UNIT – I
HIGH TEMPERATURE MATERIALS
1. CREEP :
Creep may be defied as the slow and progressive deformation of a material with time under a constant stress at a temperature approximately above 0.4 Tm, i.e.., the recrystallisation
temperature of the material (Where Tm is the melting point of the metal or alloy in degrees Kelvin).
Creep is function of temperature and time. Creep deformation is plastic in nature and occurs even though the acting stress is below the yield stress of the material. The rate of creep is very small but at higher temperatures it becomes very significant.
Certain metals such as lead and tin which have low melting temperatures creep at room temperature.
Creep behaviour is very much important when studying the behaviour of materials that are used in high temperature applications. For example steam plants, gas turbines, nuclear reactor, body of space crafts, tungsten filaments used in electric bulbs and radiator shields in furnaces are mede of molybdenum.
Creep strength of a metal is usually defined by the limiting stress below which creep is so
slow that it will not result in fracture within any finite length of time. Similarly Creep rupture
strength or rupture strength of a material is the highest stress that a material can with stand for a
given time without rupture.
Creep limit may be defined as the maximum stress that will cause creep to occur at a rate
not exceeding the specified deformation at a given temperature. In general the creep rate is higher and time to fracture shorter with increasing temperature and load.
Creep curve
The creep is tested for a material by subjecting the specimen at constant tensile stress at constant temperature and measuring the extent of strain or deformation with respect to time. (The creep test is similar to a tension test but under the influence of temperature). A typical creep curve is shown in figure.
Figure
The curve shows three stages of elongation.
I. (a) Initial instantaneous elongation after the application of load.
(b) Primary of Transient creep.
II. Secondary or viscous or steady state creep.
III. Tertiary creep or accelerated creep.
Instantaneous elongation : This a stage that is initially observed. With the first application of
load an instantaneous elastic strain occurs. If the initial load applied is higher then there is some plastic strain, in such a case the instantaneous elongation is elastic strain + plastic strain.
Primary creep or Transient creep : At the beginning of primary creep there is strain hardening
effect (i.e., The material resists deformation and becomes hard due to its own elongation ) and the deformation is slow at a decreasing rate.
For low melting temperature metals, primary creep is the predominant creep process.
Secondary creep or steady state creep : In this region the creep is constant and the creep rate is
constant. The reason for this steady state is due to an equilibrium between the strain hardening effect and the annealing effect.
Since creep occurs at an elevated temperature the annealing effect occurs to and the material tends increase in strain causes the material to resist further deformation hence there is a balance between the strain hardening effect and the annealing effect which results is a steady state creep. The constant creep rate of the secondary stage is usually assumed to be the material’s minimum rate and is called as minimum creep rate (MCR).
Tertiary creep or accelerated creep : This is the final stage of creep before fracture. The creep
occurs rapidly because of a decreases in cross-sectional area and necking of the specimen occurs, the true stress increases rapidly. During this stage there is progressive damage to the intercrystalline regions by the formation of voids and sivere oxidation of the metal (note : oxidation occurs because the material is tested for creep at elevated temperature). The material is unable to strain harden and finally fractures. Du ring tertiary creep there are changes to the microstructure, grain coarsening and recrystallisation, these factors are also responsible for the acceleration of the creep.
Creep fracture :
At high temperatures grains show more strength than grain boundaries and at low temperatures grain boundaries are stronger than the grains. The temperature at which the strength of grains equal the grain boundaries is called equicohesive temperature. The crack always initiates and propagates through weak portion and hence below equicohesive temperature. Creep is a high temperature process and hence fractures always occurs by intergranular mode.
Figure: Creep Variables
Creep resistant materials are used at high temperatures. They are capable of withstanding such temperatures without undergoing creep upto a certain limit.
The following are the factors that influence the creep property of a material.
(i) Higher creep resistance is observed with metals having high melting point. Creep
becomes significant above 0.4 Tm . Metals such as iron, cobalt, molybdenum, tungsten
that have high melting temperature are used in high temperature services.
(ii) A coarsed grained metal has high creep resistance than a fine gained metal. At creep temperatures the grain boundaries become quasi – viscous. The coarse grained materials have less total grain boundary and hence it. Developes less quasi - viscous
Single crystals have excellent creep resistance because they have no grain boundary.
(iii) Dispersion hardening improves creep resistance.
(iv) Metals having higher oxidation and scaling resistance have more creep strength.
(v) For steels, increase in carbon content increases creep resistance.
(vi) Aluminium, when added to steal acts as deoxidizer, this makes the steel to resist creep.
(vii) Creep resistance is increased by adding alloying elements such as; W, Mo, V, Cr, Ti, Nb and Co, these elements form carbides with the iron present in steel. The presence of carbides increase the resistance to soften at elevated temperatures thus resisting creep.
2. Materials for elevated temperature use :
A material suitable for high-temperature service should possess a high melting point and modulus of elasticity, and low diffusivity. In addition, such materials must possess a combination of superior creep strength, thermal fatigue resistance, and oxidation and hot corrosion resistance. As a result, alloy development has focused primarily on nickel-and cobalt-based superalloys, with earlier iron-cobalt-based alloys being replaced because of their relatively low melting point and high diffusivity. These high-temperature alloys have been produced by several methods including casting, mechanical forming, powder mechanical alloying.
For the case of nickel-based superalloys, constituent elements are introduced to enhance solid solution properties, as precipitate and carbide formers, and as grain boundary and free surface stabilizers. Tungsten (W), molybdenum (Mo), and titanium (Ti) are very effective solid solution strengtheners : W and Mo also serve to lower the diffusion coefficient of the alloy. (There is a general inverse relation between the melting point and alloy diffusivity). Though the incremental influence of chromium (Cr) on solid solution strengthening is small (i.e., dT/dc is low), the overall solid solution strengthening potential of Cr in nickel (Ni) alloys is large since large amounts of Cr can be dissolver in the Ni matrix. Cobalt (Co) provides relatively little solid
solution strengthening but serves to enhance the stability of the submicron-size Ni3(AI,X)
()precipitates within the nickel solid solution () matrix (Figure(a)). Within the phase, X corresponds to the presence of Ti, niobium (Nb),or tantalum (Ta). The difficulty of dislocation
motion through the ordered particles in these alloys is responsible for their high creep strength
at elevated temperatures. Of particular note, the phase exhibits unusual behavior in that strength increases by three to sixfold with increasing temperature from ambient to approximately 700oC.70-72
FIGURE :Electron micrographs revealing Ni3AI precipitates () in a nickel solid solution ( )
matrix. Matrix. (a) Cubic from in MAR M-200. (b) Rafted morphology in Ni-14.3 Mo-6Ta-5.8AI (Alloy 143). Tensite stress axis is in vertical direction and parallel to [001] direction. Creep tested with 210 MPa at 1040C. (Courtesy E. Thompson).
