viscous creep with a constant creep rate ðde=dtÞt¼0 ¼ ½cm=ð1 þ mtފt¼0 ¼ cm: and known as j flow (the third term in the above equation), which occurs when b ¼ 0: Hence, the addition of e0; b flow When the homologous temperature is in the range of 0.2– and j flow constitute the total constant-stress creep curve, as 0.7, two relations are found to fit the experimental results for shown in Fig. 7.5. a large number of BCC, FCC and HCP metals and alloys, Primary creep strain has been related to time by a number graphite, glass, ceramic oxides, mortar and plastics. The first of empirical relations. Garofalo (1965) summarized the work of them which describes primary creep is given by of others and showed that for a homologous temperature of e ¼ e0 þ btm ð7:7Þ 0.05–0.3 and very small creep strains up to about 2  10−3, a number of BCC, FCC and HCP metals and alloys, and Creep curve Instantaneous Transient creep, Viscous creep, strain, ε0 called β flow called К flow Strain Strain Strain Strain = ++ ε0 ε0 Time Time Time Time Fig. 7.5 Andrade analysis showing that the addition of instantaneous strain e0; transient creep, called b flow, and viscous creep, called j flow, constitute the total constant-stress creep curve

264 7 Creep and Stress Rupture where constant stress of 176 MN m−2. He found that at lower temperatures, logarithmic creep occurs, but at higher tem- e0 the instantaneous true strain observed upon loading, peratures, the creep strain at a given time is greater than that and predicted by logarithmic relationship. The creep–time rela- tion proposed by him consists of three terms, which are b; m time-independent constants logarithmic, parabolic or power law, and linear or steady-state creep. His proposed relation is: Creep behaviour in materials according to (7.7) is often called parabolic creep or b flow. Generally, b increases e ¼ a log t þ btm0 þ ct ð7:11Þ exponentially with temperature and stress. The value of m is assumed to depend on both stress and temperature and has where a, b and c are constants and m0 ’ 1=3: He found that been found to vary from 0.03 to approximately 1.0. Since low-temperature logarithmic creep followed a mechanical 0\\m\\1, (7.7) reflects a decreasing creep rate with time in equation of state. That is, the creep rate was dependent on Stage I. When m ¼ 1=3, (7.7) reduces to transient creep the instantaneous values of stress, strain and temperature and proposed by Andrade. not on the previous strain history. However, in the higher temperature range (above room temperature), creep was At stresses where steady-state creep is observed, a linear strongly dependent on prior strain and thermal history, and term jt; representing secondary creep strain, is added to hence, a mechanical equation of state was no longer valid. (7.7) and the total strain–time relation becomes Cottrell (1952) suggested a general empirical equation for e ¼ e0 þ btm þ jt ð7:8Þ the time laws of creep which can represent many creep curves. This equation expressing the variation of creep rate, where j = steady-state (secondary) creep rate. When the e_; with time, t, for a given stress is: Stage I creep is over, the primary creep rate is practically zero and creep becomes almost viscous. At temperatures e_ ¼ C1tÀp ð7:12Þ near the melting point of material, the creep curve becomes mainly linear at low stresses, which has been found for Al, where C1 and p are empirical constants and the value of Au, Cu and d-Fe at a homologous temperature of 0.96–0.99 p varies from 0 to 1. Generally, p decreases with increasing (Garofalo 1965). In such cases, primary creep basically does temperature and stress. Depending on the values of p, dif- not exist. If primary creep is non-existent or assumed to be ferent types of creep behaviour can be described by (7.12). complete, then the creep curve mainly shows steady-state When p ¼ 1; integration of (7.12) gives creep behaviour and is described by the relation e ¼ e1 þ jt ¼ e1 þ e_st ð7:9Þ e ¼ C1 ln t þ C ¼ c ln t þ C ð7:13Þ where e1 is the instantaneous true strain plus the primary e ¼ c ln t þ C ð7:4Þ creep, which is equal to the intercept made by the extrapo- lation of the linear part of the creep curve on the creep-strain where c ¼ C1; and C represents the creep strain at time t ¼ 1 axis, and j or e_s is the steady-state creep rate, which is (unit of time), since (7.13) is valid only when time t [ 0; measured by the minimum creep rate. For very long times similar to (7.4). Equation (7.13) describes logarithmic creep and low stresses, the instantaneous strain plus the primary found at low temperatures, which has been given by (7.4). In creep may be negligible compared to total steady-state creep, (7.12), at time t ¼ 0; the initial creep rate is infinitely high, es, i.e. e1 ( es; then e1 can be neglected and (7.9) can be which is unacceptable as discussed earlier. To eliminate this written in the simpler form objection, t is replaced by t þ t1 in (7.12) for p ¼ 1; which then becomes e ¼ e_st ð7:10Þ When Stage II creep lasts for thousands of hours, whereas e_ ¼ ðt C1 ð7:14Þ first and third stages of creep last only a few hours or per- þ t1Þ haps only minutes, the creep curve can be approximated by (7.10) and in such cases, the only variable required to where the initial creep rate at t ¼ 0 is C1=t1 ¼ cm (as shown describe the creep behaviour is the minimum creep rate. in 7.6), and t1 is an arbitrary time which is equal to 1=m: Now, integration of (7.14) gives Wyatt (1953) has successfully combined (7.4), (7.7) and (7.8) through his experimental findings on polycrystalline e ¼ C1 lnðt þ t1Þ þ C2 ð7:15Þ copper tested over the temperature range of 77–443 K at a

7.3 Strain–Time Relations 265 Since at t ¼ 0; the creep strain e ¼ 0; so from (7.15) we 7.4 Creep Rate–Stress–Temperature Relations get C2 ¼ ÀC1 ln t1; and (7.15) becomes the same as that given by (7.6): The total elongation and creep life of a material are strongly dependent on the magnitude of the steady-state creep rate, e_s;  which in turn is strongly influenced by the external variables, t þ1 stress and temperature. It is believed that thermal energy can e ¼ C1 ln t1 ¼ c lnð1 þ mtÞ ð7:16Þ activate or aid in the motion of dislocation. Let us consider a deformation mechanism acting locally under the influence of where c ¼ C1; and m ¼ 1=t1: When 0\\p\\1; integration of temperature but in the absence of a stress. In such case, (7.12) gives parabolic creep as shown by (7.7), which is as thermal energy is not capable to cause movement of long follows: dislocation segment, but rather is only able to activate dis- location reaction on a limited scale. Similarly in creep, e ¼ C1 p Á t1Àp þ C3 ¼ e0 þ btm ð7:17Þ thermal energy can activate the deformation mechanisms 1À that involve only a small number of atoms. So, for creep mechanism to operate the presence of stress is an important where b ¼ C1=ð1 À pÞ ¼ C1=m; m ¼ 1 À p; and e0 ¼ C3 ¼ factor and the majority of investigations have been limited to instantaneous true strain observed upon loading. When p ¼ the determination of stress dependence of steady-state creep 2=3 in (7.12), i.e. m ¼ 1=3 in (7.17), we obtain the rate. Andrade’s laws of transient creep (Andrade 1914), Hence, from a hypothetical model of the effect of stress e ¼ e0 þ bt1=3 ð7:18Þ on the activation energy of a thermally activated creep mechanism, the general form of creep equations can be When p ¼ 0; we get from (7.12) the steady-state creep as developed (Reed-Hill 1973). For a creep mechanism to shown by (7.9), which is as follows: operate, an energy barrier of the type shown in Fig. 7.6 must be overcome. For a material to undergo a small unit strain in e ¼ C1t þ C4 ¼ e1 þ jt ¼ e1 þ e_st ð7:19Þ the direction from point A to point C, an atom must be shifted from the free-energy minimum position at A over the where e_s ¼ j ¼ C1; and e1 ¼ C4 ¼ instantaneous true strain activation energy barrier at B to another free-energy mini- plus primary creep. Since steady-state creep is generally mum at C. The reversed movement of the atom would produce a unit strain in the opposite direction. Thermody- preceded by parabolic or transient creep, so replacement of namically, more correct approach would be to express the e1 in (7.19) by e from (7.17) gives (7.8). energy barrier in terms of free energy. Since the knowledge about the entropy functions of activation processes is very Garofalo (1960, 1965) proposed a creep–time relation little, so for simplicity neglecting entropy, the energy barrier that provides better fit than Andrade’s equation, although is expressed by enthalpy change, i.e. DH ¼ Q; and Q is this relation has been tested for ferritic and stainless steels in called activation energy. Now if a given stress acts in the the homologous temperature range of 0.4–0.6. The creep direction from A to C and the strain is in the direction of the strain, e; is given in the following form: stress, positive work is performed. This effectively lowers the height of the energy barrier for deformation in the e ¼ e0 þ etð1 À eÀrtÞ þ e_st ð7:20Þ direction from A to C, while raises it for deformation in the direction from C to A. Let the changes in energy, due to where stress, from state A to activated state B and from activated state B to state C are, respectively, W1 and W2: As shown in e0 the instantaneous true strain observed upon loading, Fig. 7.6b, the height of the activation energy barrier is then et the limiting transient creep strain, Q À W1 for the change of state from A to C and is Q þ W2 r the ratio of transient creep rate to the transient creep for the change of state from C to A, where Q is the height of the activation energy barrier in the absence of a stress. strain, Hence, the respective frequency with which the mechanism e_s the steady-state creep rate operates from A to C and C to A is: The second term in the right-hand side of (7.20) gives transient creep that differs from primary creep in which all of the time-dependent strain is included. The initial creep rate at t ¼ 0 is given by ðde=dtÞt¼0 ¼ ½ÀeteÀrtðÀrÞ þ e_sŠt¼0 ¼ ðreteÀrt þ e_sÞt¼0 ¼ ret þ e_s; that avoids the objection associ- ated with (7.4). Creep behaviour that satisfies (7.20) is called exponential creep.

