Modelling for Creep Cavitation Damage and Life of Three Metallic Materials

ABSTRACT

Metallic structures operating at high temperatures can have creep as a life-limiting mechanism. The limit can arise by nucleation and growth of creep cavities that eventually merge becoming the leading damage of the fracture mechanisms. Understanding and modelling for creep cavitation can therefore help in prediction of structural safety and condition of such structures. The present paper will introduce a new model to predict creep cavitation damage so that the associated creep strain is also modelled out of the predictive expression without losing accuracy in the time to critical damage and corresponding residual life. The model was verified for OFHC copper, low-alloy 0.5CMV steel and higher-alloy X20 steel, to cover a wide range of polycrystalline metals and their microstructures. The new model performed as well as or better than the classical approach which requires access to measured strain. The model was applied to predict creep cavitation damage from time, stress and temperature as predictive variables.

KEYWORDS:

• Creep
• cavitation
• damage
• metal
• model
• life

Introduction

High-temperature damage in power and process plants will emerge in metallic structures designed for creep, and this damage can accelerate by modifications in design, materials or operational modes to shorten the expected component life [1, 2]. The decreasing opportunities to baseload operation and increasing need to support intermittent renewable supply have resulted in new patterns of accumulating damage. Therefore, it is challenging to reliably detect and evaluate the leading damage so that its progression and impact can be accurately characterised for life prediction [3–12].

Good material performance will require both high creep strength and reasonable creep ductility. The latter is related to the creep cavitation damage that can be a challenge to produce similarly in the laboratory as in long-term service. For example, service-relevant creep cavitation may not appear in relatively creep ductile materials, such as P22 (10CrMo9-10 or 2.25Cr1Mo) steel within relatively modest testing times of common research exercises [13–18]. In contrast, some polycrystalline metals, like pure copper, will show such creep damage with relative ease. Unfortunately, when developing new engineering alloys with improved creep strength, it remains a persistent challenge to retain good creep ductility particularly for welded joints [13, 14].

This paper aims to establish a new analytical model of the evolution of creep cavitation damage in high temperature service and to evaluate it versus a classical model that requires access to measured strain. Examples are described on the challenges and potential solutions to characterise the creep damage for non-destructive condition monitoring and life assessment of welded structures. The damage appears at a material-dependent rate, first in apparently scattered pattern with gradually increasing cavity density, then forming chain-like patterns of growing and fusing cavities that finally start to form cracks and produce the fracture surface [19]. Here we consider as example materials polycrystalline metals with widely differing microstructures, namely oxygen-free high conductivity (OFHC) copper, a precipitate strengthened low-alloy steel (0.5CMV) and a higher-alloy ferritic-martensitic steel (11% Cr steel, X20).

Materials and methods

This paper presents first a brief summary on the features of creep cavitation in polycrystalline metals and then an analysis of cavitation damage evolution in three example metals: OFHC copper, 0.5%Cr-0.5%Mo-0.25 V low alloy steel (14CrMoV6-3, or 0.5CMV for short) and 11%Cr steel (X20CrMoV11-1, or X20 for short). Literature data [20–23] on materials, creep test results and the measures of creep damage, were used for the development of the suggested models. For the chemical composition of the test materials [20–23], see Table 1, and for the testing ranges, see Table 2. The 0.5CMV data included one series (S), for which the type IV heat-affected zone (HAZ) structure had been simulated by heat treatment [23]. Another series (CW) of 0.5CMV data included data from actual type IV HAZ from cross-weld specimens [23]. For X20 steel data, the CL series included constant load tests at a range of temperatures but at the initial stress of 80 MPa, and the CT series included constant load tests at a range of initial stresses but at a constant temperature of 600°C [21].