Also noteworthy is the fact that precipitates in single-crystal alloys tend to coarsen under
stress at 1000C and form thin parallel platelike arrays that are oriented normal to the applied stress axis. Recent studies have confirmed that alloy creep resistance is enhance by the development of this ‚rafted‛ microstructure, it is believed that the absence of dislocation climb
around the particles, due to their lenticular shape, forces dislocations to cut across the ordered
phase. As note in Section 4.4.2, this dislocation path enhances the alloy’s resistance to plastic flow.
The presence of carbides along grain boundaries in polycrystalline alloys serves to restrict grain- boundary sliding and migration. Carbide formers such as W, Mo, Nb, Ta, Ti, Cr, and
vanadium (V) lead to the formation of M7C3,M23C6, M6C, and MC, with MC carbides being most
stable (e.g., TiC). When Cr levels are relatively high, Cr23C6 particles are formed.
Surface stabilizer include Cr, Al, boron (B), zirconium (Zr), and hafnium (Hf). The presence
of Cr in solid solution allows for the formation of Cr2O3, which reduces the rate of oxidation and
hot corrosion, Aluminum contributes to improved oxidation resistance and resistance to oxide spalling. Finally, B, Zr, and Hf are added to impart improved hot strength , hot ductility, and rupture life.75Cobalt-based alloys derive their strength from a combination of solid solution
hardening and carbide dispersion strengthening. The mechanical properties of representative nickel-based and cobalt-based alloys are given in Table 5.3; references 63 to 68 provide additional information concerning these materials.
Recent efforts to improve the high-temperature performance of superalloys have tended more toward optimizing component design and making use of advance processing techniques
turbine blades of the gas turbine engine are air cooled via internal channels, the gas turbine inlet temperature can be increased markedly with a concomitant improvement in engine operating efficiency. Several processing techniques have been developed and applied to the manufacture of gas turbine components.
TABLE : Mechanical Properties of Selected Superalloys :
Date correspond to 816oC (1500oF).
Directionally solidified.
Extrapolated values.
Data courtesy Inco Alloys Inc.
One such technique involves the directional solidification of conventional superalloys to produce either highly elongated grain boundaries or single-crystal components (Figure). Helical molds are used to cast single-crystal turbine blades; multiple grains form initially and grow into the helical section of the mold. The faster growing (100) –oriented grains then crowd out other grains until a single (100) grain is left to fill the mold cavity.77-79 Current sophisticated mold
designs now allow for the simultaneous growth of two turbine blades form the same single
crystal.79 The alignment of airfoils (turbine blades) along the (100) axis parallel to the centrifugal
stress direction allows for a 40% reduction is the elastic modulus and associated lower plastic strain range during thermal fatigue cycling; a 6- to 10- fold improvement in thermal fatigue resistance is thus achieved. Since grain boundaries are eliminated, their influence on
grain-boundary sliding, cavitation, and cracking is obviated.77,78 furthermore, it is no longer necessary to
add such elements as hafnium, boron, carbon, and zirconium for the purpose of improving
grain-boundary hot strength and ductility.80 without these elements, the incipient melting temperature
of the alloy is in creased by approximately 120oC and the alloy chemistry simplified. The
development of cast superalloys turbine blades is shown in Figure (a) ; the relative ranking of the rupture lifetime for equiaxed and columnar polycrystalline alloys is compared with that of single-crystal alloys in Figure b. By applying unidirectional solidification to alloys of eutectic
composition, it has been possible to produce eutectic composite alloys possessing properties superior to those found in conventional superalloys (figure). A number of these alloys contain a
/ matrix that is reinforced with high-strength whiskers of a third phase; these strong
filamentary particles are oriented.
FIGURE: conventional and directional solidification used to prepare gas turbine blades with equiaxed, columnar, and single-crystal morphologies. (F.L.VerSnyder and E.R. Thompson, Alloys for the 80’s, R.Q. Bar, Ed., Climax Molybdenum Co., 1980, p.69 : with permission.)
(a) (b)
Figure: (a) Development of turbine blade temperature capability.
(b) Comparative high temperature strength and corrosion resistance of equiaxed, columnar, and single-crystal superalloys.79 (Reprinted with permission from Journal of Metals, 39(7),
Figure : 1000-hr strength as a function of temperature in eutectic superalloys and conventional
directionally solidified single-crystal and oxide-dispersion-strengthened superalloys. In situ
(eutectic) composites reveal generally superior stress rupture behavior. (From Lemkey81,
reprinted by permission of the publisher from F.D. Lemkey, Proceeding, MRS Conference, CISC IV, Vol. 12, F.D. Lemkey, H.E. Cline, and M. McLean, Eds., copyright by Elsevier science Publishing Co., Inc., Amsterdam, (c ) 1982.) parallel to the maximum stress direction. A though the properties of these alloys are very good, the allowable solidification rates for their manufacture are much lower than those permissible in the manufacture of directionally solidified columnar or single-crystal microstructures. One is then faced with a trade-off between the superior properties of eutectic composites and their higher manufacturing costs.
Another new fabrication technique involves forging under superplastic conditions. In this
process, the material is first hot extruded just below the solvus temperature, which causes the
material to undergo spontaneous recrystallisation. Since the precipitates in the nickel solid solution matrix tend to restrict grain diameter remains relatively stable in the size range of 1 to5
m . The part is then forged isothermally at a strain rate that enables the material to deform superplastically (recall Section). At this point, the superplastically formed component is solution treated to increases the grain size for the purpose of enhancing creep strength . The material is
then quenched and aged to optimize the / microstructure and the associated set of mechanical
properties. One major advantage of superplastic forging is its ability to produce a part closer to its final dimensions. One major advantage of superplastic forging is its ability to produce a part closer to its final dimensions, thereby reducing final machining costs.