266 (a) B 7 Creep and Stress Rupture Q Q (b) B Fig. 7.6 Hypothetical model A C showing the effect of stress on the Q – W1 Q + W2 activation energy of a thermally (c) B activated creep process; a before W1 W2 application of stress; b after A C application of stress; c continuous operation of creep mechanism by passing over a series of energy barriers D F A C E G A to C: m1 ¼ m0 exp½ÀðQ À W1Þ=kTŠ ð7:21Þ sinh W ¼ eW=kT À eÀW=kT ; kT 2 C to A: m2 ¼ m0 exp½ÀðQ þ W2Þ=kTŠ ð7:22Þ With the above definition, (7.23) becomes where m1 and m2 are respective frequencies of the mecha- e_s ¼ A0eÀQ=kT 2 sinh W ð7:24Þ nism, m0 is a constant which is assumed to have the same kT value for both directions of the mechanism, k is Boltzmann’s constant, and T is the absolute temperature. In (7.24), the temperature dependence of steady-state creep rate is represented by an exponential factor, which is eÀQ=kT ; After passing over the energy barrier at B and reaching and the stress dependence by a hyperbolic sine function whose the state C, it is possible for the deformation mechanism to argument is a function of stress, because the value of W de- operate again in the same sense and pass over a similar pends on the applied stress, r: The simplest assumption is energy barrier at D for going from state C to state E, and then continue on in this same manner over a series of bar- W ¼ vÃr ð7:25Þ riers, as indicated in Fig. 7.6c. In general, the frequency corresponding to strain in the direction of stress will be where và is a constant that depends on the nature of mech- larger than that in the direction opposite to the stress. If a anism and is called the activation volume. Activation volume large number of identical mechanisms are capable of oper- is the average volume of dislocation structure involved in the ating in this same fashion, then the net rate, e_s; at which deformation process and is given by the product of the creep strain occurs will be proportional to the difference length of dislocation segment involved in the thermal fluc- between the forward and reverse frequencies, thus tuation, the Burgers vector and the distance the atoms move during the process. Hence, (7.24) becomes hi e_s % m1 À m2 % m0 eÀðQÀW1Þ=kT À eÀðQ þ W2Þ=kT Or, e_s ¼ A0eÀQ=kT Á 2 sinh vÃr ð7:26Þ  kT e_s ¼ A0eÀQ=kT e þ W1=kT À eÀW2=kT ð7:23Þ For a very small applied stress, sinh vÃr % vÃr ; and kT kT where A0 is a constant that contains m0: Let us simplify (7.23) by assuming W1 ¼ W2 ¼ W: Recalling the definition (7.26) becomes of the hyperbolic sine function, we can write e_s ¼ A0eÀQ=kT Á 2vÃr ð7:27Þ kT

7.4 Creep Rate–Stress–Temperature Relations 267 Equation (7.27) shows that when the applied stresses are Again, at constant temperature, (7.29) can be written as very small, the creep rate is directly proportional to the applied stress. On the other hand, if Q is large compared to ðe_sÞconst: temperature ¼ A2ðsinh arÞn ð7:31Þ kT, and the stress is also large, the backward frequency m2 is negligible compared with the forward frequency m1 of the where A2 and a are constants at constant temperature. The deformation mechanism. In such case, the deformation dimension of a is inverse to that of stress r: At low stress mechanism may be assumed to act only in the forward levels, where values of ar\\0:8; it can be assumed that (stress-aided) direction and thus, the creep rate becomes sinh ar % ar; and then (7.31) reduces to a power depen- directly proportional to m1: Hence, we may write dence of stress as follows: e_s ¼ A0eÀðQÀWÞ=kT ð7:28Þ ðe_sÞconst: temperature; low stress ¼ A2ðarÞn¼ ðA2anÞrn ¼ A02rn ¼ A0eÀQ=kT eW=kT ð7:32Þ ¼ A0eÀQ=kT evÃr=kT where A02 is independent of stress, but dependent on tem- The assumption that Q ) kT is quite reasonable because perature. n is independent of stress and, to some extent, the energy barrier is around 40 times larger at room temper- depends on temperature. For annealed metals and alloys, n is ature and 10 times larger at 900 °C than the thermal energy. found to vary from 1 to 7 and seems to be independent of crystal structure. For Au, Ag, Cu and d-Fe, n approaches 1 at Garofalo (1963) proposed an empirical relation showing temperatures near the melting point under conditions where the temperature and stress dependence of steady-state creep creep is controlled by stress-induced migration of vacancies. rate, e_s; as follows: In most other cases for high-temperature creep n remains between 2 and 4 for alloys and between 4 and 6 for pure e_s ¼ Aðsinh arÞneÀQ=RT ð7:29Þ metals, although there are a few deviations from these variations. An example of the change in n with alloying is where the stress, r; is held constant. A, a; and n are exper- that n = 5.6 for both Ni and Au, but upon alloying, n drops imentally determined constants and Q is an activation rapidly from this value and reaches a minimum value of energy. Equation (7.29) is the most generally applicable roughly 3 (Garofalo 1965). relation, which was also adopted by Sellars and Tegart (1966) for hot-working studies. Let us consider (7.29) sep- Sherby (1962) has shown that the steady-state creep rate arately at constant stress and at constant temperature test for a number of pure metals follows a power dependence of conditions. At constant stress, (7.29) becomes stress, as given by (7.33). This power relation shows that n = 5, which is in agreement with the range of n-values for ðe_sÞconst: stress ¼ A1eÀQ=RT ð7:30Þ pure metals, as mentioned above. where e_s ¼ CSd2 r5 ð7:33Þ D A1 the pre-exponential term, which is constant at constant stress. It includes frequency of vibration of the flow E unit, the entropy change and a factor that depends on where the structure of the material, which includes the e_s the steady-state creep rate (h−1), number, distribution, and length of dislocations, type CS a constant (1029 cm−4), d the grain diameter (cm), and dispersion of precipitates, grain size and perhaps D the self-diffusion coefficient (cm2/s), r the creep stress (psi), other geometrical details, E the elastic modulus (psi). Q the activation energy for the rate-controlling process in At high stress levels, where values of ar [ 1:2; (7.31) creep (J mol−1), reduces to an exponential dependence of stress. Since R the universal molar gas constant, 8.314 J mol−1 K−1. sinh ar ¼ ðear À eÀarÞ=2; and at high stress levels ear ) Note that R = kN, where k is Boltzmann’s constant and eÀar; so eÀar can easily be neglected. Thus, we get N is Avogadro’s number. T the test temperature (K)

268 7 Creep and Stress Rupture sinh ar % ear=2; which may be substituted in (7.31) to 7.5 Steady-State Creep obtain the exponential relation of stress: 7.5.1 Effect of Grain Size ðe_sÞconst: temperature; high stress ¼ earn The effect of grain size on steady-state creep rate has been A2 2 studied to a great extent. Equation (7.33) indicates that steady-state creep rate increases with increasing grain ¼ A2 expðnarÞ ð7:34Þ diameter. Parker (1958) has also shown that the steady-state 2n creep rate, e_s; increases as the grain size is increased for copper at temperature of 0:5Tm; where Tm is the melting ¼ A200expðbrÞ point of material on the absolute scale. Further, this is found by other investigators (Feltham and Meakin 1959) to be true where A020 and b are independent of stress, but found to be at 0:6Tm and e_s is shown to be proportional to d2; where d is dependent on temperature. the diameter of grain. Later on, e_s / d2 is also observed for brass at 0:6Tm (Feltham and Copley 1960). But a different However, (7.31) expressing the stress dependence of behaviour is exhibited by lead at 25 °C (0:5Tm) (McKeown steady-state creep rate by a hyperbolic sine function is a 1937) that shows e_s / 1=d; i.e. as d increases e_s decreases. single stress function that has been observed to satisfy the However, in many instances, at temperatures greater than experimental results over the entire stress range from a low to about half of the homologous temperature coarse-grain-size a high level used in the tests (Garofalo 1963). This means that materials exhibit lower steady-state creep rates and higher when log e_s is plotted against log sinh ar; linearity will be creep resistances than fine-grain-size materials. In several observed over the entire stress range. For simplicity, the metals and alloys, for example, tin at 25 °C (0:6Tm) (Hanson power relation given by (7.32) can be used at low stress 1939), Fe–Cr–Ni–Mn alloy at 704 °C at various stresses levels, because the plot of log e_s versus log r shows linearity (Garofalo et al. 1964), monel at various temperatures and at low stress levels and gradually deviates from linearity as stresses (Shahinian and Lane 1953), it has been observed the stress increases. Similarly, the exponential relation given that e_s is minimum at a critical value of d and e_s increases as by (7.34) agrees well with the test results at high stress levels, d is decreased or increased from this critical value. This where the plot of log e_s versus r shows a linearity and as the general behaviour of variation e_s with d is shown schemat- stress is decreased, there is a gradual deviation from linearity. ically in Fig. 7.7 at a constant temperature at various stresses The temperature dependence of steady-state creep rate (7.30) and in Fig. 7.8 at various temperatures and stresses. The can be combined separately with the power dependence of variation of e_s with d can be described by the following stress (7.32), and the exponential dependence of stress (7.34) relation which was found to be fitted to the test results for to yield the following equations for steady-state creep rate, e_s; both Fe-alloy and monel. respectively at low and high stress levels: ðe_sÞlow stress ¼ A0rneÀQ=RT ð7:35Þ ðe_sÞhigh stress ¼ A00ebreÀQ=RT ð7:36Þ Fig. 7.7 Schematic curves Secondrary creep rate (log scale) Temperature is constant, but stresses vary from σ1 to σ4. showing variation of secondary σ1 > σ2 > σ3 > σ4 creep rate with grain size at various stresses at a constant σ1 temperature (Garofalo et al. 1964) σ2 σ3 σ4 Grain diameter (log scale)

