- Email: [email protected]
A viscoplastic constitutive model for nickel-base superalloy, part 2: modeling under anisothermal conditions
International Journal of Plasticity 18 (2002) 1111–1131 www.elsevier.com/locate/ijplas
A viscoplastic constitutive model for nickel-base superalloy, part 2: modeling under anisothermal conditions Masatsugu Yaguchi*, Masato Yamamoto, Takashi Ogata Central Research Institute of Electric Power Industry, 2-11-1 Iwado Kita, Komae-shi, Tokyo 201-8511, Japan Received in final revised form 3 April 2001
Abstract In Part 2 of this study, extensive deformation tests were carried out on the nickel-base polycrystalline superalloy IN738LC under isothermal and anisothermal conditions between 450 and 950 C. Under the isothermal conditions, the material showed almost no rate/time-dependency below 700 C, while it showed distinct rate/time-dependency above 800 C. Regarding the cyclic deformation, slight cyclic hardening behavior was observed when the temperature was below 700 C and the imposed strain rate was fast, whereas in the case of the temperature above 800 C or under slower strain rate conditions, the cyclic hardening behavior was scarcely observed. Unique inelastic behavior was observed under in-phase and out-of-phase anisothermal conditions: with an increase in the number of cycles, the stress at higher temperatures became smaller and the stress at lower temperatures became larger in absolute value although the stress range was approximately constant during the cyclic loading. In other words, the mean stress continues to evolve cycle-by-cycle in the direction of the stress at lower temperatures. Based on the experimental results, it was assumed that evolution of the variable Y that had been incorporated into a kinematic hardening rule in Part 1 of this study is active under higher temperatures and is negligible under lower temperatures. The material constants used in the constitutive equations were determined with the isothermal data, and were expressed as functions of temperature empirically. The extended viscoplastic constitutive equations were applied to the anisothermal cyclic loading as well as the monotonic tension, stress relaxation, creep and cyclic loading under the isothermal conditions. It was demonstrated that the present viscoplastic constitutive model was successful in describing the inelastic behavior of the material adequately, including the anomalous inelastic behavior observed under the anisothermal conditions, owing to the consideration of the variable Y. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: Constitutive behaviour; Viscoplastic material; Cyclic loading; Anisothermal condition
* Corresponding author. Tel.: +81-3-3480-2111; fax: +81-3-3430-2410. E-mail address: [email protected] (M. Yaguchi). 0749-6419/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0749-6419(01)00030-4
1112
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
1. Introduction Nickel-base superalloys have attracted much attention in the area of constitutive equations (Paslay et al., 1970, 1971; Moreno and Jordan, 1986; Benallal and Cheikh, 1987; James et al., 1987; Ramaswamy et al., 1987, 1990; Chan et al., 1988, 1989, 1990; Chan and Page, 1988; Walker and Jordan, 1989; Chan and Lindholm, 1990; Sheh and Stouffer, 1990; Slavik and Cook, 1990; Stouffer et al., 1990; Meric et aL, 1991; Meric and Cailletaud, 1991; Nouailhas and Chaboche, 1991; Jordan and Walker, 1992; Nouailhas and Freed, 1992; Ohno et al., 1992; Sherwood and Stouffer, 1992; Bhattachar and Stouffer, 1993a,b; Olschewski et al., 1993; Nouailhas and Cailletaud, 1995; Estrin et al, 1996; Sievert et al., 1997; Busso et al., 2000), because they have been widely used in the hot sections of jet engines and land gas turbines due to their excellent mechanical properties at high temperatures. In these studies, discussion was made on subjects such as initial anisotropy, plasticity–creep interaction, multiaxial deformation, temperature dependency and anisothermal conditions. However, in regard to stress relaxation behavior under cyclic deformation conditions, few studies have been reported thus far (Benallal and Cheikh, 1987; Chan et al., 1989; Ramaswamy et al., 1990; Slavik and Cook, 1990). Since creep damage is usually evaluated on the basis of calculated stress relaxation behavior in the life prediction procedure of the actual components, it must be described precisely by the constitutive equations used in finite element programs. In Part 1 of this study (Yaguchi et al., 2002), a series of deformation experiments including stress relaxation under cyclic deformation conditions was conducted on nickel-base polycrystalline superalloy IN738LC under a uniaxial stress condition at 850 C. The material exhibited anomalous inelastic behavior under the cyclic deformation conditions with tensile or compressive strain holding time; the stress–strain hysteresis loop continues to shift in a direction opposite the strain hold with an increase in the number of cycles while the stress range remains almost constant during cyclic straining. Consequently, the mean stress continues to grow to a great extent with the increasing number of cycles. Numerical calculations by conventional kinematic hardening rules revealed that the evolutionary behavior of the mean stress was difficult to express adequately because of the dynamic recovery property of the back stress. A kinematic hardening rule was proposed within the viscoplasticity framework to describe the growth behavior of the mean stress precisely. Numerical simulations were conducted for various loading conditions, and it was demonstrated that the proposed viscoplastic constitutive equations could adequately describe the inelastic behavior of the material at 850 C, including the anomalous behavior observed under the strain hold waveform conditions. From the viewpoint of the life prediction procedure, it is not sufficient to evaluate the stress relaxation behavior alone under cyclic deformation conditions at high temperature because the actual components are generally used under conditions in which temperature varies. Thus, the constitutive equations must be extended to anisothermal conditions. Thus far, constitutive modeling under the anisothermal conditions has been reported by many researchers (Walker, 1981; Chaboche, 1986; Moreno and Jordan, 1986; Benallal and Cheikh, 1987; Ramaswamy et al., 1987; Chan and Page, 1988;
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
1113
Freed, 1988; Chan et al, 1989; Ohno et al, 1989; Chan and Lindholm, 1990; Slavik and Cook, 1990; Lee and Krempl, 1991; Ohno and Wang, 1991; Wang and Ohno, 1991; McDowell, 1992; Moosbrugger, 1992; Bhattachar and Stouffer, 1993a,b; Chaboche, 1993a,b; Freed and Walker, 1993; Olschewski et al., 1993; Bouchou and Delobelle, 1996; Yaguchi and Takahashi, 2000). In this study, the inelastic behavior of IN738LC under anisothermal conditions as well as isothermal ones was investigated in detail. Then, the viscoplastic constitutive model proposed in Part 1 (Yaguchi et al., 2002) was extended to the anisothermal conditions, and its description capability was examined through various numerical simulations.
2. Inelastic behaviour under isothermal and anisothermal conditions 2.1. Experimental procedure The test material was the same as that used in Part 1 (Yaguchi et al., 2002), that is, solid round bars of diameter 22 mm and longitudinal length 200 mm fabricated by the precision casting method. The bars were subjected to heat treatment under conventional conditions for land gas turbine blades; 1195 C4 h for HIP, 1120 C2.5 h for solid solution and 843 C24 h aging. Test specimens were machined from the round bars in the longitudinal direction. The test specimens for monotonic tension and creep deformation were the collar-head round-bar type of diameter 6 mm and gauge length 30 mm. The test specimens for isothermal cyclic deformation were the round-bar type of diameter 10 mm and gauge length 10 mm. For anisothermal cyclic tests, tubular-type specimens were used with an outer diameter of 13 mm, an inner diameter of 10 mm and a gauge length of 12.5 mm. A conventional tensile testing machine and lever-operated creep testing facilities with an electric furnace were used for the monotonic tension tests and the creep deformation tests, respectively. An electromechanically driven fatigue testing machine was used for the isothermal and anisothermal cyclic deformation tests. In the fatigue equipment, heating was carried out by an induction heating device, and cooling was carried out by the flowing compressed air through the inside of the tubular specimen. Loading conditions were of four types: (a) monotonic tension and subsequent strain holding, (b) creep deformation, (c) isothermal cyclic deformation and (d) anisothermal cyclic deformation. In all cases, the tests were carried out under the uniaxial stress condition. In the case of the monotonic tension experiments, test temperatures were 450, 600, 700, 800, 900 and 950 C. The material was pulled until a tensile strain of 1.2% was reached at constant strain rates, and then the strain was held constant for 2000 s. Here, the strain rates for the tension portion were in the range between 106 and 103 s1. For the creep deformation tests, the temperatures were 800, 900 and 950 C, and the stresses were in the range between 150 and 400 MPa. Isothermal cyclic deformation tests were conducted at 450, 700, 800 and 900 C with strain control. Strain-time waveforms were fast tension/fast compression [Fast/
1114
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
Fast], slow tension/slow compression [Slow/Slow] and fast tension/fast compression with strain hold at the tensile strain peak [THI. Here, the strain rate for the fast loading was fixed at 103 s1 and that for the slow loading was equal to 104 s1. The strain hold time was 360 s, and the strain range was between 0.35 and 1.2%. Regarding the anisothermal experiments, temperature–straining relationships were in-phase when the variation of the temperature and the strain is the same phase, and out-of-phase when their variation differs by 180 , as shown in Fig. 1. The maximum and minimum temperatures during one cycle were 850 and 450 C, respectively. At the maximum temperature, the imposed strain is at the tensile peak in the case of the in-phase condition, and at the compressive peak in the case of the out-of-phase condition. The controlled strain rate was 104 s1, and the strain range was between 0.5 and 1.0%. 2.2. Experimental results Fig. 2 summarizes the results of the monotonic tension test by depicting the relationships between the temperatures (450, 600, 700, 800, 900 and 950 C) and applied stress at the tensile strain of 1.2% with the data at 850 C (Yaguchi et al., 2002), where the symbols represent the experimental results and the lines express those calculated by the constitutive equations described later. The monotonic tensile property significantly differed below 700 C and above 800 C as discussed below. Fig. 3 shows a representation of the monotonic tension curves below 700 C. The material showed almost the same response regardless of the temperature, that is, the temperature dependency of the monotonic tension is small between 450 and 700 C. In addition, the experimental results displayed approximately the same tension curves under the condition where there was a thousand times difference in the strain rate. Therefore, it can be concluded that the inelastic behavior is nearly rate-independent
Fig. 1. Temperature–strain relationships used for anisothermal tests: (a) in-phase, (b) out-of-phase.
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
1115
Fig. 2. Relationships between temperature and monotonic tensile stress at tensile strain of 1.2%.
