Primary and Low-Strain Creep Models for 9Cr Tempered Martensitic Steels Including the Effects of Irradiation Softening and High-Helium Re-Hardening

Abstract

Primary and low-strain creep represents a very important integrity challenge to large, complex structures, like fusion reactors. Here, we develop a predictive empirical primary creep model for 9Cr tempered martensitic steels (TMS), relating the applied stress (σ) to strain (ε), time (t) and temperature (T). The most accurate model is based on the applied σ normalized by the steel’s T-dependent ultimate tensile stress (σ_o), σ/σ_o(T). The model, fit to 17 heats of 9Cr TMS, yielded a σ root mean square error (RMSE) of ≈±11 MPa. Notably, the model also provides robust predictions for all the other TMS, when calibrated only by the fusion candidate Eurofer97 database. The model was extended to explore two possible effects of neutron irradiation, which produces both displacements per atom (dpa) and helium (He in atomic parts per million, appm) damage. These effects, which have not been previously considered, include: (a) softening, as a function of dpa, at T > ≈400–450 °C, in low-He fission environments (<1 He/dpa); and (b) subsequent re-hardening in high-He (≥10 He/dpa) fusion first-wall environments. The irradiation effect models predict (a) accelerated primary creep due to irradiation softening; and (b) fully arrested creep due to high-He re-hardening.

1. Introduction and Background

Thermal creep predictions typically focus on minimum rates and rupture times. However, rapidly accumulating primary and low-strain creep are absolutely critical for the large interconnected, gradient-dominated structures, like fusion reactors, which require both conservative safety margins and precise dimensional tolerances and stability [1]. Here, we develop an empirical primary and low-strain creep model, for 9Cr tempered martensitic steels (TMS), relating the applied stress (σ) to strains (ε) from 0.2 to 5%, times (t) from 10 h to 8 × 10⁴ h and temperatures (T) from 375 to 550 °C. Note that while the primary creep regime is generally restricted to ε of ≈1–3%, our model accurately extends to 5%, which is often in, or close to, the tertiary creep regime. However, for convenience, we will subsequently refer to our model as being for primary creep.

It is well-known that neutron irradiation extends the time-dependent accumulation of ε down to a T range that is much lower than that for thermal creep [2,3]. This, so-called, irradiation creep (IC) is not addressed in this work. However, thermal creep can also be affected by irradiation-service. Thus, for the first time, we consider neutron irradiation-enhanced softening of TMS at T >≈ 400 to 450 °C. We further show that fusion reactor environments lead to re-hardening, above a threshold transmutant He content, essentially arresting creep. Here, we focus on these irradiation effects in a fusion candidate TMS, reduced-activation ferritic–martensitic (RAFM) steel Eurofer97.

Notably, outstanding summaries of a wide range of RAFM property data and correlations, developed over many decades of international research, do not, or only minimally, address primary creep [4,5,6,7]. Thus, new, quantitatively predictive primary creep models, including the effects of irradiation, are needed for the design and operation of fission and fusion structures, which can functionally fail due to modest dimensional instabilities. These models are not only needed for assuring Design and Regulatory Code-imposed σ limits but also for constitutive models such as reported in [1]. The quantitatively predictive models developed here are intended to begin filling this major fusion energy property gap.

The primary creep model was derived in the following sequence. Various data-fitting models were first evaluated, including (i) the Larson–Miller parameter (LMP); (ii) the applied stress (σ) normalized to the temperature (T)-dependent either 0.2% tensile yield stress [σ/σ_y(T)] or ultimate tensile stress σ_o [σ/σ_o(T)], as measured in static tensile tests; and (iii) a threshold stress that scales with either σ_y or σ_o, as (σ − Cσ_y) or (σ − Cσ_o). We found that the σ/σ_o(T) model, championed by Wilshire et al. [8], performed the best. The primary creep models were least square fit (LSF) to a database for 17 heats of 9Cr TMS [9,10,11,12], resulting in a predicted σ RMSE of ≈±11 MPa and a mean error (ME) of ≈0. The effect of irradiation softening in low-He fission environments was also tested, using a simple empirical model for increases in σ/σ_o, found by fitting the corresponding decrease in σ_o, with the log (dpa). Furthermore, primary creep rates in fusion environments were predicted by fitting He-hardening data. The threshold dpa for He hardening is higher than that for softening. Thus, due to re-hardening beyond the threshold He, σ/σ_o decreases, following an initial softening increase, depending on the He/dpa ratio. The model predicts that re-hardening essentially arrests creep, at a dpa that depends on the saturation primary strain (ε_xs), σ, T and He/dpa. The fitting models and methods are described in the following section.

Typical creep behavior is schematically illustrated in Figure 1a, showing ε as a function of the log of time (t) for increasing T and σ [13,14]. Figure 1b illustrates that creep may manifest secondary either steady-state or minimum rates (ε′), with the latter behavior typical of 9Cr TMS. Figure 1c shows the σ dependence of the time (t_x) to ε_x = 1%, for Eurofer97, fit to a simple Norton power law creep model, t_x = ε_x/[Aσ^p exp (−Q/RT)], with a typically large p ≈ 18 [8,9]. We found that ε and ε′, as a function of σ/σ_o, are very similar for various TMS.

