## Abstract

This technical note describes a minimum creep rate model for 2-1/4Cr-1Mo steel that is consistent with the current creep strain equation embedded in the ASME Boiler & Pressure Vessel Code Section III, Division 5, Subsection HB, Subpart B isochronous stress–strain curves. Minimum creep rate models for all the Section III, Division 5 Class A materials are required for the development of improved high temperature design methods. Of all the Class A materials, only 2-1/4Cr-1Mo does not have a readily identifying minimum creep rate term in the current isochronous stress–strain curve model.

## 1 Background

As part of recent Code modernization efforts, the ASME Boiler & Pressure Vessel Code Subgroup on High Temperature Reactors identified the need for minimum creep rate models for the Section III, Division 5, Subsection HB, Subpart B Class A materials in a new evaluation procedure for the primary load check that does not involve stress classification. The 2021 Code edition of the Subsection HB, Subpart B rules contains isochronous stress–strain relations for these materials. The equations underlying these curves are uniaxial, monotonic constitutive models of the form
$ε=εelastic+εplastic+εcreep$
(1)

where $εelastic$ is the elastic strain, $εplastic$ is the time-independent plastic strain, and $εcreep$ is the creep strain. For all the Class A materials except 2-1/4Cr-1Mo, the creep deformation model, $εcreep$, contains a simple, definite minimum creep rate term that can be extracted from the full creep strain equation to form the required minimum creep rate model.

The new primary load design procedure explicitly calculates stress relaxation damage due to secondary stresses using an elastic-creep constitutive model and the material minimum stress to rupture. The design method then requires a conservative estimate of stress relaxation in the material, i.e., a creep model that produces higher stresses and more creep damage than the actual relaxation. Basing the relaxation analysis on the material minimum creep rate is therefore an appropriate, conservative approximation.

The creep model underlying the Subsection HB, Subpart B isochronous stress–strain curves for 2-1/4Cr-1Mo originates with Refs. [1] and [2]. The following equations describe the model:

$εc={0T<371ε2/100ε2<ε1ε1/100t
with
$ε1=C1p1t1+p1t+ε˙1(ss)t$
(2)
$ε2=C2p2t1+p2t+ε˙2(ss)t$
(3)
$ε3=C2p2t′1+p2t′+ε˙2(ss)t$
(4)

where εc, t, and T represent creep strain, time in h, and temperature in °C, respectively. The remaining symbols represent parameters, with the parameter definitions taken from Section III, Division 5 of the ASME Boiler & Pressure Vessel Code [3] and the original model references. The appendix defines these parameter values, including the definition of the modified time $t′$, which have complicated mathematical definitions.

The general model form is a fairly straightforward decreasing primary creep rate plus constant secondary creep rate model. The multiple regions of behavior capture the nonclassical shape of the creep curves observed in Ref. [2] for 2.25Cr-1Mo steel. This region-splitting means we cannot extract a simple minimum creep rate model directly from the original equations.

## 2 Model

The model minimum creep rate is the minimum of $ε˙1(ss)$ and $ε˙2(ss)$, which are the time-independent terms in the creep rate equations. In this work, we first find the region in temperature–stress space corresponding to the lower of the $ε˙1(ss)$ or $ε˙2(ss)$ such that we can use it to simplify the minimum creep-rate equation for 2-1/4Cr-1Mo. We consider temperatures between 371 and 649 °C, based on the Section III, Division 5 temperature limits expressed in table HBB-T-1834-1 and stresses between 0 and 800 MPa to identify the bounding surface separating the minimum creep rate between $ε˙1(ss)$ and $ε˙2(ss)$. The upper limit of stress value (800 MPa) is twice the maximum stress value (414 MPa) reported for the current isochronous stress–strain curves for 2-1/4Cr-1Mo and therefore bounds all operating stress conditions for 2-1/4Cr-1Mo.

This work numerically discretizes the space to 500 points in each of temperature and stress dimensions, and then identifies the minimum rate between $ε˙1(ss)$ and $ε˙2(ss)$. Figure 1 shows the boundary separating the minimum between the two creep rate expressions.

Fig. 1
Fig. 1
Close modal

To generate a simple minimum creep rate, we derive an algebraic expression for the boundary curve between the two regions. To do so, we divide the region into an approximate linear region (linear part in Fig. 2) and a polynomial region (polynomial part in Fig. 2) and fit the boundary using least square regression. Figure 2 (black curve) shows the result, which accurate capture the exact boundary. Applying the algebraic equation for the boundary, the reduced minimum creep rate formulation for 2-1/4Cr-1Mo is

$ε˙c={ε˙2(ss)if σ<60.0ε˙hif σ≥60.0$

with

$ε˙h={ε˙1(ss)if T≤13.571 σ0.68127−1.8σ+437.63ε˙2(ss)if T>13.571 σ0.68127−1.8σ+437.63$
where
$ε˙1(ss)=10(6.7475+0.011426σ+987.72U log10σ−13494T+273.15)100$
(5)
$ε˙2(ss)=10(11.498−8.2226UT+273.15−20448T+273.15+5862.4T+273.15 log10σ)100$
(6)
Fig. 2
Fig. 2
Close modal

The definitions of $ε˙1(ss)$ and $ε˙2(ss)$ come from the original references, defined in full in the appendix. Table 1 lists the values of U at different temperatures [1,2]. We use linear interpolation to obtain U values for intermediate temperatures not listed in Table 1.

