Abstract
There are many applications where the combination of stress and elevated temperature require creep to be considered during the design process. For some applications, an evaluation of rupture life for given conditions is sufficient, however, for components such as those in gas turbine aeroengines, the accumulation of creep strain over time and the effect this has on other phenomena, such as high-temperature fatigue must be considered. In this chapter, modern creep curve modelling methods are applied to alloys used in gas turbine applications over a wide range of test conditions. Also, different creep hardening modelling methods are discussed along with their application to transient creep showing the deficiencies of simplistic models. Models are related to micromechanical properties where possible, and creep damage models are evaluated and applied to different applications using finite element analysis (FEA).
Keywords
- creep
- creep damage
- stress relaxation
- finite element analysis (FEA)
1. Introduction
When designing components for high temperature applications, the effects of creep must be considered. For simple cases, prediction methods that evaluate creep rupture life,
where
Evaluating creep behaviour based on single point prediction methods alone cannot fully describe the shape of a creep curve since different curves can exhibit the same rupture life and minimum creep rate (Figure 1). Furthermore, to evaluate the creep behaviour of complex shaped components, simple calculations must be replaced by a more multifaceted approach such as finite element analysis [10]. To develop creep models suitable for finite element analysis, the full creep curve of a material must be obtainable over a wide range of conditions and an appropriate hardening model must be used. Furthermore, in order to predict rupture during variable creep conditions, a suitable damage model must be used.
2. Creep deformation
2.1. Full creep curve prediction methods
Many approaches exist for evaluating creep deformation behaviour from simple time-hardening models which only predict tertiary creep, to phenomenological models which aim to predict the micro-mechanisms of creep. Two main approaches for predicting full creep curves are investigated in this section. The first approach involves evaluating the times to given creep strain levels for a range of tests using a similar approach to stress rupture. Parameters are then derived to describe how this relationship varies with increasing creep strain. Examples of this method include an empirical relationship proposed by Gray and Whittaker [11] and a method based on the Wilshire equations [12, 13]. The former method relates times to strain,
where the
where
The Wilshire approach has been show to extrapolate stress rupture times and minimum creep rates well [4, 5, 12–15]. An extension of this method uses an equation of similar form to relate times to a given creep strain to applied stress and temperature,
where σ and
Different relationships have been proposed to relate the parameters of Eq. (5) for different values of strain. For the nickel based superalloy Alloy720Li, w was found to vary minimally as strain increased and was assumed to be constant. A power law relationship was used to relate
where
An alternative method to predicting full creep curves involves using equations to represent time verses creep strain behaviour and relating the parameters of these equations to applied stress and temperature. Examples of this method include the theta-projection method [16, 17], a model by Dyson and McLean [18], and, a true stress method proposed by Wu et al. [19]. Both methods split the creep curve in to primary and tertiary regions, using equations of similar form to characterise each region. In the case of the theta method, creep strain is related to time using:
where
for
Various methods have been proposed to relate
where
where
An alternative approach to the
where ε0 is the instantaneous strain,
2.2. Hardening methods
Various methods to predict creep curves and hence creep rates during uniaxial creep have been proposed, however, to predict creep behaviour in engineering components using FEA, a suitable hardening model is required. During uniaxial constant stress creep, creep rate,
However, in real engineering situations, the stress and/or temperature may change, resulting in the calculation of a different creep curve. Therefore, creep behaviour is now dependent on the position on the creep curve at new applied conditions and therefore more information is required to calculate creep rate. The simplest case is to consider that the new creep rate relates to the current creep time on the new curve (time-based hardening). This method, although easy to implement, fails to accurately account for strain rate changes for all but minor changes in applied conditions. For large changes in creep conditions, the shape of the creep curve changes significantly resulting in poor predictions. An alternative method is to relate creep rate to the applied conditions and creep strain history (strain-based creep hardening). For this case, the strain rate is calculated at the point on a creep curve which relates to the current total creep strain. This method, although more accurate that time-based hardening, produces inaccuracies when creep conditions change from primary creep dominated to tertiary creep dominated. To address the shortcomings of these methods, a life-fraction hardening method is often used. This method uses an effective time based on
