A Temperature-Dependent Tertiary Creep Modeling of a Ni-base Alloy
Modeling the Temperature Dependence of Tertiary Creep Damage of a Ni-Based Alloy Calvin M. Stewart Ali P. Gordon Department of Mechanical, Materials, and Aerospace Engineering, University of Central Florida, Orlando, FL 32816-2450
To capture the mechanical response of Ni-based materials, creep deformation and rupture experiments are typically performed. Long term tests, mimicking service conditions at 10,000 h or more, are generally avoided due to expense. Phenomenological models such as the classical Kachanov–Rabotnov (Rabotnov, 1969, Creep Problems in Structural Members, North-Holland, Amsterdam; Kachanov, 1958, “Time to Rupture Process Under Creep Conditions,” Izv. Akad. Nauk SSSR, Otd. Tekh. Nauk, Mekh. Mashin., 8, pp. 26–31) model can accurately estimate tertiary creep damage over extended histories. Creep deformation and rupture experiments are conducted on IN617 a polycrystalline Ni-based alloy over a range of temperatures and applied stresses. The continuum damage model is extended to account for temperature dependence. This allows the modeling of creep deformation at temperatures between available creep rupture data and the design of full-scale parts containing temperature distributions. Implementation of the Hayhurst (1983, “On the Role of Continuum Damage on Structural Mechanics,” in Engineering Approaches to High Temperature Design, Pineridge, Swansea, pp. 85–176) (tri-axial) stress formulation introduces tensile/compressive asymmetry to the model. This allows compressive loading to be considered for compression loaded gas turbine components such as transition pieces. A new dominant deformation approach is provided to predict the dominant creep mode over time. This leads to development of a new methodology for determining the creep stage and strain of parametric stress and temperature simulations over time. 关DOI: 10.1115/1.3148086兴 Keywords: gas turbine, tertiary creep damage, IN617, Ni-based alloy, constitutive model, tensile/compressive asymmetry
1
Introduction
Gas turbines generate electric power through a compressioncombustion process that forces superheated highly pressurized exhaust gas through various components. High pressure in the compression area coupled with combustion chamber exit temperatures of up to 1350° C creates a situation where material design and selection play a major role in the long term reliability of gas turbine parts. Drives to increase gas turbine efficiency require raising turbine firing temperatures, which negatively impact component life. Under these conditions high strength materials are needed to provide long life at elevated temperatures. The traditional method of determining the service life of gas turbine components employs practical experience, experimental data, and finite element analysis 共FEA兲. The boundary conditions, temperature and stress gradients, environmental conditions, and time-dependent variations induce a number of damage mechanisms, which cause crack initiation and eventual rupture. Common practice in the gas turbine industry is to conduct high temperature mechanical testing on a sample-sized specimen over a period of time and use practical experience 共from previous design兲 to extend empirical models to failure time. Creep rupture life is typically estimated using experience and secondary creep 共i.e., steady state兲 modeling with temperature dependence. Unfortunately, this method neglects the effects of both strain-softening behavior 共tertiary creep modeling兲 and tensile/compressive asymmetry. Not including these damage mechanisms creates both Contributed by the Pressure Vessel and Piping Division of ASME for publication in the JOURNAL OF PRESSURE VESSEL TECHNOLOGY. Manuscript received September 19, 2008; final manuscript received February 28, 2009; published online September 3, 2009. Review conducted by Douglas Scarth. Paper presented at the ASME Early Career Technical Conference 2008.
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strain history and component life estimates of limited accuracy. Using a more advanced tertiary creep damage model with tensile/ compressive asymmetry improves the ability to predict component life. Creep deformation and rupture experiments are used to determine the constants of the constitutive model. As a part of this investigation, the material model is evaluated and a numerical parametric study is carried out to determine the total strain experienced at various combinations of uni-axial stress, temperature, and time.
2
Tertiary Creep Damage Model
A method has been established for modeling tertiary creep at high temperatures. This method is based on the Norton power law for secondary creep, i.e., ˙ cr = A¯n
共1兲
where A and n are the creep strain coefficient and exponent, respectively, and ¯ is the von Mises effective stress. The coefficient A exhibits Arrhenian temperature dependence, e.g.,
冉 冊
A = B exp −
Qcr RT
共2兲
Here Qcr is the activation energy for creep deformation, and T is the temperature measured in K. The term R is the universal gas constant. Both A and its temperature-independent counterpart, B, have units MPa−n h−1. The exponent n may also exhibit temperature dependence. As shown in Eq. 共1兲, the creep strain rate is stress dependent due to ¯. For most materials, traditional methods tend to find A and n to be stress independent but there are exceptions 关1兴. The secondary creep constants are determined by considering the minimum strain rate versus the applied stress, and using a regression analysis. Hyde et al. 关2兴 showed for a Ni-based
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Table 1 Creep Deformation data for IN617 Stress,
Temperature, T
a
°C
°F
MPa
ksi
Stress ratio, / ys
649 649 649 649 760a 760 760 760 871 871 871 871 982a 982 982 982
1200 1200 1200 1200 1400 1400 1400 1400 1600 1600 1600 1600 1800 1800 1800 1800
310 310 345 345 145 145 158 158 62 62 69 69 24 24 28 28
45 45 50 50 22 22 23 23 9 9 10 10 3.5 3.5 4 4
1.22 1.22 1.36 1.36 0.53 0.53 0.58 0.58 0.29 0.29 0.32 0.32 0.19 0.19 0.22 0.22
Time to strain, t 共h兲
Min strain rate 共%/h兲
Rupture strain 共%兲
0.25% strain
1% strain
5% strain
Rupture
0.0006 0.0008 0.0024 0.0024 0.0098 0.0051 0.0031 0.0035 0.0022 0.0022 0.0082 0.0088 0.0002 0.0002 0.0011 0.0021
1.1 1.2 1.2 1.2 16.0 19.8 13.3 14.0 22.0 19.3 19.8 28.5 19.0 14.5 20.2 17.8
204 69 54 40 7 14 19 16 54 33 17 14 170 228 162 53
1338 904 353 296 57 138 188 171 224 200 84 74 256 384 259 240
N/A N/A N/A N/A 355 581 577 560 563 533 251 231 384 614 416 404
1485 1090 412 341 748 981 852 828 1050 1082 585 488 520 946 658 654
Removed from consideration.
alloy Waspaloy that once a critical value of applied stress is reached the relationship between minimum creep strain and stress evolves. This leads to a two-stage relationship and stress dependence of the secondary material constants. Table 1 shows that for the collected creep deformation data, only two stress levels per temperature were conducted. Any possible stress dependence of A and n is not visible in the current data. Therefore, stress dependence of the secondary material constants is neglected. Jeong et al. 关3兴 demonstrated that while under stress relaxation short term transient behavior occurs where Qcr and n are both stress dependent at relativity short times 共e.g., less than 10 min兲. After this transient stage, the Norton power law becomes independent of initial stress and strain. This transient behavior has been neglected here since the current study focuses on longer periods 共e.g., 1–100,000 h兲. During the transition from secondary to tertiary creep, stresses in the vicinity of voids and microcracks along grain boundaries of polycrystalline materials are amplified. The stress concentration due to the local reduction in cross-sectional area is the phenomenological basis of the Kachanov–Rabotnov 关4,5兴 damage variable, , that is coupled with the creep strain rate and has a stressdependent evolution, i.e., ˙ cr = ˙ cr共¯,T, , . . .兲 共3兲
˙ = ˙ 共¯,T, , . . .兲
A straightforward formulation implies that the damage variable is the net reduction in cross-sectional area due to the presence of defects such as voids and/or cracks, e.g., ¯net = ¯
A0 = Anet
冉
¯ ¯ = A0 − Anet 1− 1− A0
冊
共4兲
where ¯net is called the net 共area reduction兲 stress, A0 is the undeformed area, Anet is the reduced area due to deformation, and is the damage variable. Some models account for the variety of physically observed creep damage mechanisms with multiple damage variables 关6兴. An essential feature of damage required for application of continuum damage mechanics concepts is a continuous distribution of damage. Generally, microdefect interaction 共in terms of stress fields, strain fields, and driving forces兲 is relatively weak until impingement or coalescence is imminent. By substituting Eq. 共4兲 into Eq. 共1兲, where ¯ is replaced by ¯net, the 051406-2 / Vol. 131, OCTOBER 2009
steady-state creep strain is expressed as a modified secondary creep rate: ˙ cr = A
冉 冊 ¯ 1−
n
共5兲
The isotropic damage variable is scalar valued from 0 for an initial state 共e.g., undamaged兲 to 1 at final state 共e.g., completely ruptured兲. Its evolution was given by Rabotnov 关4兴 as
˙ =
M ¯ 共1 − 兲
共6兲
where M is the creep damage coefficient having units of MPa−. This damage evolution formulation allows the model to account for tertiary creep. In an earlier study, Kachanov 关5兴 developed a similar formulation with the exception that the creep damage exponents, and , were equivalent. Time hardening, tensile/ compressive asymmetry, and multi-axial behavior can be accounted for in the model by replacing the effective stress with one that considers multi-axial stress, e.g.,
˙ =
M r m t 共1 − 兲
共7兲
The exponent, m, accounts for time hardening and the units of M are in MPa−n h−m−1; however, m is defined to be 0.0 in most cases 关7,8兴. Here r describes the Hayhurst 共tri-axial兲 stress and is related to the principal stress, 1, and hydrostatic 共mean兲 stress, m, and ¯. The Hayhurst stress 关9兴 is given as
r = ␣1 + 3m + 共1 − ␣ − 兲¯
共8兲
where ␣ and  are the weighing factors valued between 0 and 1 and are determined from multi-axial creep experiments. This formulation disallows damage under uni-axial compression when ␣ + 2 ⱖ 1. More recent investigations have extended this damage evolution description to account for anisotropic damage resulting from multi-axial states of stress 关10兴. In these cases, damage is described by a second-rank tensor. The coupled evolution 共Eqs. 共5兲 and 共7兲兲, which make up the tertiary creep damage model, can be reverted back to secondary creep when M = 0. As a result, the damage evolution equation to model reverts back to the Norton power law for secondary creep Transactions of the ASME
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Table 2 Nominal chemical composition of IN617 Cr
Co
Ti
Al
Mo
C
B
Fe
Ni
22.5
12.5
0.6
1.0
9.6
0.08
0.004
0.98
Bal.
with the exception that ¯ is replaced by r. This is a useful property that is exploited later to determine the transition from secondary to tertiary creep dominance over time. The tertiary creep damage formulation can be applied to model the gradual reduction in load carrying capability 共e.g., secondary through tertiary creep兲. Since characterizes the microstructural evolution of the material and is dependent on stress and temperature, the damage variable is an internal state variable 共ISV兲. This formulation has been used in a variety of studies of turbine and rotor materials. The constants A, n, M, , and are considered material properties. For example, in Waspaloy at 700° C, A = 9.23⫻ 10−34 MPa−n h−1, n = 10.65, M = 1.18⫻ 10−25 MPa− h−1, = 8.13, and = 13.0 关7兴. For stainless steel at 650° C, A = 2.13 ⫻ 10−13 MPa−n h−1, n = 3.5, M = 9.0⫻ 10−10 MPa− h−1, and = = 2.8 关8兴. In these previous studies, isothermal and constant stress conditions were applied. In the current study, isothermal and constants force conditions are applied allowing for stress relaxation similar to the conducted creep experiments. It should be noted that no rule for the temperature dependence of the tertiary creep constants has been established.