Table 1

Temperature-dependent values for parameter U in creep rates

T (°C)U
371471
400468
450452
500418
550364
600284
621300
649270
T (°C)U
371471
400468
450452
500418
550364
600284
621300
649270

Figure 3 shows a fringe plot of the ratio between the creep rates obtained using the new, simplified minimum creep rate model and the exact creep rates from the full set of equations for creep times of 10, 100, 100,000, and 300,000 h, along with the boundary separating the minimum creep rate regions (black curve). The shaded regions in the plot are temperature–stress combinations where the current formulation does not provide a lower bound to the full creep rate. These regions occur only along the boundary between the linear and polynomial regions and the differences between the full and simplified models are not significant compared to the uncertainty in actual experimental measurements of the minimum creep rate. Except in these very narrow regions, the new minimum creep rate model conservatively bounds the full creep rate expression. The full creep rate and the new minimum creep rate models agree exactly for long times, which is to be expected as the primary creep contribution in the full model goes to zero.

Fig. 3
Fig. 3
Close modal

## Acknowledgment

The research was sponsored by the U.S. Department of Energy (DOE), under Contract No. DE-AC02-06CH11357 with Argonne National Laboratory, managed and operated by UChicago Argonne LLC. Programmatic direction was provided by the Office of Nuclear Reactor Deployment of the Office of Nuclear Energy (NE).

## Funding Data

• U.S. Department of Energy (Contract No. DE-AC02-06CH11357; Funder ID: 10.13039/100000015).

### Appendix: Creep Model Parameters

The parameters given in this appendix are in units consistent with the unit scheme described in the main body: strain in mm/mm, time in h, and temperature in °C. The symbol $log$ refers to the base 10 logarithm. Where the symbol U appears, it refers to the temperature-dependent values given by interpolation in Table 1.

Parameters for$ε1$
$C1=10d1$
(A1)
$d1=1.0328+168,680U(T+273.15)−0.023772U+0.0079141U log σ$
(A2)
$p1=10q1$
(A3)
$q1=7.6026+3.3396 log σ−12,323T+273.15$
(A4)
$ε˙1(ss)=10e1$
(A5)
$e1=6.7475+0.011426σ+987.72Ulog σ−13,494T+273.15$
(A6)
Parameters for$ε2$
$C2=10d2$
(A7)
$d2=−0.051086+140,730U(T+273.15)−0.01U+0.0037345U log σ$
(A8)
$p2=10q2$
(A9)
$q2=8.1242+0.0179678σ+404.63Ulog σ−11,659T+273.15$
(A10)
$ε˙2(ss)=10e2$
(A11)
$e2=11.498−8.2226UT+273.15−20448T+273.15+5862.4T+273.15log σ$
(A12)
Parameters for t1
$t1={t1(a)T≤454t1(a)+t1(b)−t1(a)56(T−454)454510$
(A13)
$t1(a)=10a$
(A14)
$a=−13.528+6.5196UT+273.15+23,349T+273.15−5693.8T+273.15log σ$
(A15)
$t1(b)=10b$
(A16)
$b=−11.098−4.0951σU+11,965T+273.15$
(A17)
Parameters for$t′$
$t′=t−(t1−tc)$
(A18)
$tc=−ε˙2(ss)−C2p2+ε1′p2+4ε1′ε˙2(ss)p2+(ε˙2(ss)+(C2−ε1′)p2)22ε˙2(ss)p2$
(A19)
$ε1′=C1p1t11+p1t1+ε˙1(ss)t1$
(A20)

## References

1.
Klueh
,
R.
, and
Hebble
,
T.
,
1976
, “
A Mathematical Description for the Stress Strain Behavior of Annealed 2-1/4 Cr-1 Mo Steel
,”
ASME J. Pressure Vessel Technol.
,
98
(
2
), pp.
118
125
.10.1115/1.3454348
2.
Booker
,
M.
,
1977
, “
Analytical Description of the Effects of Melting Practice and Heat Treatment on the Creep Properties of 2 1/4 Cr-1 Mo Steel
,”
Effects of Melting and Processing Variables on the Mechanical Properties of Steel
,
G. V.
Smith
, ed.,
The American Society of Mechanical Engineers
,
New York
, pp.
1
44
.
3.
ASME
,
2021
, “
Section III, Division 5, Subsection HB, Subpart B, Boiler and Pressure Vessel Code
,”
American Society of Mechanical Engineers
,
New York
, Standard.