Fig. 3. Monotonic tensile curves below 700 C: (a) 450 C, (b) 700 C.
in these temperature regions, although at 700 C there is a small negative ratedependency that the stress under the low strain rate condition (106 s1) yields a larger value than that under the high strain rate condition (103 s1).
1116
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
Fig. 4 shows a representation of the monotonic tension curves above 800 C. The monotonic tension property greatly depended on both the temperature and the strain rate. The stress became smaller with increasing the temperature or decreasing the imposed strain rate. Comparing the stress at the tensile strain of 1.2%, it decreased by approximately 200 MPa as the temperature increased by 100 C, and it decreased by approximately 300 MPa as the strain rate decreased by a thousand times. Examining the experimental results carefully, the monotonic tensile property at 800 C exhibited the highest value between 450 and 950 C for the strain rate of 103 s1, and the tensile property was maximum at 700 C in the case of the strain rate of 106 s1. That is, IN738LC shows an inverse temperature dependency of the flow stress, which is unusual for materials such as stainless and low alloy steels; however, it is usually observed in the case of 0 -strengthened nickel-base superalloys (Takeuchi and Kuramoto, 1973; Lall et al., 1979; Paidar et al., 1984; Pope and Ezz, 1984; Milligan and Antolovich, 1987; Lin and Wen, 1989). Fig. 5 summarizes the experimental results of the stress relaxation behavior after the monotonic tension of 1.2%, and Fig. 6 shows a representation of the stress relaxation curves. The stress relaxation behavior depended on both the temperature and the prior strain rate. When the temperature was above 800 C and the prior strain rate was high (103 s1), the material exhibited large stress relaxation behavior. On the other hand, when the temperature was below 700 C or the prior strain rate was low (106 s1), the stress relaxation behavior was relatively small. Fig. 7 shows examples of the creep deformation test results between 800 and 950 C. Being the same as that at 850 C (Yaguchi et al., 2002), the material showed little primary creep deformation, and the tertiary creep deformation tended to begin at relatively small strain. Fig. 8 shows the representative results of cyclic deformation tests under the isothermal conditions between 450 and 900 C. Cyclic hardening behavior was observed in the case that the temperature was below 700 C and the strain waveform was
Fig. 4. Monotonic tensile curves above 800 C: (a) 800 C, (b) 950 C.
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
1117
Fig. 5. Experimental results of stress relaxation behavior after monotonic tension of 1.2%: (a) prior strain rate: 103 s1, (b) prior strain rate: 106 s1.
Fig. 6. Stress relaxation curves after monotonic tension of 1.2%: (a) 450 C, (b) 800 C.
Fast/Fast, as shown in Fig. 8(a). Even if the temperature was below 700 C, cyclic hardening behavior was scarcely observed when the waveform was Slow/Slow or TH, as shown in Fig. 8(b) and (c), respectively. At temperatures higher than 800 C, the material exhibited negligible cyclic hardening behavior except under the very small strain range condition (e=0.35%), as shown in Fig. 8(d). As for the growth behavior of the mean stress observed under the TH conditions at 850 C (Yaguchi et al., 2002), it was not observed at the temperature below 700 C, as shown in Fig. 8(c). When the loading conditions were anisothermal cyclic types, the material exhibited unique inelastic behavior that was similar to that under TH and CH conditions at 850 C. Fig. 9 shows the cyclic response of peak stress observed under the anisothermal
1118
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
Fig. 7. Creep deformation curves: (a) 800 C, (b) 900 C.
in-phase tests. While the stress range was almost constant during cyclic loading, the maximum and minimum stresses became smaller with an increase in the number of cycles, and the mean stress continued to grow toward the compressive stress direction. In the case of the out-of-phase condition, the trend was exactly opposite; the peak stresses became larger with the increase in the number of cycles and the mean stress continued to grow toward the tensile stress direction. To the best of our knowledge, this behavior is scarcely observed on other materials such as stainless and low alloy steels. Fig. 10 shows the stress–strain hysteresis loop at the half-life under the in-phase condition. The maximum and minimum stresses differ by as much as 300 MPa in absolute value because of the evolutionary behavior of the mean stress in addition to the temperature dependency of the stress–strain response.
3. Simulations under isothermal conditions In this section, the viscoplastic constitutive model proposed in Part. 1 (Yaguchi et al., 2002) is applied to the experimental results under the isothermal conditions, and its description capability is examined. 3.1. Constitutive equations It was assumed that strain "t is divided additively into elastic part "e and inelastic part "in "t ¼ "e þ "in
ð1Þ
and that the elastic part obeys Hooke’s law "e ¼
1v v ðtrÞI E E
ð2Þ
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
1119
Fig. 8. Cyclic response of peak stress under isothermal conditions: (a) 450 C, Fast/Fast; (b) 450 C Slow/ Slow; (c) 700 C, TH; (d) 900 C, Fast/Fast.
where E and v indicate Young’s modulus and Poisson’s ratio, respectively, and I mean the stress tensor and the unit tensor of second-rank, respectively, and tr expresses trace. The inelastic strain rate is given by Eq. (3), 3 : 0 X0 : ; " in ¼ p 2 Jð XÞ
ð3Þ
where 0 and X0 mean the deviator of the stress tensor and the back stress tensor (second-rank) X, respectively. p is the accumulated inelastic strain, and is a function of J(X), as expressed by Eq. (4). J(X) is the second invariant overstress defined by Eq. (5). Jð XÞ n : ð4Þ p¼ K
1120
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
Fig. 9. Cyclic response of peak stress under anisothermal conditions.
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 0 Jð X Þ ¼ ð X 0 Þ : ð 0 X 0 Þ 2
ð5Þ
Here, K and n are material constants, and (:) means the inner product between second-rank tensors. The kinematic hardening rule is expressed by Eq. (6), m1 : 2 : : X; ð6Þ X ¼ C a" in ðX YÞp JðXÞ 3 where C, a, and m are material constants. The second-rank tensor Y is the internal variable that was proposed in Part 1 of this study (Yaguchi et al., 2002) to describe the evolutionary behavior of the mean stress under the TH and CH conditions. Eq. (6) is the same expression as the kinematic hardening rule proposed by Chaboche and Nouailhas (1989). However, the back stress in the present constitutive model exhibits completely different behavior than that exhibited by the Chaboche– Nouailhas model, because the evolutionary law of the variable Y differs between the Chaboche–Nouailhas model and the proposed model. The evolutionary law in the authors’ constitutive model was expressed as follows:
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
1121
Fig. 10. Stress–strain hysteresis loop at half-life under anisothermal conditions.
: X þ Y fJðXÞgm Y ¼ 0 Yst JðXÞ X þ Y fJðXÞgm ¼ Yst JðXÞ
ð7Þ
ð8Þ
where (=0 ) and Yst are material constants. In the evolutionary law, the driving force of the variable Y was assumed to be rate/time-dependent inelastic deformation, and the static recovery term of the back stress, {J(X)}m, was used to represent the rate/time-dependent deformation. 3.2. Material constants The proposed viscoplastic constitutive model has eight material constants, K, n, C, a, , m, Yst and , excluding the elastic constants (Young’s modulus E and Poisson’s ratio v). The material constants related to the inelastic behavior were determined for each temperature condition (450, 600, 700, 800, 900 and 950 C) in the following manner. When the temperatures were above 800 C, the determination procedure was the same as the one at 850 C described in Part 1 (Yaguchi et al., 2002), that is:
1122
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
1. Assume that Yst and are equal to zero, determine K, n, C, a, and m by trial and error so as to express the monotonic tensile response under the low strain rate that is slightly larger compared with the experimental results. 2. Determine Yst on the basis of the mean stress at the half-life under the TH conditions. 3. Control in order to express an evolutionary rate of the mean stress with respect to cycles under the TH conditions. 4. To obtain a balanced description capability, conduct adjustments of material constants based on simulations of various inelastic behaviors. For the temperature below 700 C, the authors assumed that the material constants were the same regardless of the temperature, because the temperature dependency of the monotonic tensile property was relatively small in this temperature region, as shown in Fig. 2. Furthermore, the static recovery term of the back stress was assumed to be negligible, because the monotonic tensile property was almost rate-independent and the stress relaxation behavior was very small, as shown in Figs. 2 and 5, respectively. Based on these assumptions, the remaining material constants, i.e. K, n, C and a, were determined so as to express the monotonic tension curve. Further, to make it possible to calculate the stress–strain relationship under the anisothermal conditions, material constants depending on the temperature were empirically expressed as functions of temperature from 450 C (723 K) to 950 C (1223 K). The final results are summarized in the following.
1. 2. 3.
4.
5.
6.
7.
8.
9.
E=1.971105+3.756101T5.998102T2 v=0.3 K=2.000102 =3.711104+6.60101T2.847102T2 =1.190103 n=9.996 =4.9371029.030101T+4.171104T2 =5.645 C=1.500103 =3.7131046.256101T+2.667102T2 =5.000102 a=7.000102 =2.302104+4.752101T2.379102T2 =2.9771032.340T =6.5421013+1.1921015T5.3361019T2 =2.5041011+4.4791014T1.9991017T2 =1.4991016 m=4.275 =8.8881021.569T+6.954104T2 =5.680 =6.5421014+1.1921016T5.3361020T2 =2.5041012+4.4791015T1.9991018T2
7234T41223 7234T41223 7234T4973 973< T41173 1173< T41223 7234T4973 973< T41123 1123< T41223 7234T4973 973< T41123 1123< T41223 7234T4973 973< T41123 1123< T41223 7234T41073 1073< T41173 1173< T41223 7234T41123 1123< T41173 1173< T41223 7234T41073 1073< T41173
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
10.