Superalloys can also be fabricated from powders produced by vacuum spray atomization of liquid or by solid-state mechanical alloying techniques (recall Sect4.5). Powders may then be placed in a container that is a geometrically larger version on the final component shape. The can is then heated under vaccum and hydrostatically compressed to yield a fully dense component
with dimensions close to the design values. The microstructure of hot isostatically pressed (HIP) Astroloy superalloys is shown in Figure.5.33a. Note the persistence of the necklace of prior particle boundary borides, carbides, and oxides that surround the atomized powder particles. Hot isostatic pressing is also being used to heal defects in conventionally cast parts and to heal certain defects in parts that experience creep damage in service.
With significant additions of formers, such as AI and Ti, mechanically alloyed oxide-dispersion-strengthened (MA/ODS) products possess attractive strength levels over a broad
temperature range.84-85 Two such alloys are Ma6000 and Alloy 51, which contain approximately 55
v/o and 75 v/o , respectively (figure 5.33b).84-85 The 1000-hr rupture strength (normalized with
respect to density) of these alloys and others is shown in figure.5.34 as a function of temperature. As expected, directionally solidified (DS MAR-M200) and single-crystal (PWA 1480) cast alloys are superior to the two mechanically alloyed products at temperatures up to 900oC with the
relative rankings being reversed above this temperature. At high temperatures near the solvus
temperature, the particles that dominate the precipitation hardening process tend to coarsen and/or go back into solution. The superiority of MA materials relative to that of directionally solidified and single-crystal cast alloys at temperatures in excess of 900oC is due to the
oxide-dispersion-strengthening influence of the Y2O3 particles that remain in the microstructure and do
not coarsen to any significant degree.
Recent attention has focused on the unusual creep rate and rupture-life stress dependence
of ODS alloys. Whereas most pure metals and associated solid solutions reveal a 4 5
dependence of (recall Equation 5-15 and 5-20), the steady-state creep rate in ODS alloys exhibits
a stress dependency of 20 or more.70.87.86 furthermore, the apparent activation energy for the creep
process is found to be two three times.
(a) Macrostructure of HIP’d Astroloy superalloys. Note persistent necklaces of prior particle boundary borides, carbides, and oxides. (Reprinted with permission from J.S. Crompton and R.w. Hertzberg, J.Mater Sci., 21,3445 (1986), Chapmen & Hall Pub.)
(b) Microstructure of MA 6000 showing precipitates (large light areas) and Y2O3 dispersoids
(small dark regions). (Photo courtesy W. Hoffelener from w. Hoffelener and R.F. singer ,
Metallurgical Transactions 16A, 393(1985).
FIGURE : Comparison of 1000-rupture strength (density corrected) in directionally solidified and
oxide-dispersion-strengthened nickel-based superalloys. Note superior properties of ODS alloys
at temperatures above 900oC .(Reprinted with permission from S.K.Kang and R.C. Benn,
Metallurgical Transactions, 16A, 1285 (1985).)
Greater than the activation energy for self-diffusion. Tien and coworkers have suggested that these apparent difference in creep response can be rationalized by considering creep to be dominated by an effective stress rather than the applied stress; the effective stress is defined as
the applied stress minus a back stress that reflects dislocation interactions with Y2O3 dispersion
strengthening particles. When the applied stress level is replaced by the effective stress value in
Equation.5-20, the stress dependency of s and apparent activation energy for creep are found to
be similar to those values corresponding to pure metals (i.e.,n4-5 and Hc HSD).
In corresponding fashion, the rupture life of ODS alloys can reveal a very strong applied stress dependency and an upward slope change with increasing rupture lifetime, opposite to that observed in many other alloy (e.g.recall Figure 5.3). Figure 5.35 reveals that MA6000 and Alloy 51 exhibit two regions of behavior; Region I corresponds to high stress levels and intermediate
temperatures and is dominated by the precipitates, At higher temperatures, lower stress levels
and longer times (Region II), stress rupture is dominated by the Y2O3 dispersoid phases. Note
that ODS alloy MA754, which contains no phase, does not exhibit Region I behavior;
Region II behavior. Recent studies have sougth to clarify the nature of the dislocation dispersoid particle interation so as to better understand the unique phenomenological behavior of ODS alloys.
In another recent thrust, researchers have focused attention of the development of a gas turbine engine using ceramic components. Since ceramics often possess higher melting points and moduli of elasticity and lower diffusivities than metal systems, they offer considerable potential in such applications. Unfortunately, ceramics suffer from low ductility and brittle behavior in tension (see Table 10.8). This serious problem must be resolved before the ceramic engine can become a reality. Progress toward this end is being made as discussed in Section 10.4.3.
Finally, fiber-reinforced superalloys are receiving increased attention as candidate materials for structural used at elevated temperatures. Tungsten fibers hold promise as
FIGURE : Stress rupture response of MA/ODS cast nickel superalloys..84 (From R.C. Benn and
s.K.Kang, superalloys 1984, American Society for Metals, Metals Park, OH, 1984, with permission.)
A suitable reinforcement for superalloys in that they possess superior high-temperature strength
and creep resistance.88 In addition, a good interface is developed between the superalloy matrix
and the tungsten fibers without excessive surface reactions that degrade W-fiber mechanical properties. Preliminary studies have shown that operation temperatures of fiber-reinforced
superalloys may be increased by 175oC over that of unreinforced superalloys.
Whatever the alloy or process used to fabricate superalloy parts, the high-temperature environments that are experienced demand that careful attention be given to the suppression of oxidation and corrosion damage. To this end, coatings such as MCrAl/Y (where M = Ni, Co, and Fe) may be placed on the component’s exterior surface; surface coatings with such compositions
promote the formation and retention of Al2O3, which serves as an effective barrier to the diffusion
of oxygen into the component interior. Unfortunately, these coatings tend to spall away during
thermal cycling and must be stabilized. Other ceramics (e.g., ZrO2) may serve as thermal barrier
coatings (approximately 0.25 mm thick) that can reduce superalloy turbine blade surface temperatures by as much as 125-2500C. Here, too the tendency for spallation due to thermally
induced strains must be suppressed.