7.5 Steady-State Creep 269 Stress, σ1 > σ2 > σ3 > σ4 levels, whereas Fig. 7.7 shows that the values of dm remain Temperature, T1 < T2 < T3 < T4 nearly constant at a constant temperature at various stress levels. These indicate that the values of dm are weakly σ1 , T1 dependent on stress, from which it can be concluded that the increase in dm in Fig. 7.8 is due to the increase in temper- Secondary creep rate (log scale) σ2 , T2 ature. From Fig. 7.8, it is seen that dm is very small at low σ3 , T3 creep temperatures. Hence, we can say e_s / d2 at low creep temperatures, since we can substitute 2dm3 þ d3 % d3 in the σ4 , T4 numerator of (7.38). On the other hand, at high creep tem- peratures, where dm is large, we may neglect d3 term in the Average grain diameter (log scale) numerator of (7.38), since 2dm3 ) d3; and thus at high creep temperatures, e_s will be essentially proportional to 1/d. At Fig. 7.8 Schematic curves showing secondary creep rate as a function intermediate creep temperatures, (7.38) will predict a mini- of grain size at various stresses and temperatures (Shahinian and Lane mum in e_s: It has been found (Pranatis and Pound 1955) that 1953) in copper, near its melting point, where stress-induced migration of vacancies controls creep, e_s / 1=d2; instead of e_s ¼ k þ Kd2 ð7:37Þ creep rate varying as 1/d. d As grain-boundary sliding can add to the creep defor- where k and K are constants at constant temperature and mation and increase the creep rate, one simple explanation for lower creep rate observed in a coarse-grain material at stress. Let dm is the critical grain diameter of the material at high creep temperatures is that a coarse-grain material has which e_s is minimum. The above relation can be more smaller grain boundary areas per unit volume available for sliding than a fine-grain material. Parker (1958) has pro- conveniently expressed by differentiating (7.37) with respect posed that the change in grain-boundary structure associated with different grain-coarsening treatments is the primary to d at the point of minimum in the curve, where d ¼ dm; reason for variation of creep rate with grain size. Since and setting the differential to zero for the minimum as high-temperature creep is dependent on dislocation climb, the creep rate increases with increase in the rate of diffusion follows: of vacancy to edge dislocation. Vacancy can diffuse more rapidly along high-energy grain boundaries than through the de_s! o2rd;m3 Àþddkm23!þ 2Kdm interior of the grains. Vacancy diffusion is rapid in a dðdÞ d fine-grain-size material having more number of high-angle ¼ 0; ¼ 0; or; k ¼ 2Kdm3 grain boundaries and so, the creep rate is higher. When the same material is subjected to high temperature for coarsen- d¼dm ing of grains, the grain growth will cause most of the high-energy grain boundaries to vanish. Hence, material ) e_s ¼ K with coarser grain will have lesser grain boundary areas per unit volume and mostly lower-energy grain boundaries for which vacancy diffusion is relatively slow. Therefore, the coarse-grain material will result in a lower creep rate and higher resistance to creep. For example, when aluminium is added for deoxidation of steels, it refines the grain size, and thus, creep properties are generally poor in aluminium-deoxidized steels than in silicon-deoxidized steels having relatively coarse grains. ð7:38Þ Since (7.38) fits to both curves shown in Figs. 7.7 and 7.8 7.5.2 Activation Energy and e_s increases with increasing stress over the whole range of grain diameter, d, these indicate that K is strongly Steady-state creep becomes dominant at temperatures above about half of the absolute melting point of the material. dependent on stress. Figure 7.8 shows that the values of dm Since creep is dependent on the thermally activated increase with increasing temperature and decreasing stress

270 7 Creep and Stress Rupture processes, so it is necessary to examine the role of temper- temperature change is considered to be the only reason for ature on creep mechanisms. More than one mechanism is the change of creep rate. If creep rates, e_1 and e_2; are mea- likely to operate in creep at a given time. If these various sured before and just after the temperature change, then mechanisms are assumed to depend on each other, the one with the slowest rate that requires the highest activation (7.30) can be applied to determine Q as follows: energy will be the rate-controlling mechanism. If, however, the operations of these mechanisms are independent of each A1 ¼ e_1eQ=RT1 ¼ e_2eQ=RðT1 þ DTÞ; other, the one with the fastest rate that requires the lowest activation energy will be the dominant mechanism. In the Or;    simplest way, creep may be assumed to be due to a single activated process. In such conditions, the creep rate can be R ln ¼ee__ 21R¼lnQee__ 12 T11ðTÀð1TTþ1 1þDþ1DTDTÞTÞÀT1T1 ! ¼  e_2T12  expressed by an Arrhenius-type rate equation, which has ) Q R ln e_1 þ T1DT already been shown by (7.30) and is repeated below for DT convenience of reference: ð7:39Þ ðe_sÞconst: stress ¼ A1eÀQ=RT ð7:30Þ Employing temperature differential tests of the type shown in Fig. 7.9, Dorn determined the activation energies For a given material with the same structure, if log e_s is plotted against 1/T a series of parallel straight lines, one for for polycrystalline aluminium over the temperature range each stress level, will result, provided (7.30) is effective. The slope of these lines will be 0.4343Q/R. from 75 K to near its melting point. It was found that Q in- creased with increasing temperature from 75 K to about Using temperature differential tests of the type shown in 233 K, where Q reached a value of 117 kJ mol−1. Between Fig. 7.9, Dorn (1957) developed method to determine acti- vation energy Q. He assumed that the term A1 in (7.30) that 233 and approximately 373 K, the activation energy includes the dislocation structure remained substantially remained approximately constant around the value of constant with small changes in temperature. The assumption 117 kJ mol−1. Above 373 K, Q increased again, reached a that the dislocation structure remains constant during the value of 146.45 kJ mol−1 at 467 K and then remained change of temperature is quite reasonable because disloca- constant up to near the melting point. The increase in acti- tions are not easily formed by thermal activation. Further, it vation energy from a value of 117–146.45 kJ mol−1 between is assumed that a single creep-rate-controlling mechanism 373 and 467 K implies that the two creep mechanisms are prevails over the small temperature range considered. A creep test is carried out at a temperature T1 at a constant both operative and interdependent in this temperature range. stress r: After a suitable creep time t, or attaining an The activation energy value of 146.45 kJ mol−1 has been equivalent creep strain e; the temperature is changed by a found to be equal to that for self-diffusion in Al (Dorn 1957). small amount DT to T1 þ DT; which causes to change the creep rate immediately from e_1 at T1 to e_2 at T1 þ DT: The It was also found that if the ratio of the activation energy for same metallurgical microstructure is assumed to be present before and after the change of creep rate. Thus, the creep to that for self-diffusion is plotted against the homol- ogous temperature, the curve obtained for polycrystalline pure aluminium coincides with that for polycrystalline OFHC copper (Landon et al. 1959). An alternative method to determine the activation energy is to carry out creep tests at two separate temperatures not too widely separated, say T1 and T2; but at the same stress (a) (b)Log10 ⑀ σ = Const. σ = Const. ⑀ 2 T1 + ΔT ⑀2 ⑀ ⑀1 T1 T1 + ΔT T1 ⑀1 t⑀ Fig. 7.9 Method to determine activation energy in creep from temperature differential tests carried out at a constant stress r; by changing temperature from T1 to T1 þ DT; which is demonstrated by a strain–time plot and by b log of strain rate versus strain plot