=1.4991017 Yst=1.000102
1123
1173< T41223 7234T41223
Here, the units for temperature, stress, strain and time are K, MPa. mm/mm and seconds, respectively. 3.3. Simulations Fig. 2 shows the calculation results of the monotonic tension property between 450 and 950 C in comparison with the experimental results, and Figs. 3 and 4 show representation of the simulations of the monotonic tension curves below 700 C and above 800 C, respectively. The constitutive model approximately describes the monotonic tensile curves below 700 C, which are nearly independent of the temperature and the strain rate, and also expresses the inelastic behavior above 800 C that greatly depends on both the temperature and the strain rate. In the case of the strain rate of 103 s1, the calculated stress showed the maximum value at approximately 750 C due to the combination of temperature-dependent material constants. This calculation result was not highly unusual because the nickel-base superalloys usually show the maximum tensile strength between 700 and 740 C (Takeuchi and Kuramoto, 1973; Lall et al., 1979; Paidar et al., 1984; Pope and Ezz, 1984; Milligan and Antolovich, 1987; Lin and Wen, 1989). Fig. 6 shows examples of predictions of the stress relaxation behavior after the monotonic tension with the corresponding experimental data. The present constitutive model simulated the large stress relaxation behavior that was observed under the condition of prior faster straining above 800 C. When the temperature was below 700 C or the prior loading strain rate was low, the calculated stress relaxation behavior was relatively small, which agreed with the experimental results. The creep deformation was also simulated with sound accuracy under all of the temperature conditions except the tertiary creep regime, as shown in Fig. 7. Fig. 8 shows simulations of the isothermal cyclic deformation. Since the proposed constitutive model includes no cyclic hardening parameter, it could not express the cyclic hardening behavior observed under the Fast/Fast waveform below 700 C, as shown in Fig. 8(a). When the imposed waveform was not the Fast/Fast or the temperature was above 800 C, both the experimental results and the calculated ones exhibit no cyclic hardening behavior and their agreements were good, although in some cases there was a difference in peak stress between the experimental and the calculated results.
4. Simulations under anisothermal conditions 4.1. Constitutive equations As for the anisothermal conditions, it is general to consider the temperature rate term in kinematic hardening rules (Walker, 1981; Freed, 1988; Ohno and Wang, 1991; Wang and Ohno, 1991; McDowell, 1992; Chaboche, 1993a,b). Considering the
1124
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
temperature rate term, the kinematic hardening rule in the proposed constitutive model is expressed as follows: : 2 : X @ðCaÞ : : T; ð9Þ X ¼ C a" in ðX YÞp fJðXÞgm1 X þ 3 Ca @T where the last term on the right-hand side in Eq. (9) is the temperature rate term. It adjusts the variable to keep the ratio of the current value to temperature-dependent asymptotic value constant under temperature variation conditions. Therefore, even if there is no inelastic deformation, the value of the back stress will change with the temperature variation. Regarding the variable Y, the temperature rate term was not introduced in its evolutionary law due to the following presumption. As the variable Y is assumed to correspond to the substructure generated by the creep deformation (Yaguchi et al., 2002), the evolution of the variable Y is thought to need time at higher temperature. Consequently, it seems not appropriate for the variable Y to evolve rate/time-independently according to the temperature rate. Thus, it was assumed that the temperature rate term was not necessary in the evolutionary law of the variable Y. Therefore, the constitutive equations for the anisothermal conditions were the same as those mentioned in the former section except for changing the kinematic hardening rule from Eqs. (6) to (9). The material constants were also the same as those given in the former section. 4.2. Simulations Fig. 9 shows simulations of the anisothermal cyclic deformation under the inphase condition, where the symbol depicts the experimental result, and the solid line represents the result calculated by the above-mentioned constitutive model. Calculation was also conducted using the constitutive model, neglecting only the variable Y, and its result is expressed by the broken line in Fig. 9. In the case of neglecting the variable Y, the stress–strain response was constant with the cycles, which does not agree with the experimental result qualitatively, On the other hand, the calculation result obtained by the constitutive model considering the variable Y revealed that the maximum and minimum stresses continued to become smaller with the increase in the number of cycles and that the mean stress continued to grow in the compressive stress direction, which is characteristic inelastic behavior of the material. Under the out-ofphase condition, the peak stress calculated by the present constitutive model shifted in the tensile stress direction with the increase in the number of cycles, which is also true for the experimental results. It is well known that it is difficult to express the shift of the stress-strain hysteresis loop under the anisothermal conditions by the conventional constitutive equations if the temperature rate term is introduced into the kinematic hardening rules (Wang and Ohno, 1991; Chaboche, 1993a,b); however, the present constitutive model considering the temperature rate term was able to simulate the inelastic behavior well due to the incorporation of the variable Y as follows.
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
1125
Fig. 11 shows an example of the behavior of the back stress and the variable Y under the in-phase condition in a uniaxial state. The variable Y evolves in the compressive stress direction during the higher temperature process, because the static recovery of the back stress occurs under tensile back stress conditions. When the temperature becomes lower, the variable Y displays no variation because the static recovery of the back stress is negligible in the lower temperature region. When the temperature again becomes higher, the variable Y evolves in the compressive direction again due to the static recovery of the tensile back stress. Therefore, the variable Y exhibits the evolutionary behavior in the compressive stress direction with the increase in the number of cycles, as shown in Fig. 12; consequently, the back stress and the applied stress shift in the same direction with the increase in the number of cycles according to the enlarged dynamic recovery property of the back stress. The variable Y was originally incorporated into the kinematic hardening rule in order to express the anomalous inelastic behavior observed under the TH and the CII conditions at high temperature. However, it was demonstrated that the variable was also required for describing the inelastic behavior under the anisothermal conditions. 4.3. Effect of temperature rate term In the former section, the authors carried out the calculations by considering the temperature rate term because it is generally used for anisothermal analyses (Walker,
Fig. 11. Calculation results of back stress and variable Y under anisothermal condition by extended constitutive model.
1126
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
Fig. 12. Cyclic response of peak stress and variable Y under anisothermal condition by extended constitutive model.
1981; Freed, 1988; Ohno and Wang, 1991; Wang and Ohno, 1991; McDowell, 1992; Moosbrugger, 1992; Chaboche, 1993a,b). However, there have been some research results which indicate that the temperature rate term is not always necessary for describing the inelastic behavior under the anisothermal conditions (Bhattachar and Stouffer, 1993a; Delobelle and Bouchou, 1996). In this section, the authors discuss the effect of the temperature rate term on the description capability under the anisothermal conditions. One of the reasons for considering the temperature rate term is that without it the shift of the stress–strain hysteresis loop may occur under the anisothermal cyclic straining due to linear or quasilinear kinematic hardening components (Wang and Ohno, 1991; Chaboche, 1993a,b). The shift of the stress-strain loop with cycles is not ordinary for materials such as stainless and low alloy steels; however, the unusual behavior was observed in IN738LC in the present study. Therefore, it was examined whether or not the inelastic behavior can be adequately expressed by the constitutive model without considering the temperature rate term. The constitutive equations and material constants were the same as those used in Section 4.2, except here we neglect the temperature rate term and the variable Y. Fig. 13 shows the peak stress calculated under the in-phase condition (450850 C), where the strain rate was 104 s1 and the strain ranges were 0.3, 0.5 and 0.7%. The shift of the stress-strain loop with cycles depended on the strain range. In
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
1127
Fig. 13. Calculation results of anisothermal cyclic deformation without temperature rate term and variable Y.
the case of the strain range of 0.3%, the calculation result exhibited a pronounced shift of the stress–strain response with the increase in the number of cycles; however, the shift was scarcely observed when the strain range was 0.7%. The reason for the absence of the shift was thought to be that the nonlinearity of the kinematic hardening variable becomes great under the strain range condition. Examining the experimental results of IN738LC, the shift of the stress-strain hysteresis loop was observed under the strain range of 0.7%, as shown in Fig. 9. Therefore, it seems that it is difficult to describe the shift of the stress-strain response with sound accuracy solely by neglecting the temperature rate term in the kinematic hardening rules. Another characteristic of the constitutive equations without the temperature rate term is that it expresses the stress as larger in absolute value than that by constitutive equations considering the temperature rate term during the increasing process of the temperature, and it expresses the stress as smaller during the decreasing process of the temperature. In order to examine the effect of this on description accuracy, a simulation was conducted for the in-phase condition (450–850 C) where the strain rate and the strain range were 104 s1 and 0.7%, respectively, Fig. 10 shows the stress–strain hysteresis loop at the 900th cycle; here, the symbol means the
1128
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
experimental result, and the solid line express the calculation result obtained using the constitutive model previously mentioned in Section 4.2, and the broken line represents that obtained using the constitutive model neglecting only the temperature rate teim. In the lower temperature region (compressive strain side) there was little difference in stress–strain response between the constitutive models with and without the temperature rate term; however, some differences were observed in the higher temperature region (tensile strain side). Compared with the experimental result, the calculated result with the temperature rate term showed better agreement than the calculated result without the temperature rate term. Judging from these results, it was considered appropriate to introduce the temperature rate term into the kinematic hardening rule in the present viscoplastic constitutive model.
5. Conclusions In this paper, the inelastic behavior of nickel-base superalloy IN738LC was investigated under isothermal and anisothermal conditions between 450 and 950 C. Based on the experimental results, the viscoplastic constitutive model proposed in Part 1 was extended to the anisothermal loading conditions. The extended constitutive model was applied to the anisothermal test results as well as to the isothermal ones, and its description capability was evaluated. A summary of the present study is given below. 1. Under the isothermal conditions above 800 C, the material exhibited both temperature- and rate/time-dependent deformation behavior, whereas inelastic behavior was almost independent of the temperature and the loading rate/time under the conditions below 700 C. As for the isothermal cyclic deformation, the material exhibited slight cyclic hardening behavior when the temperature was below 700 C and the imposed strain rate was high. On the other hand, the cyclic hardening behavior was scarcely observed in the case of the temperature above 800 C or under slower strain rate conditions. 2. Anomalous inelastic behavior was observed under the anisothermal conditions of in-phase and out-of-phase types. With an increase in the number of cycles, stress at higher temperature becomes smaller and stress at lower temperature becomes larger in absolute value, although the stress range is almost constant with the number of cycles. That is, the mean stress continues to evolve cycle-by-cycle in the direction of the stress at the lower temperature. 3. The proposed viscoplastic constitutive model for IN738LC at 850 C was extended to the anisothermal conditions. Based on the experimental results, the evolution of the variable Y was assumed to be active under higher temperatures and negligible under lower temperatures. Material constants were determined with isothermal data, and were empirically expressed as functions of temperature. 4. The extended viscoplastic constitutive model was applied to the anisothermal cyclic loading as well as monotonic tension, stress relaxation, creep and cyclic loading under the isothermal conditions. It was demonstrated that the present visco-
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
1129
plastic constitutive model was successful in describing the inelastic behavior of the material adequately, including the anomalous inelastic behavior observed under the anisothermal conditions as a result of consideration of the variable Y.
