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On the uniaxial mechanical behaviour of an advanced nickel base superalloy at high temperature
Mechanics of Materials 33 (2001) 593±600
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On the uniaxial mechanical behaviour of an advanced nickel base superalloy at high temperature L.G. Zhao *, J. Tong, B. Vermeulen, J. Byrne Department of Mechanical and Manufacturing Engineering, University of Portsmouth, Anglesea Building, Anglesea Road, Portsmouth PO1 3DJ, UK Received 3 July 2001
Abstract The uniaxial mechanical behaviour of an advanced nickel base superalloy has been studied under strain-controlled cyclic loading at two elevated temperatures, 300 and 650 °C. High-temperature related features, such as a transition from initial cyclic hardening to softening and stress relaxation during load-holding periods, were observed. The material's low cycle fatigue life experienced heavy reduction with increase in temperature and strain range. Using constitutive models for cyclic plasticity/viscoplasticity, attempts were made to simulate the stabilized cyclic loops for dierent strain ranges at both temperatures. The model parameters were determined simultaneously. Comparison of the simulated and experimental results is made for the stabilized cyclic loops and the stress relaxation behaviour. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Mechanical behaviour; Cyclic hardening/softening; Low cycle fatigue life; Constitutive models; Stabilized cyclic loops; Stress relaxation
1. Introduction Modern techniques for life assessment of structural components need a knowledge of material inelastic deformation, especially under hightemperature cyclic loading conditions. In the past three decades, a great deal of experimental and theoretical work has been done to investigate the plastic and creep deformation of various materials under uniaxial and multiaxial cyclic loading at isothermal or anisothermal conditions. The achievements have provided useful guides for the design and life assessment of structural compo-
*
Corresponding author. E-mail address: [email protected] (L.G. Zhao).
nents subjected to severe loading conditions and complex temperature histories. The material under investigation has been gradually extended from stainless steels (Miller, 1976; Chaboche and Rousselier, 1983; Ohashi et al., 1985) to advanced superalloys (Abdul-Latif, 1996; Arrell et al., 2000; Vermeulen et al., 2001). Various constitutive models have been developed to allow more accurate modelling of material behaviour. The concept of internal variables has been introduced where the actual state of the material depends on the values of a set of internal and observable variables. These variables, introduced phenomenologically from the analysis of experimental results, are also justi®ed by microstructural considerations. For time-independent plasticity, an internal-variable approach has been
0167-6636/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 6 3 6 ( 0 1 ) 0 0 0 7 1 - 0
594
L.G. Zhao et al. / Mechanics of Materials 33 (2001) 593±600
developed for the description of kinematic behaviour. This includes the use of independent multiyield surfaces (Mroz, 1967), models with two surfaces only (Krieg, 1975) and the so-called ``nonlinear kinematic hardening rule'' de®ned by a dierential equation (Armstrong and Frederick, 1966; Chaboche, 1986). A general review of the above models was given by Chaboche (1986, 1989). The time dependency can be introduced either by separating plastic and creep strains (Robinson et al., 1976), taking into account the coupling eects through the hardening rules (Kawai and Ohashi, 1987), or in the framework of uni®ed constitutive equations, considering only one inelastic strain (Miller, 1976; Walker, 1981; Chaboche and Rousselier, 1983; Chaboche, 1989). The uni®ed Chaboche constitutive model has received much attention due to its simplicity to comprehend and use. The Chaboche model is capable of simulating all the well-known phenomena of cyclic plasticity as well as the important rate eects and other time dependent processes such as creep, relaxation and recovery. Applications of the Chaboche model to various materials have been demonstrated (Chaboche and Rousselier, 1983; Dunne et al., 1992; Yang, 1997; Vermeulen et al., 2001). This work is a study of the mechanical behaviour of an advanced nickel base superalloy under uniaxial high-temperature loading conditions. Nickel base superalloys are designed to combat fatigue, creep and oxidation experienced by aeroengine turbine discs. The alloy under investigation is one of the current ranges of superalloys obtained via the powder metallurgy route. The new alloys exhibit superior tensile and creep properties to those of conventional alloys. The low cycle fatigue properties have been examined with a view to modelling the constitutive behaviour of the material. The mechanical tests were strain controlled at a strain rate e_ 0:5%/s, strain ratio R 0 and strain ranges 0:9% 6 De 6 2:2%. The loading waveform is trapezoidal with a one second holding period at the maximum and minimum strains. Two temperatures, 300 and 650 °C, were considered. The tensile curves were described using the Ramberg±Osgood model. The stabilized cyclic loops for 300 °C were simulated using a time-in-
dependent cyclic plasticity model. The uni®ed Chaboche model was applied for the simulation of stabilized cyclic loops at 650 °C to deal with the stress relaxation during the load holding period.
2. Analysis of experimental data The experimental work was carried out at the Defense and Evaluation Research Agency (DERA) in Farnborough of the UK and the test data were provided (Wilcock, 2000) for analysis purposes. Four strain ranges were considered, namely, 0.9%, 1.2%, 1.6% and 2.2%. The strain cycle is trapezoidal, i.e., ramp up to the maximum strain at a rate of e_ 0:5%/s, 1 s hold at the maximum strain, ramp down to zero strain at a rate of e_ 0:5%/s and 1 s hold at zero strain. The stresses and strains shown in the following are the values of the true stress and true strain. 2.1. Monotonic tension The monotonic tension was subtracted from the cyclic tests and corresponds to the initial ramp stage of the cyclic loading. The obtained tensile curves are plotted in Fig. 1 for both 300 and 650 °C. The Young's modulus E is about 214 GPa at 300 °C and 190 GPa at 650 °C. The 0.2% oset yielding stress is about 1010 MPa at 300 °C and 980 MPa at
Fig. 1. Monotonic tensile stress±strain behaviour at 300 and 650 °C, experimental results and ®tted Ramberg±Osgood models.
