The development of creep damage constitutive equations for high Cr steel

Figures & data

Figure 1. The variation of creep cavitation coefficient$\mathrm{A}$ with different stress and temperature.

Table 1. The variation of creep cavitation coefficient A [10].

Stress (MPa)	600°C	650°C
92		2.19E+02
104		2.92E+02
110		3.81E+02
145	1.93E+02
160	4.25E+02

Table 2. Materials and its parameters.

Material	Parameters A	Parameters B	q
0 · 5Cr–0 · 5Mo–0 · 25 V	4.12E-08	2.51E-04	2
2 · 25Cr–1Mo	5.57E-07	2.40E-04	2
P91	6.10E-07	2.14E-04	2

Figure 2. Experimental data of minimum strain rate and stress at 600°C and 650°C for P92 steel [9,11,15].

Figure 3. The comparison between the different function of minimum creep strain rate and applied stress for P92 steel at 600°C.

Table 3. The function of minimum creep strain rate with calibrated material parameters for P92 steel at 600°C.

	A	B	n	q
Power law [18,19] (${\dot{\epsilon }}_{min}=A{\sigma }^n$)	1.00E-39		16.2
Sine law [20,21] (${\dot{\epsilon }}_{min}=Asinh\left(B\sigma \right)$)	1.40E-09	8.40E-02
Novel sine law (${\dot{\epsilon }}_{min}=Asinh(B{\sigma }^q)$)	1.12E-06	2.47E-04		2

Figure 4. The comparison between the different function of minimum creep strain rate and applied stress for P92 steel at 650°C.

Table 4. The function of minimum creep strain rate with calibrated material parameters for P92 steel at 650°C.

	A	B	n	q
Power law [18,19] (${\dot{\epsilon }}_{min}=A{\sigma }^n$)	8.80E-23		9.6
Sine law [20,21] (${\dot{\epsilon }}_{min}=Asinh\left(B\sigma \right)$)	1.40E-07	9.60E-02
Novel sine law (${\dot{\epsilon }}_{min}=Asinh(B{\sigma }^q))$	1.00E-05	4.95E-04		2

Figure 5. No noticed difference of the predicted probability density function by Equation (3.1) (with α = 0.9999) and Equation (3.4) dots: experimental data from ref [12,25] curves.

Figure 6. The probability density function of cavity equivalent R for E911, dots: experimental data from ref [12,25], curve by Equation (3.5.b).

Table 5. Comparison of nucleation and growth rate coefficients.

Material	Temperature (°C)	Axial stress (MPa)	Internal stress (MPa)	Lifetime (h)	Parameters A1	Parameters A2
P91 [1]	575	52.6	23.6	10,200	7.80E-01	8.30E-05
E911	600	48.9	17.7	37,800	8.76E+00	9.47E-07

Table 6. The value of U’ for P92 at 600°C [24].

Stress (MPa)	Rupture time (h) [11]	U’ (P92 at 600°C)
120	65,363.4	1.84E-10
130	39,539.9	5.02E-10
140	25,944.6	1.17E-09
160	8219.9	1.16E-08
180	1740.7	2.59E-07
190	613.4	2.09E-06
210	112.6	6.19E-05
230	19.9	1.98E-03
250	5.1	3.02E-02

Table 7. The value of U’ for P92 at 625°C.

Stress (MPa)	Rupture time (h) [11]	U’ (P92 at 625 °C)
100	33,518.5	6.9907E-10
110	17,530	2.5558E-09
130	3886.1	5.2007E-08
140	1458.2	3.69365E-07
160	213.4	1.72465E-05

Table 8. The value of U’ for P92 at 650°C [24].

Stress (MPa)	Rupture time (h) [11]	U’ (P92 at 650 °C)
70	50,871.2	3.03E-10
80	21,717.1	1.67E-09
90	10,001.9	7.85E-09
100	3738.7	5.62E-08
110	1689.1	2.75E-07
130	194	2.09E-05
140	66	1.80E-04
160	10.5	7.12E-03

Figure 7. The trend of the values of U’ under different stress and temperature [24].