Figure 1 illustrates that there are three regimes, of primary, minimum (or secondary) and tertiary creep. Here, we mainly focus on primary creep rates, which rapidly decrease with strain, as illustrated in Figure 1d, for widely observed 9Cr TMS, such as in 9Cr-1-4W TMS [15]. TMS typically exhibit primary creep behavior, typically up to $\mathsf{\epsilon }$ ≈ 2%, followed by rapidly increasing tertiary creep, typically starting by ≈5% [15]. Hence, from a design and operation perspective, primary and low-strain creep is very important for avoiding functional failure due to dimensional instabilities, especially in large and complex structures, like fusion first-wall energy conversion systems [1].

A detailed discussion of the mechanisms controlling primary creep is beyond the scope of this paper. However, briefly, the primary ε′ down to a minimum (or secondary) creep rate (ε′_m) can be phenomenologically represented by

ε′(ε) = ε′_m + (ε′_o − ε′_m) exp (−ε/ε_xp)

(1)

Here, ε′_o is the initial creep rate as t → 0, and ε_xp is a primary creep relaxation strain, which is a phenomenologically fit parameter. The initial decrease in (ε′_o − ε′_m) is mainly due to hardening, associated with the build-up of dislocation structures, in a manner roughly analogous to strain hardening in static tensile tests [8,13,14,16]. This build-up reaches a steady-state condition, at ε′ = ε′_m, when the generation and annihilation of dislocations balance [14,16]. Precipitation hardening can also reduce the rate of primary creep with an increasing ε [15]. Note that this phenomenological description is sometimes couched in terms of decaying internal back stress [14,15].

Thus, primary creep rates are mediated by an evolving alloy microstructure, which is also, to some extent, reflected in the initial strength and strain-hardening rates observed in static tensile tests [8]. Although quantitative details certainly differ, and are a function of both T and t, normalizing σ by σ_o captures some of these physics [8]. Another advantage of using normalized σ/σ_o is that it permits the use of the model for various 9Cr TMS, with a corresponding range of static σ_o. Finally, σ/σ_o models also allow treating time-dependent changes in σ_o, here, those due to the effects of fission and fusion neutron irradiation.

2. Materials and Methods

2.1. Materials and the Creep Database

The creep database used for fitting the σ/σ_o model was composed of 4 alloy classes, including 8 heats from the Rieth et al. compilation for fusion candidate TMS 9Cr-1W Eurofer97 [9]. The database also included 4 heats of 9Cr-1Mo-V-Nb (Gr91 of T91/P91) from the National Institute for Materials Science (NIMS, Tsukuba, Japan) [10], 4 heats of NIMS 9Cr-0.5Mo-1.8W-V-Nb (Gr92 of T92/P92) [11], and 1 heat of Holmstrom’s 9Cr-1Mo-V-N (Gr91) [12].

The chemical compositions of the alloy classes used in this study are shown in

Table 1. Compositions (wt.%) of Eurofer97 [9] and Gr91/92 steels [10,11,12] used in this study.

Comp	C	Cr	Mo	V	Nb	Mn	Ni	Si	W	Ta	Cu	N	S	Ti
Eu97	0.09–0.12	8.1–9.5	0.005	0.15–0.25	0.001	0.2–0.6	0.005	0.05	1–1.2	0.05–0.09	0.005	0.03	0.005	0.01
NIMS-Gr91	0.09–0.1	8.5–8.7	0.9–0.96	0.21	0.076	0.35–0.45	0.04–0.28	0.24–0.38	-	-	0.012–0.032	0.042–0.058	0.001	-
NIMS-Gr92	0.092–0.11	8.91–9.5	0.36–0.44	0.16–0.2	0.05–0.062	0.41–0.44	0.13–0.27	0.1–0.29	1.68–1.85	-	-	0.039–0.046	0.0003–0.0037	0.003
Holms-Gr91	0.12	8.32	1.02	0.235	0.084	0.41	0.1	0.24	-	-	0.05	0.041	0.001	0.002

EU97: Eurofer97, Gr91: Grade 91, Gr92: Grade 92.

, while the details of their heat treatment procedures can be found in [9,10,11,12]. TMS are characterized by a tempered martensitic microstructure, with lath-packets in larger prior austenite grains. Heat treatments vary. For example, the reference condition of TMS Eurofer97 is austenitizing at ≈1040 °C for 4 h, air-cooling (normalizing) and tempering at ≈750 °C for 1 h, leading to a room-temperature (RT) yield stress (σ_y) of ≈550 ± 50 MPa and ultimate tensile stress (σ_o) of ≈700 ± 50 MPa [9,10,11,12,17]. More generally, Gr91/92 steels have a range of σ_y and σ_o and contain a variety of precipitates, including secondary hardening carbides, like Mo₂C, as well as other metal carbonitrides, like TaC [5,10,11,18,19,20,21]. At high temperatures and service stresses, these microstructures evolve over time in association with processes like a loss of matrix-strengthening solutes to coarse-scale precipitates, coarsening of fine-scale precipitates, dislocation and subgrain structural recovery, and, in some cases, lath-scale instabilities [15,19,21,22,23,24,25,26].