L.G. Zhao et al. / Mechanics of Materials 33 (2001) 593±600
650 °C. The material seems to have a higher tensile strength at 650 °C than at 300 °C, possibly due to the c0 precipitates which provide the high-temperature strength and stability (William, 1996). The c0 precipitates increase the material high-temperature strength by making it dicult for the dislocation pairs to move under stress in the precipitates (William, 1996). Strain hardening at 650 °C is stronger than that at 300 °C. The tensile curves may be described by the Ramberg±Osgood model n e r r ; 1 e 0 r0 r0
595
proaches Q when the cyclic saturation is reached and b controls the pace of cyclic hardening. As shown by Chaboche (1986, 1989), the evolution of R is related to the change in stress range during the cyclic tests via Dri Drs
Dr1 R 1 Dr1 Q
e
bp
;
3
where Dr1 and Drs are the stress ranges for the ®rst and saturation cycles, respectively, and Dri is the stress range in between. A demonstration of Eq. (3) is shown in Fig. 4 for the cyclic tests at 300 °C. A reasonable correlation was obtained, especially
where r0 and e0 are reference stress and strain and n is the hardening exponent. The ®tted curves are shown in Fig. 1 together with the experimental results. The corresponding values of r0 , e0 and n are listed in Table 1. 2.2. Cyclic loading For each strain range, the change of the stress range is plotted in Fig. 2 for 300 °C and Fig. 3 for 650 °C, as a function of cycle number. Cyclic hardening was clearly observed for the test at 300 °C, while at 650 °C cyclic softening was noted after the initial short-period of cyclic hardening. Similar softening behaviour was also reported by AbdulLatif (1996) for the cyclic tests of underaged Waspaloy (nickel base superalloy) at room temperature, where the phenomenon was attributed to the shearing process of the c0 precipitates during the cyclic loading. The evolution of cyclic hardening is normally described by an internal variable R, the drag stress, and expressed by (Chaboche, 1986, 1989) R Q 1
e
bp
;
Fig. 2. Average stress amplitude as a function of the number of cycles at 300 °C.
2
where p is the accumulated plastic strain, Q and b are material constants to be identi®ed. R apTable 1 Values of r0 , e0 and n 300 °C 650 °C
r0 (GPa)
e0 (%)
n
1.018 1.041
0.471 0.545
62.3 24.2
Fig. 3. Average stress amplitude as a function of the number of cycles at 650 °C.
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L.G. Zhao et al. / Mechanics of Materials 33 (2001) 593±600
Fig. 4. Modelling of cyclic hardening at 300 °C.
for the strain range De 2:2%. For the description of cyclic softening, modi®cation of the evolution rule of R is necessary, which may result in introducing of more internal variables and material constants (Abdul-Latif, 1996) and is beyond the scope of this work. For both temperatures, cyclic hardening/softening seems to stabilize after tens of cycles. The stabilized cyclic loops are shown in Fig. 5 for 300 °C and Fig. 6 for 650 °C. Some stress relaxation was observed during the hold period at 650 °C. The inelastic strains during the hold period may be a mixture of ``creep'' and ``fatigue'' strains hence justifying the creep±fatigue interaction consideration (Miller, 1976).
Fig. 5. Stabilized cyclic loops for various strain ranges at 300 °C, experimental results.
Fig. 6. Stabilized cyclic loops for various strain ranges at 650 °C, experimental results.
The low cycle fatigue life Nf against the strain range De is shown in Fig. 7 for both temperatures. It is seen that the low cycle fatigue life was reduced signi®cantly with the increased temperature and strain range. For the strain range De 2:2%, the low cycle fatigue life is only 190 cycles at 650 °C compared to 639 cycles at 300 °C. The low cycle fatigue life of the material can be described by the parameters a and C of the Manson±Con relation DeNfa C. The ®tted curves are shown in Fig. 7 together with the corresponding values of a and C.
Fig. 7. Low cycle fatigue life against the strain range at 300 and 650 °C.
L.G. Zhao et al. / Mechanics of Materials 33 (2001) 593±600
3. Simulation of stabilized cyclic loops The stabilized cyclic behaviour of a material is of particular interest since the crack initiation predictions are generally based on it (Chaboche, 1986). In this section, eorts were made to simulate the stabilized cyclic loops (see Figs. 5 and 6) using the already developed constitutive models (Chaboche, 1986, 1989). Since no stress relaxation occurs during the load-holding period at 300 °C, the classical time-independent cyclic plasticity model will be used for the simulation; while the uni®ed Chaboche model combining plasticity and creep will be adopted for the simulation at 650 °C where stress relaxation is obvious during the hold period. 3.1. Time-independent cyclic plasticity Within uniaxial small-strain hypothesis, the total strain e can be partitioned into an elastic part ee and a plastic part ep and the resulting stress r is r E e
ep :
4
For the stabilized cyclic deformation, the VonMises yield function f can be written as f r; v; k jr
vj
k 6 0;
v;
6
where the plastic multiplier k_ is identical to the accumulated plastic strain rate, i.e., k_ p_ je_p j. Evolution of the back stress v can be described through the following superposition of two nonlinear kinematic hardening rules (Chaboche, 1986, 1989): v v1 v 2 ; v_ 1 C1 a1 e_p v_ 2 C2 a2 e_p
_ v1 p; _ v2 p;
where C1 , a1 , C2 and a2 are four material and temperature dependent constants which determine the shape and amplitude of the cyclic loops. The consistency condition f f_ 0 leads to the following equations to express the plastic multiplier k_ and the hardening modulus h: 1 _ if f 0 and f_ 0; _k h hsgn r vri 8 0 otherwise; h a1 C1 a2 C2
C1 v1 C2 v2 sgn r
v; 9
where the sign function and the brackets are de®ned by 8 x > 0; < 1; sgn x 0; x 0; and : 1; x < 0; 10 x; x P 0; hxi 0; x < 0: The above constitutive equations, which do not incorporate the plastic strain range memorization and additional hardening caused by complex loading, contain six material parameters, namely, E, k, C1 , a1 , C2 and a2 . The parameter identi®cation from the experimental data is described in Section 3.3.