References Benallal, A., Cheikh, A.B., 1987. Constitutive equations for anisothermal elasto viscoplasticity. Int. Conf. on Const. Laws for Eng. Mat., 667-674. Bhattachar, V.S., Stouffer, D.S., 1993a. Constitutive equations for the thermomechanical response of Rene 80: part 1— development from isothermal data. ASME J. Eng. Mat. Tech. 115, 351–357. Bhattachar, V.S., Stouffer, D.S., 1993b. Constitutive equations for the thermomechanical response of Rene 80: part 2—effects of temperature history. ASME J. Eng. Mat. Tech. 115, 358–364. Bouchou, A., Delobelle, P., 1996. Behaviour and modelization of a 17-12SPH stainless steel under cyclic, unidirectional and bi-directional anisothermal loadings. Nuc. Eng. Design 162, 21–45. Busso, E.P., Meissonnier, F.T., O’Dowd, N.P., 2000. Gradient-dependent deformation of two-phase single crystals. J. Mech. Phys. Solids 48, 2333–2361. Chaboche, J.L., 1986. Time-independent constitutive theories for cyclic plasticity. Int. J. Plasticity 2, 149– 188. Chaboche, J.L., 1993a. Cyclic viscoplastic constitutive equations, part 1: a thermodynamically consistent formulation. ASME J. Appl. Mech. 60, 813–821. Chaboche, J.L., 1993b. Cyclic viscoplastic constitutive equations, part 2: stored energy—comparison between models and experiments. ASME J. Appi. Mech. 60, 822–828. Chaboche, J.L., Nouailhas, D., 1989. Constitutive modeling of ratchetting effects— part 2: possibilities of some additional kinematic rules. ASME J. Eng. Mat. Tech. 111, 409–416. Chan, K.S., Bodner, S.R., Lindholm, U.S., 1988. Phenomenological modeling of hardening and thermal recovery in metals. ASME J. Eng. Mat. Tech. 110, 1–8. Chan, K.S., Lindholm, U.S., 1990. Inelastic deformation under nonisothermal loading. ASME J. Eng. Mat. Tech. 112, 15–25. Chan, K.S., Lindholm, U.S., Bodner, S.R., Nagy, A., 1990. High temperature inelastic deformation of the B1900+Hf alloy under multiaxial loading: theory and experiment. ASME J. Eng. Mat. Tech. 112, 7– 14. Chan, K.S., Lindholm, U.S., Bodner, S.R., Walker, K.P., 1989. High temperature inelastic deformation under uniaxial loading: theory and experiment. ASME J. Eng. Mat. Tech. 111, 345–353. Chan, K.S., Page, R.A., 1988. Inelastic deformation and dislocation structure of a nickel alloy: effects of deformation and thermal histories. Met. Trans. A. 19A, 2477–2486. Delobelle, P., Bouchou, A., 1996. Behavior and modeling of a 1712 SPH stainless steel under cylic, uni and bi-directional, part 2: modeling. Int. J. Plasticity 12, 311–341. Estrin, Y, Braasch, H., Brechet, Y., 1996. A dislocation density based constitutive model for cyclic deformation. ASME J. Eng. Mat. Tech. 118, 441–447. Freed,A.D., 1988. Thermoviscoplastic Model with Application to Copper. NASA Report TP-2845, NASA. Freed, A.D., Walker, K.P., 1993. Viscoplasticity with creep and plasticity bounds. Int. J. Plasticity 9, 213– 242. James, G.H., Imbrie, P.K., Hill, P.S., Allen, D.H., Haisler, W.E., 1987. An experimental comparison of several current viscoplastic constitutive models at elevated temperature. ASME J. Eng. Mat. Tech. 109, 130–139. Jordan, E.H., Walker, K.P., 1992. A viscoplastic model for single crystals. ASME J. Eng. Mat. Tech. 114, 19–26. Lall, C., Chin, S., Pope, D.P., 1979. The orientation and temperature dependence of the yield stress of Ni3(Al,Nb) single crystals. Met. Trans. A 10A, 1323–1332.
1130
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
Lee, K.D., Krempl, E., 1991. An orthotropic theory of viscoplasticity based on overstress for thermomechanical deformation. Int. J. Solids Struct. 27, 1445–1459. Lin, T.L., Wen, M., 1989. Dislocation structure due to high temperature deformation in the 20 phase of a nickel-based superalloy. Mat. Sci. Eng. A113, 207–214. McDowell, D.L., 1992. A nonlinear kinematic hardening theory for cyclic thermoplasticity and thermoviscoplasticity. Int. J. Plasticity 8, 695–728. Meric, L., Cailletaud, G., 1991. Single crystal modeling for structural calculations: part 2—finite element implementation. ASME Eng. Mat. Tech. 113, 171–182. Meric, L., Poubanne, P., Cailletaud, G., 1991. Single crystal modeling for structural calculations: part 1— model presentation. ASME Eng. Mat. Tech. 113, 162–170. Milligan, W.W., Antolovich, S.D., 1978. Yielding and deformation behavior of the single crystal superalloy PWA 1480. Met. Trans. A 18A, 85–95. Moosbrugger, J.C., 1992. Nonisothermal constitutive model for the small strain behavior of 9Cr–Mo–V– Nb pressure vessel steel. ASME J. Eng. Mat. Tech. 114, 354–361. Moreno, V., Jordan, E.H., 1986. Prediction of material thermomechanical response with a unified viscoplastic constitutive model. Int. J. Plasticity 2, 223–245. Nouailhas, D., Cailletaud, G., 1995. Tension–torsion behavior of single-crystal superalloys experiment and finite element analysis. Int. J. Plasticity 11, 451–470. Nouailhas, D., Chaboche, J.L., 1991. Anisotropic constitutive modeling for single-crystal superalloys using a continuum phenomenological approach. In Desai, C.S., Krempl, E., Frantziskonis, G., Saadatmanesh, H. (Eds). 3rd Int. Conf. On Const. Laws for Eng. Mat. ASME Piess, pp. 213–218. Nouailhas, D., Freed, A.D., 1992. A viscoplastic theory for anisotropic materials. ASME J. Eng. Mat. Tech. 114, 97–104. Ohno, N., Mizuno, T., Kawaji, H., Okada, I., 1992. Multiaxial creep of a nickel-base directionally solidified alloy: anisotropy and simulation. Acta Metall. Mater. 40, 559–567. Ohno, N., Takahashi, Y., Kuwabara, K., 1989. Constitutive modeling of anisothermal cyclic plasticity of 304 stainless steel. ASME J. Eng. Mat. Tech. 111, 106–114. Ohno, N., Wang, J.D., 1991. Transformation of a nonlinear kinematic hardening rule to a multisurface form under isothermal and nonisothermal conditions. Int. J. Plasticity 7, 879–891. Olschewski, J., Sievert, R., Qi, W., Bertram, A., 1993. Prediction of inelastic response of a Ni-based superalloy under thermal-mechanical cyclic loading. In: Kussmaul, K. (Ed.), Structural Mechanics in Reactor Technology (SMiRT- 12), Vol. L. Elsevier Science, 17–22. Paidar, V., Pope, D.P., Vitek, V., 1984. A theory of the anomalous yield behavior in L12 ordered alloys. Acta Metall. 32, 435–448. Paslay, P.R., Wells, C.H., Leverant, G.R., 1970. An analysis of primary creep of nickel-base superalloy single crystals. ASME J. Appl. Mech., 759–764. Paslay, P.R., Wells, C.H., Leverant, G.R., Burck, L.H., 1971. Creep of single crystal nickelbase superalloy tubes under biaxial tension. ASME J. Appi. Mech., 623–626. Pope, D.P., Ezz, S.S., 1984. Mechanical properties of Ni3Al and nickel-base alloys with high volume fraction of 20 . Int. Metals Reviews 29, 136–167. Ramaswamy, V.G., Stouffer, D.C., Laflen, J.H., 1990. A unified constitutive model for the inelastic uniaxial response of Rene’80 at temperatures between 538 C and 982 C. ASME J. Eng. Mat. Tech. 112, 280–286. Ramaswamy, V.G., Stouffer, D.C., VanStone, R.H., Laflen, J.H., 1987. Modeling thermomechanical cyclic response with a unified state variable constitutive equation. ASME PVP 123, 57–64. Sheh, M.Y., Stouffer, D.C., 1990. A crystallographic model for the tensile and fatigue response for Rene N4 at 982 C. ASME J. Applied Mechanics 57, 25–31. Sherwood, J.A., Stouffer, D.C., 1992. A phenomenologically based constitutive model for Rene 95. ASME J. Eng. Mat. Tech. 114, 340–347. Sievert, R., Osterle, W., Olschewski, J., Ziebs, J., 1997. Physical interpretation of a viscoplastic model applied to high temperature performance of nickel-base superalloy. Z. Metallkd 88, 416–424. Slavik, D., Cook, T.S., 1990. A unified constitutive model for superalloys. Int. J. Plasticity 6, 651–664. Stouffer, D.C., Ramaswamy, V.G., Laflen, J.H., Stone, R.H.V., Williams, R., 1990. A constitutive model
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
1131
for the inelastic multiaxial response of Rene’80 at 871C and 982C. ASME J. Eng. Mat. Tech. 112, 241– 246. Takeuchi, S., Kuramoto, E., 1973. Temperature and orientation dependence of the yield stress in Ni3Ga single crystals. Acta metall. 21, 415–425. Walker, K.P., Jordan, E.H., 1989. Biaxial constitutive modeling and testing of a single crystal superalloy at elevated temperatures. Biaxial and Multiaxial Fatigue EGF 3, 145–170. Walker K.P., 1981. Research and Development Program for Nonlinear Structural Modeling with Advanced Time–temperature Dependent Constitutive Relationships. NASA Report CR- 165533. NASA. Wang, J.D., Ohno, N., 1991. Two equivalent forms of nonlinear kinematic hardening:application to nonisothermal plasticity. Int. J. Plasticity 7, 637–650. Yaguchi, M., Takahashi, Y., 2000. A viscoplastic constitutive model incorporating dynamic strain aging effect during cyclic deformation conditions. Int. J. Plasticity 16, 241–262. Yaguchi,M., Yamamoto,M., Ogata,T. 2002. A viscoplastic constitutive model for nickelbase superalloy, part 1: kinematic hardening rule of anisotropic dynamic recovery. Int. J. Plasticity, 18, 1083–1109.
