- Open Access
Fatigue crack growth in a nickel-based superalloy at elevated temperature - experimental studies, viscoplasticity modelling and XFEM predictions
© Farukh et al.; licensee Springer. 2015
- Received: 25 November 2014
- Accepted: 22 January 2015
- Published: 5 May 2015
Nickel-based superalloys are typically used as blades and discs in the hot section of gas turbine engines, which are subjected to cyclic loading at high temperature during service. Understanding fatigue crack deformation and growth in these alloys at high temperature is crucial for ensuring structural integrity of gas turbines.
Experimental studies of crack growth were carried out for a three-point bending specimen subjected to fatigue at 725°C. In order to remove the influence of oxidation which can be considerable at elevated temperature, crack growth was particularly tested in a vacuum environment with a focus on dwell effects. For simulation, the material behaviour was described by a cyclic viscoplastic model with nonlinear kinematic and isotropic hardening rules, calibrated against test data. In combination with the extended finite element method (XFEM), the viscoplasticity model was further applied to predict crack growth under dwell fatigue. The crack was assumed to grow when the accumulated plastic strain ahead of the crack tip reached a critical value which was back calculated from crack growth test data in vacuum.
Computational analyses of a stationary crack showed the progressive accumulation of strain near the crack tip under fatigue, which justified the strain accumulation criterion used in XFEM prediction of fatigue crack growth. During simulation, the crack length was recorded against the number of loading cycles, and the results were in good agreement with the experimental data. It was also shown, both experimentally and numerically, that an increase of dwell period leads to an increase of crack growth rate due to the increased creep deformation near the crack tip, but this effect is marginal when compared to the dwell effects under fatigue-oxidation conditions.
The strain accumulation criterion was successful in predicting both the path and the rate of crack growth under dwell fatigue. This work proved the capability of XFEM, in conjunction with advanced cyclic viscoplasticity model, for predicting crack growth in nickel alloys at elevated temperature, which has significant implication to gas turbine industries in terms of “damage tolerance” assessment of critical turbine discs and blades.
- Fatigue crack growth
- Crack growth rate
- Finite element analysis
- Nickel base superalloy
Nickel-based superalloys are designed to provide superior combination of properties such as strength, toughness and thermal performance which make them suitable for structural components undergoing high mechanical and thermal stresses. The typical application of these alloys are in turbine blades and discs in the hot section of gas turbine engines, which are subjected to cyclic loading at high temperature during their service life. Understanding the fatigue damage behaviour, associated with crack initiation and propagation, of nickel-based superalloys at high temperature is crucial for structural integrity assessment of gas turbines based on the “damage-tolerance” approach.
The mechanical behaviour of this type of material involves time consuming and costly tests under cyclic load with superimposed hold time at maximum or minimum load level, representative of typical service loading conditions. The material in such circumstances undergoes a combination of creep-fatigue deformation. In particular, numerous studies have been performed to investigate the creep-fatigue crack growth in nickel superalloys, focusing on the effect of various fatigue loading parameters (e.g. waveform, frequency, ratio and dwell periods) on crack propagation (Pang & Reed 2003; Dalby & Tong 2005). Crack growth rates (da/dN) have been correlated with stress intensity factor range (∆K) to quantify the damage tolerance capability of the material. This approach has been used for crack growth characterisation in various engineering materials for over four decades (Suresh 1998). However, the methodology is largely empirical and does not consider the physical mechanism of crack tip deformation, especially the cyclic plasticity, which is believed to control crack growth behaviour in metallic alloys.
Crack growth simulation using finite element has been extensively carried out to study crack-tip plasticity and associated crack growth behaviour including closure effects (Sehitoglu & Sun 1989; Pommier & Bompard 2000; Zhao et al. 2004). In terms of constitutive models, most work is limited to time-independent plasticity, with a lack of capability to predict crack growth path and rate. Recently, Zhao and Tong (Zhao & Tong 2008) used a viscoplastic constitutive model to study the fundamental crack deformation behaviour for a nickel based superalloy at elevated temperature, focusing on the stress–strain field near the crack tip. Results showed distinctive strain ratchetting behaviour near the crack tip, leading to progressive accumulation of tensile strain normal to the crack growth plane. Low frequencies and superimposed hold periods at peak loads significantly enhanced strain accumulation at crack tip, which is also the case for a growing crack. A damage parameter based on strain accumulation was used to predict the crack-growth rate for different fatigue loading conditions. The crack was assumed to grow when the accumulated strain ahead of the crack tip reaches a critical value over a characteristic distance. The average crack growth rate was then calculated by dividing the characteristic distance with the recorded number of cycles. Although predictions are in good agreement with experimental data, the work was unable to predict the process of crack growth as it was based on stationary crack analyses only.