7.5 Steady-State Creep 271 level, where it is assumed that the structure will remain the at the transition temperature, i.e. e_a ¼ 200 e_c: This correlates same at both the temperatures for a given strain during with self-diffusion coefficient in a-iron; Da; being 350 times steady-state creep. This method is based on Dorn’s obser- higher than that in c-iron; Dc; i.e. Da ¼ 350 Dc: Thus, FCC vation, which is: c-iron is more creep-resistant than BCC a-iron under com- parable conditions. Further, the self-diffusion coefficient in h ¼ teÀQ=RT ð7:40Þ alpha iron increases with carbon content, which causes to increase the creep rate and decrease the creep resistance. where h is called a temperature-compensated time parame- ter and t is the time of creep exposure. Creep curves deter- It is important to know whether the magnitude of acti- mined at several different temperatures but at the same stress vation energy for creep, Q, depends on any of the metal- can be represented by a single common creep curve when lurgical variables or not. Investigations for aluminium, creep strain, e; is plotted against h; i.e. teÀQ=RT ; provided a copper (Landon et al. 1959), magnesium, stainless steel single creep-rate-controlling mechanism prevails. An (Garofalo et al. 1963) and several other alloys have shown important characteristic of (7.40) is that equivalent structure that both at low and high temperatures, Q is independent of is obtained for the same values of e and h: To reach a given creep strain. For creep deformation at high temperatures, strain, e; if the times of creep tests carried out under the same where the activation energy for creep is very close to that for applied stress at temperatures T1 and T2 are, respectively, self-diffusion, the effect of stress on Q seems to be reason- determined as t1 and t2; then from (7.40) ably small. The activation energy is independent of defor- mation temperature, if the temperature is sufficiently high, h1 ¼ h2 ¼ t1eÀQ=RT1 ¼ t2eÀQ=RT2 ð7:41Þ but Q changes significantly at lower temperatures. At tem- peratures where the activation energy for creep is essentially Since e ¼ f ðhÞ; so for a given strain e; the values of h are equal to that for self-diffusion, Q is found to be independent of grain size and essentially the same for single crystals and equal, i.e. h1 ¼ h2: Hence, it follows from (7.41) that polycrystals of Al, Cu, Mg, Sn and Pb.  T1 Á T2  Q ¼ R ln t1 T2 À T1 t2 ð7:42Þ This method was used satisfactorily by Dorn for alu- 7.6 Creep Deformation Mechanisms minium in the temperature range 424–531 K at a constant stress of 20.7 MPa, which showed activation energy of The main deformation processes at elevated temperature are: 142 kJ mol−1. • Slip, which includes multiple slip, the formation of An extensive correlation (Sherby et al. 1954) of creep and coarse slip bands, involvement of operation of new slip diffusion data shows that at high temperatures, where system, etc. steady-state creep predominates, the activation energy in most cases is equal to or nearly equal to the activation en- • Subgrain formation as shown in Fig. 1.16, where excess ergy for self-diffusion. This indicates that the dislocations of one sign arrange themselves into a creep-rate-controlling mechanism is diffusion controlled. low-angle grain boundary by dislocation climb process Significant phenomena involved in steady-state creep are the that readily occurs at high temperature. During primary formation and movement of vacancies and the formation of a creep, the subgrain structure or cell structure forms and dislocation subgrain structure. These strongly support the the dislocation density of the subgrain network increases view that dislocation climb is the rate-controlling mecha- to a certain level, which remains essentially constant nism in high-temperature creep. In general, metals with during steady-state creep. The size of subgrain depends lower self-diffusion coefficients will show higher creep on the applied stress and deformation temperature. High resistance. Since the activation energy for self-diffusion, temperature and a low stress or creep rate produce large Q (assuming to be the same as that for creep), is the sum of subgrains. Metals of high stacking fault energy readily the energies for vacancy formation, QF; and movement, QM; form a subgrain structure. i.e. Q ¼ QF þ QM; it is expected that metals in which vacancy movement is restricted, i.e. QM is high, will have • Grain-boundary sliding. It is a shear process occurring in more resistance to creep. The higher the QM; the higher is the direction of grain boundary, causing the movement of the Q, and the lower is the self diffusion coefficient, D, since grains relative to each other in polycrystals. Grain D / eÀQ=RT : Hence, BCC metals in which diffusivity is boundaries lying at about 45° to the applied tensile stress higher than in FCC metals will have less creep resistance. will experience the maximum shear stress and slide the Sherby (1962) has indicated that the creep rate for BÀCCÁ most. It is encouraged by decreasing the strain rate and/or a-ironðe_aÞ is 200 times higher than that for FCC c-iron e_c increasing the temperature. The strain due to grain-boundary sliding may vary from only a few per

272 7 Creep and Stress Rupture cent to as high as 50% of the total strain, depending on energy (Sherby and Burke 1967; Mukherjee et al. 1969), the test conditions and the material. G is the elastic shear modulus, b is the burgers vector, k is At low homologous temperatures, the dominant defor- the Boltzmann’s constant, T is the temperature on absolute mation mechanisms in crystalline solids are slip and twin- scale, D0 is the frequency factor, Q is the activation energy ning. The mechanism of low-temperature creep is for creep, d is the grain size, p0 is the inverse grain size dislocation glide, which involves the movement of disloca- exponent, r is the applied stress, and n0 is the stress expo- tion along slip planes and the overcoming of obstacles by nent. Specific creep mechanisms have specific values of n0: thermal activation. Creep resulting from this mechanism Equation (7.45) shows that stress and temperature are the occurs at high stress where r ! 10À2G: These stress levels are higher than those generally considered in creep defor- two main external variables, on which the creep rate mation. The creep rate depends on the hindrance of dislo- cations by barriers such as other dislocations, solute atoms depends. Note that the last three-bracketed terms on the and precipitates. right-hand side in (7.45) are dimensionless quantities. It is However, at intermediate and high homologous temper- atures where creep predominates, the three basic mecha- commonly observed that the activation energy for creep is nisms that contribute to creep in metals are: equal to that for diffusion; hence, the term D0 expðÀQ=kTÞ in (7.45) is replaced by the relevant diffusion coefficient, D, (Sherby and Burke 1967; Mukherjee et al. 1969) and (7.45) takes the following form: 1. Dislocation creep that involves dislocation glide and e_s ¼ B DGb bp0  r n0 ð7:46Þ climb. kT dG 2. Diffusional creep that involves stress-assisted diffusional 7.6.1 Dislocation Creep or Climb–Glide Creep flow of atoms and vacancies. Dislocation creep involves the dislocation glide coupled with 3. Grain-boundary sliding. dislocation climb that assists to overcome barriers by a process of diffusion of vacancies or interstitials. This occurs Most often several creep mechanisms operate simulta- at intermediate and high homologous temperatures and at intermediate stresses, where normalized stress is such that neously. If more than one mechanism operates indepen- 10À4\\r=G\\10À2: The stress dependence of climb–glide creep is stronger than that of diffusional creep. dently of each other, i.e. they operate parallelly, then the When metals are deformed at elevated temperature, slip total steady-state creep rate is given by system that was not active at room temperature may become operative and dislocation glide is promoted. When a X ð7:43Þ well-annealed material is initially subjected to a stress, the e_s ¼ e_i movement of dislocations will be rapid as few obstacles are present to resist their motion. However, dislocations multi- i ply rapidly and cause strain hardening which subsequently decreases the creep rate during the primary transient creep where e_i is the creep rate for ith mechanism. For parallel stage. This strain hardening occurs as the number of dislo- mechanisms, the fastest one will control or dominate the cations increases and they start to act as barriers to glide creep deformation. If there are i number of mechanisms that motion of other dislocations. On encountering an obstacle, operate sequentially, i.e. operate in series, then the total dislocations are blocked and tend to pile up at the barrier. At steady-state creep rate is given by low applied stress levels, dislocations are unable to bow around or cut through the obstacle. However at elevated 1 ¼ X 1 ð7:44Þ temperature, an edge dislocation may climb upwards or e_s e_i downwards to a parallel slip plane. Climbing occurs by i diffusion of vacancies or interstitial atoms through a crystal lattice to or away from the edge dislocation. Compressive For series mechanisms, the slowest one will control or stress in the direction normal to the incomplete plane causes dominate the creep deformation. upwards climb of dislocation, whereas downwards climb is caused by tensile stress acting normal to the incomplete Creep of various materials exhibiting a variety of mech- plane. After climbing, the dislocation glides along the new anism can be described by the following Mukherjee–Bird– slip plane until it encounters another resisting obstacle, Dorn equation (Mukherjee et al. 1969) which expresses the steady-state creep rate in terms of stress, temperature and grain size as: e_s ¼ BGb D0  Q bp0 rn0 ð7:45Þ kT exp À kT d G where B is a dimensionless material constant that includes additional effects of microstructure such as stacking fault