A viscoplastic constitutive model for nickel-base superalloy, part 2: modeling under anisothermal conditions Masatsugu Yaguchi*, Masato Yamamoto, Takashi Ogata Central Research Institute of Electric Power Industry, 2-11-1 Iwado Kita, Komae-shi, Tokyo 201-8511, Japan Received in final revised form 3 April 2001
Abstract In Part 2 of this study, extensive deformation tests were carried out on the nickel-base polycrystalline superalloy IN738LC under isothermal and anisothermal conditions between 450 and 950 C. Under the isothermal conditions, the material showed almost no rate/time-dependency below 700 C, while it showed distinct rate/time-dependency above 800 C. Regarding the cyclic deformation, slight cyclic hardening behavior was observed when the temperature was below 700 C and the imposed strain rate was fast, whereas in the case of the temperature above 800 C or under slower strain rate conditions, the cyclic hardening behavior was scarcely observed. Unique inelastic behavior was observed under in-phase and out-of-phase anisothermal conditions: with an increase in the number of cycles, the stress at higher temperatures became smaller and the stress at lower temperatures became larger in absolute value although the stress range was approximately constant during the cyclic loading. In other words, the mean stress continues to evolve cycle-by-cycle in the direction of the stress at lower temperatures. Based on the experimental results, it was assumed that evolution of the variable Y that had been incorporated into a kinematic hardening rule in Part 1 of this study is active under higher temperatures and is negligible under lower temperatures. The material constants used in the constitutive equations were determined with the isothermal data, and were expressed as functions of temperature empirically. The extended viscoplastic constitutive equations were applied to the anisothermal cyclic loading as well as the monotonic tension, stress relaxation, creep and cyclic loading under the isothermal conditions. It was demonstrated that the present viscoplastic constitutive model was successful in describing the inelastic behavior of the material adequately, including the anomalous inelastic behavior observed under the anisothermal conditions, owing to the consideration of the variable Y. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: Constitutive behaviour; Viscoplastic material; Cyclic loading; Anisothermal condition
* Corresponding author. Tel.: +81-3-3480-2111; fax: +81-3-3430-2410. E-mail address: [email protected] (M. Yaguchi). 0749-6419/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0749-6419(01)00030-4
1112
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
1. Introduction Nickel-base superalloys have attracted much attention in the area of constitutive equations (Paslay et al., 1970, 1971; Moreno and Jordan, 1986; Benallal and Cheikh, 1987; James et al., 1987; Ramaswamy et al., 1987, 1990; Chan et al., 1988, 1989, 1990; Chan and Page, 1988; Walker and Jordan, 1989; Chan and Lindholm, 1990; Sheh and Stouffer, 1990; Slavik and Cook, 1990; Stouffer et al., 1990; Meric et aL, 1991; Meric and Cailletaud, 1991; Nouailhas and Chaboche, 1991; Jordan and Walker, 1992; Nouailhas and Freed, 1992; Ohno et al., 1992; Sherwood and Stouffer, 1992; Bhattachar and Stouffer, 1993a,b; Olschewski et al., 1993; Nouailhas and Cailletaud, 1995; Estrin et al, 1996; Sievert et al., 1997; Busso et al., 2000), because they have been widely used in the hot sections of jet engines and land gas turbines due to their excellent mechanical properties at high temperatures. In these studies, discussion was made on subjects such as initial anisotropy, plasticity–creep interaction, multiaxial deformation, temperature dependency and anisothermal conditions. However, in regard to stress relaxation behavior under cyclic deformation conditions, few studies have been reported thus far (Benallal and Cheikh, 1987; Chan et al., 1989; Ramaswamy et al., 1990; Slavik and Cook, 1990). Since creep damage is usually evaluated on the basis of calculated stress relaxation behavior in the life prediction procedure of the actual components, it must be described precisely by the constitutive equations used in finite element programs. In Part 1 of this study (Yaguchi et al., 2002), a series of deformation experiments including stress relaxation under cyclic deformation conditions was conducted on nickel-base polycrystalline superalloy IN738LC under a uniaxial stress condition at 850 C. The material exhibited anomalous inelastic behavior under the cyclic deformation conditions with tensile or compressive strain holding time; the stress–strain hysteresis loop continues to shift in a direction opposite the strain hold with an increase in the number of cycles while the stress range remains almost constant during cyclic straining. Consequently, the mean stress continues to grow to a great extent with the increasing number of cycles. Numerical calculations by conventional kinematic hardening rules revealed that the evolutionary behavior of the mean stress was difficult to express adequately because of the dynamic recovery property of the back stress. A kinematic hardening rule was proposed within the viscoplasticity framework to describe the growth behavior of the mean stress precisely. Numerical simulations were conducted for various loading conditions, and it was demonstrated that the proposed viscoplastic constitutive equations could adequately describe the inelastic behavior of the material at 850 C, including the anomalous behavior observed under the strain hold waveform conditions. From the viewpoint of the life prediction procedure, it is not sufficient to evaluate the stress relaxation behavior alone under cyclic deformation conditions at high temperature because the actual components are generally used under conditions in which temperature varies. Thus, the constitutive equations must be extended to anisothermal conditions. Thus far, constitutive modeling under the anisothermal conditions has been reported by many researchers (Walker, 1981; Chaboche, 1986; Moreno and Jordan, 1986; Benallal and Cheikh, 1987; Ramaswamy et al., 1987; Chan and Page, 1988;
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
1113
Freed, 1988; Chan et al, 1989; Ohno et al, 1989; Chan and Lindholm, 1990; Slavik and Cook, 1990; Lee and Krempl, 1991; Ohno and Wang, 1991; Wang and Ohno, 1991; McDowell, 1992; Moosbrugger, 1992; Bhattachar and Stouffer, 1993a,b; Chaboche, 1993a,b; Freed and Walker, 1993; Olschewski et al., 1993; Bouchou and Delobelle, 1996; Yaguchi and Takahashi, 2000). In this study, the inelastic behavior of IN738LC under anisothermal conditions as well as isothermal ones was investigated in detail. Then, the viscoplastic constitutive model proposed in Part 1 (Yaguchi et al., 2002) was extended to the anisothermal conditions, and its description capability was examined through various numerical simulations.
2. Inelastic behaviour under isothermal and anisothermal conditions 2.1. Experimental procedure The test material was the same as that used in Part 1 (Yaguchi et al., 2002), that is, solid round bars of diameter 22 mm and longitudinal length 200 mm fabricated by the precision casting method. The bars were subjected to heat treatment under conventional conditions for land gas turbine blades; 1195 C4 h for HIP, 1120 C2.5 h for solid solution and 843 C24 h aging. Test specimens were machined from the round bars in the longitudinal direction. The test specimens for monotonic tension and creep deformation were the collar-head round-bar type of diameter 6 mm and gauge length 30 mm. The test specimens for isothermal cyclic deformation were the round-bar type of diameter 10 mm and gauge length 10 mm. For anisothermal cyclic tests, tubular-type specimens were used with an outer diameter of 13 mm, an inner diameter of 10 mm and a gauge length of 12.5 mm. A conventional tensile testing machine and lever-operated creep testing facilities with an electric furnace were used for the monotonic tension tests and the creep deformation tests, respectively. An electromechanically driven fatigue testing machine was used for the isothermal and anisothermal cyclic deformation tests. In the fatigue equipment, heating was carried out by an induction heating device, and cooling was carried out by the flowing compressed air through the inside of the tubular specimen. Loading conditions were of four types: (a) monotonic tension and subsequent strain holding, (b) creep deformation, (c) isothermal cyclic deformation and (d) anisothermal cyclic deformation. In all cases, the tests were carried out under the uniaxial stress condition. In the case of the monotonic tension experiments, test temperatures were 450, 600, 700, 800, 900 and 950 C. The material was pulled until a tensile strain of 1.2% was reached at constant strain rates, and then the strain was held constant for 2000 s. Here, the strain rates for the tension portion were in the range between 106 and 103 s1. For the creep deformation tests, the temperatures were 800, 900 and 950 C, and the stresses were in the range between 150 and 400 MPa. Isothermal cyclic deformation tests were conducted at 450, 700, 800 and 900 C with strain control. Strain-time waveforms were fast tension/fast compression [Fast/
1114
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
Fast], slow tension/slow compression [Slow/Slow] and fast tension/fast compression with strain hold at the tensile strain peak [THI. Here, the strain rate for the fast loading was fixed at 103 s1 and that for the slow loading was equal to 104 s1. The strain hold time was 360 s, and the strain range was between 0.35 and 1.2%. Regarding the anisothermal experiments, temperature–straining relationships were in-phase when the variation of the temperature and the strain is the same phase, and out-of-phase when their variation differs by 180 , as shown in Fig. 1. The maximum and minimum temperatures during one cycle were 850 and 450 C, respectively. At the maximum temperature, the imposed strain is at the tensile peak in the case of the in-phase condition, and at the compressive peak in the case of the out-of-phase condition. The controlled strain rate was 104 s1, and the strain range was between 0.5 and 1.0%. 2.2. Experimental results Fig. 2 summarizes the results of the monotonic tension test by depicting the relationships between the temperatures (450, 600, 700, 800, 900 and 950 C) and applied stress at the tensile strain of 1.2% with the data at 850 C (Yaguchi et al., 2002), where the symbols represent the experimental results and the lines express those calculated by the constitutive equations described later. The monotonic tensile property significantly differed below 700 C and above 800 C as discussed below. Fig. 3 shows a representation of the monotonic tension curves below 700 C. The material showed almost the same response regardless of the temperature, that is, the temperature dependency of the monotonic tension is small between 450 and 700 C. In addition, the experimental results displayed approximately the same tension curves under the condition where there was a thousand times difference in the strain rate. Therefore, it can be concluded that the inelastic behavior is nearly rate-independent
Fig. 1. Temperature–strain relationships used for anisothermal tests: (a) in-phase, (b) out-of-phase.
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
1115
Fig. 2. Relationships between temperature and monotonic tensile stress at tensile strain of 1.2%.
Fig. 3. Monotonic tensile curves below 700 C: (a) 450 C, (b) 700 C.
in these temperature regions, although at 700 C there is a small negative ratedependency that the stress under the low strain rate condition (106 s1) yields a larger value than that under the high strain rate condition (103 s1).