The extended finite element method (XFEM) has received considerable attention since its inception in 1999 by researchers dealing with computational fracture mechanics (Moës et al. 1999). The method has been widely applied to a variety of crack problems involving frictional contact (Dolbow et al. 2001), crack branching (Daux et al. 2001), thin-walled structures (Dolbow et al. 2000) and dynamic loading (Belytschko et al. 2003). The approach was also capable of modelling problems such as holes and inclusions (Sukumar et al. 2001), complex microstructure geometries (Moës et al. 2003), phase changes (Chessa et al. 2002) and shear band propagation (Samaniego & Belytschko 2005). For instance, Stolarska et al. (Stolarska et al. 2001) used the extended finite element method, in conjunction with the level set method, to solve the elastic-static fatigue crack problem. The XFEM is used to compute the stress and displacement fields necessary for determining the rate of crack growth. Mariani and Perego (Mariani & Perego 2003) utilized the cubic displacement discontinuity, which is able to reproduce the typical cusp-like shape of the process zone at the tip of a cohesive crack, to study the mode I crack growth in a wedge-splitting test and the mixed mode crack growth in an asymmetric three-point bending test. Cubic displacement discontinuity was also used in (Bellec & Dolbow 2003) as enrichment functions for modeling crack nucleation, which again allowed the reproduction of the typical cusp-like shape of the crack-tip process zone. Budyn et al. (Budyn et al. 2004) used the vector level set method, developed by Ventura et al. (Ventura et al. 2003), for modeling the evolution of multiple cracks in the framework of the extended finite element method. Nagashima et al. (Nagashima et al. 2003) applied the XFEM to two-dimensional elastostatic bi-material interface cracks problem. They used an asymptotic solution to enrich the crack tip nodes, and adopted a fourth order Gauss integration for a 4-node isoparametric element with enriched nodes. Despite increasing attempts to model crack growth using XFEM, to the authors’ knowledge, no work has been carried out by adopting the criterion of strain accumulation, which is the distinctive deformation feature at a crack tip under fatigue loading conditions (Zhao & Tong 2008). The majority of the work used the maximum principal stress or strain criteria which are not suitable to model crack growth under fatigue loading conditions as such simple criteria do not consider damage accumulation during the fatigue process.
In this paper, crack growth in a nickel-based superalloy LSHR (Low Solvus High Refractory) has been studied, both experimentally and computationally, under high temperature fatigue loading conditions. Fatigue tests were carried out for a three-point bend specimen in vacuum under a trapezoidal loading waveform with different dwell times. A cyclic viscoplastic constitutive model is used to model crack-tip deformation and to predict crack growth. The constitutive model, with parameters fitted from test data, was programmed into a user-defined material subroutine (UMAT) interfaced with ABAQUS for crack tip deformation analyses. With the assistance of the extended finite element method, the viscoplasticity model was also applied to predict crack growth based on plastic strain accumulation at the crack tip that was calculated by the UMAT. Predicted crack growth was compared with that obtained experimentally for selected loading range and superimposed dwell times.
The material used in this study was powder metallurgy LSHR superalloy provided by NASA. The material possesses excellent high temperature tensile strength and creep performance as well as good characteristics due to low γ′ solvus temperature (Gabb et al. 2005). Its chemical composition is 12.5Cr-20.7Co-2.7Mo-3.5Ti-3.5Al-0.03C-0.03B-4.3W-0.05Zr-1.6Ta-1.5Nb and balance Ni in weight percentage (Jiang et al. 2014). The alloy has a two-phase microstructure consisting of - matrix and strengthening - precipitates Ni3(Al, Ti, Ta) which are responsible for the elevated temperature strength of the alloy. The material was supersolvus heat treated to yield a coarse grain microstructure, which has a wide range of grain size, i.e. 10–140 μm. The average grain size was found to be 36.05 ± 18.07 μm.