3. TEMPERATURE-STRESS-STRAIN-RELATION
Since the creep life and total elongation of a material depends strongly on the magnitude of the
steady-state creep rate s (Equation 5-1 and 5-5), much effort has been given to the identification of
those variables that strongly affect s. As mentioned in Section 5.1, the external variables,
temperature and stress, exert a strong influence along with a number of material variables. Hence the steady-state creep rate may be given by
s f T( , , , m m1, 2)
Where T = absolute temperature
= applied tensile stress
= creep strain
m = various intrinsic lattice properties, such as the elastic modulus G and the crystal 1
structure
2
m = Various metallurgical factors, such as grain and sub grain size, stacking fault energy,
and thermo mechanical history
It is important to recognize that m also depends on T, 2 ,and . For example, subgrain diameter decreases markedly with increasing stress. Consequently, there exists a subtle but important problems of separating the effect of the major test variables on the structure from the
deformation process itself that controls the creep rate. Dorn, Sherby, and coworkers10-13 suggested
that where Th 0.5 for the steady-state condition, the structure could be defined by relating the
creep strain to a parameter
f( )
Where = te-H/RT described as the temperature-compensated time parameter
t = time
H = activation energy for the rate-controlling process
T = absolute temperature
The activation energy H, shown schematically in Figure. Represents the energy barrier to be overcome so that an atom might move from A to the lower energy location at B. Upon differentiating Equation with respect to time, one finds
. /
( ) H RT
Z f e
Which describes the strain-rate-temperature relation for a given stable structure and
applied stress. When the rate process is given by the minimums creep rate .sand its logarithm
plotted against 1/T, a series of parallel straight lines for different stress levels is predicted from
Equation (Figure). The slope of theses lines, H/2.3R, then defines the activation energy for the
controlling creep process. The fact that the is stress lines were straight in figure suggests that only one process had controlled creep in the TiO2 single crystals throughout the stress and
temperature range examined. Were different mechanisms to control the creep rate at different
temperatures, the log .svs. 1/T plots would be nonlinear. When multiple creep mechanisms are
present and act in a concurrent and dependent manner, the slowest mechanism would control
. s
. The overall strain rate would take the form
. . . . . 1 2 3 1 1 1 1 1 ... T n
Where .T = overall creep rate
. 1,2,3,....,n
= creep rates associated with n mechanisms
For the simple case where only two mechanisms act interdependently
. . 1 2 . . 1 2 T
Conversely, if the n mechanisms were to act independently of one another, the fastes one would control. For this case, .T would be given by
. . . . .
1 2 3 ...
T n
Figure: Diagram revealing significance of activation energy required in moving an atom form A to B.
To determine the activation energy for creep over a small temperature interval, where the controlling mechanism would not be expected to be expected to vary, researchers often make use of the temperature differential creep test method. After a given amount of strain at temperature T1, the temperature is changed abruptly to T2, which may be slightly above or below T1. The
difference in the steady-state creep rate associated with T1 and T2 is then recorded (figure). If the
stress is held constant and the assumption made that the small change in temperature does not change the alloy
Figure. Log steady-state creep rate versus reciprocal of absolute temperature for rutile (TiO2.)
at various stress levels. (From W.M.Hirthe and J.O. Brittain;14reprinted with permission from
Figure. Incremental step test involving slight change in test temperature to produce change n steady-state creep rate in aluminum. ( From J.e. Dorn, Creep and Recovery, reprinted with permission from American Society for Metals, Metals Park, OH, copyright (c ) 1957).
Structure, then Z is assumed constant. From Equation the activation energy for creep may then be calculated by . . 1 2 2 1 / 1/ 1/ c RIn H T T
Where HC = activation energy for creep
.1 .2 = creep rates at T1 and T2, respectively
This value of HC should correspond to the activation energy determined by a data
analysis like that shown in Figure, as long as the same mechanism controls the creep process over the expanded temperature range in the latter instance. As shown in Figure, this not always the case. The activation energy for creep in aluminum is seen to increase with increasing temperature up to Th 0.5, whereupon HC remains
FIGURE. Variation of apparent activation energy for creep in aluminum as a function of temperature. (From O.D. Sherby, J.L. Lytton, and J.E. Dorn,13 reprinted with permission from
Sherby and Pergamon Press, Elmsford, NY, 1957).
Constant up to the melting point. Similar results have been found in other metals. It would appear that different processes were rate controlling over the test temperature range. furthermore, it should be recognized that HC may represent some average activation energy reflecting the integrated effect of several mechanisms operating simultaneously and interdependently (see Section ).
Dorn, Garofalo, and Weertman have compiled a considerable body of data to demonstrate that at Th, HCis most often equal in magnitude to HsD, the activation energy for self-diffusion Figure; this fact strongly suggests the latter to be the creep rate-controlling process in
this temperature regime. While the approximate equality between HC and HSD seems to hold
for many metals and ceramics at temperatures equal to and greater than half the melting point, some exceptions do exist, particularly for the case of intermetallic and nonmetallic compounds. It is found that small departures from stoichioometry of theses compounds have a pronounced
effect on HC, which in turn affects the creep rate. For example , a reduction in oxygen content
in rutile from TiO2 to TiO1.99 causes a reduction in HCfrom about 280 to 120 kJ/mol(67-29
kcal/mol)* with an associated 100-fold increase in .s14 for the more general case, however, the
creep process is found to be controlled by the diffusivity of the material
/ SD H RT o D D e Where D = diffusivity, cm2/s D0 = diffusivity constant 1 cm2/S
HSD = activation energy, J/mol R = gas constant, J/K
FIGURE. Correlation between activation energy for self-diffusion and creep in numerous metals and ceramics. (From J. Weertman,16 reprinted with permission from American society
for Metals, Metals Park, OH, copyright (c ) 1968. (ko V T)m/T
o
D D e
0
K = dependent on the crystal structure and equal to 14 for BCC lattice, 17 for FCC and HCP lattices, and 21 for diamond-cubic lattice
V = valence of the material
m
T = absolute melting temperature
The constants K0 are estimates associated with an assumed diffusivity constant 1cm2/s.
By combining Equation.5-13 and 5-14
0
( )
SD m
H RT K V
We see that activation energy for self-diffusion increases (corresponding to a reduction D) with increasing melting point, valence, packing density, and degree of convalency. Consequently, although refractory metals with high melting points, such as tungsten, molybdenum, and chromium, seem to hold promise as candidates for high-temperature service, their performance in high-temperature application is adversely affected by their open BCC lattice, which enhances diffusion rates. From Equation. Ceramics are identified as the best high-temperature materials because of their high melting point and the covalent bonding that often exists.
It is important to recognize that creep rates for all materials cannot be normalized on the basis of D alone because other test variables affect the creep process in different materials. For
example, Barrett and coworkers19 noted the important influence of elastic modulus on the creep
rate and on determination of the true activation energy for creep. A semi-empirical relationship with the form
. n skT A DGb G
Has been proposed1 to account for other factors where
.