1116
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
Fig. 4 shows a representation of the monotonic tension curves above 800 C. The monotonic tension property greatly depended on both the temperature and the strain rate. The stress became smaller with increasing the temperature or decreasing the imposed strain rate. Comparing the stress at the tensile strain of 1.2%, it decreased by approximately 200 MPa as the temperature increased by 100 C, and it decreased by approximately 300 MPa as the strain rate decreased by a thousand times. Examining the experimental results carefully, the monotonic tensile property at 800 C exhibited the highest value between 450 and 950 C for the strain rate of 103 s1, and the tensile property was maximum at 700 C in the case of the strain rate of 106 s1. That is, IN738LC shows an inverse temperature dependency of the flow stress, which is unusual for materials such as stainless and low alloy steels; however, it is usually observed in the case of 0 -strengthened nickel-base superalloys (Takeuchi and Kuramoto, 1973; Lall et al., 1979; Paidar et al., 1984; Pope and Ezz, 1984; Milligan and Antolovich, 1987; Lin and Wen, 1989). Fig. 5 summarizes the experimental results of the stress relaxation behavior after the monotonic tension of 1.2%, and Fig. 6 shows a representation of the stress relaxation curves. The stress relaxation behavior depended on both the temperature and the prior strain rate. When the temperature was above 800 C and the prior strain rate was high (103 s1), the material exhibited large stress relaxation behavior. On the other hand, when the temperature was below 700 C or the prior strain rate was low (106 s1), the stress relaxation behavior was relatively small. Fig. 7 shows examples of the creep deformation test results between 800 and 950 C. Being the same as that at 850 C (Yaguchi et al., 2002), the material showed little primary creep deformation, and the tertiary creep deformation tended to begin at relatively small strain. Fig. 8 shows the representative results of cyclic deformation tests under the isothermal conditions between 450 and 900 C. Cyclic hardening behavior was observed in the case that the temperature was below 700 C and the strain waveform was
Fig. 4. Monotonic tensile curves above 800 C: (a) 800 C, (b) 950 C.
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
1117
Fig. 5. Experimental results of stress relaxation behavior after monotonic tension of 1.2%: (a) prior strain rate: 103 s1, (b) prior strain rate: 106 s1.
Fig. 6. Stress relaxation curves after monotonic tension of 1.2%: (a) 450 C, (b) 800 C.
Fast/Fast, as shown in Fig. 8(a). Even if the temperature was below 700 C, cyclic hardening behavior was scarcely observed when the waveform was Slow/Slow or TH, as shown in Fig. 8(b) and (c), respectively. At temperatures higher than 800 C, the material exhibited negligible cyclic hardening behavior except under the very small strain range condition (e=0.35%), as shown in Fig. 8(d). As for the growth behavior of the mean stress observed under the TH conditions at 850 C (Yaguchi et al., 2002), it was not observed at the temperature below 700 C, as shown in Fig. 8(c). When the loading conditions were anisothermal cyclic types, the material exhibited unique inelastic behavior that was similar to that under TH and CH conditions at 850 C. Fig. 9 shows the cyclic response of peak stress observed under the anisothermal
1118
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
Fig. 7. Creep deformation curves: (a) 800 C, (b) 900 C.
in-phase tests. While the stress range was almost constant during cyclic loading, the maximum and minimum stresses became smaller with an increase in the number of cycles, and the mean stress continued to grow toward the compressive stress direction. In the case of the out-of-phase condition, the trend was exactly opposite; the peak stresses became larger with the increase in the number of cycles and the mean stress continued to grow toward the tensile stress direction. To the best of our knowledge, this behavior is scarcely observed on other materials such as stainless and low alloy steels. Fig. 10 shows the stress–strain hysteresis loop at the half-life under the in-phase condition. The maximum and minimum stresses differ by as much as 300 MPa in absolute value because of the evolutionary behavior of the mean stress in addition to the temperature dependency of the stress–strain response.
3. Simulations under isothermal conditions In this section, the viscoplastic constitutive model proposed in Part. 1 (Yaguchi et al., 2002) is applied to the experimental results under the isothermal conditions, and its description capability is examined. 3.1. Constitutive equations It was assumed that strain "t is divided additively into elastic part "e and inelastic part "in "t ¼ "e þ "in
ð1Þ
and that the elastic part obeys Hooke’s law "e ¼
1v v ðtrÞI E E
ð2Þ
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
1119
Fig. 8. Cyclic response of peak stress under isothermal conditions: (a) 450 C, Fast/Fast; (b) 450 C Slow/ Slow; (c) 700 C, TH; (d) 900 C, Fast/Fast.
where E and v indicate Young’s modulus and Poisson’s ratio, respectively, and I mean the stress tensor and the unit tensor of second-rank, respectively, and tr expresses trace. The inelastic strain rate is given by Eq. (3), 3 : 0 X0 : ; " in ¼ p 2 Jð XÞ
ð3Þ
where 0 and X0 mean the deviator of the stress tensor and the back stress tensor (second-rank) X, respectively. p is the accumulated inelastic strain, and is a function of J(X), as expressed by Eq. (4). J(X) is the second invariant overstress defined by Eq. (5). Jð XÞ n : ð4Þ p¼ K
1120
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
Fig. 9. Cyclic response of peak stress under anisothermal conditions.
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 0 Jð X Þ ¼ ð X 0 Þ : ð 0 X 0 Þ 2
ð5Þ
Here, K and n are material constants, and (:) means the inner product between second-rank tensors. The kinematic hardening rule is expressed by Eq. (6), m1 : 2 : : X; ð6Þ X ¼ C a" in ðX YÞp JðXÞ 3 where C, a, and m are material constants. The second-rank tensor Y is the internal variable that was proposed in Part 1 of this study (Yaguchi et al., 2002) to describe the evolutionary behavior of the mean stress under the TH and CH conditions. Eq. (6) is the same expression as the kinematic hardening rule proposed by Chaboche and Nouailhas (1989). However, the back stress in the present constitutive model exhibits completely different behavior than that exhibited by the Chaboche– Nouailhas model, because the evolutionary law of the variable Y differs between the Chaboche–Nouailhas model and the proposed model. The evolutionary law in the authors’ constitutive model was expressed as follows:
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
1121
Fig. 10. Stress–strain hysteresis loop at half-life under anisothermal conditions.
: X þ Y fJðXÞgm Y ¼ 0 Yst JðXÞ X þ Y fJðXÞgm ¼ Yst JðXÞ
ð7Þ
ð8Þ
where (=0 ) and Yst are material constants. In the evolutionary law, the driving force of the variable Y was assumed to be rate/time-dependent inelastic deformation, and the static recovery term of the back stress, {J(X)}m, was used to represent the rate/time-dependent deformation. 3.2. Material constants The proposed viscoplastic constitutive model has eight material constants, K, n, C, a, , m, Yst and , excluding the elastic constants (Young’s modulus E and Poisson’s ratio v). The material constants related to the inelastic behavior were determined for each temperature condition (450, 600, 700, 800, 900 and 950 C) in the following manner. When the temperatures were above 800 C, the determination procedure was the same as the one at 850 C described in Part 1 (Yaguchi et al., 2002), that is:
1122
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
1. Assume that Yst and are equal to zero, determine K, n, C, a, and m by trial and error so as to express the monotonic tensile response under the low strain rate that is slightly larger compared with the experimental results. 2. Determine Yst on the basis of the mean stress at the half-life under the TH conditions. 3. Control in order to express an evolutionary rate of the mean stress with respect to cycles under the TH conditions. 4. To obtain a balanced description capability, conduct adjustments of material constants based on simulations of various inelastic behaviors. For the temperature below 700 C, the authors assumed that the material constants were the same regardless of the temperature, because the temperature dependency of the monotonic tensile property was relatively small in this temperature region, as shown in Fig. 2. Furthermore, the static recovery term of the back stress was assumed to be negligible, because the monotonic tensile property was almost rate-independent and the stress relaxation behavior was very small, as shown in Figs. 2 and 5, respectively. Based on these assumptions, the remaining material constants, i.e. K, n, C and a, were determined so as to express the monotonic tension curve. Further, to make it possible to calculate the stress–strain relationship under the anisothermal conditions, material constants depending on the temperature were empirically expressed as functions of temperature from 450 C (723 K) to 950 C (1223 K). The final results are summarized in the following.
1. 2. 3.
4.
5.
6.
7.
8.
9.
E=1.971105+3.756101T5.998102T2 v=0.3 K=2.000102 =3.711104+6.60101T2.847102T2 =1.190103 n=9.996 =4.9371029.030101T+4.171104T2 =5.645 C=1.500103 =3.7131046.256101T+2.667102T2 =5.000102 a=7.000102 =2.302104+4.752101T2.379102T2 =2.9771032.340T =6.5421013+1.1921015T5.3361019T2 =2.5041011+4.4791014T1.9991017T2 =1.4991016 m=4.275 =8.8881021.569T+6.954104T2 =5.680 =6.5421014+1.1921016T5.3361020T2 =2.5041012+4.4791015T1.9991018T2
7234T41223 7234T41223 7234T4973 973< T41173 1173< T41223 7234T4973 973< T41123 1123< T41223 7234T4973 973< T41123 1123< T41223 7234T4973 973< T41123 1123< T41223 7234T41073 1073< T41173 1173< T41223 7234T41123 1123< T41173 1173< T41223 7234T41073 1073< T41173
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
10.