5
where v is the back stress and k is the radius of the yield surface. Plastic ¯ow occurs under the condition f 0 and of =or r_ > 0. Using the classical normality hypothesis, the plastic strain rate becomes of k_ sgn r e_p k_ or
597
7
3.2. Uni®ed cyclic viscoplasticity The uni®ed constitutive framework employed here is essentially that developed by Chaboche (Chaboche and Rousselier, 1983; Chaboche, 1989). In this model, the kinematic hardening rules are chosen to be the same as those in the time-independent case described by Eq. (7). And, there is no partitioning of the inelastic strain into a plastic one and a creep one. The viscoplastic strain incorporates both strains simultaneously. Consequently, the strain rate does not exhibit a discontinuity under any kind of loading. Chaboche adopted a simple power relationship for the viscopotential and the viscoplastic strain rate was expressed as n f _ep sgn r v; 11 Z
598
L.G. Zhao et al. / Mechanics of Materials 33 (2001) 593±600
where f is the yield surface de®ned by Eq. (5), v is the back stress given by Eq. (7) and Z and n are two additional material parameters responsible for the time and rate dependency. For this theory, the motion of yield surface f continues to hold but stress states in excess of the yield condition are now admissible. The so-called ``overstress'' or n viscous stress is rv Z p_ 1=n with p_ je_p j hf =Zi . The stress state can then be written easily as an additive function of the internal and viscous stress, i.e.,
rameters that minimizes the objective function. Here, the objective function is selected as (Dennis and Schnable, 1983) F h
N X M 1X wij rij 2 i1 j1
2
rij h ;
13
The above equation returns to the time-independent theory when n ! 1 or Z ! 0.
where rij and rij h denote experimental measurements and model predictions; h represents the set of parameters; N is the number of experiments; M is the number of data points in the ith experiment. The weighting factors, wij , are generally used to non-dimensionalize and scale the variables of dierent units and magnitudes (Fossum, 1997, 1998). Here, the following form was adopted (Vermeulen et al., 2001):
3.3. Parameter identi®cation and demonstration
wij
Identi®cation of the model parameters, which in mathematical terminology is an inverse problem, is generally based on the experimental data. Parameter determination can be done either step by step or simultaneously. For the step-by-step approach, the parameters are given a physical interpretation and a group of them are determined at a step from a special set of experiments to which they are sensitive. The experiments are special in the sense that they are designed to activate only a few parameters in the model. The simultaneous determination is to identify all the parameters in one step from a sucient number of independent experiments. The latter can be achieved if, by using an appropriate ®tting procedure, the dierence between theoretical and experimental variables reaches a minimum, as described by Schwertel and Schinke (1996). This approach gives an optimum description of the material behaviour (Schwertel and Schinke, 1996; Mahnken and Stein, 1996; Ekh et al., 2000; Vermeulen et al., 2001). In this work, the simultaneous determination approach was used to obtain the corresponding model parameters at 300 and 650 °C from the stabilized cyclic loops. An objective function was de®ned to measure the dierence between the experimental stress values and the corresponding calculated stress values with a given set of parameters. The goal is to determine the set of pa-
where M is the total test data points involved in the optimization, M is the number of data points in the ith test and rmax represents the greatest abi solute value within the ith test data. The above weighting factors provide a uniform weighting for each test data point. An iterative application of the gradient-based Levenberg±Marquardt algorithm (Dennis and Schnable, 1983) has been used to determine the optimum set of material parameters in the sense of nonlinear least squares. In order to ®nd the global minimum of the objective function, dierent initial estimates of the material parameters were tried to obtain a stabilized solution (Ekh et al., 2000). Then, give a perturbation to the stabilized solution and restart the Levenberg±Marquardt procedure to see if the optimization algorithm returns a better solution or the same one (Fossum, 1997, 1998). The calibrated model parameters are listed in Table 2 for both 300 and 650 °C. Note that the test corresponding to the strain range of De 0:9% was not considered for the parameter calibration due to its limited plastic deformation. The Young's modulus E was derived directly from the linear parts of the cyclic loops. The values of the parameters listed in Table 2 need to be understood as a group and any single value may be meaningless individually. The simulated results together with the experimental results are demonstrated in Fig. 8
r v k Z p_ 1=n sgn r
v:
12
M 1 ; M rmax i
14
L.G. Zhao et al. / Mechanics of Materials 33 (2001) 593±600
599
Table 2 Identi®ed model parameters for 300 and 650 °C 300 °C 650 °C
E (GPa)
k (MPa)
a1 (MPa)
C1
a2 (MPa)
C2
Z (MPa s1=n )
n
196 194
190 150
772 537
1239 1201
573 810
126 84
± 277
± 19.4
Fig. 8. Correlation between measured and simulated cyclic loops at 300 °C.
Fig. 10. Stress relaxation behaviour at 650 °C during the maximum-strain hold period, comparison of the experimental results with the simulation.
ing the maximum-strain hold period at 650 °C. It is seen that the stress relaxation behaviour was simulated reasonably well using the uni®ed Chaboche model.