Table 1. The chemical composition (wt%) of test materials [20–23]

	OFHC copper	0.5CMV steel (S and CW)	X20 steel (series CL)	X20 steel (series CT)
C	-	0.14	0.202	0.218
Si	<0.001	0.26	0.265	0.278
Mn	-	0.59	0.657	0.645
P	-	0.009	0.0185	0.0184
S	0.0021–0.0024	0.018	0.0054	0.0049
Cr	-	0.32	11.537	11.45
Ni	<0.0005	0.19	0.709	0.699
Mo	-	0.61	1.061	1.057
Cu	Balance	0.16	0.091	0.091
Al	-	<0.01	0.0289	0.0281
Nb	-	<0.01	-	-
Ti	-	<0.01	-	-
W	-	-	0.068	0.063
V	-	0.29	0.286	0.29
Co	-	-	0.022	0.017
Fe	<0.001	Balance	Balance	Balance
Sb	<0.0005	<0.01	-	-
Pb	<0.0005	-	-	-
Sn	<0.0005	0.01	0.001	0.005
Ag	0.001	-	-	-
As	<0.003	0.02	-	-
Zn	<0.01	-	-	-

Table 2. Temperature, time and stress ranges for the assessed data in creep testing [20–23]

Quantity/material	OFHC copper	0.5CMV steel (S)	0.5CMV steel (CW)	X20 steel (CL)	X20 steel (CT)
Temperature (°C)	400–550	640–700	600–640	600–670	600
Time (h)	0.5–97.5	22–1508	545–8886	92–7566	3766
Stress (MPa)	13.8–34.5	20–160	35.7–80	80	90

For creep damage, density change was measured to assess the extent of cavitation in copper after interrupted or terminated testing [20]. In case of the two steels, number density of cavities per unit area and mean cavity size were available from metallographic inspections after interrupted or terminated testing, with minimum recorded cavity size of about 0.3 µm in 0.5CMV and about 0.2 µm in X20 [21–23]. The overall damage in terms of cavity volume fraction was estimated as number density of cavities multiplied by the mean cavity size, and this was taken to be proportional to the area fraction of cavities in the metallographic assessments. In case of copper, creep damage was density change that was converted to estimates of cavity volume and area fraction. The outcome was used for the suggested predictive models for creep damage evolution in the selected metals. The performance of the models was assessed via Equation (1)(1) $\mathrm{l}\mathrm{o}\mathrm{g}\left(Z\right)=2.5\cdot \sqrt{\frac{{\sum }^{{(log\left(t_P\right)-\mathrm{l}\mathrm{o}\mathrm{g}\left(t_O\right))}^2}}{n_A-1}}$(1) with the scatter factor Z, defined so that log(Z) equals 2.5 times the standard deviation between the logarithms of predicted and observed quantity, such as testing time or cavity area fraction. For example, for the predicted and observed testing times,

(1) $\mathrm{l}\mathrm{o}\mathrm{g}\left(Z\right)=2.5\cdot \sqrt{\frac{{\sum }^{{(log\left(t_P\right)-\mathrm{l}\mathrm{o}\mathrm{g}\left(t_O\right))}^2}}{n_A-1}}$(1)

where t_P and t_O are predicted and observed times, respectively, and n_A is the total number of data points. The scatter factor is widely used for assessing creep model performance by the European Creep Collaborative Committee (ECCC). The Z value of 1 means that the model fits perfectly with the assessed data, and values under 2 are generally considered to indicate good model performance [6, 24, 25].

To demonstrate the evolving creep damage in similar types of materials, metallographic assessment was extended to two additional example materials: oxygen-free phosphorus-doped copper, creep tested under uniaxial constant load at 200°C and 95 MPa for about 27,000 hours,Figure 1(c), creep tested with CT specimen at 175°C and 35 MPa equivalent stress for about 25,000 hours [26], Figure 2(c), and welded joint of X20 steel from an ex-service (220,000 hours at 535°C) steam header [27], Figure 2(a-b).

Figure 1. (a) A schematic 2D illustration of creep cavities at grain boundaries, (b) a simplified 3D model of creep cavitation in a polycrystal adapted from [40] and (c) a faceted creep cavity shaped by surface diffusion in copper.