7.6 Creep Deformation Mechanisms 273 where the dislocation again climbs up or down to another supports the concept that the rate of power law creep is parallel slip plane and the process is repeated. Thus, creep controlled by lattice self-diffusion at homologous tempera- occurs by the sequential processes of dislocation glide and tures greater than or equal to 0.5. Hence in (7.47), the rel- evant diffusion coefficient, D, is to be substituted by the climb. During the primary transient creep stage, the process lattice or bulk self-diffusion coefficient DL: Hence, consid- ering n0 ¼ 5 (the most common value) for power law creep, of climbing makes the dislocation structure gradually orga- (7.47) can be presented as nized into low-angle boundaries that define subgrains within the grains. This substructure becomes more stable as the creep deformation approaches the steady state. Note that DLGb  r n ¼ DLGb  r 5 kT G kT G subgrain structures do not form in glide-controlled creep. In e_s ¼ B B ð7:48Þ climb–glide creep, almost all of the creep strain is produced by the glide step, whereas the average velocity of dislocation However, at lower homologous temperatures lying is controlled by the climb step. As the climb step is slower between 0.25 and 0.5, the activation energy for creep falls to than the glide, so the rate-controlling step is the dislocation lower values and generally corresponds to the activation climb, i.e. the diffusion of vacancies or interstitials. energy for vacancy diffusion along dislocation cores. In this For dislocation climb–glide creep, the value of inverse temperature range, the creep rate depends on the dislocation grain-size exponent p0 in (7.46) is zero, because this type of density that serves as diffusion paths for this creep is independent of grain size. The stress dependence of low-temperature climb. Since the density of dislocations is steady-state creep rate in climb–glide creep exhibits a power– directly proportional to the square of stress [see (1.58a)], this law relation, which is obtained by substituting p0 ¼ 0; in leads to an effective diffusion coefficient, Deff; given by (7.46):  rn0  r 2 G / e_s ¼ B DGb ð7:47Þ Deff ¼ DL þ Dcore ð7:49Þ kT G Theoretical treatments can yield a value of 3 or 4 for n0 where / is a constant which is equal to about 10, and Dcore is the self-diffusion coefficient in the dislocation core. Substi- and this is often termed the natural creep law. The value of tution of Deff from (7.49) for DL in (7.48) shows that the n0 is a function of crystal structure and stacking fault energy expected most common stress exponent in this temperature and n0 increases with decreasing stacking fault energy. It has region is n0 ¼ 7; which arises as a result of the above been seen that n0 may vary from 3 to 8. additional stress dependence. Thus, we see that the lattice At high homologous temperatures greater than or equal to diffusion is dominant at high homologous temperatures, 0.5, it is observed for a variety of materials that n0 ¼ 5 is the whereas core diffusion becomes dominant at intermediate most common in the power law region which agrees well homologous temperatures, in which the steady-state creep rate varies as rn0 þ 2 instead of rn0 : with (7.33), and accordingly, this regime is often called The plot of the steady-state creep rate, normalized with power law creep. For a wide variety of materials, it is seen lattice diffusivity, against the stress, normalized with the (Nix and Gibeling 1985) that the activation energy for creep, shear modulus, is shown in Fig. 7.10. Power law creep Q, is equal to that for lattice self-diffusion, QL at homolo- gous temperatures greater than or equal to 0.5. This evidence Fig. 7.10 Schematic plot of 1018 steady-state creep rate, normalized with diffusivity, 1014 Power-law versus stress, normalized with breakdown shear modulus, showing the effect of the stress on the steady-state 10–2 creep rate εs 1010 Harper-Dorn Power-law DL creep creep 106 5 1 102 1 10–7 1 10–4 10–6 σ/G

274 7 Creep and Stress Rupture occupies the intermediate region of the stress. At a stress of vacancies from grain boundaries experiencing tensile stres- r [ 10À3G; the creep rate starts to increase more rapidly ses to those which are subjected to compressive stresses. Simultaneously, there is a corresponding migration of atoms with applied stress and the measured creep rates are higher or ions in the opposite direction. Over a period of time, this stress-assisted flow of atoms leads to the elongation of grains than that predicted by (7.48). This is the power law break and test specimen in the direction of applied tensile stress and the contraction of grains and test specimen in the down region, where the creep rate is mathematically transverse direction resulting in creep. The grain size is important in diffusional creep because the grain boundaries described by hyperbolic sine function as shown by (7.29) or serve as sources and sinks for the diffusing vacancies. So, it is necessary to include grain-size dependence in the rate by an exponential relation as shown by (7.36). According to equation of diffusional creep. In diffusional creep, vacancy diffusion can occur either through the grain interiors or Weertman (1957), the onset of exponential creep at high through the grain boundaries. Depending on the path of diffusion of vacancies, two mechanisms operate in the region stress levels is related to accelerated diffusion, which occurs of diffusional creep, and accordingly, there are two types of diffusional creep: due to an excess vacancy concentration caused by disloca- tion–dislocation interactions. At stresses lower than r ¼ 5 Â 10À6G; but at high homologous temperatures and for large grain sizes, the steady-state creep rate for a dislocation creep mechanism is found to be proportional to the applied stress, i.e. the stress exponent, n0 ¼ 1: This was first noted in aluminium by Harper and Dorn (Harper and Dorn 1957; Harper et al. 1958) and is called Harper–Dorn creep. They observed that the rates of this linear viscous creep were much higher than • Nabarro–Herring creep. • Coble creep. those possible by diffusional creep. Harper–Dorn creep is believed to be due to climb-controlled creep under condi- tions where the dislocation density is independent of stress, 7.6.2.1 Nabarro–Herring Creep i.e. does not change with stress and remains constant at a low In Nabarro–Herring creep (Nabarro 1948; Herring 1950), value of about 108/m2. The Harper–Dorn creep rate is given diffusion of vacancies and atoms or ions occurs through the by the following relationship, which is obtained by substi- grain interiors since it is the shortest path of diffusion. tuting the inverse grain-size exponent p0 ¼ 0; diffusion coefficient D ¼ DL and the stress exponent n0 ¼ 1 in (7.46): Nabarro–Herring creep is favoured at high homologous temperatures. Figure 7.11 shows the mechanism of r Nabarro–Herring creep, in which stress-directed flow of G e_ HD ¼ BHD DLGb ð7:50Þ vacancies from tensile to compressive grain boundaries is kT indicated by solid arrow lines, whereas the dashed arrow where experimental value of BHD for aluminium is lines indicate the corresponding reverse flow of atoms or 5 Â 10−11 (Harper et al. 1958). Equation (7.50) shows that the rate of Harper–Dorn creep is directly proportional to ions. The Nabarro–Herring creep rate is given by the fol- stress. Note that at low stresses and for large grain sizes, lowing relationship, which is obtained by substituting the inverse grain size exponent p0 ¼ 2; diffusion coefficient D ¼ Harper-Dorn creep will dominate when the diffusional creep DL and the stress exponent n0 ¼ 1 in (7.46): is suppressed by a large grain size. DLGb b2  r  kT dG e_ NH ¼ BNH ð7:51Þ 7.6.2 Diffusional Creep where BNH % 7 (Hertzberg 1989). Equation (7.51) shows that the Nabarro–Herring creep rate varies as dÀ2; and is Diffusional creep is favoured and often becomes the domi- nating mechanism for fine grain sizes at relatively low directly proportional to stress. stresses where r 10À4G: The process of diffusional creep is controlled by stress-directed diffusion of atoms. A stress 7.6.2.2 Coble Creep changes the chemical potential of the atoms at the surfaces of Diffusion is very sensitive to temperature; at lower homol- the grains in a polycrystal. When a tensile stress is applied to ogous temperatures, the main diffusion path is via grain grain boundaries, it causes a decrease in energy for creation boundaries, because the activation energy for grain boundary of vacancies and thus increases vacancy concentration along diffusion is significantly lower than that for lattice diffusion grain boundaries subjected to tension while there is an and diffusion is more rapid through the grain boundary than increase in energy for formation of vacancies under com- through the grain body. When diffusion occurs along grain pression leading to less concentration of vacancies along boundaries, the mechanism is known as Coble creep (Coble compressed grain boundaries. In diffusional creep, this 1963). The mechanism of this creep is shown in Fig. 7.12, in concentration gradient of vacancies induces a flow of which the solid arrow lines indicate the grain boundary

7.6 Creep Deformation Mechanisms 275 Stress 7.6.3 Grain-Boundary Sliding Flow of The grain-boundary sliding that occurs at sufficiently high atoms or ions homologous temperatures is important in initiating inter- granular fracture (see Sect. 7.9) and this fracture-initiation Grain indicates the onset of tertiary creep. For occurrence of grain-boundary deformation without formation of cracks at Grain boundary Flow of grain boundaries, other deformation modes must be avail- vacancies able to obtain continuity of strain along the grain boundary. One of the processes to accommodate grain-boundary strain Grain boundary at elevated temperature is grain-boundary migration, in which the grain boundary moves normal to itself under the Grain action of shear stress and relieves the stress concentration. Flow of On the other hand, grain-boundary sliding is needed to atoms or ions prevent the formation of internal cracks or voids during diffusional creep of polycrystals. Hence, it may be stated that Stress grain-boundary sliding does not represent an independent deformation mechanism. In order to maintain grain conti- Fig. 7.11 Mechanism of Nabarro–Herring creep, in which guity during diffusional flow process, the grain-boundary stress-directed flow of vacancies from tensile to compressive grain sliding rate must exactly balance the diffusional creep rate. boundaries, is indicated by solid arrow lines, whereas the dashed arrow This situation is explained in Fig. 7.13. Note that the lines indicate the corresponding reverse flow of atoms or ions stress-directed atomic diffusion from grain boundaries under compression to those subjected to tension elongates grains diffusion of atoms from compressive to tensile grain and causes the grain boundaries to separate from one another. The separation of grains leads to the formation of boundaries and the corresponding reverse flow of vacancies internal cracks or voids. This can be prevented if the diffu- sional flow is accommodated by concurrent displacement of is indicated by the dashed arrow lines. Coble creep occurs at the grains via their sliding over one another so as to bring grains together. Diffusional creep and grain-boundary sliding homologous temperatures lower than and for grain sizes can therefore be considered to take place sequentially in smaller than those where Nabarro–Herring creep occurs. The which diffusional creep leading to grain separation is fol- lowed by grain-boundary sliding causing to ‘heal’ voids Coble creep rate can be obtained from (7.51), in which DL is between grains. Hence, one finds that Nabarro–Herring and replaced by dDgb=d: Coble creep models are themselves dependent on grain-boundary sliding. It may be concluded that Nabarro– e_ Co ¼ BCo DgbGb db2  r  Herring and Coble diffusional creep mechanisms are ‘iden- kT dd G tical with grain-boundary sliding with diffusional accom- db3  r  ð7:52Þ modation’ (Raj and Ashby 1971). DgbGb bd G ¼ BCo kT If the grain boundary is smooth on atomic level, the accommodation by sliding occurs readily and the creep rate where BCo % 50; d is the grain-boundary thickness, and Dgb is given by the sum of (7.51) and (7.52). However, grain is the grain-boundary diffusion coefficient. Equation (7.52) boundaries are not so uniform. Let us consider the effect of shows that the Coble creep rate is directly proportional to irregularity of grain boundary on grain-boundary sliding. stress and varies as dÀ3; i.e. it is more sensitive to grain size Figure 7.14a shows a sinusoidal form of ‘wavy’ grain than is Nabarro–Herring creep. Since smaller grain size has boundary in which perturbation height or amplitude is h and increased number of grain boundaries, so the Coble creep perturbation wavelength is k: The wavy grain boundaries that are often seen during high-temperature creep are pro- rate is expected to increase with decreasing grain size. duced due to grain-boundary migration and inhomogeneous deformation of grain boundary. Grain-boundary sliding,