=1.4991017 Yst=1.000102
1123
1173< T41223 7234T41223
Here, the units for temperature, stress, strain and time are K, MPa. mm/mm and seconds, respectively. 3.3. Simulations Fig. 2 shows the calculation results of the monotonic tension property between 450 and 950 C in comparison with the experimental results, and Figs. 3 and 4 show representation of the simulations of the monotonic tension curves below 700 C and above 800 C, respectively. The constitutive model approximately describes the monotonic tensile curves below 700 C, which are nearly independent of the temperature and the strain rate, and also expresses the inelastic behavior above 800 C that greatly depends on both the temperature and the strain rate. In the case of the strain rate of 103 s1, the calculated stress showed the maximum value at approximately 750 C due to the combination of temperature-dependent material constants. This calculation result was not highly unusual because the nickel-base superalloys usually show the maximum tensile strength between 700 and 740 C (Takeuchi and Kuramoto, 1973; Lall et al., 1979; Paidar et al., 1984; Pope and Ezz, 1984; Milligan and Antolovich, 1987; Lin and Wen, 1989). Fig. 6 shows examples of predictions of the stress relaxation behavior after the monotonic tension with the corresponding experimental data. The present constitutive model simulated the large stress relaxation behavior that was observed under the condition of prior faster straining above 800 C. When the temperature was below 700 C or the prior loading strain rate was low, the calculated stress relaxation behavior was relatively small, which agreed with the experimental results. The creep deformation was also simulated with sound accuracy under all of the temperature conditions except the tertiary creep regime, as shown in Fig. 7. Fig. 8 shows simulations of the isothermal cyclic deformation. Since the proposed constitutive model includes no cyclic hardening parameter, it could not express the cyclic hardening behavior observed under the Fast/Fast waveform below 700 C, as shown in Fig. 8(a). When the imposed waveform was not the Fast/Fast or the temperature was above 800 C, both the experimental results and the calculated ones exhibit no cyclic hardening behavior and their agreements were good, although in some cases there was a difference in peak stress between the experimental and the calculated results.
4. Simulations under anisothermal conditions 4.1. Constitutive equations As for the anisothermal conditions, it is general to consider the temperature rate term in kinematic hardening rules (Walker, 1981; Freed, 1988; Ohno and Wang, 1991; Wang and Ohno, 1991; McDowell, 1992; Chaboche, 1993a,b). Considering the
1124
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
temperature rate term, the kinematic hardening rule in the proposed constitutive model is expressed as follows: : 2 : X @ðCaÞ : : T; ð9Þ X ¼ C a" in ðX YÞp fJðXÞgm1 X þ 3 Ca @T where the last term on the right-hand side in Eq. (9) is the temperature rate term. It adjusts the variable to keep the ratio of the current value to temperature-dependent asymptotic value constant under temperature variation conditions. Therefore, even if there is no inelastic deformation, the value of the back stress will change with the temperature variation. Regarding the variable Y, the temperature rate term was not introduced in its evolutionary law due to the following presumption. As the variable Y is assumed to correspond to the substructure generated by the creep deformation (Yaguchi et al., 2002), the evolution of the variable Y is thought to need time at higher temperature. Consequently, it seems not appropriate for the variable Y to evolve rate/time-independently according to the temperature rate. Thus, it was assumed that the temperature rate term was not necessary in the evolutionary law of the variable Y. Therefore, the constitutive equations for the anisothermal conditions were the same as those mentioned in the former section except for changing the kinematic hardening rule from Eqs. (6) to (9). The material constants were also the same as those given in the former section. 4.2. Simulations Fig. 9 shows simulations of the anisothermal cyclic deformation under the inphase condition, where the symbol depicts the experimental result, and the solid line represents the result calculated by the above-mentioned constitutive model. Calculation was also conducted using the constitutive model, neglecting only the variable Y, and its result is expressed by the broken line in Fig. 9. In the case of neglecting the variable Y, the stress–strain response was constant with the cycles, which does not agree with the experimental result qualitatively, On the other hand, the calculation result obtained by the constitutive model considering the variable Y revealed that the maximum and minimum stresses continued to become smaller with the increase in the number of cycles and that the mean stress continued to grow in the compressive stress direction, which is characteristic inelastic behavior of the material. Under the out-ofphase condition, the peak stress calculated by the present constitutive model shifted in the tensile stress direction with the increase in the number of cycles, which is also true for the experimental results. It is well known that it is difficult to express the shift of the stress-strain hysteresis loop under the anisothermal conditions by the conventional constitutive equations if the temperature rate term is introduced into the kinematic hardening rules (Wang and Ohno, 1991; Chaboche, 1993a,b); however, the present constitutive model considering the temperature rate term was able to simulate the inelastic behavior well due to the incorporation of the variable Y as follows.
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
1125
Fig. 11 shows an example of the behavior of the back stress and the variable Y under the in-phase condition in a uniaxial state. The variable Y evolves in the compressive stress direction during the higher temperature process, because the static recovery of the back stress occurs under tensile back stress conditions. When the temperature becomes lower, the variable Y displays no variation because the static recovery of the back stress is negligible in the lower temperature region. When the temperature again becomes higher, the variable Y evolves in the compressive direction again due to the static recovery of the tensile back stress. Therefore, the variable Y exhibits the evolutionary behavior in the compressive stress direction with the increase in the number of cycles, as shown in Fig. 12; consequently, the back stress and the applied stress shift in the same direction with the increase in the number of cycles according to the enlarged dynamic recovery property of the back stress. The variable Y was originally incorporated into the kinematic hardening rule in order to express the anomalous inelastic behavior observed under the TH and the CII conditions at high temperature. However, it was demonstrated that the variable was also required for describing the inelastic behavior under the anisothermal conditions. 4.3. Effect of temperature rate term In the former section, the authors carried out the calculations by considering the temperature rate term because it is generally used for anisothermal analyses (Walker,
Fig. 11. Calculation results of back stress and variable Y under anisothermal condition by extended constitutive model.
1126
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
Fig. 12. Cyclic response of peak stress and variable Y under anisothermal condition by extended constitutive model.
1981; Freed, 1988; Ohno and Wang, 1991; Wang and Ohno, 1991; McDowell, 1992; Moosbrugger, 1992; Chaboche, 1993a,b). However, there have been some research results which indicate that the temperature rate term is not always necessary for describing the inelastic behavior under the anisothermal conditions (Bhattachar and Stouffer, 1993a; Delobelle and Bouchou, 1996). In this section, the authors discuss the effect of the temperature rate term on the description capability under the anisothermal conditions. One of the reasons for considering the temperature rate term is that without it the shift of the stress–strain hysteresis loop may occur under the anisothermal cyclic straining due to linear or quasilinear kinematic hardening components (Wang and Ohno, 1991; Chaboche, 1993a,b). The shift of the stress-strain loop with cycles is not ordinary for materials such as stainless and low alloy steels; however, the unusual behavior was observed in IN738LC in the present study. Therefore, it was examined whether or not the inelastic behavior can be adequately expressed by the constitutive model without considering the temperature rate term. The constitutive equations and material constants were the same as those used in Section 4.2, except here we neglect the temperature rate term and the variable Y. Fig. 13 shows the peak stress calculated under the in-phase condition (450850 C), where the strain rate was 104 s1 and the strain ranges were 0.3, 0.5 and 0.7%. The shift of the stress-strain loop with cycles depended on the strain range. In
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
1127
Fig. 13. Calculation results of anisothermal cyclic deformation without temperature rate term and variable Y.
the case of the strain range of 0.3%, the calculation result exhibited a pronounced shift of the stress–strain response with the increase in the number of cycles; however, the shift was scarcely observed when the strain range was 0.7%. The reason for the absence of the shift was thought to be that the nonlinearity of the kinematic hardening variable becomes great under the strain range condition. Examining the experimental results of IN738LC, the shift of the stress-strain hysteresis loop was observed under the strain range of 0.7%, as shown in Fig. 9. Therefore, it seems that it is difficult to describe the shift of the stress-strain response with sound accuracy solely by neglecting the temperature rate term in the kinematic hardening rules. Another characteristic of the constitutive equations without the temperature rate term is that it expresses the stress as larger in absolute value than that by constitutive equations considering the temperature rate term during the increasing process of the temperature, and it expresses the stress as smaller during the decreasing process of the temperature. In order to examine the effect of this on description accuracy, a simulation was conducted for the in-phase condition (450–850 C) where the strain rate and the strain range were 104 s1 and 0.7%, respectively, Fig. 10 shows the stress–strain hysteresis loop at the 900th cycle; here, the symbol means the
1128
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
experimental result, and the solid line express the calculation result obtained using the constitutive model previously mentioned in Section 4.2, and the broken line represents that obtained using the constitutive model neglecting only the temperature rate teim. In the lower temperature region (compressive strain side) there was little difference in stress–strain response between the constitutive models with and without the temperature rate term; however, some differences were observed in the higher temperature region (tensile strain side). Compared with the experimental result, the calculated result with the temperature rate term showed better agreement than the calculated result without the temperature rate term. Judging from these results, it was considered appropriate to introduce the temperature rate term into the kinematic hardening rule in the present viscoplastic constitutive model.
5. Conclusions In this paper, the inelastic behavior of nickel-base superalloy IN738LC was investigated under isothermal and anisothermal conditions between 450 and 950 C. Based on the experimental results, the viscoplastic constitutive model proposed in Part 1 was extended to the anisothermal loading conditions. The extended constitutive model was applied to the anisothermal test results as well as to the isothermal ones, and its description capability was evaluated. A summary of the present study is given below. 1. Under the isothermal conditions above 800 C, the material exhibited both temperature- and rate/time-dependent deformation behavior, whereas inelastic behavior was almost independent of the temperature and the loading rate/time under the conditions below 700 C. As for the isothermal cyclic deformation, the material exhibited slight cyclic hardening behavior when the temperature was below 700 C and the imposed strain rate was high. On the other hand, the cyclic hardening behavior was scarcely observed in the case of the temperature above 800 C or under slower strain rate conditions. 2. Anomalous inelastic behavior was observed under the anisothermal conditions of in-phase and out-of-phase types. With an increase in the number of cycles, stress at higher temperature becomes smaller and stress at lower temperature becomes larger in absolute value, although the stress range is almost constant with the number of cycles. That is, the mean stress continues to evolve cycle-by-cycle in the direction of the stress at the lower temperature. 3. The proposed viscoplastic constitutive model for IN738LC at 850 C was extended to the anisothermal conditions. Based on the experimental results, the evolution of the variable Y was assumed to be active under higher temperatures and negligible under lower temperatures. Material constants were determined with isothermal data, and were empirically expressed as functions of temperature. 4. The extended viscoplastic constitutive model was applied to the anisothermal cyclic loading as well as monotonic tension, stress relaxation, creep and cyclic loading under the isothermal conditions. It was demonstrated that the present visco-
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
1129
plastic constitutive model was successful in describing the inelastic behavior of the material adequately, including the anomalous inelastic behavior observed under the anisothermal conditions as a result of consideration of the variable Y.
