- Email: [email protected]
An empirical relation between strain energy and time in creep deformation
International Journal of Pressure Vessels and Piping 75 (1998) 939–943
An empirical relation between strain energy and time in creep deformation K.G. Samuel, P. Rodriguez* Metallurgy and Materials Group, Indira Gandhi Centre for Atomic Research, Kalpakkam 603 102, India Received 6 July 1998; accepted 17 August 1998
Abstract
R The stored energy of creep defined as W 10 s d1 and estimated as the area under the isochronous stress–strain curve is found to vary with time t and temperature T (in particular Wt m C, where m and C are constants). The work of creep defined as Wc s1 also varies similarly (Wct p B where p and B are constants). From the relationships, a new method for generating isochronous stress–strain curves from three or more short-term creep tests is proposed. The method is useful particularly for newly developed materials. 䉷 1998 Elsevier Science Ltd. All rights reserved. Keywords: Creep deformation; Strain energy; Isochronous curves; Design stress
1. Introduction The basic philosophy in the design of engineering components is to guard against failure of the component. The basis for establishing the stress intensity values and allowable stress values for design are described in ASME Pressure Vessel Codes [1, 2]. At any temperature below the creep range, where deformation is time-independent, the maximum allowable stress value (Sm) for ferritic steels is the lowest of the following: (i) one-fourth of the specified minimum tensile strength at room temperature; (ii) one-fourth of the tensile strength at the temperature; (iii) two-thirds of the specified minimum yield strength at room temperature; (iv) two-thirds of the yield strength at the temperature. For austenitic steels the fourth condition is modified to be 90% of the yield strength at the temperature but not to exceed two-thirds of the specified minimum yield strength at room temperature. At elevated temperatures where time-dependent creep dominates, three failure modes are identified: (a) accumulation of a specified amount of plastic strain; (b) rupture of the component; and (c) onset of tertiary creep. A stress intensity limit (St) dependent on both temperature and time is considered here. The data considered in establishing the St values are obtained from long-term, constant load, uniaxial creep tests. For each specific time t, the St values shall be the lowest of: (a) 100% of the average stress required to obtain a total strain of 1%; (b) 67% of the minimum stress to cause
rupture; and (c) 80% of the minimum stress to cause initiation of tertiary creep. The allowable limit of general primary membrane stress intensity (Smt) to be used as a reference for stress calculations for the actual service life is the lowest of the two stress intensity values, Sm (time-independent) and St (time-dependent). Isochronous stress–strain curves are used for establishing the allowable design stress intensities for condition (a) in order to avoid excessive deformation over the intended service life. An isochronous curve presents, in a plot of stress against strain, the loci of total strain accumulated when different stresses are applied for a fixed time. For relatively short service lives (say a few thousand hours as in aircraft engine components) isochronous stress–strain curves can be developed by taking constant time section through a family of creep curves. The generation of isochronous curves for long service life (for example that of power plants or for more years) requires extrapolation of shortertime test results. Several approaches have been employed in developing isochronous curves [3]. This paper describes a new procedure for utilising the traditionally employed constant load creep tests to estimate the allowable design stress intensities St for limiting deformation condition based on the work done during creep in deforming the material.
2. Procedure * Corresponding author. Tel.: ⫹ 91-4114-40267; Fax: ⫹ 91-411440360; e-mail: [email protected]. 0308-0161/98/$ - see front matter 䉷 1998 Elsevier Science Ltd. All rights reserved. PII: S0308-016 1(98)00095-7
Average isochronous stress–strain curves for various
940
K.G. Samuel, P. Rodriguez / International Journal of Pressure Vessels and Piping 75 (1998) 939–943
Fig. 1. Average isochronous stress–strain curves for type 316 stainless steel at 1200 F (649⬚C).