4. Conclusion
Fig. 9. Correlation between measured and simulated cyclic loops at 650 °C.
for 300 °C and Fig. 9 for 650 °C. Good correlation between the simulation and the experimental data was obtained for the strain ranges of 1.6% and 2.2%. Some dierence can be seen for the strain range of 1.2%, where the measured cyclic loops exhibit higher modulus than the simulated ones. In addition, Fig. 10 gives the stress relaxation behaviour for the strain ranges of 1.6% and 2.2% dur-
Under uniaxial high-temperature loading conditions, the mechanical behaviour of an advanced nickel base superalloy has been studied. The material shows enhanced yield stress and high-temperature strength. A transition from cyclic hardening to softening and some stress relaxation during the load-holding periods were observed at 650 °C. Low cycle fatigue life experienced signi®cant reduction with the increase of temperature and strain range. The stabilized cyclic loops at 300 °C were simulated using the classical time-independent cyclic plasticity model. At 650 °C, the uni®ed Chaboche model was adopted for the simulation in order to deal with the stress relaxation behaviour during the load-holding period.
600
L.G. Zhao et al. / Mechanics of Materials 33 (2001) 593±600
Model parameters were identi®ed using the simultaneous determination approach. Correlation between the simulation and the test data was demonstrated for dierent strain ranges. The current work is limited to the case of a speci®c strain rate, strain ratio and waveform. Further investigations are under way to account for the eects of loading rate, ratio and waveform and to utilize creep test data in parameter identi®cation. Acknowledgements This work was funded in part by the EPSRC of the UK (GR/M44811). The authors gratefully acknowledge the provision of the low cycle fatigue data by the Defense and Evaluation Research Agency (DERA) of the UK. References Abdul-Latif, A., 1996. Constitutive equations for cyclic plasticity of Waspaloy. International Journal of Plasticity 12, 967±985. Armstrong, P.J., Frederick, C.O., 1966. A mathematical representation of multiaxial Bauschinger eect. G.E.G.B. Report RD/B/N 731. Arrell, D.J., Bressers, J., Timm, J., 2000. Eect of cycle history on thermomechanical fatigue response of nickel-base superalloys. ASME Journal of Engineering Materials and Technology 122, 300±304. Chaboche, J.L., 1986. Tme-independent constitutive theories for cyclic plasticity. International Journal of Plasticity 2, 149±188. Chaboche, J.L., 1989. Constitutive equations for cyclic plasticity and cyclic viscoplasticity. International Journal of Plasticity 5, 247±302. Chaboche, J.L., Rousselier, G., 1983. On the plastic and viscoplastic constitutive equations (Parts I and II). ASME Journal of Pressure Vessel Technology 105, 153±164. Dennis, J.E., Schnable, R.B., 1983. Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Clis, NJ. Dunne, F.P.E., Makin, J., Hayhurst, D.R., 1992. Automated procedures for the determination of high temperature viscoplastic damage constitutive equations. Proceeding of Royal Society London 437, 527±544. Ekh, M., Johansson, A., Thorberntsson, H., Josefson, B.L., 2000. Models for cyclic ratchetting plasticity ± integration
and calibration. ASME Journal of Engineering Materials and Technology 122, 49±55. Fossum, A.F., 1997. Parameter estimation for an internal variable model using nonlinear optimization and analytical/numerical response sensitivities. ASME Journal of Engineering Materials and Technology 119, 337±345. Fossum, A.F., 1998. Rate data and material model parameter estimation. ASME Journal of Engineering Materials and Technology 120, 7±12. Kawai, M., Ohashi, Y., 1987. Coupled eect between creep and plasticity of type 316 stainless steel at elevated temperature. In: The 2nd International Conference on Constitutive Laws for Engineering Materials: Theory and Applications. Elsevier, Amsterdam. Krieg, R.D., 1975. A pratical two-surface plasticity theory. ASME Journal of Applied Mechanics 47, 641±646. Mahnken, R., Stein, E., 1996. Parameter identi®cation for viscoplastic models based on analytical derivatives of a least-squares functional and stability investigations. International Journal of Plasticity 12, 451±479. Miller, A.K., 1976. An inelastic constitutive model for monotonic, cyclic, and creep deformation (Parts I and II). ASME Journal of Engineering Materials and Technology 98, 97±113. Mroz, Z., 1967. On the description of anisotropic workhardening. Journal of Mechanics and Physics of Solids 15, 163±175. Ohashi, Y., Kawai, M., Kaito, T., 1985. Inelastic behavior of type 316 stainless steel under multiaxial nonproportional cyclic stressings at elevated temperature. ASME Journal of Engineering Materials and Technology 107, 101±109. Robinson, D.N., Pugh, C.E., Corum, J.M., 1976. Constitutive equations for describing high temperature inelastic behaviour of structural alloys. ORNL Report 766027. Schwertel, J., Schinke, B., 1996. Automated evaluation of material parameters of viscoplastic constitutive equations. ASME Journal of Engineering Materials and Technology 118, 273±280. Vermeulen, B., Tong, J., Byrne, J., 2001. Eects of viscoplasticity/cyclic plasticity on crack growth behaviour in Ni base superalloys. Report to DERA Structural Materials Centre. Walker, K.P., 1981. Research and development program for non-linear structural modeling with advanced time-temperature dependent constitutive relationships. Report PWA-5700-50, NASACR-165533. Wilcock, I.M., 2000. Cyclic test data for nickel base superalloy RR1000 at 300 °C and 650 °C. Private Communications, DERA. William, F.S., 1996. Principles of Materials Science and Engineering. McGraw-Hill, New York. Yang, X.J., 1997. Constitutive description of temperaturedependent nonproportional cyclic viscoplasticity. ASME Journal of Engineering Materials and Technology 119, 12± 19.