276 7 Creep and Stress Rupture Stress Grain boundary diffusion of vacancies Grain boundary diffusion of atoms Stress Fig. 7.12 Mechanism of Coble creep, in which the solid arrow lines indicate the grain boundary diffusion of atoms from compressive to tensile grain boundaries and the dashed arrow lines indicate the corresponding reverse flow of vacancies (a) (b) (c) X X (2) X (1) (1) (1) (2) (2) Y' Y Y Y\" (3) (3) Z (3) Z Z Fig. 7.13 a Three grains in a hexagonal array before creep deforma- voids. The amount of sliding displacement is expressed quantitatively tion. b After stress-induced diffusional creep, grains elongate leading to by the distance Y′Y″, which is the offset along the boundary between grain separation, i.e. void formation between the grains. grains (1) and (3) of the original vertical scribe line XYZ (Evans and c Grain-boundary sliding brings grains together and removes the Langdon 1976) (a) (b) Shear stress Shear stress Grain I Grain I h Flow of Flow of matter matter λ Grain II Grain II Shear stress Shear stress Fig. 7.14 a Between grains I and II, a sinusoidal form of ‘wavy’ grain grain. b After grain-boundary sliding, the irregularity of the boundary is boundary, irregularity of which is described by the perturbation height, represented by the dashed lines. The relative displacement is allowed h, and the perturbation wavelength, k: Shear stress applied to grains by the flow of matter shown by the arrows promotes relative displacement of one grain with respect to another

7.6 Creep Deformation Mechanisms 277 i.e. relative displacement of one grain with respect to another or (7.48), (7.51) and (7.52). The abscissa of a map is grain necessitates flow of matter by lattice diffusion near the homologous temperature, T/Tm, where T is the absolute grain boundary (Nabarro–Herring process), by grain- temperature of deformation and Tm is the absolute melting boundary diffusion (Coble process), or by both mecha- point of a specific material and the ordinate is the ratio of the nisms. After grain-boundary sliding, the irregularity shape of applied tensile stress r to the shear modulus G of the same the boundary is shown by the broken line in Fig. 7.14b and material, i.e. the normalized tensile stress r=G; which is the mass transfer is designated by arrows. The mass transfer typically plotted on logarithmic scale. The homologous occurs by vacancy concentration gradients in the same way temperature scale ranges from the minimum value of 0 to the as the diffusional creep occurs. It is to be noted that as the maximum value of 1. The normalized tensile stress scale irregularity of the grain boundary increases, i.e. as the ratio ranges from r=G ¼ 10À1 to downwards up to a value of h=k becomes more, volume of matter displaced per unit r=G ¼ 10À8; (note that the value r ¼ 0:1G is equivalent to sliding distance must be more. the theoretical strength of a material). These ranges of temperature and stress cover all possible values of the According to Raj and Ashby (1971), steady-state creep rate variables ever found in practice. The great advantage of the increases rapidly as the ratio k=h is increased. Furthermore, above normalization is that the maps for materials of the when the homologous temperature is high and k is large, the same crystal class and with similar bonding are reduced to a grain-boundary sliding process is controlled by lattice diffu- single group. sion (Nabarro–Herring mechanism) (Raj and Ashby 1971). On the other hand, when the homologous temperature is rel- A typical deformation mechanism map is presented atively low and k is small, grain-boundary diffusion (Coble schematically in Fig. 7.15, where grain sizes of material and mechanism) occurs. As a result, grain-boundary sliding may strain rate are constant. Fields of several dominant creep be accommodated by diffusional flow which is found to mechanisms are indicated in the map. Each field of the map depend on both the temperature and the shape of grain represents the range of stress–temperature combinations over boundary. For an irregular grain boundary, if grain-boundary which a particular mechanism is expected to dominate the sliding is fully accommodated by diffusional flow and inter- creep process. For example, at a constant deformation tem- granular voids are not formed, the grain-boundary morphol- perature T2; and at successively higher applied stresses ogy can reduce the creep rate below those predicted by (7.51) r1; r2; r3; and r4; the respective dominant deformation and (7.52) and the net creep rate is the lesser of the diffusional mechanisms are elastic deformation, Coble creep, disloca- creep rate or the boundary sliding rate since the processes are tion creep and dislocation glide. Similarly, at a constant sequential. When the irregularity of the boundary is quite applied stress r2; and at successively higher deformation high, i.e. when the ratio h=k is quite high the grain-boundary temperatures T1; T2; and T3; the respective dominant defor- sliding may limit the diffusional flow. If accommodation does mation mechanisms are elastic deformation, Coble creep and not take place, the grain-boundary sliding leads to formation Nabarro–Herring creep. The boundaries separating each of inter granular voids, which are associated with the initiation deformation field represent stress–temperature combinations of fracture in creep. at which the respective strain rates from the two deformation mechanisms are equal and both mechanisms contribute 7.7 Deformation Mechanism Map equally to the overall creep rate of the material. Similarly, triple points in the deformation map represent a particular The stress and temperature dependence of each deformation stress–temperature combination where equal strain rates are mechanism are different from another. Hence, the values of produced from three mechanisms and the contributions of all temperature, applied stress and microstructural features such mechanisms to the overall creep rate of the material are as grain size will decide the relative contribution of each equal. The deformation field boundaries are obtained by deformation process. The stress–temperature conditions equating the appropriate constitutive equations, such as under which different creep mechanisms predominate for a (7.47) or (7.48), (7.51) and (7.52), and solving for stress as a specific material have been represented by a graphical function of temperature. Note that the field of dislocation method. This graphical approach, known as deformation creep has been divided into high- and low-temperature mechanism map, has been developed by Ashby and segments corresponding to dislocation climb controlled by co-workers (Ashby 1972; Frost and Ashby 1982) based on lattice and dislocation core diffusion. The field of the original suggestion by Weertman (1968). These maps are grain-boundary sliding has not been shown in the map since constructed for specific materials using experimental data to there are uncertainties regarding the appropriate constitutive determine the necessary material properties and constants in equation for this mechanism. equations that describe each creep mechanism such as (7.47) The dominant deformation mechanism is that which provides greatest strain over the timescale of interest.

278 7 Creep and Stress Rupture Theoretical strength 10–1 σ4 σ Dislocation glide H.T. creep G (lattice L.T. creep Ratio of tensile stress to shear modulus, (core diffusion) Dislocation diffusion) (log scale) creep σ3 Coble creep Elastic regime Nabarro–Herring σ2 creep σ1 10–8 T1 T2 T3 1.0 0 Homologous temperature, T/Tm Fig. 7.15 Schematic map of deformation mechanism, where grain sizes of material and strain rate are constant. The dominant deformation mode is determined by stress–temperature combination Although elastic deformation is the dominant mechanism at then establish the third value in addition to the identification low temperatures and stresses, but on a geological timescale of the dominant deformation mechanism. diffusional creep could become the dominant mechanism by producing a greater strain than the time-independent elastic In the construction of the deformation mechanism map strain. In some cases, where only plastic deformation is discussed so far, the effect of grain size has not been con- considered significant, the field of elasticity is eliminated sidered, i.e. a constant grain size has been assumed. With from the map and the previously occupied elastic deforma- decreasing grain size, the area occupied by the diffusional tion regions in the map are replaced almost all by the flow mechanisms in a map becomes larger. For example, at a creep-dominated mechanisms. homologous temperature of 0.5, and a strain rate of 10−9/s a 100 fold decrease in grain size in pure nickel causes the It is desirable to show the strain rate associated with a transition of the creep rate-controlling process from given stress–temperature combination on the deformation low-temperature dislocation creep to Coble creep. To show mechanism map, irrespective of the rate-controlling mecha- the effect of grain-size variation, a plot of normalized stress, nism. Contours of equal strain rate (isostrain rate) can be r=G; versus grain size, d, on natural logarithmic scale at a developed from the constitutive equation relating strain rate specified temperature and strain rate is given schematically to temperature and stress and plotted on the deformation in Fig. 7.17, which portrays the occurrence of diffusional mechanism map. Such a diagram is shown schematically in and dislocation creep in the simplified map. Creep mecha- Fig. 7.16 for a specific grain size of material, which also nisms that dominate for various combinations of stress and portrays that the Coble creep has replaced the field of elastic grain size can be noticed in Fig. 7.17. The transition regime shown in Fig. 7.15. The map in Fig. 7.16 shows that boundary between the Nabarro–Herring and Coble creep is the deformation mechanism is Nabarro–Herring creep at a represented by the vertical line at d ¼ dc in Fig. 7.17, where strain rate of e_2; which is obtained at stress r1 and temper- dc is the critical grain size at which transition occurs. Note ature T1; and the same strain rate is obtained at stress–tem- that the critical grain size is dependent on temperature. The perature combination ðr2; T2Þ; where dislocation glide takes fact that the importance of diffusional mechanism decreases place. This map allows one to select any two of the three with increasing grain size is reflected by the negative slopes major variables—stress, temperature and strain rate—and (−3 for Coble creep and −2 for Nabarro–Herring creep) of