Notably, these evolutions occur at lower temperatures and shorter times under neutron irradiation [22]. This is believed to be primarily due to radiation-enhanced diffusion (RED) [22,27]. Note here that we do not include advanced variants of 9Cr TMS, which are strengthened by very-fine-scale alloy (M) carbide, nitride and oxide MX phases [28]. Rather, we mainly focus on Eurofer97, which is in an alloy class often referred to as reduced-activation ferritic–martensitic (RAFM) steels [21,29,30]. Reduced activation is achieved by substituting W for Mo and controlling the contents of various alloying elements and impurities to reduce the inventory of long-lived radioactive isotopes [30,31]. Establishing the close relationship between fusion-relevant RAFM and Gr91/92 steels is of great benefit, since the latter have been used since the early 1980s, are ASME BPVC Section 3 Division 5 [32,33] qualified and are resistant to void swelling [21,33,34]. Our 17 heat TMS primary creep database covers a wide range of strains, ε_x, for x from ≈0.2 to 5%, T from 375 to 550 °C, σ from 160 to 420 MPa (depending on the T) and t_x from <10 to 8 × 10⁴ h [9,10,11,12]. Thus, as shown below, it is particularly important that the σ/σ_o model calibrated only to the Eurofer97 database predicts the entire 17 heat TMS database with high accuracy. This enables the use of regulatory and design guidance on issues like negligible creep and deformation primary creep limit conditions, such as ε_x = 0.2% and 1%, respectively.

Note that our compiled database also includes other creep properties, which have also been modeled. Our database does not generally include the full creep curves, so discrete σ, t, T, ε_x and t_x data points were used for LSF optimization of the models. The LSF criteria include a minimum predicted (P) minus measured (M) σ RMSE, augmented by a negligible mean error (ME), the absence of residual bias for the key variables and 1/1 fitting of the P versus M σ/σ_o, with a 0/0 intercept.

2.2. The σ/σ_o Model for Primary Thermal Creep

As noted above, Wilshire et al. [8] proposed a creep model based on normalizing the applied σ with the temperature-dependent ultimate tensile stress, σ_o(T), as follows:

ε′ = A(σ/σ_o)ⁿ exp(−Q/RT).

(2)

A is a fit constant and Q an activation energy, generally related to, but not fully controlled by, self-diffusion [8,13,14]. The time increment (t_x) needed to accumulate a strain of ε_x where A(σ/σ_o)ⁿ exp(−Q/RT) is the average ε′ is as follows:

ε_x = A(σ/σ_o)ⁿ exp(−Q/RT)t_x.

(3a)

This can be re-arranged as follows:

ε_x/[Aexp(−Q/RT)t_x] = [σ/σ_o]ⁿ.

(3b)

The [σ/σ_o] can then be expressed in terms of the log of a temperature-compensated time (t_xc) as follows:

σ/σ_o = {ε_x/[Aexp(−Q/RT)t_x]}^1/n = [ε_x/A]^1/n [t_xc]^−1/n.

(3c)

where t_xc = t_x exp(−Q/RT). The normalized stress, σ/σ_o, expression can be further re-arranged as follows:

log[σ/σ_o] = [1/n]log[ε_x/A] − [1/n]log[t_x exp(−Q/RT)] = k(ε_x) − log(t_xc)/n(ε_x).

(3d)

The k and n are the LSF-optimized parameters for σ/σ_o vs. t_xc data on a log–log scale. For a given Q and T, the fitted k and n only depend on ε_x. Figure 2a shows that log(σ/σ_o) is a linear function of the log(t_xc) at various ε_x. Here, k is the fit constant at σ/σ_o = 1 and 1/n is the log–log slope, again which were found by fitting the log(σ/σ_o) and log(t_xc) data at various ε_x. The resulting functions for k(ε_x), n(ε_x) were then modeled by cross-fitting to ε_x, using a second-order polynomial function as follows:

log[σ(ε_x,T)/σ_o(T)] = {C₀ + C₁ε_x + C₂ε_x² + (C₃ + C₄ε_x + C₅ε_x²)log[t_x exp(−Q/RT)]

(4a)

σ(ε_x,T) = σ_o(T)10^{C₀ ^{+ C}₁^ε_x ^{+ C}₂^ε_x^{2 + (C}₃ ^{+ C}₄^ε_x ^{+ C}₅^ε_x^2)log[tx ^{exp(−Q/RT)]}}

(4b)

Note that the LSF also simultaneously optimizes Q. Thus, Equations (4a) and (4b) constitutes a primary creep model, relating σ(ε_x,T) to ε_x, t_x and T, with known values for C_1–6 and Q. For the σ(ε_x,T)/σ_o(T) shown in Figure 2a, the best-fit Q = 335 kJ/mole.

2.3. Static Tensile Properties

The tensile σ_o and σ_y data used in this study are shown in Figure 3 [9,10,11,12] for T from 375 to 550 °C, marking the likely use limits of standard 9Cr TMS for extended irradiation services. The σ_o and σ_y for T that were not directly available were interpolated between the two adjacent values. Figure 3 shows that both the σ_y and σ_o differ by about 80 to 180 MPa for the various TMS. These variations reflect differences in the TMS compositions and, especially, their heat treatments [9,10,11,12].