Figure 2. Creep cavitation damage (a) and (b) in HAZ of X20 steel [27], and (c) in oxygen free phosphorus doped (OFP) copper [26].

Nucleation and growth of creep cavitation damage

At typical service temperatures of creep-resistant alloys, the diffusion-assisted flow of matter can facilitate local vacancy condensation and accumulation to cavity nuclei, Figure 1(a-b). In classical models assuming spherical nuclei, stability requires growth to critical diameter of 4γ_s/σ, where γ_S is surface energy at the boundary and σ is tensile stress normal to it [28–31]. Any local decrease of surface energy, or increase in the normal stress, will facilitate easier nucleation of new stable cavities. Common features of such nature include high- and low-angle boundaries of polycrystalline metals, and boundaries to second phases like sulphides [32–34]. The value of normal stress σ is influenced by the emerging cavities that can relax this stress [29].

At the critical stable size, the cavities are much less than 100 nm in diameter, and not detectable in conventional inspections [34]. In the process of cavity growth, continuing creep will accelerate the damage accumulation, shortening the intervals between the subsequent stages of damage. This will reduce the time to the detection threshold for given inspection or monitoring technique, but it will also progressively shorten the warning time to failure.

At present, techniques such as replica inspection supported by optical microscopy can detect reasonable dense formations of creep cavitation from the approximate minimum cavity size of about 0.3–0.5 µm [35–37]. Although this resolution can be easily enhanced by using scanning electron microscopy (SEM), it is less common as a routine as it would be impractical in the field and imply more effort and cost [19, 38, 39].

On cavitation-related material weakening, one classical assumption is drastic reduction of the load-bearing capability of the grain boundary where cavity is residing [29]. Then, additional cavitation on the same grain boundary is only reflecting the imposed strain without reducing the load this boundary can take. Nevertheless, increasing cavity density would imply accompanying creep strain and therefore increasing overall creep damage in a classical sense. The same would apply for the growth of a lone cavity on such a boundary (Figure 1(c)). For example, the increasing size and increasingly facetted shape of growing singular cavities in pure metals would be a parallel indication of creep strain, although it also reflects diffusive mass transport that requires time and not only strain. The appearance of cavities will again change in commercial iron alloys such as 9–11%Cr steels, with more complex composition and microstructure. In steels, the cavities will frequently nucleate and grow at weak phase boundaries such as at sulphides [30]. Furthermore, these steels include numerous small-angle boundaries within the grains.

In addition to the associated strain at the immediate vicinity of the individual cavity, it also matters how such cavities are distributed. Sparse, evenly distributed cavities mean evenly distributed strain that is as far as possible from the material limit of ductility exhaustion. In contrast, concentrated creep cavitation either in terms of local cavity density or in terms of cavity size distribution is more prone to reach the local strength limit of the material [13].

During early stages of nucleation and growth, the mechanism of load shedding from the cavity-forming boundary will result in initial decrease of cavity growth rate [14]. This process evens out the cavity distribution until the locally increasing density or other factors start to override so that additional cavity growth will appear from interaction and coalescence of cavities [21]. The coalescence will first promote the appearance of string-like oriented cavitation, typically along boundaries that are perpendicular to the maximum tensile principal stress, or like in welds, to the weakest material zone [11–15]. The cavities will concentrate on the same grain boundaries from the very beginning, so that the appearance of oriented cavitation is also a function of cavity density and not only of the distribution in terms of orientation with respect to the stress state. Nevertheless, the combination of increasing cavity density and the appearance of oriented cavity strings will indicate increasing level of damage localisation already before crack formation [34]. Further coalescence will result in initiation and gradual growth of cracks that retain the characteristic form of creep damage, showing parallel cracking and cavitation along the growing main crack. This process is conventionally described to include the phases of initially intact material without detected cavitation, then scattered cavitation, and thereafter increasingly localised damage, first as oriented cavity formation and then merging to microcracks and macroscopic creep cracking, which are within the resolution of conventional NDT techniques [19, 35, 37–39].