7.7 Deformation Mechanism Map 279 Fig. 7.16 Schematic map of deformation mechanism for a Theoretical strength H. T. creep specific grain size of material with 10–1 (lattice diffusion) isostrain-rate contours imposed on the map. It shows that at a Ratio of tensile stress to shear modulus, σ ε1 Dislocation glide Dislocation strain rate of e_2; Nabarro–Herring (log scale) G creep creep occurs at stress r1 and ε2 temperature T1; and dislocation σ2 ,T2 glide takes place at stress– L.T. creep temperature combination ðr2; T2Þ (core diffusion) Fig. 7.17 Alternative ε1 representation of a simplified deformation mechanism map at a Coble creep Nabarro–Herring specified temperature and strain (grain-boundary diffusion) creep rate. In this figure, the predominant creep mechanisms ε1 > ε2 ε2 are distinguished for various Homologous temperature, T/Tm stress–grain size combinations σ1 ,T1 10–8 1.0 0 σ Slope = − 3 Ratio of tensile stress to shear modulus, Dislocation creep (natural logarithmic scale) G Coble creep Slope = − 2 (grain-boundary diffusion) Nabarro−Herring creep (bulk diffusion) dc (critical grain size for transition from Coble to N-H creep) Grain size, d (natural logarithmic scale) ln r versus ln d at the diffusional boundaries. Hence, resis- directional solidification technique so that virtually all grain tance to diffusional creep can be improved by increasing boundaries perpendicular and inclined to the tensile stress grain size or developing an elongated structure through axis are eliminated.

280 7 Creep and Stress Rupture 7.8 The Stress-Rupture Test transcrystalline fractures occur, while that to the right of the break corresponds to lower stresses or loads and longer During the stress-rupture test, the tensile specimen is rupture times, where intercrystalline fractures occur. It must deformed at constant load and temperature until the speci- be noted that the shift in the fracture mechanism with men undergoes fracture. This test differs from the creep test decreasing stress does not take place abruptly. Specimens in some respects, which are: corresponding to points near the break often show a mixed mode of fracture—partly transcrystalline and partly inter- • The creep test ordinarily avoids the region of tertiary crystalline. Further, it is evident from Fig. 7.18 that as the creep and obviously, the fracture of specimen, whereas test temperature is decreased the break point shifts to higher the stress-rupture test is carried out up to the point of rupture time. Ultimately at lower temperatures, no break fracture of specimen. point in the slope of stress-rupture curve is observed within the period of the test duration and the fracture observed is • To fracture the specimen within a reasonable length of fully transcrystalline. These indicate that lower temperatures time, higher load is applied leading to higher strain rate favour transcrystalline fractures, whereas higher tempera- in a stress-rupture test than in a creep test. tures favour intercrystalline fractures. Finally, it may be concluded that transcrystalline fractures are favoured by low • Since structural changes in specimen occur at a shorter temperatures, high stresses and short-time tests, i.e. high time due to higher stress and strain rate in a stress-rupture strain rates, while intercrystalline fractures are promoted by test than in a creep test, the stress-rupture test is usually high temperatures, low stresses and long-time tests, i.e. slow conducted up to 1,000 h, while creep test is frequently strain rates. Sometimes, stress-rupture fractures show both carried out for periods that range from 2,000 to 10,000 h. transcrystalline and intercrystalline fracture paths. In such cases, it is usually found that intercrystalline cracks were • Precise determination of strain, particularly the determi- initially developed, which reduced the cross-sectional area nation of minimum creep rate is the aim of creep testing, and caused an increase in the stress level and this high stress whereas the main aim of stress-rupture testing is to obtain ultimately resulted in transcrystalline fracture. the information about the time to cause rupture at a given engineering stress for a constant temperature. An important characteristic of intercrystalline fracture is that it effectively reduces the rupture life of a metal, as seen • The total strain in a creep test often does not exceed from Fig. 7.19. If the straight line in Fig. 7.19, where tran- 0.5%, while that in a stress-rupture test may be around scrystalline fractures occur, is extended to the right of the 50%, for which a simpler strain-measuring device like a break, a relationship between stress and rupture time is dial gage is sufficient. obtained which should be valid if intercrystalline fractures would not take place. This extended line shows that at any An important advantage of the stress-rupture test over the creep test is the shorter time of testing and this has lead to Engineering stress, MPa (log scale) 1000 Transcrystalline fracture the increased use of the stress-rupture test. To obtain 100 T1 stress-rupture data, tests are carried out on several series of T2 specimens at different temperatures until specimens are fractured. In these series, each specimen is subjected to a Intercrystalline fracture different load. Data for a set of specimens ruptured at the same temperature are plotted with nominal (engineering) T3 stress versus rupture time on a log-log scale. This kind of plot can be made for tests carried out at different tempera- Temperature, T1 < T2 < T3 tures. Figure 7.18 shows schematically the variation of 10 nominal (engineering) stress with rupture time at various temperatures, in which at each temperature the data are 0.001 0.01 0.1 1 10 100 1000 plotted from a series of specimens tested under different Rupture time, hr (log scale) stress levels. Plot on a log-log scale usually shows a linear variation for each test temperature. Curves plotted for higher Fig. 7.18 Schematic plot of stress-rupture diagram at various temper- temperatures show two intersecting straight lines. The most atures, in which at each temperature the data are plotted from a series of common cause for this break in the slope of stress-rupture specimens tested under different stress levels. Curves plotted for higher curve at any constant temperature is the change of fracture temperatures show break in the slope of curve due to the change from mechanism (see Sect. 7.9) from transcrystalline to inter- transcrystalline to intercrystalline fracture, whereas lower temperature crystalline fracture as conditions of the test change. The test within the stipulated test period shows only transcrystalline fracture straight line to the left of the break corresponds to higher stresses or loads and shorter rupture times, where

7.8 The Stress-Rupture Test 281 Fig. 7.19 Rupture life of a creep 1000 Constant temperature test specimen is effectively reduced Transcrystalline fracture by intercrystalline fracturing Engineering stress, MPa (log scale) 100 Intercrystalline Arbitrary stress fracture 10 Rupture time Rupture time intercrystalline fractutre transcrystalling fractutre 1 0.1 1.0 10 100 1000 10000 Rupture time, hr (log scale) arbitrary stress the rupture time for the actually observed 7.9 Concept of ECT intercrystalline facture is much shorter than that for the and Elevated-Temperature Fracture extended transcrystalline fracture. If polycrystalline metals fail by fractures passing through the Other reasons for the break in the slope of stress-rupture interiors of grains, this mode of fracture is called tran- curve are recrystallization and grain growth, internal oxi- scrystalline or transgranular or intragranular fracture. dation, or other structural changes such as graphitization, There is another mode of fracture known as intercrystalline spheroidization or some kind of phase transformation. The or intergranular fracture, where fractures run along the grain extrapolation of the stress-rupture data to longer time can boundaries of polycrystalline metals. From the work of result in serious errors if such microstructural instabilities Rosenhain and Ewen (1913), it is known that as the defor- exist, but they are not detected or known to the designer. mation temperature increases, metals undergo a fracture The time span over which a material can sustain the mechanism transition from transcrystalline to intercrystalline applied stress without fracture at different temperatures is fracture. Thus, a metal, which at low temperature fails with a found from the stress-rupture test. The rupture life normal transcrystalline fracture, is likely to fail with an obtained from stress-rupture test are used in the design of intercrystalline fracture at elevated temperature. The transi- components designated for short-time elevated temperature tion from transcrystalline to intercrystalline fracture is more uses, e.g. ‘one-shot’ rocket engine component, where abrupt in complex alloys than in metals. At low tempera- strain due to creep may be tolerated but fracture must be tures, where transcrystalline fracture usually occurs, the avoided. grains themselves are usually weaker than the grain boundaries and the deformation of polycrystalline metal Apart from noting the rupture time from the test, the occurs essentially by deformation of grains. At low tem- reduction of area at fracture and the elongation (engineering peratures, intercrystalline fractures may occur in exceptional strain to fracture) are also determined. If the test is conducted cases, where there are some structural irregularities, such as for a suitable duration and the elongation measured as a the presence of a brittle intercrystalline film or a form of function of time is converted to true strain, then from the plot of true strain versus time the minimum creep rate can be determined.