2.4. Softening Model

There is modest literature on the softening of 9Cr TMS under long-term high-temperature aging, typically at T > 550–600 °C for 1000 to 100,000 h or more [20,21,25,35,36,37,38,39]. Notably, thermal softening is recognized in ASME and other design and safety codes [32,33]. Again, however, neutron irradiation shifts thermal-aging effects to a lower T and shorter t [22]. Note, most reported data for irradiation hardening and softening are for Δσ_y, rather than Δσ_o. Both thermal-aging and irradiation effects can be expressed in terms of the irradiated (i) and unirradiated (u) yield and ultimate stress ratios, σ_yi/σ_yu and or σ_oi/σ_ou, which are a strong function of T, as derived from the data in [40,41,42,43,44,45,46,47,48,49,50]. Here, the σ_yi and σ_yu data are for similar irradiation and tensile test temperatures. Typical fission reactor irradiations, with He/dpa ratios << 1, cause hardening at T ≤ 400 °C, while again, they lead to gradual softening at T > ≈400–450 °C.

The primary irradiation-enhanced mechanism shifting softening to a lower T is believed to be radiation-enhanced diffusion (RED) [22]. Note that there are also other possible irradiation-enhanced softening effects of neutron displacement damage, including cascade ballistic mixing [51] and dislocation climb and recovery [24]. When assuming solute diffusion primarily takes place due to an increased steady-state vacancy concentration, the RED diffusion coefficient, D*, is approximately given by [27]

D* = ηϕσ_dpa/Z_v + Dth(T) = G_v/Z_v + Dth(T).

(5)

Here, Dth(T) is the thermal coefficient, G_v = ηϕσ_dpa is the vacancy production rate per atom and Z_v is the total sink strength for annihilating vacancies (here taken as 2 × 10¹⁴/m²). η (≈0.33) is the number of vacancies generated per dpa, σ_dpa is the fast fission dpa production cross-section (≈1.5 × 10⁻²¹/cm²) and ϕ is the neutron flux (n/cm²-s, E > 1 MeV) [52,53]. This D* model assumes that there is no recombination of diffusing vacancies and self-interstitials [27]. Figure 4a shows D* and Dth, taken here as the self-diffusion coefficient, as a function of 1/T, for a nominal dpa rate of 10⁻⁶/s, corresponding to fluxes of ϕ ≈ 10¹⁵ (fast fission) and 5 × 10¹⁴ n/cm²-s (fusion first wall), respectively [52,53,54,55]. In the absence of recombination, between T ≈ 450 and 550 °C, D* is approximately athermal. At higher temperatures, D*/Dth ≈ 1. At lower temperatures, irradiation hardening increasingly dominates. The softening T range where D*/Dth is >1 is marked by the dashed vertical lines. The corresponding D*/Dth irradiation-softening acceleration factor is shown in Figure 4b. Figure 4c is discussed below.

A simple softening model was fit to both irradiation and the long-term RED-adjusted thermal-aging time (t_a), producing t_aD_th/D* σ_yi/σ_yu data at 450, 500, 550 and 600 °C, as shown in Figure 5. The softening data suggest a bi-linear fit form, with a threshold dpa_t, and a subsequent σ_yi/σ_yu slope as a function of log(dpa):

σ_yi/σ_yu = 1    dpa ≤ dpa_t

(6a)

σ_yi/σ_yu = 1 − B[log dpa − log dpa_t]  dpa > dpa_t

(6b)

The filled and unfilled symbols in Figure 5 are for the irradiated and the RED acceleration factor (see Figure 4b)-adjusted t_aD_th/D* σ_yi/σ_yu data, respectively. As shown in Figure 5a–c, up to ≈200 dpa, the LSF results are very similar between 450 and 550 °C. The corresponding B and dpa_t average 0.088 ± 0.034 and 4.4 ± 2.1, respectively, while they are 0.14 and 3.4 at 600 °C. The similar behavior between 450 and 550 °C is not surprising, given the corresponding athermal D*, again as indicated by the vertical dashed lines in Figure 4a. As expected, softening is faster at 600 °C, consistent with the dominance of an increasing Dth at higher T, with Dth ≈ D* (Figure 5d). The average σ_yi/σ_yu softening trends are shown as the solid lines in the previous Figure 4c. The softened σ_yi/σ_yu RSME and ME are 0.085 and −0.007, respectively. The corresponding ultimate stresses, σ_oi/σ_ou, are assumed to be approximately equal to σ_yi/σ_yu [56].

2.5. He-Re-Hardening Model

It is well-established that high levels of He can make a significant hardening contribution (σ_He) beyond a threshold concentration (He_t) [57]. A primary source of this hardening is believed to be He bubbles, which act as obstacles to dislocation glide [57]. Spallation proton irradiation (SPI) results in a very high He (appm)/dpa, ranging from ≈50 to 87 [57,58,59,60]. The main source of SPI data is an extensive program led by Yong Dai at the Paul Sherrer Institute in Switzerland [57,59,60]. Other possible high-He-hardening mechanisms include enhanced-dislocation loop formation [61,62] and precipitation of insoluble solid transmutants generated in SPI [63,64,65]. Other approaches to studying the effects of high He on the microstructure and properties of TMS, such as Ni and B doping, isotope tailoring, in situ He injection (ISHI), single-ion He implantation and dual ion irradiation, are reviewed in [57].