= steady-state creep rate K = Boltzman’s constant T = absolute temperature D = diffusivity G = Shear modulus B = Burgers vector = applied stress
A,n = material constants
By combining Equation. 5-8 and 5.13 , the steady-state creep rate at different temperatures can be normalized with respect to D to produce a single curve, as shown in Figure. This is an important finding since it allows one to conveniently portray a great deal of data for a given material. For example, we see from a reexamination of Figure. That at the allotropic
transformation temperature, the creep rate in -ion (FCC lattice) is found to be approximately 200
time slower than that experienced by -iron (BCC lattice).6 This substantial difference is traced
directly to the 350-fold lower diffusivity in the close-packed FCC lattice in -ion. Similar findings
were reviewed by Sherby andBurke17 for the allotropic transformation from HCP to BCC in
thallium. Therefore, it is appropriate to briefly consider those factors that strongly inflorescence
magnitude of D. Sherby andsimnad18 reported an empirical correlation showing D to be a
function f the type of lattice, the valence, and the absolute melting point of the material.
Here again we see that creep is assumed to be diffusion controlled. Even after normalizing creep data with Equation, a three-decade scatter band still exists for the various metals shown in Figure. While some of this difference might be attributable to actual test scatter or relatively imprecise high-temperature measurements of D and G, other as yet unaccounted for variables most likely will account for the remaining inexactness. For example, there appears to be a trend toward higher creep rates in FCC metals and alloys possessing high stacking fault energy (SFE). Whether the SFE variable should be incorporated into either A or n id the subject of current
discussion.20-22 The role of substructure on A and n must also be identified more precisely.
FIGURE. Creep data in aluminum. (a) Stress versus steady-state creep rate .s divided by the diffusion coefficient. (From O.D. Sherby and P.M. Burke;17 reprinted with permission from
Sherby and Pergamon Press, Elmsford,Ny, 1968).
It is now generally recognized that .s varies directly with at low stresses and
temperatures near the melting point. At intermediate to high stresses and at tompet atures above
0.5Tm, where the thermally activated creep process is dominated by the activation energy for
self-diffusion, .s 4-5 (so-called power law creep). It should be noted that this stress dependency
holds for pure metals and their solid solutions. Much stronger stress dependencies of .s and ts
have been reported in oxide-dispersion-strengthened superalloys (see Section). At very high stress levels .s e . Garofalo23 showed that power law and exponential creep resented limiting
cases for a general empirical relationship
. s
(sinh )n
Equation . reduces to power law creep when <0.8, but approximates exponential creep
when < 1.2. An explanation for the changing stress dependence of .s in several operative
FIGURE. Creep data in metals. (a) Data for FCC metals; materials with high-stacking lault energy then to have higher steady-state creep rates. (b) Data for BCC metals. (From A.K. Mukherjee, J.E. Bird, and J.E. Dorn1; copyright American society for Metals, Metals Park, OH,
( c) 1969).
4. Creep laws, Factors Affecting Creep, Mechanism of Creep.
Primary, secondary and tertiary creep curves follow different creep laws for various materials.
The variation of creep strain cr with time t may be expressed as below.
Andrade’s law of transient creep for metals and some plasties expresses creep strains as, n
cr Ct
where C is constant, and n is power index constant whose value is 1/3. Logarithmic law of
transient creep for glass and rubber expresses creep strain as,
1 log 1 cr c t K t
Where K is a constant and t1 is any arbitrarily choosen time.
Hyperbolic law of transient creep for concrete expresses creep strain as,
cr t n t
Where is a constant, and n is creep-time constant.
Secondary creep law may be stated by
cr 1 v tcr
Where 1 is creep intercept (see Figure) and v is viscous or minimum creep rate. cr
Minimum creep rate increases with increasing stress and is given by
n cr
v A (n>1) Where A and n are constants.
Example: During a creep test on pure aluminium at 280oC under steady stress of 6.85 MPa, the
following data were recorded.
Plot strain-time curve, and show the extents of primary, secondary and tertiary stages on it. Determine (a) minimum creep rate, (b) the creep intercept, and (c) transient creep law.
Solution : The strain-time cries shown in Figure. Primary and secondary creep stages are market
or it. It does not have tertiary creep stage.
Time t (min) Stain (mm’mm) Time t (min) Strain (mm/mm) 0 0 24 0.094 1 0.020 32 0.109 2 0.029 40 0.122 4 0.041 48 0.136 8 0.057 60 0.156 16 0.078 72 0.176
(a) The minimum creep rate is found by taking slope of the viscous part of the curves. It is shown in above figure and is obtained as
12 0.857 14 cr v mm/mm
(b) the creep intercept marked in above figure is found to be
1
= 0.055 mm/mm
(c) As the material is the question is a metal (pure A1), we shall use Equation. Taking log on both sides of this equation.
logcr logC n logt
Considering the data for t= min and t = 4 min, the Equation. (I may be written as log 0.02 = log C + n log 1
log 0.041 = log C + n log 4 Solution of Equation. (ii) and (iii) yields
n = 0.51
Now substitution of this value n other of Equation (ii)and (iii) gives C = 0.02
Hence transient law is obtained as
0.51
0.02
cr t
Factors Affecting Creep :
It has already been pointed-out that the load (hence stress) and tempera ture influence the creep behaviour of a material. So we obtain different curve profiles as shown in figure . There separate curves marked A, Band C for the same material are shown. If the temperature is
Figure. Effect of changing temperature at constant stress and changing stress at constant temper
Constant, the curves A, B and C are obtained at stresses 1 2and 3( 3 2 1) respectively. Similarly if the stress is kept constant, the curves A, B and C are noticed at temperaturesT T1, 2 and (T T3 3 T2 T1). Although a single diagram is shown to explain two effects,
but it does not mean that the same curves are inter-replace abed in the two cases of = constant
and T = constant.
It may be concluded that the effect of increasing stress and temperature is to speed-up the rate or creep At higher stress or at higher temperature. The total strain is large and creep fracture occurs in lesser time . The duration of three creep stages also very. Consequently viscous stage It is prolonged in curved reduced in curve B and messing in curve C .
Mechanism of Creep :
Occurrence
1. Vacancy diffusion
2. Edge dislocation climb-up or climb-down. 3. Grain boundary sliding
4. Screw dislocations cross-slip 5. Elastic aftereffect.
Figure. Mechanism of creep (a) vacancy diffusion (b) dislocation climb-up or climb-down, and (c ) grain boundary sliding.