282 7 Creep and Stress Rupture (a) The ECT is not a fixed one like recrystallization tem- perature and depends on the strain rate, the applied stress and Grain boundary purity or composition of the material, etc. Since a change (a decrease) in strain rate is believed to affect (to decrease) the grain boundary strength more than the strength of grains, Grain decreasing the strain rate decreases the ECT and thus Strength increases the tendency for intercrystalline fracture, as evident from Fig. 7.20b. The elongation and particularly the reduc- Transgranular tion in area have been found to decrease with decreasing the fracture strain rate as the mode of fracture changes from transcrys- talline to intercrystalline fracture. Similarly, low applied Intergranular fracture ECT stress decreases the strain rate and favours intercrystalline Temperature fracture. Since grain-boundary sliding becomes more (b) prominent with decreasing stress, a decrease in total elon- Low strain rate gation or reduction in area is usually found with a decrease in stress at constant temperature even when the mode of High strain rate fracture is intercrystalline in all cases. Although the ECT occurs within a narrow temperature range for commercially pure metals and alloys, but for high-purity material the ECT Strength Grain exists over a wide range of temperature, where there is not much difference in the strengths of grains and grain Grain boundaries (Servi and Grant 1951), as shown in Fig. 7.20c, boundary and in such cases, transcrystalline fracture can be observed up to rather high temperatures. Vacuum melting eliminates ECT1 ECT2 trace elements and is found to increase elongation and rup- Temperature ture life and decrease the tendency for intercrystalline frac- (c) ture in many alloys. Under creep conditions, ECT is approximately 0.5 times Grain boundary the absolute melting point of materials. At approximately this temperature, the following deformation features have Grain been observed: Strength Grain • The tensile strength and creep rate of materials with a small grain size become equal to those with a large grain Range Pure metal size. Since the grain boundary area per unit volume of ECT decreases with increasing grain size, materials with a coarse grain size are stronger above the ECT and those Temperature with a fine grain size are stronger below the ECT. Fig. 7.20 a Schematic of equicohesive temperature (ECT), showing • Grain-boundary sliding takes place. transgranular fracture at temperatures\\ECT; and intergranular fracture • Most generally intercrystalline fracture occurs. at temperatures [ ECT: b Effect of strain rate on ECT. c ECT exists • Materials experience a reduction in ductility. As the over a wide range of temperature for high-purity materials temperature is increased from low temperature, corrosion that weakens the grain boundaries. The fracture is grain-boundary migration becomes more prominent and generally intercrystalline at high temperatures, where the the ductility measured by the total elongation or reduc- grain boundaries are weaker than the grains and significant tion in area begins to increases up to the temperature sliding occurs along grain boundaries. Therefore, there must where the change from transcrystalline to intercrystalline be a temperature at which the strengths of grains and grain fracture occurs, and thereafter, the ductility is found to boundaries are the same. Jeffries (1919) defined this tem- drop. perature as the equicohesive temperature (ECT) which is shown schematically in Fig. 7.20a. It is to be noted that intercrystalline fracture and a reduction in ductility, which generally occurs in the vicinity of a homologous temperature of 0.5, do not always occur. In

7.9 Concept of ECT and Elevated-Temperature Fracture 283 Tensile elongation recrystallized grains and there is an increase in ductility. One important microstructural feature of intercrystalline fracture Intermediate Temperature is that the grains appear equiaxed even after exhibiting some ductility minimum amount of plastic deformation and total elongation. It indi- cates that there is dynamic recrystallization. In contrast, Temperature severely elongated grains are often observed in the vicinity of transcrystalline fracture indicating that no recrystallization Fig. 7.21 Schematic representation of the intermediate temperature occurs. ductility minimum (Rhines and Wray 1961) The three basic modes of fracture at elevated temperature very pure metals, intercrystalline fracture may not occur at are rupture, transcrystalline fracture and intercrystalline all at any temperature even near the melting point. Sup- fracture, as illustrated in Fig. 7.22 (Courtney 1990). Rupture pression of intercrystalline fracture is partly due to occurs in a completely ductile manner, where the necked grain-boundary migration which prevents or delays inter- region of material may actually be thinned down to a line or crystalline void formation. For example, only transcrys- a point prior to fracture and the reduction in area at fracture talline fracture is observed for high-purity aluminium up to approaches 100%. High-temperature rupture may occur at its melting point (Servi and Grant 1951; Chang and Grant high stresses during hot working at high strain rates and is 1953). Further, very often entire ductility is again recovered usually associated with dynamic recovery and recrystalliza- at higher temperatures (Mullendore and Grant 1954; Gem- tion. Since rupture mode of failure is not really predominant mell and Grant 1957) in those cases where there is a in creep, it will not be further considered. reduction in ductility. If the ductility in terms of tensile elongation or reduction in area is measured and plotted with Depending on the alloy, deformation temperature, applied increasing temperature from room temperature, the ductility stress and strain rate, the mode of fracture during stress is found to increase with a ductility minimum at an inter- rupture may be either transcrystalline or intercrystalline mediate temperature, as seen from the schematic plot of fracture. Conditions that cause transcrystalline fracture tensile elongation against temperature in Fig. 7.21. Rhines or/and intercrystalline fracture in stress rupture have been and Wray (1961) designated this as the intermediate tem- narrated in Sect. 7.8. The predominant mode of fracture perature ductility minimum, which is associated with inter- during creep is intercrystalline fracture that occurs at lower crystalline fracture. The elongation minimum shown in the applied stress and longer creep times resulting in a low level plot will also be accompanied by a corresponding minimum of creep strain at fracture, although transcrystalline fracture in the reduction in area. The ductility minimum for alu- can occur during creep when the levels of stress and creep minium occurs near room temperature but that for most strain at fracture are fairly high. Intercrystalline fractures alloys, especially nickel-based, occurs within hot-working occur with very little macroscopic plastic flow and show range. The temperature range of minimum ductility normally little total elongation often with little necking and macro- falls just below the recrystallization temperature in a region scopically appear to be brittle in nature. On the other hand, where grain-boundary sliding can take place to develop transcrystalline fractures usually show more pronounced intercrystalline cracking. When the temperature is increased elongation accompanied by necking and are characterized to beyond the temperature zone of minimum ductility in a be ductile in nature. In transcrystalline fractures, voids region where recrystallization can happen to a great extent, nucleate, usually around inclusions within the grains, and the grain-boundary cracks are isolated from the newly then grow and coalesce until fracture takes place. The void-forming process is very similar to that of microvoid coalescence in room temperature ductile fracture, except that formation and growth of voids at elevated temperatures are aided by diffusion. In intercrystalline fractures, the nucle- ation, growth and subsequent coalescence of voids primarily occur only on the grain boundaries. 7.9.1 Wedge-Shaped Cracks and Round or Elliptically Shaped Cavities Any factor that tends to raise the resistance to shear inside the grains compared to that at the grain boundaries, and makes grain-boundary migration more difficult, tends to develop intercrystalline fracture. In general, strain hardening,

284 (a) (b) 7 Creep and Stress Rupture Fig. 7.22 Three modes of high (c) temperature failure: a rupture; b transcrystalline creep fracture, in which the coloured circles represent intragranular voids that form, grow and coalescence leading to failure; c intercrystalline creep fracture, in which the coloured shaded regions at grain boundaries are intercrystalline voids or cracks that nucleate, grow and coalescence to some degree followed by fracture (Courtney 1990) solid-solution hardening, precipitation hardening, etc., can proper conditions. Formation of wedge cracks at the triple make slip inside the grains more difficult and thus increase points and also ‘r-type’ cavities along grain boundaries was the resistance to shear. Grain-boundary migration can also seen in coarse-grained Al-20% Zn specimen tested at 260 °C be greatly restrained by solid-solution and precipitation (Chang and Grant 1956). hardening. Analyses of the fracture problem and/or experimental The nucleation and growth of intercrystalline cracks or investigations have indicated the following: cavities occur at an accelerated rate in the tertiary creep stage. This process is often called creep cavitation. Metal- • Grain-boundary sliding is a necessary prerequisite for lographically, it has been observed that there are at least two the initiation of both types of crack or cavity. In tests on types of intercrystalline cracking. These are wedge-shaped coarse-grained Al-20% Zn specimens, Chang and Grant cracks, often called ‘w-type’ cracks, and round or elliptically (1956) found that extensive grain-boundary sliding was shaped cavities, known as ‘r-type’ cavities, which may be visible at the grain corners where a ‘w-type’ crack had polyhedral in shape in the very early stages after nucleation. formed. For a Cu bicrystals strained in creep at Cracks of ‘w-type’ initiate mostly at grain boundary triple 2.1 MN m−2 in temperature range of 923–1173 K, points (edges where three grains meet), where the grain Intrater and Machlin (1959) found that the number of boundaries are aligned for maximum shear, and propagate ‘r-type’ cavities per unit length of grain boundary primarily along the grain boundaries which are roughly increased with the increase in the amount of normal to the applied stress (Chang and Grant 1956), grain-boundary sliding. As the temperature is raised, the although propagation along grain boundaries oblique to the number of cavities increases in the same manner as the direction of stressing is not uncommon. They are also grain-boundary sliding increases with increasing tem- referred to as grain-corner or triple-point cracks. On the perature. Though some experiments have shown that the other hand, ‘r-type’ cavities are found to form primarily density of cavities at grain boundaries decreases at very along grain boundaries that are aligned normal to the tensile high temperature, but this is because of the occurrence of stress, but their formations are not limited exclusively to grain-boundary migration which leaves behind the cavi- grain boundaries normal to the applied stress. ‘r-type’ cav- ties previously formed on the boundaries. ities grow by the joint action of grain-boundary sliding and vacancy diffusion. The formation of one type of crack or • Very large stress concentrations are required to nucleate cavity does not necessary exclude the formation of the other both types of crack or cavity. Analysis (Nix 1983) indi- type of crack or cavity, rather both types of crack or cavity cates that excessively high stresses normal to a grain may appear simultaneously in the same specimen under boundary, e.g. on the order of 1/100 times the elastic modulus, are required for nucleation of a cavity. If it is