References Benallal, A., Cheikh, A.B., 1987. Constitutive equations for anisothermal elasto viscoplasticity. Int. Conf. on Const. Laws for Eng. Mat., 667-674. Bhattachar, V.S., Stouffer, D.S., 1993a. Constitutive equations for the thermomechanical response of Rene 80: part 1— development from isothermal data. ASME J. Eng. Mat. Tech. 115, 351–357. Bhattachar, V.S., Stouffer, D.S., 1993b. Constitutive equations for the thermomechanical response of Rene 80: part 2—effects of temperature history. ASME J. Eng. Mat. Tech. 115, 358–364. Bouchou, A., Delobelle, P., 1996. Behaviour and modelization of a 17-12SPH stainless steel under cyclic, unidirectional and bi-directional anisothermal loadings. Nuc. Eng. Design 162, 21–45. Busso, E.P., Meissonnier, F.T., O’Dowd, N.P., 2000. Gradient-dependent deformation of two-phase single crystals. J. Mech. Phys. Solids 48, 2333–2361. Chaboche, J.L., 1986. Time-independent constitutive theories for cyclic plasticity. Int. J. Plasticity 2, 149– 188. Chaboche, J.L., 1993a. Cyclic viscoplastic constitutive equations, part 1: a thermodynamically consistent formulation. ASME J. Appl. Mech. 60, 813–821. Chaboche, J.L., 1993b. Cyclic viscoplastic constitutive equations, part 2: stored energy—comparison between models and experiments. ASME J. Appi. Mech. 60, 822–828. Chaboche, J.L., Nouailhas, D., 1989. Constitutive modeling of ratchetting effects— part 2: possibilities of some additional kinematic rules. ASME J. Eng. Mat. Tech. 111, 409–416. Chan, K.S., Bodner, S.R., Lindholm, U.S., 1988. Phenomenological modeling of hardening and thermal recovery in metals. ASME J. Eng. Mat. Tech. 110, 1–8. Chan, K.S., Lindholm, U.S., 1990. Inelastic deformation under nonisothermal loading. ASME J. Eng. Mat. Tech. 112, 15–25. Chan, K.S., Lindholm, U.S., Bodner, S.R., Nagy, A., 1990. High temperature inelastic deformation of the B1900+Hf alloy under multiaxial loading: theory and experiment. ASME J. Eng. Mat. Tech. 112, 7– 14. Chan, K.S., Lindholm, U.S., Bodner, S.R., Walker, K.P., 1989. High temperature inelastic deformation under uniaxial loading: theory and experiment. ASME J. Eng. Mat. Tech. 111, 345–353. Chan, K.S., Page, R.A., 1988. Inelastic deformation and dislocation structure of a nickel alloy: effects of deformation and thermal histories. Met. Trans. A. 19A, 2477–2486. Delobelle, P., Bouchou, A., 1996. Behavior and modeling of a 1712 SPH stainless steel under cylic, uni and bi-directional, part 2: modeling. Int. J. Plasticity 12, 311–341. Estrin, Y, Braasch, H., Brechet, Y., 1996. A dislocation density based constitutive model for cyclic deformation. ASME J. Eng. Mat. Tech. 118, 441–447. Freed,A.D., 1988. Thermoviscoplastic Model with Application to Copper. NASA Report TP-2845, NASA. Freed, A.D., Walker, K.P., 1993. Viscoplasticity with creep and plasticity bounds. Int. J. Plasticity 9, 213– 242. James, G.H., Imbrie, P.K., Hill, P.S., Allen, D.H., Haisler, W.E., 1987. An experimental comparison of several current viscoplastic constitutive models at elevated temperature. ASME J. Eng. Mat. Tech. 109, 130–139. Jordan, E.H., Walker, K.P., 1992. A viscoplastic model for single crystals. ASME J. Eng. Mat. Tech. 114, 19–26. Lall, C., Chin, S., Pope, D.P., 1979. The orientation and temperature dependence of the yield stress of Ni3(Al,Nb) single crystals. Met. Trans. A 10A, 1323–1332.
1130
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
Lee, K.D., Krempl, E., 1991. An orthotropic theory of viscoplasticity based on overstress for thermomechanical deformation. Int. J. Solids Struct. 27, 1445–1459. Lin, T.L., Wen, M., 1989. Dislocation structure due to high temperature deformation in the 20 phase of a nickel-based superalloy. Mat. Sci. Eng. A113, 207–214. McDowell, D.L., 1992. A nonlinear kinematic hardening theory for cyclic thermoplasticity and thermoviscoplasticity. Int. J. Plasticity 8, 695–728. Meric, L., Cailletaud, G., 1991. Single crystal modeling for structural calculations: part 2—finite element implementation. ASME Eng. Mat. Tech. 113, 171–182. Meric, L., Poubanne, P., Cailletaud, G., 1991. Single crystal modeling for structural calculations: part 1— model presentation. ASME Eng. Mat. Tech. 113, 162–170. Milligan, W.W., Antolovich, S.D., 1978. Yielding and deformation behavior of the single crystal superalloy PWA 1480. Met. Trans. A 18A, 85–95. Moosbrugger, J.C., 1992. Nonisothermal constitutive model for the small strain behavior of 9Cr–Mo–V– Nb pressure vessel steel. ASME J. Eng. Mat. Tech. 114, 354–361. Moreno, V., Jordan, E.H., 1986. Prediction of material thermomechanical response with a unified viscoplastic constitutive model. Int. J. Plasticity 2, 223–245. Nouailhas, D., Cailletaud, G., 1995. Tension–torsion behavior of single-crystal superalloys experiment and finite element analysis. Int. J. Plasticity 11, 451–470. Nouailhas, D., Chaboche, J.L., 1991. Anisotropic constitutive modeling for single-crystal superalloys using a continuum phenomenological approach. In Desai, C.S., Krempl, E., Frantziskonis, G., Saadatmanesh, H. (Eds). 3rd Int. Conf. On Const. Laws for Eng. Mat. ASME Piess, pp. 213–218. Nouailhas, D., Freed, A.D., 1992. A viscoplastic theory for anisotropic materials. ASME J. Eng. Mat. Tech. 114, 97–104. Ohno, N., Mizuno, T., Kawaji, H., Okada, I., 1992. Multiaxial creep of a nickel-base directionally solidified alloy: anisotropy and simulation. Acta Metall. Mater. 40, 559–567. Ohno, N., Takahashi, Y., Kuwabara, K., 1989. Constitutive modeling of anisothermal cyclic plasticity of 304 stainless steel. ASME J. Eng. Mat. Tech. 111, 106–114. Ohno, N., Wang, J.D., 1991. Transformation of a nonlinear kinematic hardening rule to a multisurface form under isothermal and nonisothermal conditions. Int. J. Plasticity 7, 879–891. Olschewski, J., Sievert, R., Qi, W., Bertram, A., 1993. Prediction of inelastic response of a Ni-based superalloy under thermal-mechanical cyclic loading. In: Kussmaul, K. (Ed.), Structural Mechanics in Reactor Technology (SMiRT- 12), Vol. L. Elsevier Science, 17–22. Paidar, V., Pope, D.P., Vitek, V., 1984. A theory of the anomalous yield behavior in L12 ordered alloys. Acta Metall. 32, 435–448. Paslay, P.R., Wells, C.H., Leverant, G.R., 1970. An analysis of primary creep of nickel-base superalloy single crystals. ASME J. Appl. Mech., 759–764. Paslay, P.R., Wells, C.H., Leverant, G.R., Burck, L.H., 1971. Creep of single crystal nickelbase superalloy tubes under biaxial tension. ASME J. Appi. Mech., 623–626. Pope, D.P., Ezz, S.S., 1984. Mechanical properties of Ni3Al and nickel-base alloys with high volume fraction of 20 . Int. Metals Reviews 29, 136–167. Ramaswamy, V.G., Stouffer, D.C., Laflen, J.H., 1990. A unified constitutive model for the inelastic uniaxial response of Rene’80 at temperatures between 538 C and 982 C. ASME J. Eng. Mat. Tech. 112, 280–286. Ramaswamy, V.G., Stouffer, D.C., VanStone, R.H., Laflen, J.H., 1987. Modeling thermomechanical cyclic response with a unified state variable constitutive equation. ASME PVP 123, 57–64. Sheh, M.Y., Stouffer, D.C., 1990. A crystallographic model for the tensile and fatigue response for Rene N4 at 982 C. ASME J. Applied Mechanics 57, 25–31. Sherwood, J.A., Stouffer, D.C., 1992. A phenomenologically based constitutive model for Rene 95. ASME J. Eng. Mat. Tech. 114, 340–347. Sievert, R., Osterle, W., Olschewski, J., Ziebs, J., 1997. Physical interpretation of a viscoplastic model applied to high temperature performance of nickel-base superalloy. Z. Metallkd 88, 416–424. Slavik, D., Cook, T.S., 1990. A unified constitutive model for superalloys. Int. J. Plasticity 6, 651–664. Stouffer, D.C., Ramaswamy, V.G., Laflen, J.H., Stone, R.H.V., Williams, R., 1990. A constitutive model
M. Yaguchi et al. / International Journal of Plasticity 18 (2002) 1111–1131
1131
for the inelastic multiaxial response of Rene’80 at 871C and 982C. ASME J. Eng. Mat. Tech. 112, 241– 246. Takeuchi, S., Kuramoto, E., 1973. Temperature and orientation dependence of the yield stress in Ni3Ga single crystals. Acta metall. 21, 415–425. Walker, K.P., Jordan, E.H., 1989. Biaxial constitutive modeling and testing of a single crystal superalloy at elevated temperatures. Biaxial and Multiaxial Fatigue EGF 3, 145–170. Walker K.P., 1981. Research and Development Program for Nonlinear Structural Modeling with Advanced Time–temperature Dependent Constitutive Relationships. NASA Report CR- 165533. NASA. Wang, J.D., Ohno, N., 1991. Two equivalent forms of nonlinear kinematic hardening:application to nonisothermal plasticity. Int. J. Plasticity 7, 637–650. Yaguchi, M., Takahashi, Y., 2000. A viscoplastic constitutive model incorporating dynamic strain aging effect during cyclic deformation conditions. Int. J. Plasticity 16, 241–262. Yaguchi,M., Yamamoto,M., Ogata,T. 2002. A viscoplastic constitutive model for nickelbase superalloy, part 1: kinematic hardening rule of anisotropic dynamic recovery. Int. J. Plasticity, 18, 1083–1109.