In the previous sections, we discussed the different mechanisms of creep that have been observed in various materials. The stress, temperature and microstructural dependence of each mechanism was described and the steady-state strain rate of deformation of each mechanism was correlated to these parameters. We also outlined the different regions, through defor­mation mechanism maps, where a given mechanism would be dominant over others. In all these sections, more emphasis was laid on the second­ary creep region and the mechanism maps were also constructed taking into account these steady-state creep-rates. However, such a methodology would be based on the premise that the secondary creep stage accounts for a significant fraction of the useful creep life. While such a method is not entirely wrong, it is unsuitable for several materials which tend to have larger primary or tertiary creep regimes. For example, Ni-based superalloys have been found to exhibit primary creep strains of the order of 1% or more.81 These alloys are used as materials for fan and compressor blades of aero-engines. The dimensional tolerance for these components is very small and plastic strains in the order of 1% are sufficient to wreck the stability of the engine. Hence under such conditions, modeling by considering only the steady-state creep rates will grossly overestimate the useful creep life of the material. Furthermore some of the mechanisms of creep, for example in
the power-law creep regime, have been proposed following microstructural studies on the crept specimens. For instance observation of subgrains in the crept microstructure is considered evidence for creep controlled by climb of edge dislocations. Similarly observation of jogged screw dislocations is believed to indicate deformation controlled by the Barrett-Nix model or its recent modification proposed by Mills and co-workers.

In most cases the deformation microstructures are investigated through TEM studies. Hence the sample studied, due to its very small volume, may not be a real representation of the condition of the material. Thus there is some uncertainty associated with the rate controlling mechanism. While the physically based mechanisms discussed in the previous sections are impor­tant for understanding and predicting deformation rates, an equally large number of studies has been carried out to predict creep life using math­ematical models and empirical correlations. The Larson-Miller parameter (LMP), Monkman-Grant constant, в-projection concept and a host of other graphical and mathematical methods have been utilized to predict the creep life of various engineering materials.

Generally engineering components are designed for a stress level below which there is no danger of rupture or excess deformation during the ser­vice life of the component. The stress level is decided by one of the follow­ing two criteria: (a) stress level at which rupture/failure would be caused in 100 000 or 200 000 h, whichever period is appropriate and (b) stress level which produces a nominal strain of 0.1%, 0.2% or 0.5% in a certain period, say 100 000 h.82 However there are not many tests carried out till 100 000 h even for established materials and hence it is necessary to extrapolate data from much shorter tests, say 103-104 h. This is especially important for new materials where it is necessary to understand their long term behavior within a short span of time. Hence the extrapolation techniques become important and in this section we discuss some of the existing extrapolation techniques for predicting long term creep behaviors. Penny and Marriott82 provide an excellent review of the various extrapolation methods and also the advantages and disadvantages associated with each method. They divide the extrapolation techniques into three main groups:

1 Parametric methods

2 Graphical methods

3 Algebraic methods.

Equations correlating time-temperature or stress-time fall under the para­metric method. Functional relationships between time, temperature and stress are established and it is believed that when stress is plotted against a function of time and temperature, a single master curve will be obtained. This master curve can be constructed by performing short term tests at

higher temperatures. It is then assumed to be equally valid for longer times and lower temperatures thus allowing for extrapolation. The Larson-Miller method83 is based on this logic. The original Larson-Miller equation is given by the following:

LMP = T (C + logic tr), [3.49]

where LMP is the Larson-Miller parameter and C is a constant which was assumed to be equal to 20 and was found to be reasonably accurate for many materials. Plots of applied stress versus the LMP would then allow extrapolation of short term data for long term predictions. Figure 3.19 shows a LMP obtained from short term tests for a variety of materials. It is interesting to note the change in slope as lower stresses are approached. Some of the other parameters which fall under the category of paramet­ric methods are by Dorn and Shepherd,84 Manson and Haferd,85 Murry,86 etc. However, LMP is quite commonly used in creep life predictions and extrapolations.