In welded ferritic steels, the susceptible region for cavity formation is often the Type IV region of the heat affected zone (HAZ) [9–17]. Figure 2(a-b) show examples of in-service creep damage in the HAZ of X20 steel, in the form of scattered and orientated creep cavities. The 9–11%Cr steels exhibit complex microstructure including multiple levels of small-angle boundaries within the grains, so that creep strain and damage tend to become more even and no longer limited to the grain (high-angle) boundaries [13, 21]. Figure 2(c) shows cavities in the grain boundaries of the other example metal (copper). In conventional polycrystalline metals the high temperature strain and creep damage tend to concentrate at relatively narrow weak zones, such as grain boundaries. The resulting strain to failure, i.e. creep ductility, will be therefore limited and lower than in materials of microstructurally ideally (evenly) distributed damage.

Evolution of life-limiting damage in high temperature service

A common approach to the indicators of creep damage, such as microstructural degradation, creep cavitation or creep strain, is to account for them only indirectly, for example in predicting the time to failure, or safe loading level for given lifetime, through material-dependent functions of time, stress and temperature [41]. The standard approach implies that creep life and creep design can be largely managed with these variables, without expressing the levels of strain or cavitation damage. However, the measured level of strain or cavitation damage can usefully indicate the structural condition at in-service inspections [42, 43], and the cavitation damage can be modelled in more detail for such or other purposes with or without the expressed strain [29, 44–46].

Here we define the extent of creep cavitation damage (Φ’) as volume fraction of cavities, or inverse of relative density change, taken to equal the product of volumetric number density N_v and mean diameter D_m of cavities. Assuming that both N_v and mean cavity volume, i.e. πD_m³/6, are proportional to creep strain ε, the volumetric measure of damage Φ’ would be proportional to ε^4/3 [47]. Alternatively, assuming that the resulting damage is also related to evolution in time, from Norton type of expression for creep strain as tσ^n’, could suggest damage proportional to ε^xt^yσ^n’.

From creep testing data to estimate the values of x, y and n’ for the evolution of cavitation damage after time t and strain ε, the classical model of the Equation (2)(2) ${\mathrm{\Phi }}^{\prime }=A^{\prime }\epsilon t{\sigma }^{n{ }^{\hspace{0.17em}}}{e}^{-Q/RT}$(2) has found experimental support [29, 48–50].

(2) ${\mathrm{\Phi }}^{\prime }=A^{\prime }\epsilon t{\sigma }^{n{ }^{\hspace{0.17em}}}{e}^{-Q/RT}$(2)

where A’ is material dependent parameter, σ is stress, n´ is typically 2 ±0.3, Q is the apparent activation energy, close to that for grain boundary diffusion [39, 51], R is the gas constant, and T is absolute temperature.

Here, time, stress and temperature are the natural independent variables, resulting in creep damage including strain and cavitation. Similarly, as above, the creep strain after time t is assumed to be proportional to tσⁿ, where n is the creep exponent. Normalising the stress term with (temperature-dependent) shear modulus G and adopting the formulation for power-law creep from [51] gives strain as Equation (3)(3) $\epsilon =B\left(T,\sigma \right)t(\sigma /G{)}^n$(3) .

(3) $\epsilon =B\left(T,\sigma \right)t(\sigma /G{)}^n$(3)

where B(T,σ) = AD_eGb/kT, A is Dorn constant, D_e is temperature and stress dependent effective diffusion coefficient, b is Burgers vector and k is Boltzmann constant. Rearrangement of the Equation (2)(2) ${\mathrm{\Phi }}^{\prime }=A^{\prime }\epsilon t{\sigma }^{n{ }^{\hspace{0.17em}}}{e}^{-Q/RT}$(2) allows prediction of the volumetric creep damage as:

(4) ${\mathrm{\Phi }}^{\prime }=B\left(T,\sigma \right)t\left(t/t_{ref}\right){\left(\sigma /G\right)}^{n+n{ }^{\hspace{0.17em}}}{e}^{-Q/RT}$(4)

where the reference time t_ref = 1 h. Strictly speaking, this applies to microstructurally stable metals such as copper at constant stress. In case of uniaxial constant load creep testing, the initial stress increases during creep. Further weakening can take place in alloys such as steels with microstructural change during creep. Converting here from volume fraction ${\mathrm{\Phi }}^{\prime }$ to area fraction of damage $\mathrm{\Phi }$ gives N_AD_m = N_vD_m² [47], where N_A is a real number density of cavities. Taking that the cavity volume according to the mean cavity diameter is πD_m³/6 = a₂ε, where a₂ is a proportionality factor, we get D_m² = (6a₂ε /π)^2/3. Since N_v is proportional to strain, we can write $\mathrm{\Phi }$ = a₃ε^5/3 where a₃ is the corresponding proportionality factor.

Taking strain to be proportional to tσⁿ, we can write $\mathrm{\Phi }=$ a₄ · t^5/3· σ^n+5/3 [47], where a₄ is yet another proportionality factor. Assuming that the weakening processes are as above related to time and strain, the new analytical model gives the a real damage fraction $\mathrm{\Phi }$ as in Equation (5)(5) $\mathrm{\Phi }=B\left(T,\sigma \right)t(t/t_{ref}{)}^{1+m}{\left(\sigma /G\right)}^{n+\omega }{e}^{-Q/RT}$(5) .

(5) $\mathrm{\Phi }=B\left(T,\sigma \right)t(t/t_{ref}{)}^{1+m}{\left(\sigma /G\right)}^{n+\omega }{e}^{-Q/RT}$(5)

where m (idealised as 2/3 above) and ω (idealised as 5/3 above) are material parameters covering microstructural damage [52] and, e.g. stress increase in the case of constant load testing.

Equations (2)(2) ${\mathrm{\Phi }}^{\prime }=A^{\prime }\epsilon t{\sigma }^{n{ }^{\hspace{0.17em}}}{e}^{-Q/RT}$(2) and (5(5) $\mathrm{\Phi }=B\left(T,\sigma \right)t(t/t_{ref}{)}^{1+m}{\left(\sigma /G\right)}^{n+\omega }{e}^{-Q/RT}$(5) ) can be solved to give predicted creep cavity area fractions as a function of time. Figure 3 presents the predicted versus observed cavity area fractions for selected data sets that included test specimens of identical materials pedigree and constant or relatively similar ranges of loading conditions (temperature and stress), and as many comparable specimens in the same test series as possible. Figure 4 shows the predicted versus observed time to given cavity area fractions for the available experimental data according to Equations (2)(2) ${\mathrm{\Phi }}^{\prime }=A^{\prime }\epsilon t{\sigma }^{n{ }^{\hspace{0.17em}}}{e}^{-Q/RT}$(2) and (5(5) $\mathrm{\Phi }=B\left(T,\sigma \right)t(t/t_{ref}{)}^{1+m}{\left(\sigma /G\right)}^{n+\omega }{e}^{-Q/RT}$(5) ). Both in Figures 3 and 4 the sections (a) and (b) present the results for OFHC copper, sections (c) and (d) for simulated type IV HAZ specimens of 0.5CMV steel, and sections (e) and (f) for X20 steel. Figure 5 presents the results with the new Φ model for 0.5CMV cross-weld specimens, for which the predictions with classical Φ’ model could not be calculated because of the missing strain data. Figure 5(a) shows the predicted versus observed time to given cavity area fraction for 0.5CMV type IV cross-weld HAZ, for which the strain data was not available, and Figure 5(b) the predicted versus observed cavity area fraction for selected data. In general, the new model Equation (5)(5) $\mathrm{\Phi }=B\left(T,\sigma \right)t(t/t_{ref}{)}^{1+m}{\left(\sigma /G\right)}^{n+\omega }{e}^{-Q/RT}$(5) provided improved fitting (reduced values of Z) for the assessed materials. Figure 6 shows the predicted and experimental damage levels as a function of time for a selected test sets of copper in (a), simulated type IV HAZ of 0.5CMV in (b), cross-weld 0.5CMV in (c), and X20 in (d). The predicted time to given damage appears to agree well with the experimental evidence in (a) and (b), and occurs somewhat early (conservatively) but with similar trendline as the corresponding experiments in (c) and (d). This is because all fitting has been done for all relevant test series and not optimised for those shown in Figure 6.