3
Creep Experiments
Polycrystalline 共PC兲 nickel-chromium-cobalt-molybdenum alloys are a modern class of materials commonly used in gas turbine components. One such material is wrought Ni-based alloy IN617. This nonproprietary sheet material has several equivalent monikers 共e.g., Inconel-617, INCO-617, Nicrofer® 5520 Co-Alloy 617, and Haynes 617兲. The nominal chemical composition is given in Table 2. The alloying composition of the material allows it to withstand a range of conditions critical to gas turbine components. The combination of various elements produces a strong atomic structure that imparts strength at high temperatures. Although it is suitable below 1000° C, IN617 displays exceptionally high levels of creep rupture strength, even at temperatures of 980° C 共1800° F兲. An inherent feature of the material is it has good corrosion resistance compared with other standard Ni-based alloys such as alloy 556, alloy 230, and alloy Hastelloy-X 关11兴. It carries a good combination of resistance to oxidation and carburizing atmospheres that are important for components that undergo long term loading. Thermal shock resistance is another key feature that forward the uses of Ni-based materials 关12兴. All of these features result in a material that has good resistance to creep deformation. The material also has relatively good weldability allowing field repair; this is another important requirement of gas turbines. One major detractor for this material is that it is susceptible to intense nitridation at elevated temperatures while undergoing thermal cycling. This susceptibility is due to the development of crack defects in the oxide scale, which perpetuates the nitridation process and allows surface intrusion of oxides, thus
leading to crack initiation and eventual failure 关13兴. Additionally, alloy 617 encounters deep internal defects through the development of aluminum nitrides, which severely hinder the load carrying capability of the material. Despite this condition, alloy 617 is still found to be the best combination of overall material properties. Proper turbine operation has been shown to produce gas turbine transition pieces of alloy 617 that last up to 15,000 total operating hours 关14兴. The tensile properties of the material are given in Table 3 关15兴. An attractive property of alloy 617 is that it has a nonlinear yield strength, which is enhanced at temperatures between 700° C and 800° C 关16兴. This yield strength anomaly makes the material an excellent candidate for components that are heated within that temperature range 共gas turbine transition pieces兲. This phenomenon has the added effect of making the creep deformation and rupture response at temperatures higher than 800° C more sensitive to temperature change 共due a drop in yield strength兲. Creep deformation and rupture experiments were conducted according to an ASTM standard E-139 关17兴 at 649° C, 760° C, 871° C, and 982° C on samples incised from wrought slabs of IN617. The combination of conditions is given in Table 1. From this variety of creep deformation and rupture experiments, the subject PC material exhibits primary, secondary, and tertiary creep strains depending on the combination of test temperature and stress, as shown in Fig. 1. Each test was conducted twice per stress level to account for variability in alloy samples due to slight differences in heat treatment. This method showed that inconsistencies are found at 760° C with 148 MPa and 982° C with 24 MPa, shown in Table 1. To account for this, these two creep curves were removed from the analysis. Based on these strain histories, the deformation mechanisms can be inferred from investigations that complemented mechanical experimentation with microscopic analysis 关18,19兴. Deformation at 649° C at high stress is mostly due to plastic strain, along with primary and secondary creeps. At stress levels of 345 and 310 MPa, the stress ratios, applied stress over yield strength, are found to be 1.35 and 1.22, respectively. Upon loading, a substantial time-independent plastic strain was produced before the creep test began. During primary creep, the strain rate, which is initially high, decreases significantly. This transient phenomenon is due to the ease at which dislocations can glide upon initial loading 共due to initially present dislocations in untreated workpieces兲 and the saturation of the dislocation density, which inhibits continued creep deformation. Transient creep also drops due to pinning of the dislocations when they encounter various obstacles after the easy glide period. Secondary creep arises due to the nucleation of grain boundary 共GB兲 cavities and grain boundary sliding. The deformation mode of tertiary creep was not observed in these cases. At higher temperatures 共760° C and above兲, deformation is dominated by elastic strain, and secondary and tertiary creeps are facilitated by the coalescence of grain boundary voids into microcracks. Sharma et al. 关20兴 showed that at temperatures above 800° C, dynamic recrystallization is found to occur. This process is a part of the hot deformation process, which decreases average grain size as stress increases. This has the effect of rapidly reducing dislocation density in the alloy at higher temperatures and thus
Table 3 IN617 material properties Temperature, T
a
0.2% tensile yield strength,a y
Tensile strength
Young’s modulus
°C
°F
MPa
ksi
MPa
ksi
GPa
103 ksi
Elongation 共%兲
649 760 871 982
1200 1400 1600 1800
254 272.7 215.6 128.4
37 40 31 19
586.2 525.5 341 181.6
85 76.2 49 26
170 161.2 151.6 141
24.7 23.4 22 20.4
56.6 50.2 69.3 85
Yield strength assumed symmetric.
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Fig. 1 Stress and temperature combinations of available creep experiments „does not include elastic strain…
reducing primary creep deformation. Notwithstanding, primary creep was still observed, but its contribution to the overall deformation was negligible compared with that of secondary and tertiary creeps; therefore for the case of this study primary creep was neglected. For all cases except those at 649° C, time-independent plasticity is ignored. Creep deformation in the current study is assumed to also encompass time-dependent viscoplasticity. Later studies will evaluate the response of a coupled plasticity and creep model.
The creep rupture data at 649° C, 760° C, 871° C, and 982° C compare well to an earlier comprehensive study 关11兴 of creep rupture at temperature between 600° C and 1000° C, from previous studies 关21–23兴. Figure 2 shows stress versus rupture time of both the obtained and study creep rupture data. Generally, creep rupture time increases with either decreasing applied stress or decreasing temperature. The plot shows that the obtained creep rupture data are positioned within the bounds of creep rupture data at
Fig. 2 Creep rupture data at various temperatures †11‡
051406-4 / Vol. 131, OCTOBER 2009
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Z
Y
X Fig. 3 Single element FEM geometry used with force and displacement applied
adjacent temperature from the comprehensive study. It is important to note that in Fig. 2 the creep tests performed at 649° C or less experienced a time-independent plastic strain. In all the creep rupture data, scatter is present. This scatter is common in creep rupture charts. It is due to inconsistency in commercial material quality and inadequate control of the heat treatment process. It also is a result of differences in testing procedure and equipment. In the comprehensive study, multiple data sources were used where the testing procedure and equipment are not known. Manufacturers commonly use least-squares linear regression analysis to determine the minimum creep rupture life of an alloy while neglecting scatter. Advancements by Zuo et al. 关24兴 show that the maximum likelihood method produces more accurate predictions of creep rupture life.