high-temperature materials are available in the literature (Code case N-47) [4]. A typical set of curves for AISI 316 stainless steel is shown in Fig. 1. The areas under the isochronous stress–strain curves W at different strain levels were evaluated by counting the number of squares under the curve for 304, 316 and 2.25Cr–1Mo steels for various duration of time, t and found to obey a functional relation of the form Wtm C Constant;
1
where C and m are constants depending on the strain level and temperature. The value of m is estimated as d log W/ d log t. The functional relation is typically represented graphically in Fig. 2(a–c) for these materials. It is to be noted that unlike the stress–rupture time relation where normally a two slope behaviour is observed making the extrapolation of stress for long-term life from the shortterm accelerated tests unrealistic, W–t relationship obeyed a linear relation in a double logarithmic plot without any break in the relation up to 10 5 h for the strain levels and temperatures investigated. The values of C and m for various strain levels at different test temperatures are also summarised in Tables 1–3 for these materials. The relationship between W and t leads us to a new method to generate isochronous stress–strain curves from a few short-time creep tests as described below. This
Fig. 2. Variation of stored energy of creep as a function of time for various strain levels for (a) 2.25Cr–1Mo steel; (b) 316 stainless steel; and (c) 304 stainless steel.
method will be especially useful for newly developed materials with limited creep test data. Let us consider that a material has undergone creep deformation at a stress s to a strain 1 at time t. The work of creep Wc s1 . We also reach the point (s ,1 ) through the isochronous stress–strain curve for time t. The area under the R1 isochronous stress–strain curve W defined earlier is 0 s d1; W s1 ⫺ Wc ⫺
Zs 0
Zs 0
1 ds 3
1 d s:
If Wc is the work of creep, then W can be considered the stored energy of creep, the second term on the right hand side of Eq. (3) is the dissipated component of the work of creep. Assuming that a Hollomon type stress–strain relationship of the form s K1 n is valid for the isochronous stress– strain curves, where K is the work hardening coefficient and n is the work hardening exponent of the isochronous stress– strain curve, ds K·n·1n⫺1 d1:
4
K.G. Samuel, P. Rodriguez / International Journal of Pressure Vessels and Piping 75 (1998) 939–943 Table 1 Empirical constants (C and m) derived from isochronous stress–strain curves for 304 stainless steel
941
Table 2 Empirical constants (C and m) derived from isochronous stress–strain curves for 316 stainless steel
Temperature (F/C)
Strain (%)
C (kJ/m 3)
m
Correlation coefficient
Temperature (F/C)
Strain (%)
C (kJ/m 3)
m
Correlation coefficient
950/510
0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0
228.07 547.69 872.84 1208.17 1552.69 208.89 506.87 828.85 1164.30 1509.83 176.71 436.78 728.91 1044.86 1379.74 149.29 368.24 612.11 882.36 1178.90 121.43 296.95 498.97 721.71 958.84 104.08 250.28 416.86 597.11 787.77
0.06809 0.05807 0.04878 0.04266 0.03834 0.07480 0.07404 0.06979 0.06565 0.06213 0.08076 0.08389 0.08382 0.08341 0.08306 0.08715 0.09721 0.09935 0.10130 0.10397 0.10877 0.11432 0.11763 0.12026 0.12198 0.12795 0.13418 0.13729 0.13924 0.14053
0.93 0.94 0.95 0.96 0.96 0.97 0.98 0.99 1.00 1.00 0.99 0.99 1.00 1.00 1.00 0.99 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.99 0.98 0.99 0.99 0.99 0.99
1000/538