Optimised parameter values for the viscoplastic constitutive model
a 1 (MPa)
a 2 (MPa)
Finite element modelling
Stationary crack analysis
Analysis of the actual crack growth is hard to achieve using approaches such as cohesive zone element (CZE) and virtual crack closer technique (VCCT) due to the well-known fact that in these schemes the crack path has to be defined in advance. However, with the extended finite-element method (XFEM), a crack-propagation process can be modelled based on a solution-dependent criterion without introduction of a predefined path. In the XFEM, a crack is represented by enriching the classical displacement-based finite element approximation through the framework of partition of unity (Melenk & Babuska 1996). A crack is modelled by enriching the nodes whose nodal shape function support intersects the interior of the crack by a discontinuous function, and enriching the nodes whose nodal shape function supports intersect the crack-tip by the two-dimensional linear elastic asymptotic near-tip fields.
Fatigue crack growth behaviour
Fatigue crack-Tip deformation
Crack tip deformation was studied by applying cyclic load to the 3-point bend stationary crack model (Figure 3). A total of ten cycles was simulated by considering a triangular loading waveform with a load ratio of R = 0.1, a maximum load of 4kN and a frequency of 0.5Hz. The load corresponds to a stress intensity factor range of ΔK = 31.6 MPa√m.
XFEM prediction of fatigue crack growth
To simulate a growing crack, three-point-bend loading conditions were applied to the specimen according to experiments. The load was kept constant and the value of ΔK was constantly increasing with the increase of crack length. Simulations of crack growth were performed by considering three different dwell times (1s, 20s and 90s). During the simulation, the crack growth length was obtained by image processing. Accumulated plastic strain obtained by Eq. (8) was used as crack growth criterion. It was assumed that crack starts to grow when the plastic strain accumulation at the crack tip reached a critical value of 0.23. This critical strain value was back calculated using finite element analysis, which predicts the same crack growth rate as the experimental results for a 1-20-1-1 fatigue test (Pmax = 2.62kN and R = 0.1) at 725°C in vacuum.
Whilst the applied stress and instantaneous crack length are usually considered in the form of stress intensity factor range, the role of a characteristic fracture strain has not been well explored (Zhao & Tong 2008). The strains local to a crack tip are of multiaxial nature such that an equivalent strain, which accounts for all strain components, is often adopted as a damage parameter which is also the case for the current work. The strain-based crack growth criterion is according to plastic strain accumulation at the crack tip, hence is more relevant to crack tip mechanics and the physical process of material damage. This criterion in combination with XFEM predicts reasonably well the crack growth rate. The present work is the first time that the accumulated plastic strain in combination with XFEM is used to predict the crack growth in superalloys at high temperature under fatigue loading. However, the XFEM prediction is limited to continuum level which is incapable of dealing with material micro-structural features and the influence of these features on the mechanical behaviour of the material. Alloy LSHR is a face-centred-cubic (f.c.c.) polycrystalline metallic alloy with a wide range of grain size and random grain orientation, which may also increase, decrease or arrest the crack growth (Pang & Reed 2003; Suresh 1998). Further work is under way to predict crack growth in the presence of explicit grain microstructure by using a crystal plasticity model combined with XFEM technique.
The current work considered three fatigue loading conditions with a frequency range of 0.01Hz to 0.25Hz, for which the model works well. However, with the decrease of loading frequency, time-dependent creep deformation tends to become dominant and leads to the prevalence of intergranular cracking. In this case, the model prediction may not work well. Due to the lack of test data at very low frequencies, it is very difficult to tell at which frequency the model prediction may break down, and further work is required in future study.
Experimental study showed that dwell time enhances fatigue crack propagation in nickel alloy LSHR, but only slightly for vacuum conditions. Dwell effects are more considerable for tests carried out in air. This is largely due to the detrimental effect of oxidation damage which attacks the grain boundaries, especially at the crack tip region, and induces accelerated crack propagation along the grain boundaries, i.e., intergranular crack growth. Computational analyses of stationary crack showed the progressive accumulation of plastic strain near the crack tip, which has been subsequently used as a fracture criterion to predict crack growth using the extended finite element method (XFEM). This crack growth criterion was successful in predicting both the path and the rate of crack growth at selected loading range and dwell period. This work proved the capability of XFEM, in conjunction with advanced cyclic viscoplasticity constitutive model, for predicting crack growth in nickel alloys at elevated temperature, which has significant implication to gas turbine industries in terms of “damage tolerance” assessment of critical turbine discs and blades made of nickel alloys.