4
Numerical Methodology
The constitutive model described in Eqs. 共5兲, 共7兲, and 共8兲 has been implemented in a general-purpose FEA software in order to determine the constants for the constitutive model used in the secondary-tertiary creep formulation. The formulation was implemented into a FORTRAN subroutine in the form of a userprogrammable feature 共UPF兲 in ANSYS 关25兴. The subroutine is incorporated with an implicit integration algorithm. This backward Euler integration algorithm is more accurate over long time periods than other practical numerical integration methods. This allows larger time steps that reduce the numerical solve time. Since the viscoplastic/creep behavior of materials is significant at extended histories, the backward Euler method is the desired method for integration of creep constitutive models 关26兴. This implementation allows for the update of the internal state variable, . Initially, = 0.0 and during loading, increases. To prevent the singularity that is caused by rupture 共e.g., = 1.0兲, the damage is restricted to a maximum of 0.90. Comparison of creep rupture data to numerical simulations shows that rupture occurs at between 0.4 and 0.6; therefore, rupture can be achieved before damage reaches = 1.0. To prevent an excessive model solve time and large deformation errors, a simulation was terminated once the total strain reached 100%. If needed, this model can be applied with time-independent and time-dependent plasticity models in a straightforward manner. The UPF has been used to simulate the temperature and stress loading conditions of a series of uni-axial creep and rupture experiments. A single, solid, three-dimensional, eight-noded element was used, and the appropriate initial and boundary conditions were applied to numerically simulate the uni-axial creep deformation curves shown in Fig. 1. The loads and temperatures applied numerically simulated those of the obtained creep rupture experiments. Figure 3 demonstrates how these conditions are applied on Journal of Pressure Vessel Technology
the FEM geometry. The constant axial force load experience during tensile testing was applied as a force load on the top surface of the element. A uniform temperature was applied across the element. Displacement controls were set to match the constraints experienced during tensile testing. These conditions lead to an accurate assessment of the creep deformation within a single element model. Histories of creep deformation, cr共t兲, were recorded to a data file and subsequently compared with experiments. Using this method, expansion beyond a single element to multi-element simulations will produce an accurate measure of creep deformation. A goal of the research was to provide a uni-axial creep deformation formulation that was capable of matching the material response up to 5% of the total strain determined from experiments. The time required for each experimentally tested specimen to deform to 5% is shown in Table 1. The secondary creep constants 共e.g., A and n兲 were evaluated directly from the available creep deformation and rupture data for IN617 at each temperature. As described early, A and n were found by plotting the minimum strain rate versus applied stress and using regression analysis in the form of Eq. 共1兲 to determine the values of A and n. Sharma et al. 关20兴 found for IN617 an apparent activation energy, Qcr, equal to 271 kJ/mol similar to what has been observed in a bulk specimen tested in a He+ O2 environment 关27兴. Using this Qcr value as an initial guess produced a B constant of 1.27⫻ 10−19 MPa−n h−1. A value of B that is less than 10−12 is not suitable for application in FEM. Such numbers experience numerical cutoff where trailing significant figures are discarded resulting in limited accuracy. To overcome this problem, the guess value of Qcr was doubled to 542 kJ/mol. Based on these constants and Eq. 共2兲, B and Qcr were approximated as 2.95⫻ 10−3 MPa−n h−1 and 542 kJ/mol, respectively. The tertiary creep constants 共e.g., M, , and 兲 were determined for each temperature by an iterative trial and error method. This trial and error method consisted of parametrically changing each constant to determine its effect on the resulting creep deformation curve and manually refining the combined selection of constants until a best fit was determined. Further development using objective function based methods 关28兴 can lead to better material constants, which match the creep deformation response up to creep rupture. For simplicity, both and were taken to be equivalent similar to what Altenbach et al. 关8兴 considered for stainless steel. The constants ␣ and  were set to zero to produce a Hayhurst stress equivalent to uni-axially applied stress. Coupling Norton’s law for secondary creep with the Kachanov– Rabotnov formulation for tertiary creep allows the finite element model to display a strain history that approximates the experimental results well. The strain histories simulated with the Norton– Rabotnov formulation have very good correlation with the analogous experimentally obtained 共t兲. Tertiary creep constants were optimized for every combination of temperature and stress. An example of the simulated creep deformation response is shown in Fig. 4. Temperature dependence is carried by the material constants while stress dependence is carried by the Hayhurst stress, r. In each case, the trajectories of each of the correlated creep curves correspond to experimental data.
5
Temperature Dependence
All of the creep curves exhibiting creep induced strain softening can be generated by varying M, , and . The creep damage rate coefficient, M, displays temperature dependence; however, the damage rate exponents and were assumed as temperature independent. The constants at each temperature are plotted in Fig. 5. Based on the experiments, the temperature dependence for each of the constants is mathematically represented. The elastic modulus of IN617 was written as a third order polynomial, e.g., OCTOBER 2009, Vol. 131 / 051406-5
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Fig. 5 Temperature dependence of elastic, secondary creep, and creep damage constants of IN617
TRAN subroutine so that predictions of creep deformation could be made through a range of temperatures.
6
Fig. 4 Comparison of experimental and numerically simulated creep deformation of IN617: „a… 760° C and 982° C, and „b… 871° C
E共T兲 = E3T3 + E2T2 + E1T1 + E0
共9兲
with T measured in K. In a similar manner, regression analyses were performed to model the temperature-dependence of the secondary creep and creep damage parameters. The secondary creep coefficient was determined using the Arrhenius relation as a function of temperature, where B and Qcr are independent of temperature. In Eq. 共2兲, T is measured in K. The secondary creep exponent was related by a polynomial function, e.g., n共T兲 = 具n3T3 + n2T2 + n1T1 + n0典 + nmin
共10兲
where T is measured in K. The McCauley brackets and nmin term are used in this case so that the minimum value of n is nmin, which is valued at 5. This was implemented to ensure numerical convergence of the UPF at low temperatures. The creep damage coefficient is expressed as a two-parameter exponential power law function, e.g., M共T兲 = M 1 exp共M 0T兲
共11兲
where T is measured in K, and M 1 and M 0 are constants. The fit of Eqs. 共9兲–共11兲 is compared with optimized constants for each temperature in Fig. 5. The square of the Pearson product-moment correlation coefficient, R2, indicates that each of the relations model the temperature dependence of the elastic and creep properties of IN617 with an accuracy greater than or equal to 0.90. The temperature dependence of the constants was coded into the FOR051406-6 / Vol. 131, OCTOBER 2009
Tensile/Compressive Asymmetry
It has been observed that for a number Ni-based alloys, there is a difference in the resulting mechanical response of a body undergoing compressive versus tensile loading 关29–31兴. This is due to the tendency for tensile loads to enhance the degradation of grain structure, which produces voids, crack propagation, and leads to rupture. Capturing the compressive behavior can be just as important as accurately modeling the tensile behavior, especially for materials used in gas turbine transition pieces. Transition pieces undergo multi-axial stress at high temperature, which can cause creep induced buckling 关32兴. An accurate estimate of service intervals could allow the preemptive patched repair maintenance of transition pieces, which has been found to extend the life of these components up to 8000 h 关14兴. To establish tensile/compressive asymmetric behavior in the constitutive model, a number of material constants were modified. The weight constants ␣ and  in Eq. 共8兲 modify Hayhurst stress and allow compression to occur when ␣ + 2 ⬍ 1. Hayhurst stress can be determined analytically for bi-axial stress states, as shown in Fig. 6. Hayhurst 关33兴 showed that material constants of ␣ = 0 and  = 0.25 produce a comparable tensile/compressive asymmetry in Ni-based alloys. The state of stress with y = and x = 0 yields a Hayhurst stress equal to . Applying the state of stress y = − and x = 0 yields a Hayhurst stress that is / 2. The constitutive model was modified to use this set of constants. It should be noted that the ␣ and  constants were not optimized for the IN617 alloy due to the lack of availability of compressive creep data. Additional control of tensile/compressive asymmetry is provided in the damage rate equation 共7兲 by the constants and . Increasing the exponent increases the effect that the Hayhurst stress has on the damage rate. The constant modifies the effect that the existing damage has on the damage rate. The constant can have a profound effect particularly when the damage nears = 1.0. For this study = = 3.0, based on guidance from other studies. The tensile/compressive creep deformation response at 815.5° C and various stresses is shown in Fig. 7. This plot demonstrates that in terms of secondary creep, the material is modeled Transactions of the ASME
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Fig. 6 Stress ratios resulting from various bi-axial „Hayhurst… stress configurations „a… 1 – 3m, 2 – 1, and 3 – . „b… All lines satisfy incompressibility and uni-axial tensile conditions. „c… all curves satisfy compressible and uni-axial tensile conditions. For all cases z.
to exhibit symmetric tensile and compressive behaviors; however; as tertiary creep damage accumulates, the effects of void formation and coalescence are predicted to occur more rapidly under tensile conditions. The tertiary creep model shows an overall reduction in the creep strain rate with respect to decreasing applied stress levels. This is attributed to the modified Hayhurst stress facilitating a smaller effective stress to the damage rate. Furthermore, due to the constants and , the damage rate is modified causing a lower creep strain rate to be produced. Using the tertiary creep model, Fig. 7 shows a factor of 4.7 in the time to 5% total strain between tension and compression at 100 MPa. At 10% total strain, this factor is increased to 5.5. The and constants have great significance due to the time dependence of damage and strain and the extensive amount of time required to reach rupture at lower stress levels. When applied stress is increased, the factors at 5% and 10% strain are reduced. This demonstrates that as applied stress is increased the tensile/ compressive asymmetry found in the Ni-based alloy is reduced. Increased applied stress causes a reduction in rupture time. The secondary creep stage has less time to develop and in some cases is bypassed. This results in the tertiary creep stage becoming
dominant earlier during creep deformation. The rapid deformation, which occurs during the tertiary creep stage, allows less time for tensile/compressive divergence. It should be emphasized that due to compressive creep data not being available for IN617, the compressive results could not be verified. The compressive results should, therefore, be used qualitatively.