Creep Resistant Materials :
Machine and structural parts functioning at higher temperatures must be creep resistant. Pressure vessels and heat exchangers in oil refinery and chemical industries operate at elevated temperatures. Heat engines need to operate at higher operating temperatures to achieve enhanced thermal efficiency. This necessitates the creep resistant materials to have high melting points. Some of the probable materials may be as
follows :
1. Refractories,
2. Tungsten bases alloys,
3. Nickel based alloys and nickel superalloys,
4. Cobalt based alloys
5. Steel based alloys,
6. Monocrystal titanium, and
Of these the Refractories are brittle and cannot take purposeful tensile load. Tungsten and titanium are costly metals. Tungsten is also heavy. Nickel based alloys, cobalt based alloys and steel based alloys are suitable for use from different view-points.
Nickel using Thoria by dispersion hardening method is a very good creep resistant material. It can maintain its strength upto a temperature of about 0.9 Tm. some of the latest
materials as given below are also useful.
1. Silicon nitride (Si3N4) for piston rings and cylinder heads.
2. Sialons (alloys of Si3N4 and Al2O3) for gas turbine blades upto 1300oC
UNIT – II
1. HARDENING (CONVENTIONAL HARDENING) :
By Hardening process a new hard & brittle structure called Martensite is formed.
Hardening can be explained y drawing the appropriate cooling curve in the TTT diagram. Every steel/alloy steel used in heavy engineering industry must undergo hardening. This may be understood from the following example. The needle that we use to stitch clothes is actually a highly flexible steel wire, it is only after the hardening process that it obtains the necessary hardness and does not bend.
Purpose of hardening (i) develop high hardness (ii) Improve mechanical properties (strength, elasticity, ductility and toughness) (iii) Improve wear resistance. Consider the following cooling curve drawn on a TTT diagram for a hypoeutectiod steel.
Figure. Heat treatment cycle for conventional hardening process
It is obvious that the surface and the centre of the specimen will have slight different cooling curves, which depends on the cross section of the specimen, both the curves must come under the same region in order to have the same structure on the surface and the core.
Hardening process :
The steel is heated to the austenitic temperature above A3 for hypoeutectoid steel and above
A31 for hypereutectoid steel (see figure (b) and kept in that region (soaking for the complete
transformation of the structure to austenite. It is then drastically cooled to room temperature (Note : It is not cooled below room temperature). And much of the austenite will transform to a new needle like or have like or acicular structure called Martensite. The cooling (quenching) may be performed by using a salt bath (molten KCN or NaCN, salt) or oil bath or brine solution.
The mechanism by which Martensite is formed has already been explained in section figure.
It is important to note that even the drastic cooling is not sufficient to convert the entire austenite to Martensite, hence some unstable austenite remains even after cooling as shown in figure. This austenite is called retained austenite.
Figure. Scheme showing the formation of martensitic structure
The transformation that takes place for a hypoeutectoid steel is :
(FCC structure) slowcooling (bcc structure ) + Fe3C
Drastic cooling (quenching)
M (BCT Body Centered Tretragonal structure).
In most steels, the amount of Martensite that forms is a function of the temperature of which the austenite is cooled and not a function of time.
Figure: Representation of percentage of Martensite formed as a function of temperature
The martensitic transformation occurs with out a change in composition, it occurs by a process of shear and is not caused by diffusion of carbon.
The hardness of Martensite depends on the carbon percentage present. It increases rapidly with increases in carbon content. The maximum value reached is around 64 Rc (Rockwell hardness on ‘C’ scale) at about 0.6% carbon.
Figure:
Martensitic structure is extremely hard and brittle and the steel becomes too brittle to be used in engineering applications. Martensite is said to be in a metastable phase.
The steel as quenched (i.e., after quenching and without further heat treatment process) may even crack at room temperature, such is the unstability and thermal stresses created within the martensitic structure, and further heat treatment is required to remove these stresses and avoid cracking (called quench cracks)
The steel is reheated to reduce its brittleness, without much loss of hardness. This heat treatment process is called Tempering.
It may be observed in Figure that not all the austenite converts to Martensite after quenching. Some of the austenite remains and is called as retained austenite.
During tempering the steel is reheated to a lower temperature above the Ms (Martensite start formation) temperature see figure and is cooled to room temperature for a transformation to complete.
Depending on the tempering temperature some small percentage of retained austenite and a soft structure which is generally called tempered Martensite (the detailed structre will be explained in unit) is obtained. The formation of the soft structure lowers the hardness. In the case of alloy steel such as high speed. steel the hardness actually increased due to tempering and then drops. In this the retained austenite converts to Martensite there by increasing the hardness(this is called secondary hardening) and there is also the formation of complex carbides with the alloying elements present in steel. The Martensite already present becomes tempered Martensite. Two to three tempering may be required in order to completely transform the
retained austenite to Martensite. Each tempering stage must be followed by cooling to room temperature as the transformation from retained austenite to Marten site takes place after cooling below Marten site transformation temperature.
Figure Effect of tempering temperature on hardness
The following figure explains a typical heat treatment cycle for high speed steel. Preheating is required in order to avoid stress cracks that may be formed if the steel is directly heated to the austenitic temperature form room temperature. The heating time and temperature depends on the cross section of the component heat treated.
Note on retained austenite :
Austenite to Martensite transformation depends on temperature. The transformation is never completed to 100% Martensite.
The amount of retained austenite varies from surface of the component to centre. It is less at or near the surface and more in the centre. This is because the surface cools first and then the centre. The amount of retained austenite also depends of the quenching temperature. Drastic low temperature quenching results in the formation of more percentage of Martensite. Retained austenite has certain advantages;
(i) Austenite reduces the tendency of cracking during hardening and hence about 10%
retained austenite is desirable.
(ii) If the retained austenite is more say 30-40% the steel can be easily cold worked to some extent without cracking.
Retained austenite has certain disadvantages ;
(i) Austernite is a soft unstable phase and the presence of retained austenite reduces the
hardness of hardened steel.
(ii) Small amount of retained austenite does not decrease the hardness much, but it may
increases the brittleness of steel. This is because of the fact that the austenite may get transformed to Martensite if the material is subjected to plastic deformation. This deformation (strain) induces transformation of austenite to Martensite and increases stress and as a result of which the mechanical properties decreases.
(iii) The retained austenite may slowly transform to bainite even at room temperature. This liner expansion may be 0.0001 cu/cm for every 0.3% retained austenite by volume and may cause increase in dimensions especially in sensitive gauges and tools.