Assessing the isolated He hardening requires σ_He de-superposition from other contributions to the net σ_yi from various dislocation obstacles, existing both prior to and following irradiation. Superposition is the way various hardening contributions are combined to result in a net σ_y [66,67]. In practice, superposition is complex, and the net σ_y generally falls between the limits of the linear sum (LS) and root square sum (RSS) laws, which properly weight the various individual contributions. The LS-RSS balance depends on the strength of the individual obstacles [66,67]. The de-superposition process is even more complex.

Estimates of σ_He as a function of He at the same T_i and dpa, based on the differences between SPI (high He) and neutron irradiation (low He) Δσ_y after de-superpositioning, are shown in Figure 6a. An LSF of σ_He suggests a threshold He_t ≈ 433 appm, yielding the following:

σ_He = 28.4(√He − 20.8) MPa   He ≥ 433 appm

(7)

While the data are scattered, with a RSME and ME of 110 and −0.1 MPa, respectively, the mean trend is reasonable. These results are also supported by a microstructure-based assessment of Δσ_y for neutron-irradiated Eurofer97, which was simultaneously subjected to in situ helium injection (ISHI) from a NiAl implanter coating, as shown by the red open triangle symbol. Figure 6b shows the T dependence of Δσ_y for both low-He neutron irradiations of TMS F82H with loop-hardening saturation at >10–20 dpa (green line and diamond symbols) and for SPI irradiations at 15 ± 1 dpa and 1320 ± 100 appm He (red squares). The de-superimposed σ_He (purple circles) are approximately athermal between 250 and 520 °C. The isolated σ_He, from Equation (7), must be re-superimposed with the other contributions in assessing the net hardening, shown as the red squares. The combined softening and re-hardening predictions’ effects as a function of the log of dpa, used in the following analysis, are shown as the dashed lines in the previous Figure 4c.

He re-hardening would be expected to greatly retard creep in fusion environments, if softening failure (by whatever metric) did not occur sooner. We applied the same approach for re-hardening as we used for softening, but in this case, based on the dpa- and He/dpa-dependent σ/σ_o. The primary creep curves as a function of dpa were obtained by first dividing ε_x by ε_x′ to determine the creep time under irradiation, t_x, which was then multiplied by the dpa rate, here taken as 10⁻⁶ dpa/s. The results of irradiation softening and He re-hardening are described in the next section.

3. Results

3.1. Primary Creep Models for 375 to 550 °C

We first LSF the σ/σ_o primary creep model to the datasets for all 17 heats 9Cr TMS for primary strains (ε_x) from 0.2 to 2%, temperatures (T) from 375 to 550 °C and times (t_x) from ≈10 h to 8 × 10⁴ h [9,10,11,12]. Fits to the temperature-compensated t_xc = t_x exp(−Q/RT) for the 17-heat database at various ε_x (Equation (3d)) are shown in Figure 2a. The corresponding polynomial fit cross-plots for C_i as a function of ε_x are shown in Figure 2b. In this case, the LSF-optimized Q = 335 kJ/mole. This is higher than that for self-diffusion in iron and steels, which ranges from ≈250 to 300 kJ/mol in a wide range of references. However, this is not abnormal. For instance, Wilshire et al. [8] used a Q value of 300 kJ/mol for their σ/σ_o creep model analysis of Gr91/92 steels. Our creep model predicts a slightly higher Q, based on a least square fit to the data, which we do not choose to defy, and a model which is clearly described as being “data-driven”. The activation energy for creep alloys is influenced by an interplay of multiple factors and mechanisms. For example, the higher Q may reflect deviations from pure diffusion control due to factors such as the approximation in using the temperature dependence of σ_o(T) to normalize σ.

The predicted versus measure (P/M) fit results shown in Figure 7a have an RMSE and ME of 11.3 MPa and −0.004 MPa, respectively. The 1:1 line P/M intercept is 0/0. P-M residual plots are well-centered and show no systematic bias in the independent variables (T, ε_x and t_x). The LSF lines in the σ/σ_o versus t_xc data for ε_x = 1% in Figure 7b show only small differences between the various TMS [9,10,11]. For an ε = 1% in Eurofer97 at t_x = 8670 h (1 year), the predicted σ/σ_o is ≈0.44 at 550 °C (σ ≈ 150 MPa). We also LSF the data for the eight heats of Eurofer97 over a narrower T range of 450 to 550 °C [9], as shown in Figure 7c. The LSF Q is again 335 kJ/mole. The RMSE and ME are 10.6 and −0.005 MPa, with a P/M slope of 1/1 and a P/M intercept of 0/0. Again, the residual plots are well-centered and show no systematic bias. The fit parameters for the data in Figure 7a,c are summarized in

Table 2. LSF parameters, RMSEs and MEs for normalized σ/σ_o primary creep models.