7
Dominant Deformation Modes
One of the key goals of this research is to demonstrate the time dependence of the dominance of elasticity, secondary creep, and tertiary creep strain as stress and temperature change 共primary creep is ignored兲. The dominant deformation mode was determined by creating dominance contours. The ratio of partitioned strain over total strain for each stage was found. The partition with the largest ratio was considered dominant. The following equations were used to determine the curves of dominance transition:
total =
+ cr = E + 共SC + TR兲 E
Fig. 7 Creep deformation predictions in tension and compression at 815.5° C for both secondary creep „dotted lines… and tertiary creep models „solid lines…
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Fig. 8 Comparison of „„a… and „c…… total strain and „„b… and „d…… dominant deformation mode for „„a… and „b…… 100 MPa and 815.5° C and „„c… and „d…… 250 MPa and 649° C
再
dominant mode = max
E SC TR , , total total total
冎
共12兲
where total is total strain, E is elastic strain, SC is secondary creep strain, and TR is tertiary creep strain. Based on this dominant deformation formulation, Fig. 8 shows the transition of the various deformation modes. Figure 8共a兲 depicts the timedependent evolution of each form of strain at 815.5° C and 100 MPa. It clearly shows that the secondary creep model produces a linear strain-time function, which continues at the same rate until failure, while the tertiary creep model follows the expected experimental trend of an increase deformation rate, which leads to eventual rapid failure. Figure 8共b兲 displays the dominant deformation modes using the data from Fig. 8共a兲. The figure shows the transition from elasticity to secondary creep and eventually tertiary creep mode. This demonstrates that the formulation could be used to determine the creep mode a component encounters during operation. Engineers could apply this approach as a means to avoid modes beyond secondary creep, which tend to produce rapid deformation. Figure 8共c兲 depicts the growth of each form of strain over time at 649° C and 250 MPa. It shows that tertiary creep surpasses secondary creep before either surpasses elastic strain. Figure 8共d兲 plots the dominant deformation modes using the data from Fig. 8共c兲. It shows that the secondary creep model does not reach a situation where it can become dominant; it is essentially bypassed due to the strong effects of the tertiary creep model. Dominance transition is considered the point where one form of deformation mode dominance ends and another mode takes over. This can be seen in Eq. 共12兲 by the maximum value changing to a different component of the column. Two such dominance transitions are found in this paper: the transition from elastic to secondary creep dominance and the transition from secondary creep to tertiary creep dominance. In Fig. 9, the time where a dominance transition occurs, 5% strain and 10% strain are plotted. Six simulations were conducted for each temperature between high and low stresses to create trend curves. To compare the results on an 051406-8 / Vol. 131, OCTOBER 2009
equivalent value of stress, a stress ratio, / ys, applied stress over yield strength was used. Figure 9 shows that at low stress the secondary and tertiary creep dominance lines converge. At some minimum stress, the difference between secondary and tertiary creeps becomes negligible. This minimum stress is temperature dependent. As applied stress approaches, the value of the yield strength deformation occurs more rapidly. When the stress ratio is very near unity, plastic deformation begins to occur. This demonstrates the need for a plasticity model to account for inelastic mechanical behavior. Using temperature levels from 649° C to 982° C at 8.3° C intervals and stress levels from 500 MPa to 500 MPa at 25 MPa intervals, the simulations were carried out in an increasing magnitude of hours from 1 h to 100,000 h. The temperature range was restricted to 649– 982° C to match the range used in experimental test. At lower temperatures, creep deformation disappears. The assumption is made that there is no stress dependence exhibited in the material constants such that two stress levels are sufficient to fit our material constants for the entire range of stresses. Stresses ranging from ⫾500 MPa are used to allow a clear picture of the relationship between stress and creep strain. It is particular helpful in making the large gradient of creep strain observed at low time periods 共1 h, 10 h, and 100 h兲 visible. Simulations were conducted in elasticity only, secondary creep 共e.g., M = 0兲, and tertiary creep including tensile/compressive asymmetry conditions 共20,172 simulations total兲. With parametric boundary conditions established, a closer look at the response of the model can be achieved. Figure 10 shows the trends of creep stage dominance overlaid upon total strain for the complete set of simulations. It demonstrates the importance of implementing a tertiary creep model that includes effects neglected in a secondary creep model. In the figure, the region between the solid white lines signifies simulations where the elastic strain is dominant. The region between the solid white and black lines represents secondary creep strain dominance, and anything outside of the black lines produce tertiary creep strain dominance. Transactions of the ASME
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Fig. 9 Contours of constant total deformation and constant deformation mode at „a… 760° C, „b… 871° C, and „c… 982° C. „Note: / ys is the applied stress over yield strength.…
The dashed white and black line represents the 0.2% yield strength from Table 3. At stress levels above this line, a coupled plasticity and creep model is necessary to determine the total strain. Therefore, the simulated total strain found beyond this line should not be considered. The figure shows that at extended histories, creep strain becomes dominantly tertiary following expected experimentally reproducible trends. It should be noted at more than 1000 h under tensile loading, the secondary creep dominance begins to disappear. This is signified by the tertiary dominance line passing the secondary dominance line in effect of a dominance shift. This is a result of mechanical load, time, and temperature producing a situation where the secondary creep stage is not strong enough to produce dominance. This results in tertiary creep becoming the primary source of creep deformation until failure. A similar trend is observed when stress is low and temperature is increasing; the secondary creep strain dominance transition line converges with the tertiary transition line. The tensile/compressive relationship is demonstrated in Fig. 10 using the established ␣, , , and constants. It shows that for compressive loading the tertiary creep transition line lies at higher stresses than tensile loading. This causes secondary creep to be Journal of Pressure Vessel Technology
dominant in a large range of stress and temperatures while under compression. This is a product of using the Hayhurst stress equation that produces a lower effective stress while under the same strain and the damage rate equation, which compounds these effects. Additionally the dominance shift noticed at 1000 h under tensile loading is delayed on the compressive side to 10,000 h. There is a significant difference in the creep dominance behavior of tensile/compressive asymmetric materials. A major feature of the dominant deformation mode formulation is that it can be used as a design process tool for gas turbine components, as shown in Fig. 10. When considering new gas turbine component designs, it can be used as a method to eliminate boundary condition combinations leading to premature failure. The eliminated conditions could be determined based on yield strength, deformation mode, or allowable strain. Simple modifications could allow inclusion of additional criteria 共e.g., tensile strength and compressive strength兲.
8
Conclusions
The materials used in gas turbine components deform by a number of modes. The focus of this investigation is modeling the OCTOBER 2009, Vol. 131 / 051406-9
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Fig. 10 Total strain deformation maps from creep simulations
creep deformation of materials used to fabricate turbine parts that exhibit strain softening at high temperatures 共i.e., above 649° C兲. Creep cavitation and void coalescence along and within the boundaries of grains lead to area-reduction transverse to the applied loading axis. This leads to a gradual loss of stress-carrying 051406-10 / Vol. 131, OCTOBER 2009
ability. The Kachanov–Rabotnov type creep damage formulation was used to account for this behavior found in creep deformation and rupture experiments carried out on the Ni-based alloy IN617. To account for temperature dependence, phenomenological models were developed for several mechanical properties. The Transactions of the ASME
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damage formulation was modified and directly implemented into the constitutive model routine so that predictions of creep deformation could be made through a range of temperatures. The correlated curves show excellent agreement with trends from creep deformation experiments. Utilizing the Hayhurst 共tri-axial兲 stress formulation and the damage rate equation, the model was shown to be able to produce tensile/compressive asymmetry. The deformation mode dominance over an extensive range of mechanical load, temperature, and time configurations was determined using a new dominant deformation formulation. The difference between tensile and compressive dominances was also determined.
Acknowledgment The authors are thankful for the support of Siemens in conducting this research. Calvin Stewart is thankful for the support of a Mcknight Doctoral Fellowship through the Florida Education Fund.
Nomenclature A,B,n M ,,,m ␣, Qcr R cr ˙ cr r 1 m ¯ ˙
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
secondary creep constants tertiary creep constants weighing factors in the Hayhurst stress apparent activation energy universal gas constant creep strain creep strain rate stress Hayhurst 共tri-axial兲 stress first principal stress hydrostatic 共mean兲 stress von Mises stress damage damage rate
References 关1兴 Hoff, N. J., 1953, “Necking and Rupture of Rods Subjected to Constant Tensile Loads,” ASME J. Appl. Mech., 20共1兲, pp. 105–108. 关2兴 Hyde, T. H., Xia, L., and Becker, A. A., 1996, “Prediction of Creep Failure in Aeroengine Materials Under Multiaxial Stress States,” Int. J. Mech. Sci., 38共4兲, pp. 385–403. 关3兴 Jeong, C. Y., Nam, S. W., and Ginsztler, J., 1999, “Stress Dependence on Stress Relaxation Creep Rate During Tensile Holding Under Creep-Fatigue Interaction in 1Cr-Mo-V Steel,” J. Mater. Sci., 34共11兲, pp. 2513–2517. 关4兴 Rabotnov, Y. N., 1969, Creep Problems in Structural Members, NorthHolland, Amsterdam. 关5兴 Kachanov, L. M., 1958, “Time to Rupture Process Under Creep Conditions,” Izv. Akad. Nauk SSSR, Otd. Tekh. Nauk, Mekh. Mashin., 8, pp. 26–31. 关6兴 McGaw, M. A., 1993, “Cumulative Damage Concepts in Thermomechanical Fatigue,” Thermomechanical Fatigue Behavior of Materials, H. Sehitoglu, ed., American Society of Testing and Materials, Philadelphia, PA, ASTM STP 1186, pp. 144–156. 关7兴 Hyde, T. H., Becker, A. A., and Sun, W., 2001, “Creep Damage Analysis of Short Cracks Using Narrow Notch Specimen Made From a Ni-Base Superalloy at 700° C ,” 10th International Conference on Fracture, Honolulu, HI. 关8兴 Altenbach, J., Altenbach, H., and Naumenko, K., 2004, “Edge Effects in Moderately Think Plates Under Creep-Damage Conditions,” Technische Mechanik, 24共3–4兲, pp. 254–263. 关9兴 Hayhurst, D. R., 1983, “On the Role of Continuum Damage on Structural Mechanics,” Engineering Approaches to High Temperature Design, B. Wilshire and D. R. Owen, eds., Pineridge, Swansea, pp. 85–176.