www.elsevier.com/locate/mechmat
On the uniaxial mechanical behaviour of an advanced nickel base superalloy at high temperature L.G. Zhao *, J. Tong, B. Vermeulen, J. Byrne Department of Mechanical and Manufacturing Engineering, University of Portsmouth, Anglesea Building, Anglesea Road, Portsmouth PO1 3DJ, UK Received 3 July 2001
Abstract The uniaxial mechanical behaviour of an advanced nickel base superalloy has been studied under strain-controlled cyclic loading at two elevated temperatures, 300 and 650 °C. High-temperature related features, such as a transition from initial cyclic hardening to softening and stress relaxation during load-holding periods, were observed. The material's low cycle fatigue life experienced heavy reduction with increase in temperature and strain range. Using constitutive models for cyclic plasticity/viscoplasticity, attempts were made to simulate the stabilized cyclic loops for dierent strain ranges at both temperatures. The model parameters were determined simultaneously. Comparison of the simulated and experimental results is made for the stabilized cyclic loops and the stress relaxation behaviour. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Mechanical behaviour; Cyclic hardening/softening; Low cycle fatigue life; Constitutive models; Stabilized cyclic loops; Stress relaxation
1. Introduction Modern techniques for life assessment of structural components need a knowledge of material inelastic deformation, especially under hightemperature cyclic loading conditions. In the past three decades, a great deal of experimental and theoretical work has been done to investigate the plastic and creep deformation of various materials under uniaxial and multiaxial cyclic loading at isothermal or anisothermal conditions. The achievements have provided useful guides for the design and life assessment of structural compo-
*
Corresponding author. E-mail address: [email protected] (L.G. Zhao).
nents subjected to severe loading conditions and complex temperature histories. The material under investigation has been gradually extended from stainless steels (Miller, 1976; Chaboche and Rousselier, 1983; Ohashi et al., 1985) to advanced superalloys (Abdul-Latif, 1996; Arrell et al., 2000; Vermeulen et al., 2001). Various constitutive models have been developed to allow more accurate modelling of material behaviour. The concept of internal variables has been introduced where the actual state of the material depends on the values of a set of internal and observable variables. These variables, introduced phenomenologically from the analysis of experimental results, are also justi®ed by microstructural considerations. For time-independent plasticity, an internal-variable approach has been
0167-6636/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 6 3 6 ( 0 1 ) 0 0 0 7 1 - 0
594
L.G. Zhao et al. / Mechanics of Materials 33 (2001) 593±600
developed for the description of kinematic behaviour. This includes the use of independent multiyield surfaces (Mroz, 1967), models with two surfaces only (Krieg, 1975) and the so-called ``nonlinear kinematic hardening rule'' de®ned by a dierential equation (Armstrong and Frederick, 1966; Chaboche, 1986). A general review of the above models was given by Chaboche (1986, 1989). The time dependency can be introduced either by separating plastic and creep strains (Robinson et al., 1976), taking into account the coupling eects through the hardening rules (Kawai and Ohashi, 1987), or in the framework of uni®ed constitutive equations, considering only one inelastic strain (Miller, 1976; Walker, 1981; Chaboche and Rousselier, 1983; Chaboche, 1989). The uni®ed Chaboche constitutive model has received much attention due to its simplicity to comprehend and use. The Chaboche model is capable of simulating all the well-known phenomena of cyclic plasticity as well as the important rate eects and other time dependent processes such as creep, relaxation and recovery. Applications of the Chaboche model to various materials have been demonstrated (Chaboche and Rousselier, 1983; Dunne et al., 1992; Yang, 1997; Vermeulen et al., 2001). This work is a study of the mechanical behaviour of an advanced nickel base superalloy under uniaxial high-temperature loading conditions. Nickel base superalloys are designed to combat fatigue, creep and oxidation experienced by aeroengine turbine discs. The alloy under investigation is one of the current ranges of superalloys obtained via the powder metallurgy route. The new alloys exhibit superior tensile and creep properties to those of conventional alloys. The low cycle fatigue properties have been examined with a view to modelling the constitutive behaviour of the material. The mechanical tests were strain controlled at a strain rate e_ 0:5%/s, strain ratio R 0 and strain ranges 0:9% 6 De 6 2:2%. The loading waveform is trapezoidal with a one second holding period at the maximum and minimum strains. Two temperatures, 300 and 650 °C, were considered. The tensile curves were described using the Ramberg±Osgood model. The stabilized cyclic loops for 300 °C were simulated using a time-in-
dependent cyclic plasticity model. The uni®ed Chaboche model was applied for the simulation of stabilized cyclic loops at 650 °C to deal with the stress relaxation during the load holding period.
2. Analysis of experimental data The experimental work was carried out at the Defense and Evaluation Research Agency (DERA) in Farnborough of the UK and the test data were provided (Wilcock, 2000) for analysis purposes. Four strain ranges were considered, namely, 0.9%, 1.2%, 1.6% and 2.2%. The strain cycle is trapezoidal, i.e., ramp up to the maximum strain at a rate of e_ 0:5%/s, 1 s hold at the maximum strain, ramp down to zero strain at a rate of e_ 0:5%/s and 1 s hold at zero strain. The stresses and strains shown in the following are the values of the true stress and true strain. 2.1. Monotonic tension The monotonic tension was subtracted from the cyclic tests and corresponds to the initial ramp stage of the cyclic loading. The obtained tensile curves are plotted in Fig. 1 for both 300 and 650 °C. The Young's modulus E is about 214 GPa at 300 °C and 190 GPa at 650 °C. The 0.2% oset yielding stress is about 1010 MPa at 300 °C and 980 MPa at
Fig. 1. Monotonic tensile stress±strain behaviour at 300 and 650 °C, experimental results and ®tted Ramberg±Osgood models.
