A modified θ projection model for constant load creep curves-I. Introduction of the model

doi:10.1016/j.jmst.2018.09.024

Abstract

Abstract:

Estimating long-term creep deformation and life of materials is an effective way to ensure the service safety and to reduce the cost of long-term integrity evaluation of high temperature structural materials. Since the 1980s, the θ projection model has been widely used for predicting creep lives due to its ability to capture the characteristic transitions observed in creep curves obtained under constant true stress conditions. However, the creep rupture behavior under constant load or engineering stress conditions cannot be simulated accurately using this model because of the different stress states. In this paper, creep curves obtained under constant load conditions were analyzed using a modified θ projection model by considering the increase in true stress with creep deformation during the creep tests. This model is expressed as $ε = θ 1 1 - e - θ 2 t + θ 3 e θ 4 e θ 5 ε t - 1$ , and was validated using the creep curves of K465 and DZ125 superalloys tested at a range of temperatures and engineering stresses. Moreover, it was shown that the predictive capability of the modified θ projection model was significantly improved over the original one, as it reduces the prediction uncertainty from a range of 10% to 20% to below 5%. Meanwhile, it was shown that the model can be reasonably used for predicting constant stress creep conditions, when appropriate parameters are used. The prediction performance of the modified model will be discussed in another paper. The results of this study show great potential for the evaluation and assessment of the service safety of structural materials used in applications where designs are limited by creep deformation.

Key words: Superalloy, Creep, Life prediction, Modified θ projection model;, Constant load creep

Chao Fu, Yadong Chen, Xiaofei Yuan, Sammy Tin, Stoichko Antonov, Koichi Yagi, Qiang Feng. A modified θ projection model for constant load creep curves-I. Introduction of the model[J]. J. Mater. Sci. Technol., 2019, 35(1): 223-230.

Figures/Tables 17

Fig. 1. Calculated creep curves obtained under the constant load in comparison with the corresponding creep curves obtained under the constant stress of 250?MPa and 50?MPa at 565?°C for 1/2Cr1/2Mo/1/4?V steel. [7].

Fig. 2. Physical representation of each parameter for the original θ projection model.

Table 1 Chemical composition of the experimental K465 and DZ125 superalloys (wt%).

Alloy	Ni	W	Co	Cr	Al	Ti	Mo	Nb	Ta	Fe	Hf	C	B
K465	Bal.	10.4	10.0	8.4	5.7	2.7	1.5	1.1	-	-	-	0.2	-
DZ125	Bal.	7.0	10.2	8.8	5.2	1.0	2.2	-	3.6	0.5	1.4	0.1	0.016

Fig. 3. Geometry of creep specimens used in this research.

Fig. 4. Creep curves obtained under different conditions for DZ125 superalloy and K465 superalloy: (a) creep strain vs time curve under 900?°C/350?MPa for DZ125 superalloy, (b) creep strain vs time curve under 900?°C/300?MPa for K465 superalloy, (c) strain rate vs time curve under 900?°C/350?MPa for DZ125 superalloy, and (d) strain rate vs time curve under 900?°C/300?MPa for K465 superalloy.

Table 2 θi values obtained in DZ125 superalloy by fitting the creep curves at different strain ranges under 900?°C/350?MPa, using the original θ projection model.

Strain range	θ₁	θ₂	θ₃	θ₄
0-1%	0.244	0.468	2.404	0.0048
0-2%	0.252	1.67e6	0.726	0.0127
0-4%	0.292	174.8	0.464	0.0162
0?6%	0.329	3171.7	0.370	0.0178
0?8%	0.359	4525	0.315	0.0189
0-10%	0.419	23132	0.244	0.0205

Fig. 5. Experimental creep curve and the predicted creep curves for DZ125 superalloy at 900?°C/350?MPa using θi parameters obtained by original θ projection model.

Table 3 θi values obtained in K465 superalloy by fitting the creep curves at different strain ranges under 975?°C/200?MPa, using the original θ projection model.

Strain range	θ₁	θ₂	θ₃	θ₄
0?1.5%	0.250	0.090	0.281	0.0175
0-2%	0.379	0.0567	0.117	0.0247
0?2.3%	0.109	14259.9	0.455	0.0149
0-2.6%	0.129	12922800	0.353	0.0170
0?3%	0.139	18814.6	0.255	0.0198

Table 4 θi and Ai values obtained in DZ125 superalloy by fitting the creep curves at different strain ranges, under 900?°C/350?MPa, using the modified θ projection model (6-parameter model).