Thus tempering eliminates the presence of retained austenite to some extent. Repeated tempering (atleast two ) transforms more retained austenite to Martensite. Each tempering stage must be followed by cooling to room temperate see figure.
An effective way of eliminating retained austenite is sub zero treatment, where the steel
component is cooled to very low temperatures, substances such as acetone and dry ice (-100oF) or
liquefied gases such as nitrogen (-321oF), oxygen (-297oF) or helium (-4530F) may be used as
quenching medium. After subzero treatment the steel is quenched to room temperature in conventional quenching mediums (air, oil, water). (It is beyond the scope of this book to deal with sub zero treatment in detail). It is also possible to eliminate austenite by plastic deformation above Ms Temperature. The phenomenon is called induced martensitic transformation. This method is suitable for steels with large amount of retained austenite.
Calculation for the time required to Harden :
It is useful to calculate the length of time required for hardening. Newton’s law of heating is a means of calculating the hardening time. The rate of heating of metals which are good conductors is limited to the transfer of heat from the surroundings i.e., furnace atmosphere to the surface of the metal piece being heat treated and not by internal resistance to heat flow in the metal piece being heated. Temperature difference within the piece are small when compared to those between surface and surroundings. For the purpose of describing the rate of heating it is assumed that the metal piece obtains uniform temperature throughout the piece.
The rate of heat absorption can be represented as :
P P
dT Q V C
dt
Where V = Volume P density = mass
C = Specificheat P
dT
dt = rate of change of temperature with time
The rate of heat transfer from the surrounding to the surface (by convection and radiation ) can be represented by :
Q = h A ( Tf – T )
Where h = heat transfer coefficient A = Total surface area
Tf = furnace temperature
T = Temperature to which the piece is being heated Q = V P Cp dT
dt Also
Q = h A(Tf - T)
Equating these two expressions we get :
P dT Q dt V C ( f ) P hA T T dT dt V C ( ) P f dtc V C dT hA T T ( ) P f V C dT dt hA T T
0 ) f T P f T V C dT t hA T T
To – Initial temperature of the piece when it is placed in the furnace
Tf – Furnace temperature 0 2.3 log f P f T T V C t hA T T The value f 0 f T T T T
is direct by proportional to the volume to total surface area ratio (V/A).
this ration depends on the size and shape of the part.
2. Strain (work) Hardening
Stain hardening (also referred to as work hardening or cold working) dates back to the Bronze
Age and is perhaps the first widely used strengthening mechanism for metals. Artisans hammered and bent metals to desired shapes and achieved superior strength in the process. Typical cold-worked commercial products that find used today include cold-drawn piano wire and cold-rolled sheet metal. Strain hardening results from a dramatic increases in the number of dislocation-dislocation interactions and which reduces dislocation mobility. As a result, larger stresses must be applied in order that additional deformation deformation may take place. It is interesting to note that the strength of a metal approaches extremely high levels when there are either no dislocations present (recall Equation) or when the number of dislocation is extremely high (10 / cm10 2
); low strength levels correspond to the presence of moderate numbers of
dislocation ( 103 – 105/cm2) (Figure).
To characterize more clearly the strain – hardening behavior of metal crystals, it is helpful to examine the stress-strain response of single crystals. From Figure. the
Resolved shear stress- shear strain curve is seen to contain several distinct regions: an initial
region of elastic response where the resolved shear stress is less than T CRSS; stage I, a region of
easy glide; Stage II, a region of linear hardening; and Stage III, a region of dynamic recovery or parabolic hardening. The latter three regions involve different aspects of the plastic deformation process for a given crystal. It is known that the extent of Stages I, II, and depends on such factors as the test temperature, crystal purity, initial dislocation density, and initial crystal orientation.12 It
should be noted that Stage III Closely resembles the stress-strain response of the polycrystals form of the same material.
FIGURE : Shear stress-strain curve for single crystal revealing elastic behavior when T<Tcrss and
Stage I,II,III plastic response when T > Tcrss. ,
I II III
measure the strain hardening rate in each region.
A number of theories of theories have been proposed to explain the strain-hardening process in crystals, including the reason for the dramatic changes in strain-hardening rate associated with the three stages of plastic deformation. An extensive literature3 has developed
regarding these theories, al of which have focused on some of the dislocation interaction mechanisms described in the previous chapter. Seeger4 and Friedel,5 for example, argued that
rapid strain hardening in Stage II resulted from extensive formation of dislocation pileups at strong obstacles such as Cottrell-Lomer locks.6.7 The latter represents a sessile (nonmobile )
dislocation that impedes the motion of other dislocation on their respective slip planes. An example of such a barrier is given by
The 011and
110
dislocations, which move along their slip planes, (111) and (11 1 ),respectively, join to produce the sessile dislocation 110, which cannot move along either plane.
Note that this dislocation reaction is permissible since the total elastic energy is reduced (recall Equation). Mott8 proposed that heavily jogged dislocation produced by dislocation-dislocation interactions(see Section) would be more resistant to movement, there by enhancing the hardening rate. Unfortunately, a certain degree of confusion has arisen in this field because of the varying importance of certain dislocation interactions in different alloy crystals. One wonders then why the three distinct stages of deformation are so reproducible from one material to another and why
the work hardening coefficient II associated with Stage II deformation is almost universally
constant at G/300. For these reasons the ‚mesh length‛ theory of strain hardening proposed by
Kuhlmann-Wilsdorf 9,10 is appealing pedagogically, since it does not depend on any specific
dislocations model that might be appropriate for one material but not for another. Her theory may be summarized as follows: In stage I a heterogeneous distribution of low-density dislocation exists in the crystal. Since these dislocations can move along their slip planes with little interference from other dislocations, the strain hardening rate I is low. The easy glide region
(Stage I) is considered to end when a fairly uniform dislocation distribution of moderate density is developed but not necessarily in lockstep with the onset of conjugate slip where a marked increase in dislocation – dislocation interactions would be expected. At this point Kuhlmann-Wilsdorf theorizes the existence of a quasi-uniform dislocation density array with clusters of dislocations surrounding cells of relatively low dislocation density figure. It is believed that such cell structures represent a minimum energy and, hence, preferred dislocation configuration within the crystal. Studies have shown that high stacking fault energy metals (e.g., aluminum) exhibit cell walls that are narrower and cell interiors that are more dislocation-free than lower stacking fault energy metals (e.g., coper) figure. (In very low stacking fault energy metals (e.g., Cu-7% Al) the crystal substructure is characterized by dislocation planar arrays, consistent with the tendency for these materials to exhibit restricted cross slip.