Figure 3. Observed versus predicted cavity area fraction (for copper converted from density) with classical Φ’ model and the new Φ model for OFHC copper [20] in (a) and (b), for simulated type IV HAZ of 0.5CMV steel [23] in (c) and (d), and for X20 steel [21, 22] in (e) and (f).

Figure 4. Observed versus predicted time to cavity area fraction (for copper converted from density) with classical Φ’ model and the new Φ model for OFHC copper [20] in (a) and (b), for simulated type IV HAZ of 0.5CMV steel [23] in (c) and (d), and for X20 steel [21, 22] in (e) and (f).

Figure 5. The predicted versus observed time to given cavity area fraction for 0.5CMV type IV cross-weld HAZ [23] in (a), and the predicted versus observed cavity area fraction for selected 0.5CMV type IV cross-weld HAZ [23] data in (b).

Indicative values for parameters in Equations (2)(2) ${\mathrm{\Phi }}^{\prime }=A^{\prime }\epsilon t{\sigma }^{n{ }^{\hspace{0.17em}}}{e}^{-Q/RT}$(2) –(5) are presented in Table 3 [20–23, 51, 53]. The values for D_e , G and b were obtained from [51].

Figure 6. Predicted (with the new Φ model) and measured evolution of creep cavitation damage for restricted test series in (a) OFHC copper [20], (b) 0.5CMV simulated type IV HAZ [23], (c) cross-weld 0.5CMV steel [23] and (d) X20 steel base material [21, 22].

Table 3. Indicative values for the model parameters in Equations (2)–(5) [20–23, 51, 53]

Parameter	OFHC copper	0.5CMV steel ^1)	0.5CMV steel ^2)	X20 steel	Unit
A’	26.11	115.53	-	0.0015	1/h·MPa^2.3
n’	2.3	2.3	-	4.8	-
Q	104	174	174	174	kJ/mol·K
B(T,σ) ^3)	4.13 ·10^17 (20.7MPa/773 K) 4.75 ·10^17 (24.1MPa/773 K)	6.20 ·10^24 (60MPa/913 K)	2.35 ·10^15 (60MPa/893 K)	9.69 ·10^23 (80MPa/908 K)	1/h
m	−0.24	1.89	1.51	1.51	-
n	4.79	5.0	5.0	5.0	-
ω	0	7.56	3.69	7.34	-

^a Simulated type IV HAZ specimens in [23]

^b Type IV HAZ of cross-weld specimens in [23]

^c Values for the example test series in Figures 3 and 5

Discussion

Figures 3 to 6 indicate that the evolution of creep damage appears to agree well with the suggested models for the assessed range of materials, stresses and temperatures. Because in structures operating in the creep regime the effective stresses and temperatures are likely to vary less than in the experimental range considered in this study, good agreement could be mostly expected even when unplanned changes occur for example in the structural supports or other drift in the mechanical, control or instrumentation systems. Therefore, it appears also reasonable to expect that the suggested models would work well for a wide range of operational conditions and histories of high temperature plants [13, 35]. However, note that the times to given creep damage and failure in the experiments of the source data [20–23] were short in comparison to the expected service times of most engineering structures. Therefore, the model performance should be additionally tested against for example in-service inspection statistics to describe longer-term damage evolution.