0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0
276.49 636.52 1004.89 1384.38 1773.26 149.69 434.88 764.67 1114.03 1477.46 163.53 449.84 783.24 1146.88 1531.63 147.30 380.80 646.81 936.44 1241.51 105.75 264.93 444.89 643.06 852.80 63.72 160.54 276.05 405.64 543.22
0.05110 0.04444 0.03977 0.03683 0.03480 0.04718 0.05079 0.04899 0.04735 0.04613 0.08884 0.10088 0.10526 0.10758 0.10897 0.13378 0.14219 0.14529 0.14705 0.14805 0.16441 0.17090 0.17253 0.17438 0.17552 0.16837 0.18049 0.18914 0.19501 0.19812
0.92 0.91 0.90 0.90 0.90 0.90 0.92 0.93 0.97 0.98 0.96 0.99 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.99 1.00 1.00 1.00 1.00
1050/565
1150/621
1250/677
1350/732
1450/788
1100/593
1200/649
1300/704
1400/760
1500/815
Substituting in Eq. (3) and simplifying, W s1 ⫺ K·n= n ⫹ 11n⫹1
5
W s1= n ⫹ 1 Wc = n ⫹ 1:
6
It is seen from Eq. (6) that in a creep experiment at an applied stress s , the total work of creep WC for deformation up to a strain 1 is proportional to the stored energy of creep W corresponding to an isochronous stress–strain curve of time t, the time taken to attain the strain level 1 under the applied stress s . It is to be expected that Wc will also be related to time by a relation similar to Eq. (1) Wc tp B;
7
where B (n ⫹ 1)C and Wc s 1 (n ⫹ 1)W. The value of n for isochronous stress–strain curves of duration corresponding to the life-time of the component at a given temperature is difficult to estimate experimentally. The value of n is found to vary in the range 0.10–0.30 for various isochronous stress–strain curves and is expected to be always less than 0.5. Some of the NRIM creep data sheets [5–7] provide the data for the time taken to obtain a certain amount of total
strain at different creep testing conditions for various materials and their different heats. These data for 1% strain were analysed whenever more than three data points were available and found to obey Eq. (7) as shown in Fig. 3. The values of B and p are tabulated in Table 4. This procedure is applied for the literature data available at NRIM creep data sheets for different high-temperature materials where the time for 1% strain at different temperatures and stress levels are tabulated for different heats of the same class of materials. Table 5 shows the empirical constants and the stress for a limiting deformation of 1% strain in 10 5 h. 3. Conclusions An empirical relation between the stored energy of creep (W) of a creeping material to a specified strain level and the corresponding time taken (t) is found to exist in the form Wt m Constant. The stored energy of the creeping material pertaining to a specified strain level is found to decrease with increasing temperature and time. The empirical relation is also useful in the estimation of the stress intensity (St) for the limiting strain condition for
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K.G. Samuel, P. Rodriguez / International Journal of Pressure Vessels and Piping 75 (1998) 939–943
Table 3 Empirical constants (C and m) derived from isochronous stress–strain curves for 2.25Cr–1Mo steel Temperature (F/C)
Strain (%)
C (kJ/m 3)
800/427
0.16 0.33 0.50 0.67 0.83 1.00 0.16 0.33 0.50 0.67 0.83 1.00 0.16 0.33 0.50 0.67 0.83 1.00 0.16 0.33 0.50 0.67 0.83 1.00 0.20 0.40 0.60 0.80 1.00
247.21 643.67 1053.35 1487.45 1941.94 2422.60 288.01 726.76 1179.29 1703.32 2226.34 2780.02 204.63 537.16 957.36 1400.99 1859.71 2321.89 115.84 295.00 490.19 707.82 919.68 1149.32 87.44 233.31 398.57 579.21 791.04
900/482
1000/538
1100/593
1200/649
m