Under graphical methods, there are procedures which seek to extrapo­late rupture curves by direct manipulation of the plotted data. Grant and Bucklin,87 Glen,88 Mendelsohn and Manson89 and others proposed methods

that fall under this category. Here we provide a brief description of the Grant — Bucklin method. Grant and Bucklin considered the fact that creep rupture would be influenced by several time — and temperature-dependent effects and hence mode of failure might not be uniform over the whole range of time and temperature. They identified distinct segments of the rupture curve where one mode of failure might be dominant. These segments were later described by linear relations (Fig. 3.20). By plotting the slopes of like seg­ments against temperature, it is possible to extrapolate to temperatures out­side the experimental range. Secondly the positions of the transition points may be plotted on axes of temperature versus tr for extrapolation. However Penny and Marriott87 indicate that such extensions are subjective and sensi­tive to the ability or judgment of the analyst, albeit Grant and Bucklin imply that reliable extrapolations of the rupture curves are not critically dependent on the accurate determination of either slopes or transition points.

The algebraic methods are similar in a way to the parametric methods. The difference lies in finding functions which can combine the effects of stress, temperature and time into a single relation such as

f (c, tr, T )c = constant. [3.50]

Any function f (a, t„ T)c which can be separated into two functions such as

f(a, tr, T )c = f (a) f2 (tr, T) = constant [3.51]

is similar to the time-temperature method of parametric types. In addition to these methods, there are several other methods which have been proposed and found to provide reasonable predictions. Monkman and Grant90 pro­posed a relationship between the steady-state strain rate and rupture time:

estr = к, [3.52]

where к is a material constant known as the Monkman-Grant constant. Figure 3.21 depicts such a plot for internally pressurized cp-Ti tubing.91

Other methods of extrapolation include the в-projection method advo­cated by Wilshire and co-workers9293 where the total creep curve is described by a series of в parameters. Wilshire92 suggests that for most materials the secondary creep region is only an inflection that appears to be a constant over a limited strain range. Hence it was emphasized that creep life mod­eling should take into account the total creep curve including the tertiary creep regime rather than just focusing on the secondary creep rates. On this premise, Wilshire and co-workers advocated the в — projection concept where the total creep curve would be described as

е=вг(1 — exp (-вгі))+ 0з(ехр(04О -1), [3.53]

where 61 scales the primary creep regime, в2 is a rate parameter govern­ing the curvature of the primary stage, в3 scales the tertiary creep regime and в4 is a rate parameter quantifying the shape of the tertiary curve. These parameters are found to change with stress and temperature conditions and accordingly influence a change in the shape of the creep curve. A determina­tion of the stress and temperature dependencies of the в parameters would allow the prediction of long term creep properties. Furthermore Wilshire counters the widely accepted view of transitions in creep mechanisms with changing stress and temperature conditions. The creep characteristics of a 0.5Cr-0.5Mo-0.25V ferritic steel could thus be described by the в-projection over a wide range of stress values based on a single dislocation-based mech­anism. However, as shown in Fig. 3.22 , there are definite changes in stress exponent values with changing stress. Wilshire argues that if different mech­anisms operate in different stress and temperature regimes, data collected in one mechanism regime should not be able to predict the creep behav­ior in a different mechanism regime. Furthermore Wilshire contends that the в-projection approach can be utilized to quantify material behavior in complex, non-steady stress-temperature conditions encountered in service conditions.

In addition to these methods, creep life predictions are also guided by damage mechanics. The irreversible material damage caused by mechan­ical loading and environmental features during creep eventually leads to very high strain rates of deformation and failure. Damage could be due to cavity formation, microcracks and gross deformation such as strain — or ageing-induced. A materials scientist viewpoint on micromechanical causes of damage is given by Le May.94 In addition to creep damage, other mechanisms of damage such as fatigue, surface oxidation and internal cor­rosion are also important. Although some of these phenomena are not temperature dependent, their interactions with creep, such as creep-fatigue interaction, can have significant effects on high temperature damage accu­mulation. The different damage processes constitute ductile creep rupture, intergranular cavitation during creep, continuum creep rupture, continuum fatigue damage, environmental damage and age — and strain-induced hard­ening and softening. In contrast to creep life predictions based on mecha­nistic models, continuum damage mechanics (CDM) attempts to provide a holistic view of the damage process and accordingly models the useful creep life of a material. By accepting the fact that damage is a result of the complex interactions between different mechanisms, CDM provides greater accuracy in creep life estimation in comparison to models based on a single mechanism of creep, namely grain boundary sliding or dislo­cation creep. While there have been many continuum damage mechanics models advocated over the years, a unique model is the one proposed by Kachanov,95 later elaborated by Rabotnov96 and commonly referred to as the Kachanov-Rabotnov model. A brief review of the Kachanov-Rabotnov model is presented below.