Fit Parameters	All 17 Heats, tx > 10 h, 375–550 °C	8 Eurofer97 Heats, tx ≥ 10 h; T = 450–550 °C
C0	1.464642	1.4194
C1	20.2	20.2
C2	−81	−81
C3	5.5886	5.367485
C4	−9	−10
C5	282.31	286
Q (kJ/mol)	335	335
P/M Slope	1	1
Intercept (MPa)	0	0
RMSE (MPa)	11.3	10.6
ME (MPa)	−0.004	−0.005

. Figure 7d shows the σ P/M plot for all 17 heats in our database using the LSF model for the eight Eurofer97 heats, again over a narrower range of creep conditions. In that case, the RMSE and ME are 11.7 MPa and 1.148 MPa, respectively, with a P/M slope of 1.036 and a P/M intercept of −7.3 MPa. While the fit to all the data is slightly worse, in this case, it is tremendously significant that the Eurofer97 model predicts the σ/σ_o so well for a much wider range of TMS heats and creep conditions.

3.2. Primary Creep Curve ε(σ, T, t) Predictions

Most data sources do not include continuous creep curves [9,10,11,12]. In our work, the primary creep results were used to model continuous creep curves, ε(σ,T,t) and ε′(σ,T,t), by first fitting the discrete ε_x − t_x data points and then taking the corresponding fitted ε(σ,T,t) curve derivative to evaluate ε′(σ,T,t). Further, Equation (2) can be rewritten with ε_x′ at t_x as

ε′(x) = (A₁ + A₂ε_x +A₃ε_x²)(σ/σ_o)^[n₁ ^{+ n}₂^ln(εx^)] exp(−Q/RT)

(8)

Here, the A_i and n_i are LSF parameters that were optimized for ε′ rather than σ. The corresponding t_x(h) is found by dividing ε(x) by ε′(x).

The P/M t_x for Eurofer97 ε′(x) curves [9] in Figure 8a, for σ from 300 MPa to 360 MPa at 450 °C, show the results for t_x down to 0.2 h. While the range of σ is small, the corresponding t_x varies by a factor of >10⁴ due to the large n ≈ 20. As a result of the large n, the predicted t_x (t_xp) uncertainties are larger than those for σ/σ_o. For the t_x > 10 h calibration, the mean bias of t_xm/t_xp ranges from 0.46 to 2.63 with a mean of ≈1.09 and RMSE ≈±0.48, corresponding to a 2 RMSE minimum t_xm ≈ 0.13t_xp. Notably, however, for n ≈ 20, reducing σ/σ_o by a modest factor of ≈0.9 leads to t_xm ≈ t_xp.

Figure 8b–d show the predicted ε(t), ε′_x (t_x) and ε′_x (ε_x) curves, respectively, for σ/σ_o = 0.7 at 450 °C. Here, the red and blue data symbols are for the predicted and measured values, respectively. Figure 8e shows the P/M t_x and the ε(t_x) curve for σ/σ_o = 0.57 at 500 °C. Here, the mean t_xm/t_xp ≈ 1.1 with an RMSE ± 0.51, and the 2 RMSE minimum t_xm ≈ 0.09t_xp. Again, however, reducing σ/σ_o by a factor of slightly less than 0.9 results in a similar increase in t_xp. Note that these estimated margins do not account for the uncertainty in σ_o. Overall, we estimate that a σ/σ_o safety factor of 0.85 will conservatively bind all the Eurofer97 t_xm and account for reasonable values of other sources’ uncertainties, including T. In any event, the effects of such estimated uncertainties on σ/σ_o and t_x can be quantitatively evaluated, as discussed in Section 6. Finally, Figure 8f shows the 500 °C creep curve for σ/σ_o = 0.57 (σ = 240 MPa).

4. A Primary Creep Model with Irradiation-Enhanced Softening

The solid blue line in Figure 4c shows the estimated average athermal σ_o(dpa) softening between 450 and 550 °C. Here, softening is treated by replacing the constant σ/σ_o with the corresponding σ/σ_o(dpa) ratio from Equations (6a) and (6b). Thus, softening increases σ/σ_o(dpa) beyond the dpa_t threshold. The ε′ in Equation (8) was adjusted in small ε increments of 0.01% from 0.1 to 5%, to determine the t_x at ε_x as t_x = ε_x/ε_x′. The discrete t_x(ε_x) was integrated to derive the continuous creep curves. The t_x was then converted to dpa_x by multiplying by 10⁻⁶ dpa/s. The dpa_x for at least 10 to 15 σ at each T was evaluated. Figure 9a shows an example of thermal (black) and irradiation-softening (green) creep curves at 550 °C and a pre-softening σ/σ_o = 0.4 (σ = 140 MPa). A larger t_x and higher dpa (longer t) increase the effect of softening. This is shown in Figure 9b in the log–log plot of the predicted thermal (dpa_t) versus the irradiation-softening (dpa_s) from ε_x = 0.2 to 5% for the Eurofer97 database [9]. The data points fall in a reasonably tight band, with dpa_s ≈ 2.13 (±0.1) dpa_t^2/3, for ε_x ≈ 1 to 5%. For example, softening only decreases for dpa_t = 10 to dpa_s ≈ 9.7, while for dpa_t = 200, the softened dpa_s is ≈70.