0.051204 0.038101 0.031909 0.029049 0.027477 0.026965 0.087308 0.087667 0.085738 0.084274 0.082329 0.083300 0.133926 0.136288 0.142489 0.144653 0.145325 0.143939 0.174413 0.163837 0.156012 0.154021 0.148630 0.146730 0.106460 0.117727 0.121434 0.124153 0.130913
Correlation coefficient 0.89 0.95 0.93 0.92 0.92 0.92 0.89 0.87 0.86 0.86 0.84 0.86 0.99 0.99 0.99 0.99 0.99 0.99 0.99 1.00 1.00 1.00 1.00 1.00 0.99 1.00 1.00 1.00 1.00
Table 4 Empirical constants (B and p) for the NRIM creep data Material
Temperature (⬚C)
B (kJ/m 3)
p
225Cr–1Mo (NRIM 3B)
500 550 600 500 600 650 700 750
3176 1708 11 025 14 978 5858 1976 1582 1210
0.12364 0.07227 0.33183 0.21134 0.20492 0.06502 0.09206 0.10586
1Cr–0.5Mo (NRIM 1B) 18Cr–12Ni–Mo (NRIM 42)
new and emerging class of high-temperature materials where creep and tensile data for empirical extrapolation to design life is lacking. Also it will help in assessing the stress intensity for the same class of material where there is a possibility of slight variation in mechanical properties and
Table 5 Empirical constants (B and p) for 1% strain and the stress for 1% strain in 10 5 h for the NRIM creep data
s
4256
0.06258
207
RBB RBC RBD RBE RBF RBG RBH RBJ RBA
4269 4589 3304 2776 3717 4369 4414 4184 2837
0.07546 0.09405 0.02882 0.02664 0.04400 0.07739 0.06566 0.06130 0.06394
179 155 237 204 224 179 207 207 136
RBB RBC RBG RBH RBJ VbA
2876 3072 2728 3031 3095 16 313
0.06902 0.09843 0.05422 0.06579 0.08300 0.14187
130 99 146 142 119 318
VbB VbD VbF VbG VbH VbJ VbM VbN VbA
11 209 22 356 23 135 5897 13 575 11 001 9127 6596 7725
0.12232 0.20309 0.17659 0.05283 0.11621 0.11785 0.10711 0.06922 0.12909
274 216 303 321 356 283 266 297 175
VbB VbD VbF VbG VbH VbJ VbM VbN
6003 6135 125 381 2813 9345 6977 5068 4791
0.09649 0.10752 0.41207 0.00888 0.12506 0.10903 0.08180 0.07231
198 178 109 254 221 199 198 208
Heat no.
12 Cr stainless steel, NRIM 13B/450⬚C
RBA
12 Cr stainless steel, NRIM 13B/500⬚C
1Cr–1Mo–0.25V steel, NRIM 31B/ 450⬚C
1Cr–1Mo–0.25V steel, NRIM 31B/ 500⬚C
Fig. 3. Variation of work of creep as a function of time at various temperatures for (a) 2.25Cr–1Mo steel; (b) 1Cr–0.5Mo steel; and (c) 18Cr–12Ni– Mo steel.
p
Material/temperature
B (kJ/m 3)
K.G. Samuel, P. Rodriguez / International Journal of Pressure Vessels and Piping 75 (1998) 939–943
the empirical constants due to chemistry, grain size, inclusion content, etc. Acknowledgements The authors wish to thank Dr C. Phaniraj for stimulating discussions during the course of this investigation. References [1] ASME Pressure Vessel Code—Nuclear Plants and Components, Div I. New York: ASME, 1989 [2] ASME Pressure Vessel Code—Section III, Div I—Class 1 Components in Elevated Temperature Service. New York: ASME, 1995.
943
[3] Schaefer AO, editor. The Generation of Isochronous Stress Strain Curves, Papers presented at the Winter Annual Meeting of the American Society of Mechanical Engineers, NY, 26–30 November 1972. New York: ASME. [4] ASME Code case N-47. New York: American Society of Mechanical Engineers. [5] NRIM Creep Data Sheet No 42. Data Sheet on the Elevated Temperature Stress Relaxation Properties of 18Cr–l2Mo Hot Rolled Stainless Steel Plates (SUS 316-HP). National Research Institute of Metals, 1996. [6] NRIM Creep Data Sheet No 1B. Data Sheet on the Elevated Properties of 1Cr–0.5Mo Steel Tubes for Boilers and Heat Exchangers (STBA 22). National Research Institute of Metals, 1996. [7] NRIM Creep Data Sheet No 3B. Data Sheet on the Elevated Temperature Stress Relaxation Properties of 2.25Cr–1Mo (Tubes). National Research Institute of Metals, 1996.