An alternative method of creep hardening is to base creep rate on the applied conditions and various material state variables
Where
where
The initial effective creep rate and proportionality constants can be related to applied condition using the
If damage is assumed to be purely a tertiary process, the effective creep rate,
For virgin material,
Transient uniaxial creep tests have been used to test the ability of different hardening methods to predict creep at non-constant stress and temperature for the nickel based superalloy, Waspaloy [8]. These creep tests applied temperature and stress which changed between two sets of conditions, one at high stress and low temperature, the other at low stress and high temperature, with both sets of conditions predicted to give similar lives approximately 10 days. The applied conditions were changed every 24 hours with a 1 hour transition period to allow sufficient time for the temperature to change, during which the stress was held at the lower level of the two conditions. Figure 7 shows the creep rates observed from a transient creep test in which the temperature was changed from 600 to 750°C with applied stresses of 880 MPa and 390 MPa respectively, along with the test data for the equivalent isothermal constant stress creep tests. Immediately after each applied condition change, it can be seen that the creep rates are initially high before decreasing in what appears to be regions of ‘pseudo’ primary creep. In general, the creep rates observed during transient creep are higher than those observed for isothermal constant stress creep. The creep rates for the test at 600°C with an applied load of 880 MPa display a large proportion of primary creep, shown by the initially high creep rate followed by a gradual decrease, whereas the test at 750°C and 390 MPa is dominated by tertiary creep, with a minimum creep rate achieved after only 10% of life.
Eq. (8) was used to numerically describe the isothermal constant stress creep data to allow different hardening methods to be evaluated against transient creep rates (Figure 8). Time hardening strain hardening and the hardening method based on internal state variables all predict the first cycle well, however for subsequent cycles the predictions using time and strain based hardening fail to predict the overall trend in creep rate, as well as the local peaks in rate after each load change. In this case, since the predicted rupture lives at each set of conditions is similar, life-fraction hardening would display a similar trend to time-based hardening. The creep rates predicted for a hardening method based on internal state variables describes the trend in rate more accurately but also fails to predict the ‘pseudo’ primary creep at the beginning of each cycle.
At both sets of creep conditions, creep occurs by diffusion controlled movement of dislocations, however, the dominant mechanism by which this occurs is varies between the high-stress/low-temperature and low-stress/high-temperature cases. At the high stress condition, the stress exceeds the yield point resulting in the formation of new dislocations. This leads to higher dislocations in the material at high stress than in the low stress material which only contains dislocations present after forming. Furthermore, at the higher temperature state, more thermal energy is available for diffusion controlled creep mechanisms such as climb. Whereas at low temperatures but higher stresses, precipitation cutting becomes more dominant. Therefore, when changing between two sets of creep conditions, the creep rate is dependent on the loading history of the material.
3. Creep damage and rupture
For simple uniaxial models failure can be creep rupture times may be predicted using simple equations such as those proposed to Norton [1], Larson and Miller [2]. However, for cases where the stress and temperature evolve over time a different approach is required. Creep damage models allow the accumulation of damage to be predicted regardless of applied conditions. Kachanov [21] proposed that the increase in creep rate during tertiary creep could be related to an increase in stress caused by a decrease effective cross sectional area due to the nucleation and growth of grain boundary cavities and triple point cracking. Later, Rabotnov [22] extended this idea by introducing a continuum damage parameter, ω, to represent this decrease in effective cross section. Leckie and Hayhurst [23] generalised this approach and Othman and Hayhurst [24] extended it to include the effects of primary creep. The constitutive model based on the
where
where
Where multiaxial creep data is available, the dependence of
where σTS is the temperature dependent tensile strength [17]. For other alloys such as the titanium alloy Ti6246, less dependence on loading direction was found and
4. Finite element analysis
Once suitable methods to interpolate creep curves for any given condition and suitable hardening models have been derived, it is possible to compile models for use in finite element analyses (FEA). The constitutive model based on the
4.1. Stress relaxation
An application to consider is stress relaxation due to creep. This can be predicted using a simple single element finite element model with a prescribed displacement boundary condition. Figure 10 shows a good correlation of the predicted and experimentally obtained results of a two stage stress relaxation experiment in the nickel based superalloy Alloy720Li using a model based on the strain hardening formulation of the Wilshire creep curve extrapolation method.