Journal of Pressure Vessel Technology
关10兴 Murakami, S., and Ohno, N., 1980, “A Continuum Theory of Creep and Creep Damage,” Third IUTAM Symposium Creep in Structures, Leicester, UK, pp. 422–444. 关11兴 Kim, W. G., Yin, S. N., Ryu, W. S., and Jang, J. H., 2007, “Analysis of the Creep Rupture Data of Alloy 617 for a High Temperature Gas Cooled Reactor,” Eighth International Conference on Creep Fatigue at Elevated Temperatures, San Antonio, TX. 关12兴 Aghaie-Khafri, M., and Hajjavady, M., 2008, “The Effect of Thermal Exposure on the Properties of a Ni-Base Superalloy,” Mater. Sci. Eng., A, 487, pp. 388–393. 关13兴 Lai, G. Y., 1994, “Nitridation Attack in Simulated Gas Turbine Combustion Environment,” Conference of Materials for Advanced Power Engineering 1994, Liège, Belgium, pp. 1263–1272. 关14兴 Misseijer, R. C., Thabit, T. I., and Mattheij, J. H. G., 1996, “Operational Evaluation of Patch Repaired Combustion Gas Turbine Transition Pieces,” J. Qual. Maint. Eng., 2共4兲, pp. 59–70. 关15兴 ThyssenKrupp VDM GmbH, 1994, “Nicrofer® 5520 Co-Alloy 617,” Material Data Sheet No. 4019, Werdohl, Germany. 关16兴 Venkatesh, A., Marthandam, V., and Roy, A. K., 2007, “Tensile Deformation and Fracture Toughness Characterization of Alloys 617 and 718,” Proceedings of the SEM Annual Conference and Exposition on Experimental and Applied Mechanics, 2共1兲, pp. 690–696. 关17兴 “Standard Test Methods for Conducting Creep, Creep-Rupture, and StressRupture Tests of Metallic Materials,” ASTM E-139, No. 03.01, West Conshohocken, PA. 关18兴 Cook, R. H., 1984, “Creep properties of Inconel-617 in Air and Helium at 800 to 1000° C,” Nucl. Technol., 66共2兲, pp. 283–288. 关19兴 Laanemae, W. M., Bothe, K., and Gerold, V., 1989, “High Temperature Mechanical Behaviour of Alloy 617. 2. Lifetime and Damage Mechanisms,” Z. Metallkd., 80共12兲, pp. 847–857. 关20兴 Sharma, S. K., Ka, G. D., Li, F. X., and Kang, K. J., 2008, “Oxidation and Creep Failure of Alloy 617 Foils at High Temperature,” J. Nucl. Mater., 378共2兲, pp. 144–152. 关21兴 European Alloy-DB, Licensed CD, European Commission, Joint Research Centre 共JRC兲. 关22兴 Schubert, F., Bruch, U., Cook, R., Diehl, H., Ennis, P. J., Jakobeit, W., Penkalla, H. J., Heesen, E. T., and Ulirich, G., 1984, “Creep Rupture Behavior of Candidate Materials for Nuclear Process Heat Applications,” Nucl. Technol., 66共2兲, pp. 227–239. 关23兴 Natesan, K., Purohit, A., and Tam, S. W., 2003, “Materials Behavior in HTGR Environments,” Argonne National Laboratory, Report No. NUREG/CR-6824. 关24兴 Zuo, M., Chiovelli, S., and Nonaka, Y., 2000, “Fitting Creep-Rupture Life Distribution Using Accelerated Life Testing Data,” ASME J. Pressure Vessel Technol., 122共4兲, pp. 482–487. 关25兴 Weber, J., Klenk, A., and Rieke, M., 2005, “A New Method of Strength Calculation and Lifetime Prediction of Pipe Bends Operating in the Creep Range,” Int. J. Pressure Vessels Piping, 82共2兲, pp. 77–84. 关26兴 Kouhia, R., Marjamaki, P., and Kivilahti, J., 2005, “On the Implicit Integration of Rate-Dependent Inelastic Constitutive Models,” Int. J. Numer. Methods Eng., 62共13兲, pp. 1832–1856. 关27兴 Shankar, P. S., and Natesan, K., 2007, “Effect of Trace Impurities in Helium on the Creep Behavior of Alloy 617 for Very High Temperature Reactor Applications,” J. Nucl. Mater., 366共1–2兲, pp. 28–36. 关28兴 Mustata, R., and Hayhurst, D. R., 2005, “Creep Constitutive Equations for a 0.5Cr 0.5 Mo 0.25V Ferritic Steel in the Temperature Range 565° C – 675° C,” Int. J. Pressure Vessels Piping 82共5兲, pp. 363–372. 关29兴 Kakehi, K., 1999, “Tension/Compression Asymmetry in Creep Behavior of a Ni-Based Superalloy,” Scr. Mater., 41共5兲, pp. 461–465. 关30兴 Sondhi, S. K., Dyson, B. F., and McLean, M., 2004, “Tension–Compression Creep Asymmetry in a Turbine Disc Superalloy: Roles of Internal Stress and Thermal Ageing,” Acta Mater., 52共7兲, pp. 1761–1772. 关31兴 Yamashita, M., and Kakehi, K., 2006, “Tension/Compression Asymmetry in Yield and Creep Strengths of Ni-Based Superalloy With a High Amount of Tantalum,” Scr. Mater., 55共2兲, pp. 139–142. 关32兴 Viswanathan, R., and Scheirer, S. T., 2001, “Materials Technology for Advanced Land Based Gas Turbines,” Creep: Proceedings of the International Conference on Creep and Fatigue at Elevated Temperatures, Tsukuba, Japan, No. 01-201 共20010603兲, pp. 7–21. 关33兴 Hayhurst, D. R., 1972, “Creep Rupture Under Multiaxial States of Stress,” J. Mech. Phys. Solids, 20, pp. 381–390.
OCTOBER 2009, Vol. 131 / 051406-11
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To capture the mechanical response of Ni-based materials, creep deformation and rupture experiments are typically performed. Long term tests, mimicking service conditions at 10,000 h or more, are generally avoided due to expense. Phenomenological models such as the classical Kachanov–Rabotnov (Rabotnov, 1969, Creep Problems in Structural Members, North-Holland, Amsterdam; Kachanov, 1958, “Time to Rupture Process Under Creep Conditions,” Izv. Akad. Nauk SSSR, Otd. Tekh. Nauk, Mekh. Mashin., 8, pp. 26–31) model can accurately estimate tertiary creep damage over extended histories. Creep deformation and rupture experiments are conducted on IN617 a polycrystalline Ni-based alloy over a range of temperatures and applied stresses. The continuum damage model is extended to account for temperature dependence. This allows the modeling of creep deformation at temperatures between available creep rupture data and the design of full-scale parts containing temperature distributions. Implementation of the Hayhurst (1983, “On the Role of Continuum Damage on Structural Mechanics,” in Engineering Approaches to High Temperature Design, Pineridge, Swansea, pp. 85–176) (tri-axial) stress formulation introduces tensile/compressive asymmetry to the model. This allows compressive loading to be considered for compression loaded gas turbine components such as transition pieces. A new dominant deformation approach is provided to predict the dominant creep mode over time. This leads to development of a new methodology for determining the creep stage and strain of parametric stress and temperature simulations over time. 关DOI: 10.1115/1.3148086兴 Keywords: gas turbine, tertiary creep damage, IN617, Ni-based alloy, constitutive model, tensile/compressive asymmetry
1
Introduction
Gas turbines generate electric power through a compressioncombustion process that forces superheated highly pressurized exhaust gas through various components. High pressure in the compression area coupled with combustion chamber exit temperatures of up to 1350° C creates a situation where material design and selection play a major role in the long term reliability of gas turbine parts. Drives to increase gas turbine efficiency require raising turbine firing temperatures, which negatively impact component life. Under these conditions high strength materials are needed to provide long life at elevated temperatures. The traditional method of determining the service life of gas turbine components employs practical experience, experimental data, and finite element analysis 共FEA兲. The boundary conditions, temperature and stress gradients, environmental conditions, and time-dependent variations induce a number of damage mechanisms, which cause crack initiation and eventual rupture. Common practice in the gas turbine industry is to conduct high temperature mechanical testing on a sample-sized specimen over a period of time and use practical experience 共from previous design兲 to extend empirical models to failure time. Creep rupture life is typically estimated using experience and secondary creep 共i.e., steady state兲 modeling with temperature dependence. Unfortunately, this method neglects the effects of both strain-softening behavior 共tertiary creep modeling兲 and tensile/compressive asymmetry. Not including these damage mechanisms creates both Contributed by the Pressure Vessel and Piping Division of ASME for publication in the JOURNAL OF PRESSURE VESSEL TECHNOLOGY. Manuscript received September 19, 2008; final manuscript received February 28, 2009; published online September 3, 2009. Review conducted by Douglas Scarth. Paper presented at the ASME Early Career Technical Conference 2008.
Journal of Pressure Vessel Technology
strain history and component life estimates of limited accuracy. Using a more advanced tertiary creep damage model with tensile/ compressive asymmetry improves the ability to predict component life. Creep deformation and rupture experiments are used to determine the constants of the constitutive model. As a part of this investigation, the material model is evaluated and a numerical parametric study is carried out to determine the total strain experienced at various combinations of uni-axial stress, temperature, and time.
2
Tertiary Creep Damage Model
A method has been established for modeling tertiary creep at high temperatures. This method is based on the Norton power law for secondary creep, i.e., ˙ cr = A¯n
共1兲
where A and n are the creep strain coefficient and exponent, respectively, and ¯ is the von Mises effective stress. The coefficient A exhibits Arrhenian temperature dependence, e.g.,
冉 冊
A = B exp −
Qcr RT
共2兲
Here Qcr is the activation energy for creep deformation, and T is the temperature measured in K. The term R is the universal gas constant. Both A and its temperature-independent counterpart, B, have units MPa−n h−1. The exponent n may also exhibit temperature dependence. As shown in Eq. 共1兲, the creep strain rate is stress dependent due to ¯. For most materials, traditional methods tend to find A and n to be stress independent but there are exceptions 关1兴. The secondary creep constants are determined by considering the minimum strain rate versus the applied stress, and using a regression analysis. Hyde et al. 关2兴 showed for a Ni-based
Copyright © 2009 by ASME
OCTOBER 2009, Vol. 131 / 051406-1
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Table 1 Creep Deformation data for IN617 Stress,
Temperature, T
a
°C
°F
MPa
ksi
Stress ratio, / ys
649 649 649 649 760a 760 760 760 871 871 871 871 982a 982 982 982
1200 1200 1200 1200 1400 1400 1400 1400 1600 1600 1600 1600 1800 1800 1800 1800
310 310 345 345 145 145 158 158 62 62 69 69 24 24 28 28
45 45 50 50 22 22 23 23 9 9 10 10 3.5 3.5 4 4
1.22 1.22 1.36 1.36 0.53 0.53 0.58 0.58 0.29 0.29 0.32 0.32 0.19 0.19 0.22 0.22
Time to strain, t 共h兲
Min strain rate 共%/h兲
Rupture strain 共%兲
0.25% strain
1% strain
5% strain
Rupture
0.0006 0.0008 0.0024 0.0024 0.0098 0.0051 0.0031 0.0035 0.0022 0.0022 0.0082 0.0088 0.0002 0.0002 0.0011 0.0021
1.1 1.2 1.2 1.2 16.0 19.8 13.3 14.0 22.0 19.3 19.8 28.5 19.0 14.5 20.2 17.8
204 69 54 40 7 14 19 16 54 33 17 14 170 228 162 53
1338 904 353 296 57 138 188 171 224 200 84 74 256 384 259 240
N/A N/A N/A N/A 355 581 577 560 563 533 251 231 384 614 416 404
1485 1090 412 341 748 981 852 828 1050 1082 585 488 520 946 658 654
Removed from consideration.