An empirical relation between strain energy and time in creep deformation K.G. Samuel, P. Rodriguez* Metallurgy and Materials Group, Indira Gandhi Centre for Atomic Research, Kalpakkam 603 102, India Received 6 July 1998; accepted 17 August 1998
Abstract
R The stored energy of creep defined as W 10 s d1 and estimated as the area under the isochronous stress–strain curve is found to vary with time t and temperature T (in particular Wt m C, where m and C are constants). The work of creep defined as Wc s1 also varies similarly (Wct p B where p and B are constants). From the relationships, a new method for generating isochronous stress–strain curves from three or more short-term creep tests is proposed. The method is useful particularly for newly developed materials. 䉷 1998 Elsevier Science Ltd. All rights reserved. Keywords: Creep deformation; Strain energy; Isochronous curves; Design stress
1. Introduction The basic philosophy in the design of engineering components is to guard against failure of the component. The basis for establishing the stress intensity values and allowable stress values for design are described in ASME Pressure Vessel Codes [1, 2]. At any temperature below the creep range, where deformation is time-independent, the maximum allowable stress value (Sm) for ferritic steels is the lowest of the following: (i) one-fourth of the specified minimum tensile strength at room temperature; (ii) one-fourth of the tensile strength at the temperature; (iii) two-thirds of the specified minimum yield strength at room temperature; (iv) two-thirds of the yield strength at the temperature. For austenitic steels the fourth condition is modified to be 90% of the yield strength at the temperature but not to exceed two-thirds of the specified minimum yield strength at room temperature. At elevated temperatures where time-dependent creep dominates, three failure modes are identified: (a) accumulation of a specified amount of plastic strain; (b) rupture of the component; and (c) onset of tertiary creep. A stress intensity limit (St) dependent on both temperature and time is considered here. The data considered in establishing the St values are obtained from long-term, constant load, uniaxial creep tests. For each specific time t, the St values shall be the lowest of: (a) 100% of the average stress required to obtain a total strain of 1%; (b) 67% of the minimum stress to cause
rupture; and (c) 80% of the minimum stress to cause initiation of tertiary creep. The allowable limit of general primary membrane stress intensity (Smt) to be used as a reference for stress calculations for the actual service life is the lowest of the two stress intensity values, Sm (time-independent) and St (time-dependent). Isochronous stress–strain curves are used for establishing the allowable design stress intensities for condition (a) in order to avoid excessive deformation over the intended service life. An isochronous curve presents, in a plot of stress against strain, the loci of total strain accumulated when different stresses are applied for a fixed time. For relatively short service lives (say a few thousand hours as in aircraft engine components) isochronous stress–strain curves can be developed by taking constant time section through a family of creep curves. The generation of isochronous curves for long service life (for example that of power plants or for more years) requires extrapolation of shortertime test results. Several approaches have been employed in developing isochronous curves [3]. This paper describes a new procedure for utilising the traditionally employed constant load creep tests to estimate the allowable design stress intensities St for limiting deformation condition based on the work done during creep in deforming the material.
2. Procedure * Corresponding author. Tel.: ⫹ 91-4114-40267; Fax: ⫹ 91-411440360; e-mail: [email protected]. 0308-0161/98/$ - see front matter 䉷 1998 Elsevier Science Ltd. All rights reserved. PII: S0308-016 1(98)00095-7
Average isochronous stress–strain curves for various
940
K.G. Samuel, P. Rodriguez / International Journal of Pressure Vessels and Piping 75 (1998) 939–943
Fig. 1. Average isochronous stress–strain curves for type 316 stainless steel at 1200 F (649⬚C).