The work was funded by the EPSRC (Grants EP/K026844/1, EP/K027271/1 and EP/K027344/1) of the UK and in collaboration with Nasa, Alstom, E.On and Dstl. Research data for this paper is available on request from the project principal investigator Dr Liguo Zhao at Loughborough University (email: L.Zhao@Lboro.ac.uk).
- Pang HT, Reed PAS (2003) Fatigue crack initiation and short crack growth in nickel-base turbine disc alloys – the effects of microstructure and operating parameters. Int J Fatigue 25:1089–1099View ArticleGoogle Scholar
- Dalby S, Tong J (2005) Crack growth in a new nickel-based superalloy at elevated temperature, part I: effects of loading waveform and frequency on crack growth. J Mater Sci 40:1217–1228View ArticleGoogle Scholar
- Suresh S (1998) Fatigue of Materials. Cambridge University Press, Cambridge, UKView ArticleGoogle Scholar
- Sehitoglu H, Sun W (1989) The significance of crack closure under high temperature fatigue crack growth with hold periods. Eng Fract Mech 33:371–388View ArticleGoogle Scholar
- Pommier S, Bompard P (2000) Bauschinger effect of alloys and plasticity-induced crack closure: A finite element analysis. Fatigue Fract Eng Mater Struct 23:129–139View ArticleGoogle Scholar
- Zhao LG, Tong J, Byrne J (2004) The evolution of the stress–strain fields near a fatigue crack tip and plasticity-induced crack closure revisited. Fatigue Fract Eng Mater Struct 27:19–29View ArticleGoogle Scholar
- Zhao LG, Tong J (2008) A viscoplastic study of crack-tip deformation and crack growth in a nickel-based superalloy at elevated temperature. J Mech Phys Solids 56:3363–3378View ArticleMATHMathSciNetGoogle Scholar
- Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Meth Eng 46:131–150View ArticleMATHGoogle Scholar
- Dolbow J, Moës N, Belytschko T (2001) An extended finite element method for modeling crack growth with frictional contact. Comp Meth Appl Mech Eng 190:6825–6846View ArticleMATHGoogle Scholar
- Daux C, Moës N, Dolbow J, Sukumar N, Belytschko T (2001) Arbitrary branched and intersecting cracks with the extended finite element method. Int J Numer Meth Eng 48:1741–1760View ArticleGoogle Scholar
- Dolbow J, Moës N, Belytschko T (2000) Modeling fracture in Mindlin- Reissner plates with the extended finite element method. int J Solids Structures 37:7161–7183Google Scholar
- Belytschko T, Chen H, Xu J, Zi G (2003) Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment. Int J Numer Meth Eng 58:1873–1905View ArticleMATHGoogle Scholar
- Sukumar N, Chopp DL, Moës N, Belytschko T (2001) Modeling holes and inclusions by level sets in the extended finite element method. In J Numer Meth Eng 50:993–1013View ArticleGoogle Scholar
- Moës N, Cloirec M, Cartraud P, Remacle J-F (2003) A computational approach to handle complex microstructure geometries. Comput Meth Appl Mech Eng 192:3163–3177View ArticleMATHGoogle Scholar
- Chessa J, Smolinski P, Belytschko T (2002) The extended finite element method (XFEM) for solidification problems. Int J Numer Meth Eng 53:1959–1977View ArticleMATHMathSciNetGoogle Scholar
- Samaniego E, Belytschko T (2005) Continuum-discontinuum modelling of shear bands. Int J Numer Meth Eng 62:1857–1872View ArticleMATHMathSciNetGoogle Scholar
- Stolarska M, Chopp DL, Moës N, Belytschko T (2001) Modelling crack growth by level sets in the extended finite element method. Int J Numer Meth Eng 51:943–960View ArticleMATHGoogle Scholar
- Mariani S, Perego U (2003) Extended finite element method for quasi-brittle fracture. Int J Numer Meth Eng 58:103–126View ArticleMATHMathSciNetGoogle Scholar