When using density measurement to indicate the extent of creep cavitation damage, the result will differ from those indicated by metallographic cavitation assessment in several ways. First, at the stages of early damage, particularly for primary creep a significant part of the measured density reduction will result from multiplication of dislocation structures of the metal, independently of the cavitation damage. Secondly, density measurement will automatically account for volumetric sampling, and unlike metallography, practically independently of modifications such as surface etching. Density measurement is also likely to provide higher sensitivity than conventional metallography that is often limited by sample preparation. However, such advantage for density measurement is less clear when the damage shows strong local gradients, for example in welds. Most importantly, metallography is possible to conduct in a non-destructive manner through direct in-site inspection or replica testing. Other methods such as tomography using hard X-rays from large particle accelerators are in principle able to characterise the volumetric creep cavitation damage [54, 55] but do not seem to be sensitive to cavity sizes much smaller than 1 µm and are not in practise non-destructive for on-site inspections.

For the assessed data the minimum size of observed cavities (0.3 µm) was relatively large in comparison to the resolution of current microscopy techniques. However, the in-service inspection techniques are also limited by practicalities in using, e.g. portable and reasonably convenient equipment that are not comparable to the best methods available for laboratory operations. Nevertheless, the current resolution particularly in SEM is far superior to the levels included in the common guidelines for replica inspections [24, 36, 43]. Therefore, it is potentially possible to consider extending the recommended practices in evaluating creep damage from evidence provided by in-service inspections so that smaller scale and earlier damage is included.

The assessment of creep and creep damage in metallic components is usually based on direct or indirect measurements to indicate displacement (strain) or its local concentration, including discontinuities such as creep cavities and cracks. For reasonable sensitivity, localisation of damage like creep cavitation will help to provide detectability but in the same time represents damage that one would prefer to avoid because localised damage may carry reduced ductility [13, 56]. In the development of new high temperature alloys, a consistent challenge is to improve strength while retaining good ductility, and ensuring success tends to be slow because the required creep testing will obviously take time. Nevertheless, successful materials should be ductile and resist early localised cavitation and cracking. Success may mean the need for improved resolution for detecting such early damage. Therefore, improving the material performance may increase the challenge in maintenance of critical equipment for condition and life management, and affect the targeted sensitivity and specificity of inspection and monitoring. This challenge can be seen in the assessment of class 3 (orientated or chain-like cavity formations) in materials such as X20 [43].

On the other hand, there are reasons why very small cavities may not be of much interest in practice. First, small cavities need not to be followed when the associated growth rate remains below detectable levels between typical inspections, and for sufficiently small cavities in structures designed for decades of service this would be initially very likely or inevitable. Secondly, the defects that need to be detected should be reasonably likely to characterise the future condition so that it could be properly re-examined in the follow-up measurements (monitoring or inspections). For example, early creep cavitation damage in weld metal and coarse grained HAZ may be observed after fabrication due to local relaxation damage, while the damage with actual long-term significance appears later in the fine grained or intercritical HAZ. It may well happen, therefore, that the early damage is initially irrelevant for long term service and it becomes necessary to wait until the damage at the relevant location will appear and demonstrate its stage of significance. Regardless of such complications in welds of ferritic steels, more straightforward process of damage evolution can be expected in structures such as steam piping elbows where relevant damage can appear in base material.

The currently common minimum criterion for creep cavity density is about 100–400/mm² (early class 2 or scattered creep cavitation), or perhaps slightly lower for 9–11% Cr steels [36, 37, 43]. This applies in practice for replica inspections of high temperature steel structures when assessed using light optical microscopy. For 9–11%Cr F/M steels, this stage may appear relatively late in life in comparison to most low alloy steels [43, 54, 55]. The same criteria may not strictly apply when using higher resolution techniques such as metallographic assessment with scanning electron microscope (SEM) that was applied in the above example for X20 steel [21].