L.G. Zhao et al. / Mechanics of Materials 33 (2001) 593±600
650 °C. The material seems to have a higher tensile strength at 650 °C than at 300 °C, possibly due to the c0 precipitates which provide the high-temperature strength and stability (William, 1996). The c0 precipitates increase the material high-temperature strength by making it dicult for the dislocation pairs to move under stress in the precipitates (William, 1996). Strain hardening at 650 °C is stronger than that at 300 °C. The tensile curves may be described by the Ramberg±Osgood model n e r r ; 1 e 0 r0 r0
595
proaches Q when the cyclic saturation is reached and b controls the pace of cyclic hardening. As shown by Chaboche (1986, 1989), the evolution of R is related to the change in stress range during the cyclic tests via Dri Drs
Dr1 R 1 Dr1 Q
e
bp
;
3
where Dr1 and Drs are the stress ranges for the ®rst and saturation cycles, respectively, and Dri is the stress range in between. A demonstration of Eq. (3) is shown in Fig. 4 for the cyclic tests at 300 °C. A reasonable correlation was obtained, especially
where r0 and e0 are reference stress and strain and n is the hardening exponent. The ®tted curves are shown in Fig. 1 together with the experimental results. The corresponding values of r0 , e0 and n are listed in Table 1. 2.2. Cyclic loading For each strain range, the change of the stress range is plotted in Fig. 2 for 300 °C and Fig. 3 for 650 °C, as a function of cycle number. Cyclic hardening was clearly observed for the test at 300 °C, while at 650 °C cyclic softening was noted after the initial short-period of cyclic hardening. Similar softening behaviour was also reported by AbdulLatif (1996) for the cyclic tests of underaged Waspaloy (nickel base superalloy) at room temperature, where the phenomenon was attributed to the shearing process of the c0 precipitates during the cyclic loading. The evolution of cyclic hardening is normally described by an internal variable R, the drag stress, and expressed by (Chaboche, 1986, 1989) R Q 1
e
bp
;
Fig. 2. Average stress amplitude as a function of the number of cycles at 300 °C.
2
where p is the accumulated plastic strain, Q and b are material constants to be identi®ed. R apTable 1 Values of r0 , e0 and n 300 °C 650 °C
r0 (GPa)
e0 (%)
n
1.018 1.041
0.471 0.545
62.3 24.2
Fig. 3. Average stress amplitude as a function of the number of cycles at 650 °C.
596
L.G. Zhao et al. / Mechanics of Materials 33 (2001) 593±600
Fig. 4. Modelling of cyclic hardening at 300 °C.
for the strain range De 2:2%. For the description of cyclic softening, modi®cation of the evolution rule of R is necessary, which may result in introducing of more internal variables and material constants (Abdul-Latif, 1996) and is beyond the scope of this work. For both temperatures, cyclic hardening/softening seems to stabilize after tens of cycles. The stabilized cyclic loops are shown in Fig. 5 for 300 °C and Fig. 6 for 650 °C. Some stress relaxation was observed during the hold period at 650 °C. The inelastic strains during the hold period may be a mixture of ``creep'' and ``fatigue'' strains hence justifying the creep±fatigue interaction consideration (Miller, 1976).
Fig. 5. Stabilized cyclic loops for various strain ranges at 300 °C, experimental results.
Fig. 6. Stabilized cyclic loops for various strain ranges at 650 °C, experimental results.
The low cycle fatigue life Nf against the strain range De is shown in Fig. 7 for both temperatures. It is seen that the low cycle fatigue life was reduced signi®cantly with the increased temperature and strain range. For the strain range De 2:2%, the low cycle fatigue life is only 190 cycles at 650 °C compared to 639 cycles at 300 °C. The low cycle fatigue life of the material can be described by the parameters a and C of the Manson±Con relation DeNfa C. The ®tted curves are shown in Fig. 7 together with the corresponding values of a and C.
Fig. 7. Low cycle fatigue life against the strain range at 300 and 650 °C.
L.G. Zhao et al. / Mechanics of Materials 33 (2001) 593±600
3. Simulation of stabilized cyclic loops The stabilized cyclic behaviour of a material is of particular interest since the crack initiation predictions are generally based on it (Chaboche, 1986). In this section, eorts were made to simulate the stabilized cyclic loops (see Figs. 5 and 6) using the already developed constitutive models (Chaboche, 1986, 1989). Since no stress relaxation occurs during the load-holding period at 300 °C, the classical time-independent cyclic plasticity model will be used for the simulation; while the uni®ed Chaboche model combining plasticity and creep will be adopted for the simulation at 650 °C where stress relaxation is obvious during the hold period. 3.1. Time-independent cyclic plasticity Within uniaxial small-strain hypothesis, the total strain e can be partitioned into an elastic part ee and a plastic part ep and the resulting stress r is r E e
ep :
4
For the stabilized cyclic deformation, the VonMises yield function f can be written as f r; v; k jr
vj
k 6 0;
v;
6
where the plastic multiplier k_ is identical to the accumulated plastic strain rate, i.e., k_ p_ je_p j. Evolution of the back stress v can be described through the following superposition of two nonlinear kinematic hardening rules (Chaboche, 1986, 1989): v v1 v 2 ; v_ 1 C1 a1 e_p v_ 2 C2 a2 e_p
_ v1 p; _ v2 p;
where C1 , a1 , C2 and a2 are four material and temperature dependent constants which determine the shape and amplitude of the cyclic loops. The consistency condition f f_ 0 leads to the following equations to express the plastic multiplier k_ and the hardening modulus h: 1 _ if f 0 and f_ 0; _k h hsgn r vri 8 0 otherwise; h a1 C1 a2 C2
C1 v1 C2 v2 sgn r
v; 9
where the sign function and the brackets are de®ned by 8 x > 0; < 1; sgn x 0; x 0; and : 1; x < 0; 10 x; x P 0; hxi 0; x < 0: The above constitutive equations, which do not incorporate the plastic strain range memorization and additional hardening caused by complex loading, contain six material parameters, namely, E, k, C1 , a1 , C2 and a2 . The parameter identi®cation from the experimental data is described in Section 3.3.