alloy Waspaloy that once a critical value of applied stress is reached the relationship between minimum creep strain and stress evolves. This leads to a two-stage relationship and stress dependence of the secondary material constants. Table 1 shows that for the collected creep deformation data, only two stress levels per temperature were conducted. Any possible stress dependence of A and n is not visible in the current data. Therefore, stress dependence of the secondary material constants is neglected. Jeong et al. 关3兴 demonstrated that while under stress relaxation short term transient behavior occurs where Qcr and n are both stress dependent at relativity short times 共e.g., less than 10 min兲. After this transient stage, the Norton power law becomes independent of initial stress and strain. This transient behavior has been neglected here since the current study focuses on longer periods 共e.g., 1–100,000 h兲. During the transition from secondary to tertiary creep, stresses in the vicinity of voids and microcracks along grain boundaries of polycrystalline materials are amplified. The stress concentration due to the local reduction in cross-sectional area is the phenomenological basis of the Kachanov–Rabotnov 关4,5兴 damage variable, , that is coupled with the creep strain rate and has a stressdependent evolution, i.e., ˙ cr = ˙ cr共¯,T, , . . .兲 共3兲
˙ = ˙ 共¯,T, , . . .兲
A straightforward formulation implies that the damage variable is the net reduction in cross-sectional area due to the presence of defects such as voids and/or cracks, e.g., ¯net = ¯
A0 = Anet
冉
¯ ¯ = A0 − Anet 1− 1− A0
冊
共4兲
where ¯net is called the net 共area reduction兲 stress, A0 is the undeformed area, Anet is the reduced area due to deformation, and is the damage variable. Some models account for the variety of physically observed creep damage mechanisms with multiple damage variables 关6兴. An essential feature of damage required for application of continuum damage mechanics concepts is a continuous distribution of damage. Generally, microdefect interaction 共in terms of stress fields, strain fields, and driving forces兲 is relatively weak until impingement or coalescence is imminent. By substituting Eq. 共4兲 into Eq. 共1兲, where ¯ is replaced by ¯net, the 051406-2 / Vol. 131, OCTOBER 2009
steady-state creep strain is expressed as a modified secondary creep rate: ˙ cr = A
冉 冊 ¯ 1−
n
共5兲
The isotropic damage variable is scalar valued from 0 for an initial state 共e.g., undamaged兲 to 1 at final state 共e.g., completely ruptured兲. Its evolution was given by Rabotnov 关4兴 as
˙ =
M ¯ 共1 − 兲
共6兲
where M is the creep damage coefficient having units of MPa−. This damage evolution formulation allows the model to account for tertiary creep. In an earlier study, Kachanov 关5兴 developed a similar formulation with the exception that the creep damage exponents, and , were equivalent. Time hardening, tensile/ compressive asymmetry, and multi-axial behavior can be accounted for in the model by replacing the effective stress with one that considers multi-axial stress, e.g.,
˙ =
M r m t 共1 − 兲
共7兲
The exponent, m, accounts for time hardening and the units of M are in MPa−n h−m−1; however, m is defined to be 0.0 in most cases 关7,8兴. Here r describes the Hayhurst 共tri-axial兲 stress and is related to the principal stress, 1, and hydrostatic 共mean兲 stress, m, and ¯. The Hayhurst stress 关9兴 is given as
r = ␣1 + 3m + 共1 − ␣ − 兲¯
共8兲
where ␣ and  are the weighing factors valued between 0 and 1 and are determined from multi-axial creep experiments. This formulation disallows damage under uni-axial compression when ␣ + 2 ⱖ 1. More recent investigations have extended this damage evolution description to account for anisotropic damage resulting from multi-axial states of stress 关10兴. In these cases, damage is described by a second-rank tensor. The coupled evolution 共Eqs. 共5兲 and 共7兲兲, which make up the tertiary creep damage model, can be reverted back to secondary creep when M = 0. As a result, the damage evolution equation to model reverts back to the Norton power law for secondary creep Transactions of the ASME
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Table 2 Nominal chemical composition of IN617 Cr
Co
Ti
Al
Mo
C
B
Fe
Ni
22.5
12.5
0.6
1.0
9.6
0.08
0.004
0.98
Bal.
with the exception that ¯ is replaced by r. This is a useful property that is exploited later to determine the transition from secondary to tertiary creep dominance over time. The tertiary creep damage formulation can be applied to model the gradual reduction in load carrying capability 共e.g., secondary through tertiary creep兲. Since characterizes the microstructural evolution of the material and is dependent on stress and temperature, the damage variable is an internal state variable 共ISV兲. This formulation has been used in a variety of studies of turbine and rotor materials. The constants A, n, M, , and are considered material properties. For example, in Waspaloy at 700° C, A = 9.23⫻ 10−34 MPa−n h−1, n = 10.65, M = 1.18⫻ 10−25 MPa− h−1, = 8.13, and = 13.0 关7兴. For stainless steel at 650° C, A = 2.13 ⫻ 10−13 MPa−n h−1, n = 3.5, M = 9.0⫻ 10−10 MPa− h−1, and = = 2.8 关8兴. In these previous studies, isothermal and constant stress conditions were applied. In the current study, isothermal and constants force conditions are applied allowing for stress relaxation similar to the conducted creep experiments. It should be noted that no rule for the temperature dependence of the tertiary creep constants has been established.
3
Creep Experiments
Polycrystalline 共PC兲 nickel-chromium-cobalt-molybdenum alloys are a modern class of materials commonly used in gas turbine components. One such material is wrought Ni-based alloy IN617. This nonproprietary sheet material has several equivalent monikers 共e.g., Inconel-617, INCO-617, Nicrofer® 5520 Co-Alloy 617, and Haynes 617兲. The nominal chemical composition is given in Table 2. The alloying composition of the material allows it to withstand a range of conditions critical to gas turbine components. The combination of various elements produces a strong atomic structure that imparts strength at high temperatures. Although it is suitable below 1000° C, IN617 displays exceptionally high levels of creep rupture strength, even at temperatures of 980° C 共1800° F兲. An inherent feature of the material is it has good corrosion resistance compared with other standard Ni-based alloys such as alloy 556, alloy 230, and alloy Hastelloy-X 关11兴. It carries a good combination of resistance to oxidation and carburizing atmospheres that are important for components that undergo long term loading. Thermal shock resistance is another key feature that forward the uses of Ni-based materials 关12兴. All of these features result in a material that has good resistance to creep deformation. The material also has relatively good weldability allowing field repair; this is another important requirement of gas turbines. One major detractor for this material is that it is susceptible to intense nitridation at elevated temperatures while undergoing thermal cycling. This susceptibility is due to the development of crack defects in the oxide scale, which perpetuates the nitridation process and allows surface intrusion of oxides, thus
leading to crack initiation and eventual failure 关13兴. Additionally, alloy 617 encounters deep internal defects through the development of aluminum nitrides, which severely hinder the load carrying capability of the material. Despite this condition, alloy 617 is still found to be the best combination of overall material properties. Proper turbine operation has been shown to produce gas turbine transition pieces of alloy 617 that last up to 15,000 total operating hours 关14兴. The tensile properties of the material are given in Table 3 关15兴. An attractive property of alloy 617 is that it has a nonlinear yield strength, which is enhanced at temperatures between 700° C and 800° C 关16兴. This yield strength anomaly makes the material an excellent candidate for components that are heated within that temperature range 共gas turbine transition pieces兲. This phenomenon has the added effect of making the creep deformation and rupture response at temperatures higher than 800° C more sensitive to temperature change 共due a drop in yield strength兲. Creep deformation and rupture experiments were conducted according to an ASTM standard E-139 关17兴 at 649° C, 760° C, 871° C, and 982° C on samples incised from wrought slabs of IN617. The combination of conditions is given in Table 1. From this variety of creep deformation and rupture experiments, the subject PC material exhibits primary, secondary, and tertiary creep strains depending on the combination of test temperature and stress, as shown in Fig. 1. Each test was conducted twice per stress level to account for variability in alloy samples due to slight differences in heat treatment. This method showed that inconsistencies are found at 760° C with 148 MPa and 982° C with 24 MPa, shown in Table 1. To account for this, these two creep curves were removed from the analysis. Based on these strain histories, the deformation mechanisms can be inferred from investigations that complemented mechanical experimentation with microscopic analysis 关18,19兴. Deformation at 649° C at high stress is mostly due to plastic strain, along with primary and secondary creeps. At stress levels of 345 and 310 MPa, the stress ratios, applied stress over yield strength, are found to be 1.35 and 1.22, respectively. Upon loading, a substantial time-independent plastic strain was produced before the creep test began. During primary creep, the strain rate, which is initially high, decreases significantly. This transient phenomenon is due to the ease at which dislocations can glide upon initial loading 共due to initially present dislocations in untreated workpieces兲 and the saturation of the dislocation density, which inhibits continued creep deformation. Transient creep also drops due to pinning of the dislocations when they encounter various obstacles after the easy glide period. Secondary creep arises due to the nucleation of grain boundary 共GB兲 cavities and grain boundary sliding. The deformation mode of tertiary creep was not observed in these cases. At higher temperatures 共760° C and above兲, deformation is dominated by elastic strain, and secondary and tertiary creeps are facilitated by the coalescence of grain boundary voids into microcracks. Sharma et al. 关20兴 showed that at temperatures above 800° C, dynamic recrystallization is found to occur. This process is a part of the hot deformation process, which decreases average grain size as stress increases. This has the effect of rapidly reducing dislocation density in the alloy at higher temperatures and thus
Table 3 IN617 material properties Temperature, T
a
0.2% tensile yield strength,a y
Tensile strength
Young’s modulus
°C
°F
MPa
ksi
MPa
ksi
GPa
103 ksi
Elongation 共%兲
649 760 871 982
1200 1400 1600 1800
254 272.7 215.6 128.4
37 40 31 19
586.2 525.5 341 181.6
85 76.2 49 26
170 161.2 151.6 141
24.7 23.4 22 20.4
56.6 50.2 69.3 85
Yield strength assumed symmetric.