4.2. Notched bars
The value of FEA comes from its ability to predict mechanical behaviour in engineering components with complex geometries. Components such as those found in gas turbine aeroengines operate at high temperatures for extended periods of time and any geometric feature of the component that concentrates stress will exhibit higher creep rates than surrounding regions resulting in a redistribution of stress. Accurate predictions of this stress redistribution are particularly important for cases where component lives are influenced by fatigue as the stress field affects subsequent fatigue calculations. Furthermore, creep failure can initiate in subsurface locations due to a combination of stress redistribution and subsurface damage [10]. Predictions of stress redistribution in a Waspaloy round circumferentially notched bar after 1000 s of creep deformation using the
The constitutive model based on the
4.3. Small punch creep
The small punch creep test is often used to characterise the mechanical properties of materials where only small quantities exist, such as during novel alloy development or for remnant life assessment [25–27]. The test consists of applying load to the centre of a small disc of material, typically 0.5 mm thick with a diameter of 9.5 mm, using a hemispherical punch. Punch load and displacement are then related to uniaxial stress and strain using conversion factor,
4.4. High-temperature fatigue
The effects of creep in engineering components that are exposed to cyclic loading at high temperatures cannot be ignored. A fatigue crack propagating through material exhibiting creep damage will advance more rapidly as creep cavities and microvoids provide a preferential path for growth. In this case fatigue lives are negatively affected by creep damage and cracks propagate along grain boundaries (intergranular). However, redistribution of stress due to creep can reduce stress at stress concentration features such as cracks and notches, reducing the driving force for crack propagation, hence increasing fatigue lives. This has been clearly shown in the titanium alloy Ti-6Al-4V whereby increasing the temperature from 450 to 500°C increases the fatigue lives of notched specimens, whereas a further increase to 550°C has a negative effect on life [28]. Therefore, during high temperature fatigue, component life is a dependent on time-dependent (creep) and time-independent (fatigue) damage mechanisms. FEA can be used to predict stress relaxation around notches and cracks, as well as to evaluate damage ahead of an advancing crack. Using a quasi-static model, the accumulation of creep deformation and damage ahead of an advancing fatigue crack in the titanium alloy, Ti-6246 has been predicted (Figure 14) [10].
5. Conclusions
Although simple single point prediction methods can be useful for certain applications, accurate predictions of the mechanical response of some engineering components that operate at elevated temperatures require creep models that predict the full shape of the creep curve. These models, implemented in FEA have been used to predict creep behaviour in cases where stress and/or temperature are not constant, such as during stress relaxation. Furthermore, attempts have been made to relate model parameters to observed micromechanical behaviour. Creep damage models have been used to provide useful predictions of creep life which can be used to evaluate time dependent damage during other load cases, such as during high temperature fatigue and thermo-mechanical fatigue.
References
- 1.
Norton FH. The Creep of Steels at High Temperatures. New York: McGraw-Hill; 1929. 112 p - 2.
Larson FR, Miller J. A time-temperature relationship for rupture and creep stresses. Trans. ASME. 1952; 74 :765-775 - 3.
Williams SJ, Bache MR, Wilshire B. 25 Year Perspective Recent developments in analysis of high temperature creep and creep fracture behaviour. Materials Science and Technology. 2010; 26 (11):1332-1337 - 4.
Wilshire B, Battenbough AJ. Creep and fracture of polycrystalline copper. Materials Science and Engineering A. 2007; 443 :156-166 - 5.
Whittaker MT, Harrison WJ, Lancaster RJ, Williams S. An analysis of modern creep lifing methodologies in the titanium alloy Ti6-4. Materials Science and Engineering: A. 2013; 577 :114-119 - 6.
Abdallah Z, Gray V, Whittaker M, Perkins K. A critical analysis of the conventionally employed creep lifing methods. Materials. 2014; 7 (5):3371-3398 - 7.
Monkman FC, Grant NJ. An empirical relationship between rupture life and minimum creep rate. In: Grant NJ, Mullendore AW, editors. Deformation and Fracture at Elevated Temperatures. Boston: MIT Press; 1965 - 8.