5
where v is the back stress and k is the radius of the yield surface. Plastic ¯ow occurs under the condition f 0 and of =or r_ > 0. Using the classical normality hypothesis, the plastic strain rate becomes of k_ sgn r e_p k_ or
597
7
3.2. Uni®ed cyclic viscoplasticity The uni®ed constitutive framework employed here is essentially that developed by Chaboche (Chaboche and Rousselier, 1983; Chaboche, 1989). In this model, the kinematic hardening rules are chosen to be the same as those in the time-independent case described by Eq. (7). And, there is no partitioning of the inelastic strain into a plastic one and a creep one. The viscoplastic strain incorporates both strains simultaneously. Consequently, the strain rate does not exhibit a discontinuity under any kind of loading. Chaboche adopted a simple power relationship for the viscopotential and the viscoplastic strain rate was expressed as n f _ep sgn r v; 11 Z
598
L.G. Zhao et al. / Mechanics of Materials 33 (2001) 593±600
where f is the yield surface de®ned by Eq. (5), v is the back stress given by Eq. (7) and Z and n are two additional material parameters responsible for the time and rate dependency. For this theory, the motion of yield surface f continues to hold but stress states in excess of the yield condition are now admissible. The so-called ``overstress'' or n viscous stress is rv Z p_ 1=n with p_ je_p j hf =Zi . The stress state can then be written easily as an additive function of the internal and viscous stress, i.e.,
rameters that minimizes the objective function. Here, the objective function is selected as (Dennis and Schnable, 1983) F h
N X M 1X wij rij 2 i1 j1
2
rij h ;
13
The above equation returns to the time-independent theory when n ! 1 or Z ! 0.
where rij and rij h denote experimental measurements and model predictions; h represents the set of parameters; N is the number of experiments; M is the number of data points in the ith experiment. The weighting factors, wij , are generally used to non-dimensionalize and scale the variables of dierent units and magnitudes (Fossum, 1997, 1998). Here, the following form was adopted (Vermeulen et al., 2001):
3.3. Parameter identi®cation and demonstration
wij
Identi®cation of the model parameters, which in mathematical terminology is an inverse problem, is generally based on the experimental data. Parameter determination can be done either step by step or simultaneously. For the step-by-step approach, the parameters are given a physical interpretation and a group of them are determined at a step from a special set of experiments to which they are sensitive. The experiments are special in the sense that they are designed to activate only a few parameters in the model. The simultaneous determination is to identify all the parameters in one step from a sucient number of independent experiments. The latter can be achieved if, by using an appropriate ®tting procedure, the dierence between theoretical and experimental variables reaches a minimum, as described by Schwertel and Schinke (1996). This approach gives an optimum description of the material behaviour (Schwertel and Schinke, 1996; Mahnken and Stein, 1996; Ekh et al., 2000; Vermeulen et al., 2001). In this work, the simultaneous determination approach was used to obtain the corresponding model parameters at 300 and 650 °C from the stabilized cyclic loops. An objective function was de®ned to measure the dierence between the experimental stress values and the corresponding calculated stress values with a given set of parameters. The goal is to determine the set of pa-
where M is the total test data points involved in the optimization, M is the number of data points in the ith test and rmax represents the greatest abi solute value within the ith test data. The above weighting factors provide a uniform weighting for each test data point. An iterative application of the gradient-based Levenberg±Marquardt algorithm (Dennis and Schnable, 1983) has been used to determine the optimum set of material parameters in the sense of nonlinear least squares. In order to ®nd the global minimum of the objective function, dierent initial estimates of the material parameters were tried to obtain a stabilized solution (Ekh et al., 2000). Then, give a perturbation to the stabilized solution and restart the Levenberg±Marquardt procedure to see if the optimization algorithm returns a better solution or the same one (Fossum, 1997, 1998). The calibrated model parameters are listed in Table 2 for both 300 and 650 °C. Note that the test corresponding to the strain range of De 0:9% was not considered for the parameter calibration due to its limited plastic deformation. The Young's modulus E was derived directly from the linear parts of the cyclic loops. The values of the parameters listed in Table 2 need to be understood as a group and any single value may be meaningless individually. The simulated results together with the experimental results are demonstrated in Fig. 8
r v k Z p_ 1=n sgn r
v:
12
M 1 ; M rmax i
14
L.G. Zhao et al. / Mechanics of Materials 33 (2001) 593±600
599
Table 2 Identi®ed model parameters for 300 and 650 °C 300 °C 650 °C
E (GPa)
k (MPa)
a1 (MPa)
C1
a2 (MPa)
C2
Z (MPa s1=n )
n
196 194
190 150
772 537
1239 1201
573 810
126 84
± 277
± 19.4
Fig. 8. Correlation between measured and simulated cyclic loops at 300 °C.
Fig. 10. Stress relaxation behaviour at 650 °C during the maximum-strain hold period, comparison of the experimental results with the simulation.
ing the maximum-strain hold period at 650 °C. It is seen that the stress relaxation behaviour was simulated reasonably well using the uni®ed Chaboche model.
