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Abstract

Several continuum damage mechanics (CDM) modelling approaches for predicting creep deformation of tempered ferritic steels have been developed in the literature, which have evolved from efforts to extend the operability of power plant components. Few of these models, however, focus on damage assessment of ex-service states of power plant steels through the extraction of damage parameters. Furthermore, few CDM approaches leverage the high density of creep curve data available through full-field strain measurement techniques such as digital image correlation (DIC). This work uses multiple creep curves obtained from DIC computed strain data at several stresses and temperatures from individual specimens of X20CrMoV12-1 (X20) piping steel. These curves serve as input data to a modified Oruganti continuum damage mechanics (CDM) model whereby microstructural-specific damage parameters can be extracted. Good agreement is noted between CDM-extracted parameters and microstructural, creep cavity density and hardness damage indicators. Damage parameters based on subgrain growth are particularly sensitive to the ex-service state of the X20 steel. The proposed CDM approach using DIC computed creep curves is shown to be a material efficient alternative to traditional damage assessment methods of ex-service material.

Keywords

12% Cr steel creep damage digital image correlation continuum damage mechanics high temperature testing

Introduction

The operation of coal-fired power stations beyond their design lifetimes has highlighted the need for improved assessment techniques to better measure the accumulated creep damage present within critical components. Steam piping steels often used in older generation plants, such as the 12% Cr steel X20CrMoV12-1 (X20), exhibit complicated degradation mechanisms which involve interacting microstructural factors; creep strength of X20 and other 9–12% Cr tempered martensitic-ferritic steels stems from a fine substructure with dense dislocation structures stabilised by fine precipitates, such as MX carbonitrides and M₂₃C₆ carbides. Plant service exposure drives the evolution of these thermodynamically unstable features, including the coarsening of subgrains, recovery of dislocations, coarsening of precipitates and the development of new precipitates such as Laves and Z phase (Aghajani et al., 2009).

The last half of the century has seen the promulgation of several damage assessment techniques in an effort to extend thermal power plant operability. Methods include, among others, physical evaluations, such as hardness and surface cavity replica measurements, constitutive relations of creep (Cardoso et al., 2015), such as the description of minimum creep rates through Norton’s power law (Abdallah et al., 2014), rupture life predictions such as the Larson-Miller parameter (Larson and Miller, 1952) and full-curve descriptions including the $θ$ -projection and $Ω$ models (Kim et al., 2011). Decreases in Vickers hardness measurements have also been employed as crude empirical assessments of the remnant life of low- and high-alloy CrMo steels through correlation with time-temperature relations such as the Larson-Miller parameter (Cardoso et al., 2015; Masuyama, 2009). These methods, however, have limited applicability to service-exposed materials due to their inability to account for the aforementioned microstructural intricacies (Gupta et al., 2015).

Some improved creep relations have demonstrated difficulty in isolating differences between service-aged materials. In the previous work of the authors (van Rooyen et al., 2019) for example, insignificant differences were found between ex-service X20 states in terms of precipitate-dependent threshold stress calculated using a modified power law relation, despite large differences in the corresponding creep curves. Classical creep models also do not universally capture the different creep mechanisms that dominate as the material state, testing temperature and stress conditions are altered. Parameters, such as activation energy and stress exponents, have been found to be insensitive to the state of material exposure and therefore do not lend themselves as effective damage indicators (El Rayes and El-Danaf, 2017; van Rooyen et al., 2019).

Continuum damage mechanics (CDM) models provide a framework by which several coupled damage laws phenomenologically track the evolution of microstructural and cavitation factors during creep by including sets of dimensionless damage parameters. The advantage of physically based CDM methods is the adaptability of these damage descriptors to physical microstructural characterisation through electron microscopy, allowing the entire creep curve to be modelled within creep regimes with different loading conditions and operating mechanisms (McLean and Dyson, 2000).

A large body of literature has focused on the application of CDM to 9–12% Cr steels, which evolved from the works of Dyson and McLean (Dyson and McLean, 1998; Dyson, 2000; McLean and Dyson, 2000) who pioneered the incorporation of several damage parameters into the fundamental hyperbolic sine creep rate equation that lies at the root of CDM. Since then, research has adapted various combinations of these damage parameters, with most of the approaches focusing on long-term creep modelling of power plant alloys. For 10% Cr steels, Kadoya et al. (1997) proposed a model that combined the effects of MX particle coarsening, dislocation accumulation and Mo solute depletion (due to Laves phase Fe₂Mo precipitation) to predict creep strain-time curves for lives in excess of 6 000 h. This model was later adapted by Semba et al. (2008) who recognised that a dislocation recovery damage parameter is more appropriate for tempered martensitic-ferritic steels and included this as an indirect means of accounting for subgrain coarsening in the modelling of 9% Cr steel. The model formulation was later expanded in the work of Basirat et al. (2012) to include a cavitation damage parameter. However, the authors did emphasise the technical challenges associated with quantifying the dislocation densities required by a dislocation-based creep model.

