The creep–rupture data for alloy S-590 plotted in Fig. 15.8 are given in Table E15.4(a), where, for each test, the temperature T, stress σ, and rupture time tr are listed. Employ these data as follows: (a) On the basis of Eq. 15.19, obtain a fitted equation relating the Sherby–Dorn parameter to

Question

The creep–rupture data for alloy S-590 plotted in Fig. 15.8 are given in Table E15.4(a), where, for each test, the temperature T, stress σ, and rupture time [latex]t_{r}[/latex] are listed. Employ these data as follows:

(a) On the basis of Eq. 15.19,

[latex]\begin{gathered}\log t_r=P_{S D}+\frac{0.217 Q}{T} \quad \quad (a) \ P_{S D}=a_0+a_1 x+a_2 x^2+a_3 x^3 \quad \quad (b)\end{gathered}[/latex] (15.19)

obtain a fitted equation relating the Sherby–Dorn parameter to stress. Consider Q = 85,000 cal/mole from Table 15.2 to be given.

(b) Similarly, use Eq. 15.20

[latex]\begin{gathered}\log t_r=\frac{P_{L M}}{T}-C \quad \quad (a) \ P_{L M}=b_0+b_1 x+b_2 x^2+b_3 x^3 \quad \quad (b)\end{gathered}[/latex] (15.20)

to obtain a fitted equation relating the Larson–Miller parameter to stress, with C = 17 log (h) from Table 15.3 given.

Accepted Answer

(a) First calculate x = log σ and [latex]P_{SD}[/latex] for each data point—that is, for each T, σ, [latex]t_{r}[/latex] combination in Table E15.4(a). The [latex]P_{SD}[/latex] values are obtained by substituting [latex]t_{r}[/latex] and T into Eq. 15.16,

[latex]P_{S D}=\log t_r-\frac{0.217 Q}{T}[/latex] (15.16)

with the first few calculation results shown in Table E15.4(b). Then employ the full set of data to fit [latex]P_{SD} = f (x)[/latex] to the cubic polynomial of Eq. 15.19(b). The result is

[latex]P_{S D}=-12.35+4.42 x-2.287 x^2-0.1292 x^3, \quad x=\log \sigma[/latex] (units: h, K, MPa)

A reasonable fit to the data is obtained, with the curve plotted in Fig. 15.21 corresponding to this equation.

(b) The fit for the Larson–Miller parameter proceeds similarly, and the first few values of [latex]P_{LM}[/latex] from Eq. 15.17

[latex]P_{L M}=T\left(\log t_r+C\right)[/latex] (15.17)

are also shown in Table E15.4(b). However, in Fig. 15.23, where [latex]P_{LM}[/latex] is plotted against σ on a log scale, the data appear to lie along a straight line. It was thus assumed that [latex]b_{3} = b_{4} = 0[/latex] in Eq. 15.20(b), resulting in a simple linear equation for the fit.

[latex]P_{L M}=38,405-8206 x, \quad x=\log \sigma[/latex] (units: h, K, MPa)

This line is the one plotted in Fig. 15.23, where it is seen to represent the data well.

Table E15.4(a)
T K	σ MPa	$t_{r}$ hours	T K	σ MPa	$t_{r}$ hours	T K	σ MPa	$t_{r}$ hours
811	690	22	922	345	93	1005	121	16 964
811	621	109	922	310	192	1089	172	25
811	552	433	922	276	756	1089	155	88
811	483	1677	922	241	2 243	1089	138	267
866	552	25	922	207	11 937	1089	121	719
866	517	44	922	172	43 978	1089	103	1 354
866	483	109	1005	234	59	1089	86	5 052
866	414	264	1005	193	342	1089	69	15 335
866	345	3149	1005	172	809	1089	69	11 257
922	414	26	1005	172	1028	—	—	—
922	379	63	1005	138	9 529	—	—	—
Source: Data in [Goldhoff 59a].

Table 15.2 Activation Energies for the Sherby–Dorn Parameter
Material	1Cr-1Mo-0.25V steel	A-286 Fe-Ni-Cr alloy	S-590 Fe-Cr-Ni-Co alloy	Nimonic 80A Ni-base alloy
Q, cal/mole	110,000	91,000	85,000	91,000
Source: Data in [Conway 69], [Goldhoff 59a], and [Goldhoff 59b].

Table 15.3 Constants for the Larson–Miller Parameter
	C	Polynomial Fit (units of hours, K, and MPa)				Range of L-M Fit
Material	log (hours)	$b_{0}$	$b_{1}$	$b_{2}$	$b_{3}$	σ, Mpa	T, K
1Cr-1Mo-0.25V steel	22	128,200	−141,500	64,380	−9,960	69–621	755–1005
AISI 310 stainless steel	10	20,470	-4,655	0	0	3.4–31	1255–1366
A-286 (Fe-Ni-Cr alloy)	20	116,400	−120,500	53,460	-8,188	69–758	811–1089
S-590 (Fe-Cr-Ni-Co alloy)	17	38,405	−8,206	0	0	69–690	811–1089
Nimonic 80A (Ni-base alloy)	18	16,510	11,040	−4,856	403	54–486	923–1089
Source: Values and data in [Conway 69], [Goldhoff 59a], [Goldhoff 59b], [Larson 52], [Orr 54], and [van Echo 67].

Question 15.4: The creep–rupture data for alloy S-590 plotted in Fig. 15.8 ...

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