high-temperature materials are available in the literature (Code case N-47) [4]. A typical set of curves for AISI 316 stainless steel is shown in Fig. 1. The areas under the isochronous stress–strain curves W at different strain levels were evaluated by counting the number of squares under the curve for 304, 316 and 2.25Cr–1Mo steels for various duration of time, t and found to obey a functional relation of the form Wtm C Constant;
1
where C and m are constants depending on the strain level and temperature. The value of m is estimated as d log W/ d log t. The functional relation is typically represented graphically in Fig. 2(a–c) for these materials. It is to be noted that unlike the stress–rupture time relation where normally a two slope behaviour is observed making the extrapolation of stress for long-term life from the shortterm accelerated tests unrealistic, W–t relationship obeyed a linear relation in a double logarithmic plot without any break in the relation up to 10 5 h for the strain levels and temperatures investigated. The values of C and m for various strain levels at different test temperatures are also summarised in Tables 1–3 for these materials. The relationship between W and t leads us to a new method to generate isochronous stress–strain curves from a few short-time creep tests as described below. This
Fig. 2. Variation of stored energy of creep as a function of time for various strain levels for (a) 2.25Cr–1Mo steel; (b) 316 stainless steel; and (c) 304 stainless steel.
method will be especially useful for newly developed materials with limited creep test data. Let us consider that a material has undergone creep deformation at a stress s to a strain 1 at time t. The work of creep Wc s1 . We also reach the point (s ,1 ) through the isochronous stress–strain curve for time t. The area under the R1 isochronous stress–strain curve W defined earlier is 0 s d1; W s1 ⫺ Wc ⫺
Zs 0
Zs 0
1 ds 3
1 d s:
If Wc is the work of creep, then W can be considered the stored energy of creep, the second term on the right hand side of Eq. (3) is the dissipated component of the work of creep. Assuming that a Hollomon type stress–strain relationship of the form s K1 n is valid for the isochronous stress– strain curves, where K is the work hardening coefficient and n is the work hardening exponent of the isochronous stress– strain curve, ds K·n·1n⫺1 d1:
4
K.G. Samuel, P. Rodriguez / International Journal of Pressure Vessels and Piping 75 (1998) 939–943 Table 1 Empirical constants (C and m) derived from isochronous stress–strain curves for 304 stainless steel
941
Table 2 Empirical constants (C and m) derived from isochronous stress–strain curves for 316 stainless steel
Temperature (F/C)
Strain (%)
C (kJ/m 3)
m
Correlation coefficient
Temperature (F/C)
Strain (%)
C (kJ/m 3)
m
Correlation coefficient
950/510
0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0
228.07 547.69 872.84 1208.17 1552.69 208.89 506.87 828.85 1164.30 1509.83 176.71 436.78 728.91 1044.86 1379.74 149.29 368.24 612.11 882.36 1178.90 121.43 296.95 498.97 721.71 958.84 104.08 250.28 416.86 597.11 787.77
0.06809 0.05807 0.04878 0.04266 0.03834 0.07480 0.07404 0.06979 0.06565 0.06213 0.08076 0.08389 0.08382 0.08341 0.08306 0.08715 0.09721 0.09935 0.10130 0.10397 0.10877 0.11432 0.11763 0.12026 0.12198 0.12795 0.13418 0.13729 0.13924 0.14053
0.93 0.94 0.95 0.96 0.96 0.97 0.98 0.99 1.00 1.00 0.99 0.99 1.00 1.00 1.00 0.99 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.99 0.98 0.99 0.99 0.99 0.99
1000/538