Harrison WJ, Whittaker MT, Deen C. Creep behaviour of Waspaloy under non-constant stress and temperature. Materials Research Innovations. 2013; 17 (5):323-326 - 9.
Whittaker M, Lancaster R, Harrison W, Pretty C, Williams S. An empirical approach to correlating thermo-mechanical fatigue behaviour of a polycrystalline Ni-base superalloy. Materials. 2013; 6 (11):5275-5290 - 10.
Harrison W. Creep modelling of Ti6246 and Waspaloy using ABAQUS [thesis]. UK: University of Wales Swansea; 2007 - 11.
Gray V, Whittaker M. Development and assessment of a new empirical model for predicting full creep curves. Materials. 2015; 8 (7):4582-4592 - 12.
Abdallah Z, Perkins K, Williams S. Advances in the Wilshire extrapolation technique—Full creep curve representation for the aerospace alloy Titanium 834. Materials Science and Engineering: A. 2012; 550 :176-182 - 13.
Harrison W, Whittaker M, Williams S. Recent advances in creep modelling of the nickel base superalloy, alloy 720Li. Materials. 2013; 6 (3):1118-1137 - 14.
Whittaker MT, Wilshire B. Creep and creep fracture of 2.25Cr-1W steels (Grade 23). Materials Science and Engineering A. 2010; 527 (18):4932-4938 - 15.
Whittaker MT, Harrison WJ, Deen C, Rae C, Williams S. Creep deformation by dislocation movement in Waspaloy. Materials. 2017; 10 (61):1-14 - 16.
Evans RW, Parker JD, Wilshire B. An extrapolation procedure for long-term creep strain and creep life prediction, with special reference to 0.5Cr0.5Mo0.25V ferritic steels. In: Wilshire B, Owen DRJ, editors. Recent Advances in Creep and Fracture of Engineering Materials and Structures. Swansea: Pineridge Press; 1982. p. 135 - 17.
Evans RW. A constitutive model for the high-temperature creep of particle-hardened alloys based on the θ projection method. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. 2000;456 (1996):835-868 - 18.
Dyson BE, McLean M. Microstructural evolution and its effects on the creep performance of high temperature alloys. In: Microstructural Stability of Creep Resistant Alloys for High Temperature Plant Applications. Institution of Metals, London; 1998, pp. 371-393 - 19.
Wu X, Williams S, Gong D. A true-stress creep model based on deformation mechanisms for polycrystalline materials. Journal of Materials Engineering and Performance. 2012; 21 (11):2255-2262 - 20.
Harrison W, Abdallah Z, Whittaker M. A model for creep and creep damage in the γ-titanium aluminide Ti-45Al-2Mn-2Nb. Materials. 2014 Mar 14; 7 (3):2194-2209 - 21.
Kachanov LM. Rupture time under creep conditions. International journal of fracture. 1999 Apr 1; 97 (1-4):11-18 - 22.
Rabotnov YN. Creep Problems in Structural Members. Amsterdam: North-Holland; 1969 - 23.
Leckie FA, Hayhurst DR. Constitutive equations for creep rupture. Acta Metallurgica. 1977; 25 (9):1059-1070 - 24.
Othman AM, Hayhurst DR. Multi-axial creep rupture of a model structure using a two parameter material model. International Journal of Mechanical Sciences. 1990; 32 (1):35-48 - 25.
Lancaster RJ, Harrison WJ, Norton G. An analysis of small punch creep behaviour in the γ titanium aluminide Ti–45Al–2Mn–2Nb. Materials Science and Engineering: A. 2015; 626 :263-274 - 26.
Kobayashi KI, Kajihara I, Koyama H, Stratford GC. Deformation and fracture mode during small punch creep tests. Journal of Solid Mechanics and Materials Engineering. 2010; 4 (1):75-86 - 27.
Blagoeva D, Li YZ, Hurst RC. Qualification of P91 welds through small punch creep testing. Journal of Nuclear Materials. 2011; 409 (2):124-130 - 28.
Whittaker MT, Harrison W, Hurley PJ, Williams S. Modelling the behaviour of titanium alloys at high temperature for gas turbine applications. Materials Science and Engineering: A. 2010; 527 (16):4365-4372