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Fig. 1 Stress and temperature combinations of available creep experiments „does not include elastic strain…
reducing primary creep deformation. Notwithstanding, primary creep was still observed, but its contribution to the overall deformation was negligible compared with that of secondary and tertiary creeps; therefore for the case of this study primary creep was neglected. For all cases except those at 649° C, time-independent plasticity is ignored. Creep deformation in the current study is assumed to also encompass time-dependent viscoplasticity. Later studies will evaluate the response of a coupled plasticity and creep model.
The creep rupture data at 649° C, 760° C, 871° C, and 982° C compare well to an earlier comprehensive study 关11兴 of creep rupture at temperature between 600° C and 1000° C, from previous studies 关21–23兴. Figure 2 shows stress versus rupture time of both the obtained and study creep rupture data. Generally, creep rupture time increases with either decreasing applied stress or decreasing temperature. The plot shows that the obtained creep rupture data are positioned within the bounds of creep rupture data at
Fig. 2 Creep rupture data at various temperatures †11‡
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Z
Y
X Fig. 3 Single element FEM geometry used with force and displacement applied
adjacent temperature from the comprehensive study. It is important to note that in Fig. 2 the creep tests performed at 649° C or less experienced a time-independent plastic strain. In all the creep rupture data, scatter is present. This scatter is common in creep rupture charts. It is due to inconsistency in commercial material quality and inadequate control of the heat treatment process. It also is a result of differences in testing procedure and equipment. In the comprehensive study, multiple data sources were used where the testing procedure and equipment are not known. Manufacturers commonly use least-squares linear regression analysis to determine the minimum creep rupture life of an alloy while neglecting scatter. Advancements by Zuo et al. 关24兴 show that the maximum likelihood method produces more accurate predictions of creep rupture life.
4
Numerical Methodology
The constitutive model described in Eqs. 共5兲, 共7兲, and 共8兲 has been implemented in a general-purpose FEA software in order to determine the constants for the constitutive model used in the secondary-tertiary creep formulation. The formulation was implemented into a FORTRAN subroutine in the form of a userprogrammable feature 共UPF兲 in ANSYS 关25兴. The subroutine is incorporated with an implicit integration algorithm. This backward Euler integration algorithm is more accurate over long time periods than other practical numerical integration methods. This allows larger time steps that reduce the numerical solve time. Since the viscoplastic/creep behavior of materials is significant at extended histories, the backward Euler method is the desired method for integration of creep constitutive models 关26兴. This implementation allows for the update of the internal state variable, . Initially, = 0.0 and during loading, increases. To prevent the singularity that is caused by rupture 共e.g., = 1.0兲, the damage is restricted to a maximum of 0.90. Comparison of creep rupture data to numerical simulations shows that rupture occurs at between 0.4 and 0.6; therefore, rupture can be achieved before damage reaches = 1.0. To prevent an excessive model solve time and large deformation errors, a simulation was terminated once the total strain reached 100%. If needed, this model can be applied with time-independent and time-dependent plasticity models in a straightforward manner. The UPF has been used to simulate the temperature and stress loading conditions of a series of uni-axial creep and rupture experiments. A single, solid, three-dimensional, eight-noded element was used, and the appropriate initial and boundary conditions were applied to numerically simulate the uni-axial creep deformation curves shown in Fig. 1. The loads and temperatures applied numerically simulated those of the obtained creep rupture experiments. Figure 3 demonstrates how these conditions are applied on Journal of Pressure Vessel Technology
the FEM geometry. The constant axial force load experience during tensile testing was applied as a force load on the top surface of the element. A uniform temperature was applied across the element. Displacement controls were set to match the constraints experienced during tensile testing. These conditions lead to an accurate assessment of the creep deformation within a single element model. Histories of creep deformation, cr共t兲, were recorded to a data file and subsequently compared with experiments. Using this method, expansion beyond a single element to multi-element simulations will produce an accurate measure of creep deformation. A goal of the research was to provide a uni-axial creep deformation formulation that was capable of matching the material response up to 5% of the total strain determined from experiments. The time required for each experimentally tested specimen to deform to 5% is shown in Table 1. The secondary creep constants 共e.g., A and n兲 were evaluated directly from the available creep deformation and rupture data for IN617 at each temperature. As described early, A and n were found by plotting the minimum strain rate versus applied stress and using regression analysis in the form of Eq. 共1兲 to determine the values of A and n. Sharma et al. 关20兴 found for IN617 an apparent activation energy, Qcr, equal to 271 kJ/mol similar to what has been observed in a bulk specimen tested in a He+ O2 environment 关27兴. Using this Qcr value as an initial guess produced a B constant of 1.27⫻ 10−19 MPa−n h−1. A value of B that is less than 10−12 is not suitable for application in FEM. Such numbers experience numerical cutoff where trailing significant figures are discarded resulting in limited accuracy. To overcome this problem, the guess value of Qcr was doubled to 542 kJ/mol. Based on these constants and Eq. 共2兲, B and Qcr were approximated as 2.95⫻ 10−3 MPa−n h−1 and 542 kJ/mol, respectively. The tertiary creep constants 共e.g., M, , and 兲 were determined for each temperature by an iterative trial and error method. This trial and error method consisted of parametrically changing each constant to determine its effect on the resulting creep deformation curve and manually refining the combined selection of constants until a best fit was determined. Further development using objective function based methods 关28兴 can lead to better material constants, which match the creep deformation response up to creep rupture. For simplicity, both and were taken to be equivalent similar to what Altenbach et al. 关8兴 considered for stainless steel. The constants ␣ and  were set to zero to produce a Hayhurst stress equivalent to uni-axially applied stress. Coupling Norton’s law for secondary creep with the Kachanov– Rabotnov formulation for tertiary creep allows the finite element model to display a strain history that approximates the experimental results well. The strain histories simulated with the Norton– Rabotnov formulation have very good correlation with the analogous experimentally obtained 共t兲. Tertiary creep constants were optimized for every combination of temperature and stress. An example of the simulated creep deformation response is shown in Fig. 4. Temperature dependence is carried by the material constants while stress dependence is carried by the Hayhurst stress, r. In each case, the trajectories of each of the correlated creep curves correspond to experimental data.
5
Temperature Dependence
All of the creep curves exhibiting creep induced strain softening can be generated by varying M, , and . The creep damage rate coefficient, M, displays temperature dependence; however, the damage rate exponents and were assumed as temperature independent. The constants at each temperature are plotted in Fig. 5. Based on the experiments, the temperature dependence for each of the constants is mathematically represented. The elastic modulus of IN617 was written as a third order polynomial, e.g., OCTOBER 2009, Vol. 131 / 051406-5
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Fig. 5 Temperature dependence of elastic, secondary creep, and creep damage constants of IN617
TRAN subroutine so that predictions of creep deformation could be made through a range of temperatures.
6
Fig. 4 Comparison of experimental and numerically simulated creep deformation of IN617: „a… 760° C and 982° C, and „b… 871° C
E共T兲 = E3T3 + E2T2 + E1T1 + E0
共9兲
with T measured in K. In a similar manner, regression analyses were performed to model the temperature-dependence of the secondary creep and creep damage parameters. The secondary creep coefficient was determined using the Arrhenius relation as a function of temperature, where B and Qcr are independent of temperature. In Eq. 共2兲, T is measured in K. The secondary creep exponent was related by a polynomial function, e.g., n共T兲 = 具n3T3 + n2T2 + n1T1 + n0典 + nmin
共10兲
where T is measured in K. The McCauley brackets and nmin term are used in this case so that the minimum value of n is nmin, which is valued at 5. This was implemented to ensure numerical convergence of the UPF at low temperatures. The creep damage coefficient is expressed as a two-parameter exponential power law function, e.g., M共T兲 = M 1 exp共M 0T兲
共11兲
where T is measured in K, and M 1 and M 0 are constants. The fit of Eqs. 共9兲–共11兲 is compared with optimized constants for each temperature in Fig. 5. The square of the Pearson product-moment correlation coefficient, R2, indicates that each of the relations model the temperature dependence of the elastic and creep properties of IN617 with an accuracy greater than or equal to 0.90. The temperature dependence of the constants was coded into the FOR051406-6 / Vol. 131, OCTOBER 2009
Tensile/Compressive Asymmetry
It has been observed that for a number Ni-based alloys, there is a difference in the resulting mechanical response of a body undergoing compressive versus tensile loading 关29–31兴. This is due to the tendency for tensile loads to enhance the degradation of grain structure, which produces voids, crack propagation, and leads to rupture. Capturing the compressive behavior can be just as important as accurately modeling the tensile behavior, especially for materials used in gas turbine transition pieces. Transition pieces undergo multi-axial stress at high temperature, which can cause creep induced buckling 关32兴. An accurate estimate of service intervals could allow the preemptive patched repair maintenance of transition pieces, which has been found to extend the life of these components up to 8000 h 关14兴. To establish tensile/compressive asymmetric behavior in the constitutive model, a number of material constants were modified. The weight constants ␣ and  in Eq. 共8兲 modify Hayhurst stress and allow compression to occur when ␣ + 2 ⬍ 1. Hayhurst stress can be determined analytically for bi-axial stress states, as shown in Fig. 6. Hayhurst 关33兴 showed that material constants of ␣ = 0 and  = 0.25 produce a comparable tensile/compressive asymmetry in Ni-based alloys. The state of stress with y = and x = 0 yields a Hayhurst stress equal to . Applying the state of stress y = − and x = 0 yields a Hayhurst stress that is / 2. The constitutive model was modified to use this set of constants. It should be noted that the ␣ and  constants were not optimized for the IN617 alloy due to the lack of availability of compressive creep data. Additional control of tensile/compressive asymmetry is provided in the damage rate equation 共7兲 by the constants and . Increasing the exponent increases the effect that the Hayhurst stress has on the damage rate. The constant modifies the effect that the existing damage has on the damage rate. The constant can have a profound effect particularly when the damage nears = 1.0. For this study = = 3.0, based on guidance from other studies. The tensile/compressive creep deformation response at 815.5° C and various stresses is shown in Fig. 7. This plot demonstrates that in terms of secondary creep, the material is modeled Transactions of the ASME
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Fig. 6 Stress ratios resulting from various bi-axial „Hayhurst… stress configurations „a… 1 – 3m, 2 – 1, and 3 – . „b… All lines satisfy incompressibility and uni-axial tensile conditions. „c… all curves satisfy compressible and uni-axial tensile conditions. For all cases z.
