Data-based selection of creep constitutive models for high-Cr heat-resistant steel

Figures & data

Table 1. Some previous studies of creep curves in which the theta or modified theta method was applied to Gr.91 steel

Papers	Steady (modified theta)	Non-steady (theta or revised theta)	Other models	Test conditions	Minimum creep rate	Secondary (steady- state) creep
Factors influencing creep model equation selection Holdsworth et al. (2007) [2]	Applied	Not applied	–	–	–	–
Long-term creep characterization of Gr.91 steel by modified creep constitutive equations Kim et al. (2011) [3]	Not applied	Applied	Omega	873 K 130–200 MPa	${10}^{-1}-{10}^{-4}$/h	Yes
Creep-rupture life prediction for 9Cr-1Mo-Nb-V weld metal Miyakita et al. (2015) [4]	Not applied	Applied	Omega	883 K 150 MPa	$3.85\times {10}^{-5}$/h	Yes
				898 K 110–170 MPa	$3.25\times {10}^{-3}$–$1.15\times {10}^{-5}$/h
				923 K 90–150 MPa	$6.10\times {10}^{-3}$–$2.16\times {10}^{-5}$/h

Figure 1. (a) Schematic diagram of model selection method with grid search. (b) shows how our algorithm worked in the case of applying the theta method to the creep strain curve for 873 K/160 MPa. The four heat maps show the distribution of Bayesian free energy by greyscale, where the point of the lowest free energy in each search area is indicated by the yellow cross mark. The green, red, and purple rectangles inserted in the top three maps show the next search area, respectively. The right-bottom plot shows the transition from the first search area to the final search area

Table 2. Chemical composition of the 9Cr-1Mo-Nb-V (Gr.91) steel

	C	Si	Mn	P	S	Cu	Ni	Cr	Mo	V	Nb	Al	N
(mass %)	0.10	0.25	0.43	0.006	0.002	0.012	0.06	8.87	0.93	0.19	0.07	0.014	0.06

Table 3. Observed rupture time and creep rate, Bayesian free energy, posterior probability of the two models, ratio of fitting parameters $\alpha$ for the modified theta method (${\alpha }_{Mod\theta }$) and theta method (${\alpha }_{\theta }$), and the ratio of the steady-state region to the rupture time for the modified theta method in the present ASME Gr.91 steel creep tests. Values for the Bayesian free energy are rounded to integers. Values for the posterior probability, the ratio of fitting parameters $\alpha$, and the ratio of the steady-state region are rounded to three significant figures

Test conditions				Free energy			Posterior probability
Temperature (K)	Stress (MPa)	Rupture (h)	Minimum creep rate(1/h)	Modified Theta	Theta	Modified theta		Theta	$\frac{{\alpha }_{\theta }}{{\alpha }_{Mod\theta }}$	Steady-state region ratio
823	240	290.7	1.76$\times {10}^{-4}$	−2322	−2002	1		0	0.0359	0.563
	220	2392.5	1.05$\times {10}^{-5}$	−2980	−2473	1		0	0.0646	0.709
	200	10,432.9	3.58$\times {10}^{-6}$	−3883	−3370	1		0	0.0546	0.585
	190	18,514.7	1.44$\times {10}^{-6}$	−11,016	−9453	1		0	0.0651	0.554
	170	47,104.5	3.46$\times {10}^{-7}$	−24,936	−22,427	1		0	0.00813	0.500
873	160	501.4	9.65$\times {10}^{-5}$	−2625	−2270	1		0	0.0167	0.579
	140	2550.2	1.48$\times {10}^{-5}$	−3992	−3432	1		0	0.0271	0.579
	130	6036.8	5.33$\times {10}^{-6}$	−2826	−2449	1		0	0.0232	0.483
	110	19,907.0	9.60$\times {10}^{-7}$	−11,258	−10,693	1		0	0.00107	0.419
	100	40,307.4	5.48$\times {10}^{-7}$	−12,410	−11,586	1		0	0.00107	0.503
923	90	73,960.9	2.83$\times {10}^{-7}$	−9612	−9585	1		0	0.00124	0.342
	90	928.0	3.50$\times {10}^{-5}$	−3141	−3026	1		0	0.000102	0.469
	80	2726.1	1.18$\times {10}^{-5}$	−2757	−2688	1		0	0.0000820	0.460
	70	8385.6	4.75$\times {10}^{-6}$	−3537	−3428	1		0	0.000129	0.382
	50	60,181.2	6.19$\times {10}^{-7}$	−16,654	−15,000	1		0	0.0159	0.618

Figure 2. Creep strain curve of Gr.91 steel at 873 K/160 MPa

Figure 3. Fitting of Gr.91 steel creep curve at 823 K/240 MPa with (a) modified theta method and (b) theta method. Log of the creep strain rate against time for 823 K/240 MPa fitted with (c) modified theta method and (d) theta method. The insets in (a) and (b) are enlargements of the primary region. The formulae in (c) and (d) are the derivatives of each term of equations (1) and (2)

Figure 4. (a) Schematic figure of log of the creep strain rate against time with its fitting by modified theta method. Blue broken, red dot-dashed, and green dotted lines are the asymptotes of the corresponding terms of the equation (similar to FIG. 3.) The gray horizontal line shows the region where the steady-state term (equation (12)) is more than 10 times larger than the other terms (equations (11) and (13)), (b) Steady-state length ratio of Gr.91 steel creep curve estimated by modified theta method. Each length is normalized by the rupture time of the creep test

Figure 5. Double logarithmic plot of the creep rate in the steady state, ${\dot{\epsilon }}_s$ versus applied stress. The standard deviation of the obtained posterior probability of each creep rate is given by the error bars

Holdsworth SR, Askins M, Baker A, et al. Factors influencing creep model equation selection. Int J Press Vessels Pip. 2008;85:80–88.

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Kim WG, Kim SH, Lee CB. Long-term creep characterization of Gr. 91 steel by modified creep constitutive equations. Met Mater Int. 2011;17:497–504.

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Miyakita A, Yamashita K, Taniguchi G, et al. Creep-rupture life prediction for 9Cr-1Mo-Nb-V weld metal. ISIJ Int. 2015;55(10):2189–2197.

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