Above we have introduced a new model (Equation 5(5) $\mathrm{\Phi }=B\left(T,\sigma \right)t(t/t_{ref}{)}^{1+m}{\left(\sigma /G\right)}^{n+\omega }{e}^{-Q/RT}$(5) ) of creep cavitation damage for predicting the evolution of creep cavitation damage, or time to given damage or time to failure without strain as an input variable. In general, similar or better fitting performance was observed with the new model in comparison to corresponding predictions using the classical model (Equation 2(2) ${\mathrm{\Phi }}^{\prime }=A^{\prime }\epsilon t{\sigma }^{n{ }^{\hspace{0.17em}}}{e}^{-Q/RT}$(2) , including strain) with creep testing and damage assessment data of OFHC copper, low-alloy 0.5CMV steel and higher-alloy X20 steel, i.e. for a wide range of polycrystalline metals and microstructures. The new model does not require the potentially inconvenient access to the value of strain as an input variable for predicting safe life from the observed creep cavitation damage.

Conclusions

For useful early warning of the life-limiting creep damage, non-destructive indications of the material condition would require enough sensitivity and specificity for the applied measurement. Such techniques like replica inspections have developed to discern smaller details than what are required in conventional in-field light optical metallography as it is applied in common practices of maintenance service. On the other hand, the practices during such inspections may influence the outcome, for example grinding of the surface to be inspected can affect the observed cavity density under strong damage gradient in the thickness direction, or when etching of the polished surface will increase the apparent number and size of the indicated cavities. It is not inconceivable that such errors become potentially more significant when considering smaller and less dense cavitation for early warning of the life limits. This is one reason why the common interpretation of creep damage is not aiming to predict longer life or consider smaller damage than what is the conventional practice.

The early interpretation rules of the observed creep cavitation damage, as they appeared in the guidelines for replica inspections [24], were essentially material independent. In reality there are also significant differences in the details between metallic materials. Such details include for example material specific model parameters, but also the important aspect of creep ductility, i.e. strain to critical events like damage accumulation to start crack growth.

The classical expression for the critical size of stable nucleating cavities implies that this size only depends on the local surface energy γ_S and the normal stress to the relevant nucleation surface. Both may vary locally quite a lot, so that while typical quoted values for γ_S are of the order of 1–2 J/m² for pure metals, they could be considerably reduced at phase boundaries of, e.g. sulphides [28, 34, 51]. Similarly, the local normal stress can be strongly reduced by cavitation [26]. The ranges of γ_S and σ will hence considerably widen the resulting distribution of cavity sizes already at the stage of stable nucleation. Although the minimum size of a stable cavity is not well established, theoretical work has indicated it to be in the range of 2–5 nm [57, 58].

The early growth rate of stable cavities will also depend on whether this stage represents primary or later stages of creep. At typical service conditions of commercial alloys, such as steels, the primary stage is short and of relatively minor significance [13–15], and the cavity growth rates tend to remain consistent throughout the component life.

To assess models for the progression of creep cavitation damage, the well performing classical model (Equation 2(2) ${\mathrm{\Phi }}^{\prime }=A^{\prime }\epsilon t{\sigma }^{n{ }^{\hspace{0.17em}}}{e}^{-Q/RT}$(2) ) was extended by modelling strain out of the predictive expression (Equation 5(5) $\mathrm{\Phi }=B\left(T,\sigma \right)t(t/t_{ref}{)}^{1+m}{\left(\sigma /G\right)}^{n+\omega }{e}^{-Q/RT}$(5) ) with verification for OFHC copper, low-alloy 0.5CMV steel and higher-alloy X20 steel, to cover a wide range of polycrystalline metals and microstructures. The new model performed at least equally well, or better, in comparison to the classical model using time, stress and temperature as predictive variables for the creep cavitation damage. The advantage of the new model stems from avoiding measured creep strain that can be inaccessible in many practical structures.

Acknowledgements

The authors wish to acknowledge the financial support of the Academy of Finland, via the EARLY project No. 325108, and Finnish Research Programme on Nuclear Waste Management, via the CRYCO project No. KYT15/2020.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Academy of Finland, Luonnontieteiden ja Tekniikan Tutkimuksen Toimikunta [325108] [325108]; Finnish Research Programme on Nuclear Waste Management (KYT2022) 2019-2022 [KYT15/2020].