It would be useful to compare the predicted effects of softening with data obtained from in-reactor pressurized tube creep experiments. Unfortunately, while there is a substantial in-reactor creep database for austenitic stainless steels, this is not the case for 9Cr TMS. Furthermore, most in-reactor TMS experiments are intended to measure both IC and stress-enhanced void swelling at T less than 500 °C [3]. Perhaps the most relevant data were reported by Kohyama for irradiations of F82H at 520 °C, where creep occurs at a much greater rate than at a lower T [68].

Here, the in-reactor creep ε was given by

ε = Bσ^1.5 dpa

(9)

where the 550 °C B ≈ 3 × 10⁻⁶%/MPa^3/2-dpa, compared to ≈5 × 10⁻⁷%/MPa^3/2-dpa at 450 °C. The in-reactor creep rate tends to increase at a higher T and σ [41,69,70,71,72,73,74,75,76], as expected, in the thermal creep regime. The purple line in Figure 9c shows that the 520 °C in-reactor ε rate of F82H is reasonably consistent with the softening predictions. Note that this may be a somewhat apples-and-oranges comparison due to differences in the steels and their corresponding unirradiated properties, as well as uncertainties in the creep parameters in Equation (9). Furthermore, the σ power n dependence of 1.5 is inconsistently low compared to the thermal primary creep p ≈ 18 ±1 (note that the hardening exponents p and n are σ^p for creep and (σ/σ_o)ⁿ for normalized creep(σ/σ_o)ⁿ). The low value of p may indicate that some component of IC is present.

More generally, the in-reactor creep database is extremely limited, and it is insufficient to reliably predict n. Finally, an equation in reference [6], shown by the blue line in Figure 9c, predicts an even lower thermal primary creep rate in F82H than either in-reactor or thermal primary creep in Euofer97. In principle, this might reflect strengthening, due to the higher W content in F82H. However, the empirical and physical basis for the F82H primary creep equation is not known. Nevertheless, these observations are qualitatively consistent with the higher T irradiation softening model concept.

5. Primary Creep with Irradiation Softening Followed by High-He Re-Hardening

The dashed blue line in the previous Figure 4c shows the athermal re-hardening effect on σ_y/σ_yu at a He/dpa = 10, between 450 and 550 °C. The effects of softening are reversed due to a decreasing σ/σ_o beyond the re-hardening threshold at He_t ≈ 432 appm. Given the high (σ/σ_o)ⁿ sensitivity of ε′ (n ≈ 20), re-hardening is expected to greatly reduce the creep rate. To assess the effect of re-hardening, a subsequently decreasing σ/σ_o(dpa) in Equation (6b) was again used in the ε_x(dpa) creep model. Notably, re-hardening not only slows but also effectively arrests creep at a saturated ε_xs, which is a function of σ and T. Figure 10a shows the thermal ε_t(dpa) (black), softened ε_s(dpa) (green) and re-hardened ε_h(dpa) (red) curves, at 550 °C and σ/σ_o = 0.4 (σ = 140 MPa). Figure 10b shows the corresponding ε′(dpa) curves. Softening reduces the minimum creep condition from ≈480 to ≈125 dpa, while increasing the corresponding ε′_m to 1.43 × 10⁻⁶/h from 3.75 × 10⁻⁷/h. The high-He-re-hardening creep rate starts to decrease rapidly near ≈43 dpa, and it effectively saturates at 1% by ≈120 dpa. Figure 10c plots the softening dpa_s and re-hardening dpa_h versus the thermal dpa_t at ε_x = 1%. Initially, the dpa_s and dpa_h are much less than dpa_t, but the re-hardening dpa_h increases rapidly, starting at about 120 dpa_t, marking the effective arrest of creep. Similar behavior was generally observed at other ε_x, with creep arrest between ≈120 and 150 dpa. Figure 10d summarizes the re-hardening σ as a function of ε_x (from 0.2 to 5%) and T. Figure 10e shows ε_xs as a function of σ/σ_o from 0.2 to 2%, and T at 450, 500 and 550 °C. σ/σ_o(ε_xs) are similar at 500 and 550 °C, but they are shifted to much higher values at 450 °C.

6. Negligible Creep and Strain Limits for Eurofer97

Figure 11 shows Eurofer97 σ versus T plots, for times of 10⁴, 5 × 10⁴ and 10⁵ h, at what might be considered negligible creep and deformation limit conditions, here taken as ε_x = 0.2% and 1%, respectively. The Eurofer97 8-heat σ/σ_o model was used to predict σ, for σ_o of 455 ± 10, 418 ± 9 and 356 ± 12 MPa at 450, 500 and 550 °C, respectively, as the average σ_o of Eurofer97 heats fitted to a second-order polynomial was used to estimate σ_o at 5 °C intervals. As noted previously, the model can also be used to estimate the effects of uncertainties or variations in σ_o, as well as service conditions, like T and irradiation. For example, the effects of variations in T on σ/σ_o are primarily controlled by Q; thus,

σ(T₁)/σ(T₂) = {exp(−[Q/R](1/T₁ − 1/T₂)}^1/n

(10)

Assuming Q = 335 and a nominal n ≈20, an increase in T by 20° C would reduce σ_/σ_o by about 4%. The same creep rates would occur when fractional reductions σ_o were matched by similar reductions in σ. Other such assessments are left to the reader. Overall, we estimated that reductions in σ by a factor of 0.85 provide conservative evaluations of primary creep.