Strain range	θ₁	θ₂	θ₃	θ₄	A₁	A₂
0-1%	0.312	1.207	56.208	2.79E-4	-7.059	0.0918
0-2%	0.311	0.418	0.793	0.0102	-1.622	0.0933
0-4%	0.319	0.346	0.553	0.0139	-1.103	0.0263
0?6%	0.313	0.318	0.543	0.0143	-0.627	0.0159
0?8%	0.316	0.306	0.526	0.0146	-0.545	0.0139
0-10%	0.282	0.672	0.924	0.0116	-3.939	0.0186

Table 5 θi values obtained in DZ125 superalloy by fitting the creep curves at different strain ranges, under 900?°C/350?MPa, using the modified θ projection model (5-parameter model).

Strain range	θ₁	θ₂	θ₃	θ₄	θ₅
0-1%	0.240	0.486	35.183	3.27E-4	0.1305
0-2%	0.289	0.322	0.929	0.0094	0.0795
0-4%	0.308	0.280	0.556	0.0142	0.0164
0?6%	0.317	0.263	0.511	0.0149	0.0121
0?8%	0.320	0.256	0.498	0.0151	0.0112
0-10%	0.303	0.297	0.551	0.0144	0.0136

Fig. 6. Experimental creep curves and predicted creep curves by the original and two modified models under different conditions of DZ125 superalloy: (a) 900?°C/350?MPa, (b) 980?°C/220?MPa.

Table 6 Error in creep time to 10% strain between experimental data and the data predicted by two θ projection models by fitting the creep curve of DZ125 superalloy to a 4% cut off strain.

Creep condition	Experimental time to 10% strain (h)	Time to 10% strain by θ projection model (h)	Relative error (%)	Time to 10% strain by 5-parameter model (h)	Relative error (%)
900?°C/350?MPa	177.8	190.6	7.2	174.8	-1.7
900?°C/370 MPa	113.3	123.9	9.4	115.6	1.9
980?°C/207?MPa	119.9	130.6	8.9	117.0	-2.4
980?°C/220?MPa	82.2	87.9	6.9	80.0	-2.7
1000?°C/180 MPa	102.3	108.3	5.9	100.0	-2.2

Table 7 θi and Ai values obtained in K465 superalloy by fitting the creep curves at different strain ranges, under 975?°C/200?MPa, using the modified θ projection model (6-parameter model).

Strain range	θ₁	θ₂	θ₃	θ₄	A₁	A₂
0?1.5%	0.190	0.147	39.480	1.750E-4	-1.539	0.466
0-2%	0.208	0.127	1.056	0.00622	-1.175	0.203
0?2.3%	0.242	0.105	0.592	0.00976	-0.979	0.138
0-2.6%	0.279	0.088	0.391	0.01286	-0.809	0.099
0?3%	0.267	0.095	0.438	0.01204	-0.942	0.108

Table 8 θi values obtained in K465 superalloy by fitting the creep curves at different strain ranges, under 975?°C/200?MPa, using the modified θ projection model (5-parameter model).

Strain range	θ₁	θ₂	θ₃	θ₄	θ₅
0?1.5%	0.161	0.144	41.603	1.776E-4	0.413
0-2%	0.176	0.129	1.389	0.00509	0.209
0?2.3%	0.201	0.110	0.764	0.00833	0.144
0-2.6%	0.234	0.093	0.484	0.01150	0.103
0?3%	0.220	0.100	0.558	0.01051	0.112

Fig. 7. Experimental creep curves and predicted curves by the 5-parameter model for K465 superalloy at different creep conditions.

Table 9 Error in creep time to 3% strain between experimental data and the data predicted by two θ projection models by fitting the creep curve of K465 superalloy up to a 2% cut off strain.

Creep condition	Experimental time to 3% strain (h)	Time to 3% strain by θ projection model (h)	Relative error (%)	Time to 3% strain by modified θ projection model (h)	Relative error (%)
900?°C/300?MPa	247.5	263.1	6.3	236.5	-4.4
900?°C/320?MPa	161.4	170.1	5.4	159.5	-1.2
950?°C/300?MPa	32.3	34.3	6.2	31.0	-4.0
975?°C/200?MPa	121.5	128.5	5.8	117.0	-3.7
1000?°C/137?MPa	215.7	238.1	10.4	212.4	-1.5

Fig. 8. The predicted creep curve under constant stress and the corresponding constant load experimental data: (a) 900?°C/350?MPa for DZ125 superalloy, (b) 975?°C/200?MPa for K465 superalloy.

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