- Bellec J, Dolbow J (2003) A note on enrichment functions for modeling crack nucleation. Commun Numer Meth Eng 19:921–932View ArticleMATHGoogle Scholar
- Budyn E, Zi G, Moës N, Belytschko T (2004) A method for multiple crack growth in brittle materials without remeshing. Int J Numer Meth Eng 61:1741–1770View ArticleMATHGoogle Scholar
- Ventura G, Budyn E, Belytschk T (2003) Vector level sets for description of propagating cracks in finite elements. Int J Numer Meth Eng 58:1571–1592View ArticleMATHGoogle Scholar
- Nagashima T, Omoto T, Tani S (2003) Stress intensity factor analysis of interface cracks using XFEM. Int J Numer Meth Eng 56:1151–1173View ArticleMATHGoogle Scholar
- T.P. Gabb, J. Gayda, J. Telesman (2005) Thermal and mechanical property characterization of the advanced disk alloy LSHR, NASA report NASA/TM- 2005–213645Google Scholar
- Jiang R, Everitt S, Lewandowski M, Gao N, Reed PAS (2014) Grain size effects in a Ni-based turbine disc alloy in the time and cycle dependent crack growth regimes. Int J Fatigue 62:217–227View ArticleGoogle Scholar
- Chaboche JL (1989) Constitutive equations for cyclic plasticity and cyclic viscoplasticity. Int J Plasticity 5:247–302View ArticleMATHGoogle Scholar
- Melenk JM, Babuska I (1996) The partition of unity finite element method: Basic theory and applications. Comput Meth Appl Mech Eng 139:289–314View ArticleMATHMathSciNetGoogle Scholar
- ABAQUS, Version 6.12 User’s manual, SIMULIA, Providence, RI, 2012 Google Scholar
- T. L. Anderson (2005) Fracture Mechanics: Fundamentals and Applications, Taylor & Francis, USAGoogle Scholar
- Lin B, Zhao LG, Tong J (2011) A crystal plasticity study of cyclic constitutive behaviour, crack-tip deformation and crack-growth path for a polycrystalline nickel-based superalloy. Eng Fract Mech 78:2174–2192View ArticleGoogle Scholar
- Kapoor A (1994) A re-evaluation of the life to rupture of ductile metals by cyclic plastic strain. Fatigue Fract Eng Mater Struct 17:201–219View ArticleGoogle Scholar
- Yaguchi M, Takahashi Y (2005) Ratchetting of viscoplastic material with cyclic softening, Part I: experiments on modified 9Cr-1Mo steel. Int J Plast 21:43–65View ArticleMATHGoogle Scholar
- Kang G, Liu Y, Li Z (2006) Experimental study on ratcheting–fatigue interaction of SS304 stainless steel in uniaxial cyclic stressing. Mater Sci Eng A 435–436:396–404Google Scholar
- Riedel H, Rice JR (1980) Tensile cracks in creeping solids. ASTM STP 700:112–130Google Scholar
- Yatomi M, Nikbin KM, O’Dowd NP (2003) Creep crack growth prediction using damage based approach. Int J Press Vessels Piping 80:573–583View ArticleGoogle Scholar
- Zhao LG, O’Dowd NP, Busso EP (2006) A coupled kinetic-constitutive approach to the study of high temperature crack initiation in single crystal nickel-base superalloys. J Mech Phys Solids 54:288–309View ArticleMATHGoogle Scholar
- Ghonem H, Zheng D (1992) Depth of intergranular oxygen diffusion during environment-dependent fatigue crack growth in alloy 718. Mater Sci Eng A150:151–160View ArticleGoogle Scholar
- Molins R, Hochestetter G, Chassaigne JC, Andrieu E (1997) Oxidation effects on the fatigue crack growth of alloy 718 at high temperature. Acta Mater 45:663–674View ArticleGoogle Scholar
- Carranza FL, Haber RB (1999) A numerical study of intergranular fracture and oxygen embrittlement in an elastic-viscoplastic solid. J Mech Phys Solids 47:27–58View ArticleMATHGoogle Scholar
- Zhao LG, Tong J, Hardy MC (2010) Prediction of crack growth in a nickel-based superalloy under fatigue-oxidation conditions. Eng Fract Mech 77:925–938View ArticleGoogle Scholar
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.