to exhibit symmetric tensile and compressive behaviors; however; as tertiary creep damage accumulates, the effects of void formation and coalescence are predicted to occur more rapidly under tensile conditions. The tertiary creep model shows an overall reduction in the creep strain rate with respect to decreasing applied stress levels. This is attributed to the modified Hayhurst stress facilitating a smaller effective stress to the damage rate. Furthermore, due to the constants and , the damage rate is modified causing a lower creep strain rate to be produced. Using the tertiary creep model, Fig. 7 shows a factor of 4.7 in the time to 5% total strain between tension and compression at 100 MPa. At 10% total strain, this factor is increased to 5.5. The and constants have great significance due to the time dependence of damage and strain and the extensive amount of time required to reach rupture at lower stress levels. When applied stress is increased, the factors at 5% and 10% strain are reduced. This demonstrates that as applied stress is increased the tensile/ compressive asymmetry found in the Ni-based alloy is reduced. Increased applied stress causes a reduction in rupture time. The secondary creep stage has less time to develop and in some cases is bypassed. This results in the tertiary creep stage becoming
dominant earlier during creep deformation. The rapid deformation, which occurs during the tertiary creep stage, allows less time for tensile/compressive divergence. It should be emphasized that due to compressive creep data not being available for IN617, the compressive results could not be verified. The compressive results should, therefore, be used qualitatively.
7
Dominant Deformation Modes
One of the key goals of this research is to demonstrate the time dependence of the dominance of elasticity, secondary creep, and tertiary creep strain as stress and temperature change 共primary creep is ignored兲. The dominant deformation mode was determined by creating dominance contours. The ratio of partitioned strain over total strain for each stage was found. The partition with the largest ratio was considered dominant. The following equations were used to determine the curves of dominance transition:
total =
+ cr = E + 共SC + TR兲 E
Fig. 7 Creep deformation predictions in tension and compression at 815.5° C for both secondary creep „dotted lines… and tertiary creep models „solid lines…
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Fig. 8 Comparison of „„a… and „c…… total strain and „„b… and „d…… dominant deformation mode for „„a… and „b…… 100 MPa and 815.5° C and „„c… and „d…… 250 MPa and 649° C
再
dominant mode = max
E SC TR , , total total total
冎
共12兲
where total is total strain, E is elastic strain, SC is secondary creep strain, and TR is tertiary creep strain. Based on this dominant deformation formulation, Fig. 8 shows the transition of the various deformation modes. Figure 8共a兲 depicts the timedependent evolution of each form of strain at 815.5° C and 100 MPa. It clearly shows that the secondary creep model produces a linear strain-time function, which continues at the same rate until failure, while the tertiary creep model follows the expected experimental trend of an increase deformation rate, which leads to eventual rapid failure. Figure 8共b兲 displays the dominant deformation modes using the data from Fig. 8共a兲. The figure shows the transition from elasticity to secondary creep and eventually tertiary creep mode. This demonstrates that the formulation could be used to determine the creep mode a component encounters during operation. Engineers could apply this approach as a means to avoid modes beyond secondary creep, which tend to produce rapid deformation. Figure 8共c兲 depicts the growth of each form of strain over time at 649° C and 250 MPa. It shows that tertiary creep surpasses secondary creep before either surpasses elastic strain. Figure 8共d兲 plots the dominant deformation modes using the data from Fig. 8共c兲. It shows that the secondary creep model does not reach a situation where it can become dominant; it is essentially bypassed due to the strong effects of the tertiary creep model. Dominance transition is considered the point where one form of deformation mode dominance ends and another mode takes over. This can be seen in Eq. 共12兲 by the maximum value changing to a different component of the column. Two such dominance transitions are found in this paper: the transition from elastic to secondary creep dominance and the transition from secondary creep to tertiary creep dominance. In Fig. 9, the time where a dominance transition occurs, 5% strain and 10% strain are plotted. Six simulations were conducted for each temperature between high and low stresses to create trend curves. To compare the results on an 051406-8 / Vol. 131, OCTOBER 2009
equivalent value of stress, a stress ratio, / ys, applied stress over yield strength was used. Figure 9 shows that at low stress the secondary and tertiary creep dominance lines converge. At some minimum stress, the difference between secondary and tertiary creeps becomes negligible. This minimum stress is temperature dependent. As applied stress approaches, the value of the yield strength deformation occurs more rapidly. When the stress ratio is very near unity, plastic deformation begins to occur. This demonstrates the need for a plasticity model to account for inelastic mechanical behavior. Using temperature levels from 649° C to 982° C at 8.3° C intervals and stress levels from 500 MPa to 500 MPa at 25 MPa intervals, the simulations were carried out in an increasing magnitude of hours from 1 h to 100,000 h. The temperature range was restricted to 649– 982° C to match the range used in experimental test. At lower temperatures, creep deformation disappears. The assumption is made that there is no stress dependence exhibited in the material constants such that two stress levels are sufficient to fit our material constants for the entire range of stresses. Stresses ranging from ⫾500 MPa are used to allow a clear picture of the relationship between stress and creep strain. It is particular helpful in making the large gradient of creep strain observed at low time periods 共1 h, 10 h, and 100 h兲 visible. Simulations were conducted in elasticity only, secondary creep 共e.g., M = 0兲, and tertiary creep including tensile/compressive asymmetry conditions 共20,172 simulations total兲. With parametric boundary conditions established, a closer look at the response of the model can be achieved. Figure 10 shows the trends of creep stage dominance overlaid upon total strain for the complete set of simulations. It demonstrates the importance of implementing a tertiary creep model that includes effects neglected in a secondary creep model. In the figure, the region between the solid white lines signifies simulations where the elastic strain is dominant. The region between the solid white and black lines represents secondary creep strain dominance, and anything outside of the black lines produce tertiary creep strain dominance. Transactions of the ASME
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Fig. 9 Contours of constant total deformation and constant deformation mode at „a… 760° C, „b… 871° C, and „c… 982° C. „Note: / ys is the applied stress over yield strength.…
The dashed white and black line represents the 0.2% yield strength from Table 3. At stress levels above this line, a coupled plasticity and creep model is necessary to determine the total strain. Therefore, the simulated total strain found beyond this line should not be considered. The figure shows that at extended histories, creep strain becomes dominantly tertiary following expected experimentally reproducible trends. It should be noted at more than 1000 h under tensile loading, the secondary creep dominance begins to disappear. This is signified by the tertiary dominance line passing the secondary dominance line in effect of a dominance shift. This is a result of mechanical load, time, and temperature producing a situation where the secondary creep stage is not strong enough to produce dominance. This results in tertiary creep becoming the primary source of creep deformation until failure. A similar trend is observed when stress is low and temperature is increasing; the secondary creep strain dominance transition line converges with the tertiary transition line. The tensile/compressive relationship is demonstrated in Fig. 10 using the established ␣, , , and constants. It shows that for compressive loading the tertiary creep transition line lies at higher stresses than tensile loading. This causes secondary creep to be Journal of Pressure Vessel Technology
dominant in a large range of stress and temperatures while under compression. This is a product of using the Hayhurst stress equation that produces a lower effective stress while under the same strain and the damage rate equation, which compounds these effects. Additionally the dominance shift noticed at 1000 h under tensile loading is delayed on the compressive side to 10,000 h. There is a significant difference in the creep dominance behavior of tensile/compressive asymmetric materials. A major feature of the dominant deformation mode formulation is that it can be used as a design process tool for gas turbine components, as shown in Fig. 10. When considering new gas turbine component designs, it can be used as a method to eliminate boundary condition combinations leading to premature failure. The eliminated conditions could be determined based on yield strength, deformation mode, or allowable strain. Simple modifications could allow inclusion of additional criteria 共e.g., tensile strength and compressive strength兲.
8
Conclusions
The materials used in gas turbine components deform by a number of modes. The focus of this investigation is modeling the OCTOBER 2009, Vol. 131 / 051406-9
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Fig. 10 Total strain deformation maps from creep simulations
creep deformation of materials used to fabricate turbine parts that exhibit strain softening at high temperatures 共i.e., above 649° C兲. Creep cavitation and void coalescence along and within the boundaries of grains lead to area-reduction transverse to the applied loading axis. This leads to a gradual loss of stress-carrying 051406-10 / Vol. 131, OCTOBER 2009
ability. The Kachanov–Rabotnov type creep damage formulation was used to account for this behavior found in creep deformation and rupture experiments carried out on the Ni-based alloy IN617. To account for temperature dependence, phenomenological models were developed for several mechanical properties. The Transactions of the ASME
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damage formulation was modified and directly implemented into the constitutive model routine so that predictions of creep deformation could be made through a range of temperatures. The correlated curves show excellent agreement with trends from creep deformation experiments. Utilizing the Hayhurst 共tri-axial兲 stress formulation and the damage rate equation, the model was shown to be able to produce tensile/compressive asymmetry. The deformation mode dominance over an extensive range of mechanical load, temperature, and time configurations was determined using a new dominant deformation formulation. The difference between tensile and compressive dominances was also determined.
Acknowledgment The authors are thankful for the support of Siemens in conducting this research. Calvin Stewart is thankful for the support of a Mcknight Doctoral Fellowship through the Florida Education Fund.
Nomenclature A,B,n M ,,,m ␣, Qcr R cr ˙ cr r 1 m ¯ ˙
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
secondary creep constants tertiary creep constants weighing factors in the Hayhurst stress apparent activation energy universal gas constant creep strain creep strain rate stress Hayhurst 共tri-axial兲 stress first principal stress hydrostatic 共mean兲 stress von Mises stress damage damage rate
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Journal of Pressure Vessel Technology
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