0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0
276.49 636.52 1004.89 1384.38 1773.26 149.69 434.88 764.67 1114.03 1477.46 163.53 449.84 783.24 1146.88 1531.63 147.30 380.80 646.81 936.44 1241.51 105.75 264.93 444.89 643.06 852.80 63.72 160.54 276.05 405.64 543.22
0.05110 0.04444 0.03977 0.03683 0.03480 0.04718 0.05079 0.04899 0.04735 0.04613 0.08884 0.10088 0.10526 0.10758 0.10897 0.13378 0.14219 0.14529 0.14705 0.14805 0.16441 0.17090 0.17253 0.17438 0.17552 0.16837 0.18049 0.18914 0.19501 0.19812
0.92 0.91 0.90 0.90 0.90 0.90 0.92 0.93 0.97 0.98 0.96 0.99 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.99 1.00 1.00 1.00 1.00
1050/565
1150/621
1250/677
1350/732
1450/788
1100/593
1200/649
1300/704
1400/760
1500/815
Substituting in Eq. (3) and simplifying, W s1 ⫺ K·n= n ⫹ 11n⫹1
5
W s1= n ⫹ 1 Wc = n ⫹ 1:
6
It is seen from Eq. (6) that in a creep experiment at an applied stress s , the total work of creep WC for deformation up to a strain 1 is proportional to the stored energy of creep W corresponding to an isochronous stress–strain curve of time t, the time taken to attain the strain level 1 under the applied stress s . It is to be expected that Wc will also be related to time by a relation similar to Eq. (1) Wc tp B;
7
where B (n ⫹ 1)C and Wc s 1 (n ⫹ 1)W. The value of n for isochronous stress–strain curves of duration corresponding to the life-time of the component at a given temperature is difficult to estimate experimentally. The value of n is found to vary in the range 0.10–0.30 for various isochronous stress–strain curves and is expected to be always less than 0.5. Some of the NRIM creep data sheets [5–7] provide the data for the time taken to obtain a certain amount of total
strain at different creep testing conditions for various materials and their different heats. These data for 1% strain were analysed whenever more than three data points were available and found to obey Eq. (7) as shown in Fig. 3. The values of B and p are tabulated in Table 4. This procedure is applied for the literature data available at NRIM creep data sheets for different high-temperature materials where the time for 1% strain at different temperatures and stress levels are tabulated for different heats of the same class of materials. Table 5 shows the empirical constants and the stress for a limiting deformation of 1% strain in 10 5 h. 3. Conclusions An empirical relation between the stored energy of creep (W) of a creeping material to a specified strain level and the corresponding time taken (t) is found to exist in the form Wt m Constant. The stored energy of the creeping material pertaining to a specified strain level is found to decrease with increasing temperature and time. The empirical relation is also useful in the estimation of the stress intensity (St) for the limiting strain condition for
942
K.G. Samuel, P. Rodriguez / International Journal of Pressure Vessels and Piping 75 (1998) 939–943
Table 3 Empirical constants (C and m) derived from isochronous stress–strain curves for 2.25Cr–1Mo steel Temperature (F/C)
Strain (%)
C (kJ/m 3)
800/427
0.16 0.33 0.50 0.67 0.83 1.00 0.16 0.33 0.50 0.67 0.83 1.00 0.16 0.33 0.50 0.67 0.83 1.00 0.16 0.33 0.50 0.67 0.83 1.00 0.20 0.40 0.60 0.80 1.00
247.21 643.67 1053.35 1487.45 1941.94 2422.60 288.01 726.76 1179.29 1703.32 2226.34 2780.02 204.63 537.16 957.36 1400.99 1859.71 2321.89 115.84 295.00 490.19 707.82 919.68 1149.32 87.44 233.31 398.57 579.21 791.04
900/482
1000/538
1100/593
1200/649
m