Figure 8. The trend of the values of U’ under different stress and temperature.

Table 9. The value of U’ for P92 at 600°C.

Stress (MPa)	Rupture time (h) [9]	U’ (Yin P92 at 600 °C)
110	200,000	1.9635E-11
115	95,000	8.70247E-11
120	65,000	1.85893E-10
135	30.000	8.72665E-10
150	10,000	7.85398E-09
170	3000	8.72665E-08
185	1000	7.85398E-07

Table 10. The value of U’ for P92 at 650°C.

Stress (MPa)	Rupture time (h) [9]	U’ (Yin P92 at 650 °C)
60	200,000	1.9635E-11
65	95,000	8.70247E-11
70	65,000	1.85893E-10
80	30,000	8.72665E-10
90	10,000	7.85398E-09
105	3000	8.72665E-08
115	1000	7.85398E-07

Figure 9. Comparison of Yin’s U’ with experiment [11] for P92 steel [24].

Figure 10. The trend of cavity nucleation rate coefficient A₂ and stress.

Table 11. The number of cavities at failure and the individual value of A₂ under a range of stress levels.

Stress (MPa)	Lifetime (h)	Number density of cavities (10^−5um^−3)	A2
70	17200	3.03	2.05E-08
80	12981	6.95	8.25E-08
100	8682	7.33	1.94E-07
130	3433	2.95	5.01E-07

Yadav SD, Sonderegger B, Stracey M, et al. Modelling the creep behavior of tempered martensitic steel based on a hybrid approach. Mater Sci Eng A662. 2016;330–341. DOI:10.1016/j.msea.2016.03.071.

 PubMed Web of Science ®Google Scholar

Yin YF, Faulkner RG. Continuum damage mechanics modelling based on simulations of microstructural evolution kinetics. Mater Sci Technol. 2006;22:929–936.

 Web of Science ®Google Scholar

Creep and rupture data of heat resistant steels, National Institute for Materials Science (NIMS). Available from: http://smds.nims.go.jp/creep/index_en.html, accessed May 2018

 Google Scholar

Panait CG. Metallurgical evolution and creep strength of 9–12% Cr heat resistant steels at 600°C and 650°C [doctor thesis]. Paris Tech Institut Des Sciences Et Technologies Paris Institute of Technology. Mines ParisTech; 2010. Available from: https://pastel.archives-ouvertes.fr/pastel-00579983

 Google Scholar

Bailey RW. Creep of steel under simple and compound stress. Engineering. 1930;121:129–265.

 Google Scholar

Norton FH. The creep of steel at high temperature. New York (NY): McGraw-Hill Book; 1929.

 Google Scholar

Dyson BF. Use of CDM in materials modelling and component creep life prediction. J Press Vessels Technol. 2000;122:281–296.

 Web of Science ®Google Scholar

Dyson B, McLean M. Microstructural evolution and its effects on the creep performance of high temperature alloys. In: Strang A, Cawley J, Greenwood GW, editors. Microstructural stability of creep resistant alloys for high temperature plant applications. 1997. p. 371–393, IOM Communications, London, United Kindom.

 Google Scholar

Renversade L, Ruoff H, Maile K, et al. Microtomographic assessment of damage in P91 and E911 steels after long- term creep. Int J Mater Res. 2014;105(7):621–627.

 Web of Science ®Google Scholar

Xu Q, Yang X, Lu ZY. On the development of creep damage constitutive equations: modified hyperbolic sine law for minimum creep strain rate and stress and creep fracture criterion based on cavity area fraction along grain boundaries. Mater High Temp. 2017;34(56):323–332.

 Google Scholar

Xu Q, Zheng XM, Okpa M, et al. Poster: the development of creep damage constitution equations for high Cr alloys. Power Plant Operation & Flexibility. 2018 July 4 –6; London: IOM3.

 Google Scholar