4. Conclusion
Fig. 9. Correlation between measured and simulated cyclic loops at 650 °C.
for 300 °C and Fig. 9 for 650 °C. Good correlation between the simulation and the experimental data was obtained for the strain ranges of 1.6% and 2.2%. Some dierence can be seen for the strain range of 1.2%, where the measured cyclic loops exhibit higher modulus than the simulated ones. In addition, Fig. 10 gives the stress relaxation behaviour for the strain ranges of 1.6% and 2.2% dur-
Under uniaxial high-temperature loading conditions, the mechanical behaviour of an advanced nickel base superalloy has been studied. The material shows enhanced yield stress and high-temperature strength. A transition from cyclic hardening to softening and some stress relaxation during the load-holding periods were observed at 650 °C. Low cycle fatigue life experienced signi®cant reduction with the increase of temperature and strain range. The stabilized cyclic loops at 300 °C were simulated using the classical time-independent cyclic plasticity model. At 650 °C, the uni®ed Chaboche model was adopted for the simulation in order to deal with the stress relaxation behaviour during the load-holding period.
600
L.G. Zhao et al. / Mechanics of Materials 33 (2001) 593±600
Model parameters were identi®ed using the simultaneous determination approach. Correlation between the simulation and the test data was demonstrated for dierent strain ranges. The current work is limited to the case of a speci®c strain rate, strain ratio and waveform. Further investigations are under way to account for the eects of loading rate, ratio and waveform and to utilize creep test data in parameter identi®cation. Acknowledgements This work was funded in part by the EPSRC of the UK (GR/M44811). The authors gratefully acknowledge the provision of the low cycle fatigue data by the Defense and Evaluation Research Agency (DERA) of the UK. References Abdul-Latif, A., 1996. Constitutive equations for cyclic plasticity of Waspaloy. International Journal of Plasticity 12, 967±985. Armstrong, P.J., Frederick, C.O., 1966. A mathematical representation of multiaxial Bauschinger eect. G.E.G.B. Report RD/B/N 731. Arrell, D.J., Bressers, J., Timm, J., 2000. Eect of cycle history on thermomechanical fatigue response of nickel-base superalloys. ASME Journal of Engineering Materials and Technology 122, 300±304. Chaboche, J.L., 1986. Tme-independent constitutive theories for cyclic plasticity. International Journal of Plasticity 2, 149±188. Chaboche, J.L., 1989. Constitutive equations for cyclic plasticity and cyclic viscoplasticity. International Journal of Plasticity 5, 247±302. Chaboche, J.L., Rousselier, G., 1983. On the plastic and viscoplastic constitutive equations (Parts I and II). ASME Journal of Pressure Vessel Technology 105, 153±164. Dennis, J.E., Schnable, R.B., 1983. Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Clis, NJ. Dunne, F.P.E., Makin, J., Hayhurst, D.R., 1992. Automated procedures for the determination of high temperature viscoplastic damage constitutive equations. Proceeding of Royal Society London 437, 527±544. Ekh, M., Johansson, A., Thorberntsson, H., Josefson, B.L., 2000. Models for cyclic ratchetting plasticity ± integration
and calibration. ASME Journal of Engineering Materials and Technology 122, 49±55. Fossum, A.F., 1997. Parameter estimation for an internal variable model using nonlinear optimization and analytical/numerical response sensitivities. ASME Journal of Engineering Materials and Technology 119, 337±345. Fossum, A.F., 1998. Rate data and material model parameter estimation. ASME Journal of Engineering Materials and Technology 120, 7±12. Kawai, M., Ohashi, Y., 1987. Coupled eect between creep and plasticity of type 316 stainless steel at elevated temperature. In: The 2nd International Conference on Constitutive Laws for Engineering Materials: Theory and Applications. Elsevier, Amsterdam. Krieg, R.D., 1975. A pratical two-surface plasticity theory. ASME Journal of Applied Mechanics 47, 641±646. Mahnken, R., Stein, E., 1996. Parameter identi®cation for viscoplastic models based on analytical derivatives of a least-squares functional and stability investigations. International Journal of Plasticity 12, 451±479. Miller, A.K., 1976. An inelastic constitutive model for monotonic, cyclic, and creep deformation (Parts I and II). ASME Journal of Engineering Materials and Technology 98, 97±113. Mroz, Z., 1967. On the description of anisotropic workhardening. Journal of Mechanics and Physics of Solids 15, 163±175. Ohashi, Y., Kawai, M., Kaito, T., 1985. Inelastic behavior of type 316 stainless steel under multiaxial nonproportional cyclic stressings at elevated temperature. ASME Journal of Engineering Materials and Technology 107, 101±109. Robinson, D.N., Pugh, C.E., Corum, J.M., 1976. Constitutive equations for describing high temperature inelastic behaviour of structural alloys. ORNL Report 766027. Schwertel, J., Schinke, B., 1996. Automated evaluation of material parameters of viscoplastic constitutive equations. ASME Journal of Engineering Materials and Technology 118, 273±280. Vermeulen, B., Tong, J., Byrne, J., 2001. Eects of viscoplasticity/cyclic plasticity on crack growth behaviour in Ni base superalloys. Report to DERA Structural Materials Centre. Walker, K.P., 1981. Research and development program for non-linear structural modeling with advanced time-temperature dependent constitutive relationships. Report PWA-5700-50, NASACR-165533. Wilcock, I.M., 2000. Cyclic test data for nickel base superalloy RR1000 at 300 °C and 650 °C. Private Communications, DERA. William, F.S., 1996. Principles of Materials Science and Engineering. McGraw-Hill, New York. Yang, X.J., 1997. Constitutive description of temperaturedependent nonproportional cyclic viscoplasticity. ASME Journal of Engineering Materials and Technology 119, 12± 19.