0.051204 0.038101 0.031909 0.029049 0.027477 0.026965 0.087308 0.087667 0.085738 0.084274 0.082329 0.083300 0.133926 0.136288 0.142489 0.144653 0.145325 0.143939 0.174413 0.163837 0.156012 0.154021 0.148630 0.146730 0.106460 0.117727 0.121434 0.124153 0.130913
Correlation coefficient 0.89 0.95 0.93 0.92 0.92 0.92 0.89 0.87 0.86 0.86 0.84 0.86 0.99 0.99 0.99 0.99 0.99 0.99 0.99 1.00 1.00 1.00 1.00 1.00 0.99 1.00 1.00 1.00 1.00
Table 4 Empirical constants (B and p) for the NRIM creep data Material
Temperature (⬚C)
B (kJ/m 3)
p
225Cr–1Mo (NRIM 3B)
500 550 600 500 600 650 700 750
3176 1708 11 025 14 978 5858 1976 1582 1210
0.12364 0.07227 0.33183 0.21134 0.20492 0.06502 0.09206 0.10586
1Cr–0.5Mo (NRIM 1B) 18Cr–12Ni–Mo (NRIM 42)
new and emerging class of high-temperature materials where creep and tensile data for empirical extrapolation to design life is lacking. Also it will help in assessing the stress intensity for the same class of material where there is a possibility of slight variation in mechanical properties and
Table 5 Empirical constants (B and p) for 1% strain and the stress for 1% strain in 10 5 h for the NRIM creep data
s
4256
0.06258
207
RBB RBC RBD RBE RBF RBG RBH RBJ RBA
4269 4589 3304 2776 3717 4369 4414 4184 2837
0.07546 0.09405 0.02882 0.02664 0.04400 0.07739 0.06566 0.06130 0.06394
179 155 237 204 224 179 207 207 136
RBB RBC RBG RBH RBJ VbA
2876 3072 2728 3031 3095 16 313
0.06902 0.09843 0.05422 0.06579 0.08300 0.14187
130 99 146 142 119 318
VbB VbD VbF VbG VbH VbJ VbM VbN VbA
11 209 22 356 23 135 5897 13 575 11 001 9127 6596 7725
0.12232 0.20309 0.17659 0.05283 0.11621 0.11785 0.10711 0.06922 0.12909
274 216 303 321 356 283 266 297 175
VbB VbD VbF VbG VbH VbJ VbM VbN
6003 6135 125 381 2813 9345 6977 5068 4791
0.09649 0.10752 0.41207 0.00888 0.12506 0.10903 0.08180 0.07231
198 178 109 254 221 199 198 208
Heat no.
12 Cr stainless steel, NRIM 13B/450⬚C
RBA
12 Cr stainless steel, NRIM 13B/500⬚C
1Cr–1Mo–0.25V steel, NRIM 31B/ 450⬚C
1Cr–1Mo–0.25V steel, NRIM 31B/ 500⬚C
Fig. 3. Variation of work of creep as a function of time at various temperatures for (a) 2.25Cr–1Mo steel; (b) 1Cr–0.5Mo steel; and (c) 18Cr–12Ni– Mo steel.
p
Material/temperature
B (kJ/m 3)
K.G. Samuel, P. Rodriguez / International Journal of Pressure Vessels and Piping 75 (1998) 939–943
the empirical constants due to chemistry, grain size, inclusion content, etc. Acknowledgements The authors wish to thank Dr C. Phaniraj for stimulating discussions during the course of this investigation. References [1] ASME Pressure Vessel Code—Nuclear Plants and Components, Div I. New York: ASME, 1989 [2] ASME Pressure Vessel Code—Section III, Div I—Class 1 Components in Elevated Temperature Service. New York: ASME, 1995.
943
[3] Schaefer AO, editor. The Generation of Isochronous Stress Strain Curves, Papers presented at the Winter Annual Meeting of the American Society of Mechanical Engineers, NY, 26–30 November 1972. New York: ASME. [4] ASME Code case N-47. New York: American Society of Mechanical Engineers. [5] NRIM Creep Data Sheet No 42. Data Sheet on the Elevated Temperature Stress Relaxation Properties of 18Cr–l2Mo Hot Rolled Stainless Steel Plates (SUS 316-HP). National Research Institute of Metals, 1996. [6] NRIM Creep Data Sheet No 1B. Data Sheet on the Elevated Properties of 1Cr–0.5Mo Steel Tubes for Boilers and Heat Exchangers (STBA 22). National Research Institute of Metals, 1996. [7] NRIM Creep Data Sheet No 3B. Data Sheet on the Elevated Temperature Stress Relaxation Properties of 2.25Cr–1Mo (Tubes). National Research Institute of Metals, 1996.