7. Summary, Caveats, Context and Implications

Empirical thermal creep models typically focus on minimum rates and rupture times. However, the accumulation of ε during rapid primary creep cannot be neglected, especially for large, interconnected, gradient-dominated, complex, high-precision structures, like in fusion reactors. The main objective of this research was to calibrate a new, easy-to-use primary creep model for 9Cr TMS. An important secondary objective was to evaluate the consistency of the 17 TMS heat fitted model with primary creep in 8 heats of RAFM Eurofer97. Several predictive approaches were evaluated, but the most accurate was provided by a model based on the applied stress (σ) normalized by the steel’s temperature-dependent ultimate tensile stress (σ_o), σ/σ_o(T). We showed that the σ/σ_o approach, championed by Wilshire et al. [8], is a robust method of predicting primary creep σ(ε_x) for a wide range of TMS with different unirradiated properties and conditions. We believe that the new σ/σ_o(T) model is unique and, for the first time as far as we know, uses a convenient cross-fitting procedure to accurately predict σ as a function of ε_x, for times t_x from 10 to 8 × 10⁴ h, ε_x from 0.2 to 5% and T from 375 to 550 °C. The σ/σ_o(T) model, which was fit to primary creep data for 17 heats of 9Cr TMS, resulted in a predicted σ root mean square error (RMSE) of ≈±11 MPa and a mean error (ME) of ≈0. We also showed that the σ/σ_o model can also be used to predict continuous creep curves, ε(σ, T).

The σ/σ_o(T) model predictions should be verified for any specific heat of 9Cr TMS. For example, here, we focused on a narrower calibration for the eight-heat Eurofer97 database [9], with a σ RSME ≈ 10.6 MPa. Recalibration for other new alloys intended for actual service will also be required. But it is both encouraging and important that the predictions for Eurofer97 are very consistent with those of other TMS and wide ranges of T, ε_x and t_x. Additional comparisons and recalibrations are also warranted, especially for newer, advanced TMS variants.

A second objective was to explore the potential effects of irradiation service phenomena, which were not previously considered. Notably, we showed that the σ/σ_o(T) model can be used to treat possible effects of irradiation service which result in time-dependent σ_o(T) as a function of dpa and He. The irradiation effects include (a) softening, as a function of dpa, at T above ≈400–450 °C, in low-He fission (<<1 He/dpa) environments; and (b) subsequent re-hardening, in high-He (≈10 He/dpa) fusion first-wall environments. While these irradiation effects on primary creep are not yet experimentally verified, so we do not claim that our predictions are quantitatively accurate, they highlight currently unrecognized, but potentially very important phenomena. These include the acceleration of primary creep at a low He/dpa (fission) and arrest of creep at a high He/dpa (fusion).

Our treatment of irradiation effects involves a number of assumptions, approximations and limitations. Most notably, there are almost no direct data on the softening and He effects on primary thermal creep in 9Cr TMS, from in-reactor pressurized tube studies including post irradiation hardness tests. We further assume that the complex, multiscale, multi-mechanism physics of softening and re-hardening can be captured in simple, analytic LSF empirical models. Furthermore, data on irradiation softening and high-He re-hardening are themselves scattered and limited. Thus, it is encouraging it appears that irradiation-enhanced softening can be reasonably understood in terms of a shift in thermal aging effects to lower T and t, primarily due to RED. Notably, however, the severe effects of σ on softening [25,35,36,37,77] and, especially, cyclic loading [75,76] are not treated. Furthermore, while the predicted effects of high-He re-hardening are reasonably consistent with known microstructure–property relations, they require further refinement and verification. Thus, we categorize the predictions of these irradiation effects on thermal primary creep, due to an evolving σ_o(T, dpa, He), as qualitatively reasonable and pointing in the right direction. However, they are likely underestimates of primary creep rates for actual in-service creep conditions, which are subject to synergistic effects of irradiation, σ and low, most notably, low cycle fatigue. Note that these σ/σ_o(T, dpa) models of irradiation effects apply to dislocation creep, and not to lower T IC at lower T, or low σ–high T diffusion creep. Furthermore, calibration involved only the base metal (plate) product form.

Nevertheless, recognizing possible effects of an evolving σ_o(T, dpa) brings into focus the research needed to develop physical models of primary creep under irradiation, which are more quantitative and predictive. For example, this research could involve extending detailed modeling studies of thermal softening mechanisms [20,23] to include the effects of RED. Post-irradiation hardness measurements on crept in-reactor pressurized tubes would be very useful to characterize the effect of σ on softening [36]. Softening under ion irradiation might also be explored. Refinement of high-He-re-hardening mechanisms, like targeted studies of He bubbles, enhanced dislocation loops and SPI solid transmutants’ strengthening, are also needed.

Finally, from a structural integrity perspective, it is important to consider all potential sources of dimensional instabilities, including void swelling and IC [1]. For example, if high He suppresses thermal creep in fusion environments, then swelling, which is likely stress-assisted, will present a major alternative life-limiting concern.