The stress necessary for further plastic deformation is then seen to depend on the mean free dislocation length l in a manner similar to that necessary for the activation of a Frank-Read source where
Gb T
l
Figure. Dislocation substructures in metals : (a)aluminium; (b) copper; (c) copper-7% aluminium. (d) Variation in dislocation cell size with percentage reduction of area in polycrystalline niobium steel alloy.
Since the dislocation density is proportional to 1/ l2, Equation may be written in the form
Gb
Where = dislocation density
T= incremental shear stress necessary to overcome dislocation barriers
The relationship has been verified experimentally for an impressive number of materials and represents a necessary requirement for any strain hardening theory. With increasing plastic deformation, increases resulting in a decrease in the mean free dislocation length l . From equation, the stress necessary for further deformation then increases. Kuhlmann-Wilsdorf
suggests9 that there is a continued reduction in cell size and an associated increase in flow stress
throughout the linear hardening region. In other words, the character of the dislocation distribution remains unchanged, only the scale of the distribution changes (see region AB in figure). With further deformation, the number of free dislocations within the cell interior decreases to the point where glide dislocations can move relatively unimpeded from one cell wall to another. Since the formation of new cell walls (and hence a reduction in l ) is believed to depend on such interations, a point would be reached where the cell size l would stabilize or at
best decrease slowly with further deformation. According to Kuhlmann-Wilsdorf,10 this condition
signals the onset of Stage III and a lower strain hardening rate, since l would not decrease. Recently Bassin and Klassen provided experimental confirmation that Stage III behaviour corresponds to strain levels where l remains constant (see region BC in figure). Of particular note, the data reported in figure are measurements taken from a polycrystalline niobium steel alloy; as such, the mesh length theory of strain hardening is applicable for both single crystal and polycrystalline commercial alloys.
Stacking fault energy is considered to be important to the onset of Stage III. Seeger4 has
argued that Stage III begins when dislocations can cross-slip around their barriers, a view initially supported by kuhlmann-Wilsdorf. From Seeger’s point of view, Stage III would occur sooner for high stacking fault energy materials since cross-slip would be activated at a lower stress. Conversely, a low stacking fault energy material, such as brass, would require a larger stress necessary to force the widely separated partial dislocations to recombine a larger stress necessary to force the widely separated partial dislocations to recombine and hence cross-slip. More recently, Kuhlman-wils dorf10,11 suggested that the mesh length theory could also explain the
sensitivity of TIII to stacking fault energy by proposing that enhanced cross-slip associated with a
high cvalue of stacking fault energy would accelerate the dislocation rearrangement process. Consequently, l would become stabilized at a lower stress level. Setting aside for the moment the question of the correctness of the seeger versus Kuhlmann. Wilsdorf interperetations, interpretains, is is sufficient for us to note that both theories account for the inverse dependence of TIII on stacking fault energy.
In discussing the deformation structure of metals, it is important to keep in mind the temperature of the operation. In is know that the highly oriented grain structure in a wrought product, which has a very high dislocation density (1011to1013 dislocations/cm2), remains stable
only when the combination of stored strain energy (related to the dislocation substructure) is below a certain level. If not, the microstructure becomes unstable and new strain-free equiaxed grains are formed by combined recovery, recrystallisation, and grain growth processes. These
new grains will have a much lower dislocation density (in the range of 104 to 106
dislocations/cm2). When mechanical deformation at a given temperature causes the
microstructure to recrystallize spontaneously, the material is said to have been hot worked. If the microstructure were stable at that temperature, the metal experienced cold working. The temperature at which metals undergo hot working varies widely from one alloy to another but is generally found to occur at about one-third the absolute melting temperature. Accordingly, lead
is hot worked at room temperature, while tungsten may be cold worked at 15000C.
Before concluding the discussion of single-crystal stress-strain curves, it is appropriate to consider whether one can relate qualitative and quantitative aspects of the stress-strain Response of single-crystal and polycrystalline specimens of the same material. For one thing, the early stages of single-crystal deformation would not be expected in a polycrystalline sample because of the large number of slip systems that would operate (especially near grain boundary regions) and interact with one another. Consequently, the tensile stress-strain responses of the polycrystalline sample is found to be similar only to the Stage III single-crystal shear stress-strain plot. A number of attempts have been made to relate these two stress-strain curves. From Equation.
1 cos cos M A
where M = 1/(coscos)
Assuming the individual grains in a polycrystalline aggregate to be randomly oriented, M would very with each grain such that some average orientation factor M would have to be defined. Since there are 384 combination of the five necessary slip systems to accomplish an arbitrary shape change, M is not easy to compute. From section 3.1, Taylor14 determined the
preferred combination to be the one for which the sum of the glide shears west minimized. As a result it may be shown15 that
M
By combining Equation 1 and 2 it is seen that
2 d d M d d
For the case of
111 110 slip in FCC metals and
110 111 slip in BCC metals, Taylor14and Groves and Kelly16 showed M equal to 3.07. Subsequently. Chin et al.17.18 analyzed the more
difficult case of
110 111 +
112 111 +
123 111 slip in BCC crystals and found M = 2.75. In either case, one can see from Equation. That the strain-hardening rate of a polycrystalline material is many times greater than its single-crystal counterpart.3 Rupture life of Creep
The Larson-Miller parameter is, perhaps, most widely used. Larson and Miller57correctly
surmised creep to be thermally activated with the creep rate described by an Arrhenius-type expression of the form
/ H RT r Ae
Where r = creep process rate H
= activation energy for the creep process
T = absolute temperature R = gas constant
A = constant
Equation 5-24 also can be written as H
Inr
RT
After rearranging and multiplying by T, Equation becomes
/ ( )
H R T InA Inr
Since r (l/t) (also suggested by Equation), Equation can be written as
/ 1 ' H RT A e t Therefore, ' H Int InA RT
And after rearranging Equation, multiplying by T, and converting Int to logt
/ ( )
H R T Iogt
Which represents the most widely used form of the Larson-Miller relation. Assuming H
to be independent of applied stress and temperature(not always true as demonstrated earlier)
the material is thought to exhibit a particular Larson-Miller parameter
T c( log )t
for a given applied stress. That is to say, the rupture life of a sample at a given stress level will very with test temperature in such a way that the Larson-Miller parameter (T clog )t remains unchanged. Forexample, if the test temperature for a particular material with c = 20 were increased from 8000C to