There has also been evidence in the literature (Aghajani et al., 2009; van Rooyen et al., 2019; Yin and Faulkner, 2006) to suggest that instead of MX growth, the main particle damage parameter in X20 steel is related to the coarsening of the subgrain-pinning (intergranular) M₂₃C₆ precipitates. This was recently implemented in the multi-precipitate CDM model developed by Christopher and Choudhary (2019) in modelling long-term creep of Grade 91 steel. In addition to thermally induced coarsening, straining has also been shown to accelerate the growth kinetics of precipitates. A strain component is included in a particle damage evolution law by Murchú et al. (2017), who revealed that such an approach does not improve parameter correlation in short-term tests but does become significant when considering creep strain contributions across weld regions (Murchú et al., 2019). Instead, the need for subgrain damage evolution was identified.

Oruganti et al. (2011) focused on essential microstructural factors that contribute to creep degradation in CDM modelling of 9 – 10% Cr steels, including subgrain and particle coarsening. One of the overarching benefits of this model is the reduction in the number of parameters fitted to 12, of which 9 are physically bounded. Even more parameters can be combined under certain testing conditions such as for isothermal testing. Furthermore, the work was one of the first efforts to directly model subgrain growth as a damage parameter, which is considered vital to X20 modelling due to the sensitivity of this parameter to ageing through service exposure (Aghajani et al., 2009; van Rooyen et al., 2019).

The literature is currently sparse in terms of the application of CDM to service-exposed power alloys. Dyson and Osgerby (1993) and later McLean and Dyson (2000) applied CDM models to service-exposed 1Cr0.5Mo steel by adjustment of two non-damage parameters. More recently, efforts have been made to capture elements of creep degradation of P9 steel subjected to different heat treatments through a comparison of fitted microstructural rate constants (Christopher and Choudhary, 2018; Pan et al., 2017) successfully applied the Oruganti model to the boiler steels T91 and T92, although it is unclear whether ex-service or virgin material was used.

One of the main challenges in using CDM models is the requirement for large datasets at different stress or temperature states to fit parameters (Dyson, 2000; Yin and Faulkner, 2006). Service-exposed steel is often in limited supply and can only be removed in small sections from repaired components during planned station maintenance periods (Gupta et al., 2015). Measurement of multiple creep curves at various temperatures and stresses requires multiple specimens and substantial amounts of material and testing time. There has been little attempt to interface CDM with modern image correlation methods that presently lie at the heart of full-field experimental mechanics, let alone the application to service-exposed power engineering alloys.

In contrast to conventional deformation measurement solutions such as extensometers, digital image correlation (DIC) allows full-field displacement measurement by spatially tracking unique speckle features within subset windows discretised from sequential digital images (Schreier et al., 2009). Previously, the authors developed a technique for obtaining multiple tensile (van Rooyen and Becker, 2018) and creep (van Rooyen et al., 2019) curves at different temperatures and different stresses (van Rooyen et al., 2022) from single specimens of ex-service X20 using DIC. The high spatial density of deformation measurement points obtainable through DIC satisfies the material economy required by the mechanical testing of ex-service X20.

This work builds on previous studies (van Rooyen et al., 2019, 2020, 2022) in which multiple creep data sets were obtained from single-specimen tests using DIC and variable stress and temperature profiles. These data sets serve as input for damage parameter extraction using a CDM model adapted from Oruganti et al. (2011) of various aged forms of ex-service X20 with the main aim of establishing a relationship between the ageing level of ex-service X20 and initial-state (prior to testing) damage parameter magnitudes. A comparative assessment of the fitted parameters between material states is coupled with microstructural analysis and alternative damage assessment methods to confirm the damage category of the tested ex-service X20.

Materials and experiments

The service-exposed X20 steel used in this work was supplied in the form of thick-walled piping from local power stations. The original categorisation of legacy steel in terms of service history is based on cavity density measurement through surface replication with higher damage¹ associated with more severe operating conditions, longer service exposure and higher cavity densities. Figure 1 shows backscatter electron scanning electron microscope (BS-SEM) micrographs highlighting the main microstructural features of the various levels of service exposure of the X20 steel (labelled from here on as virgin and low, medium and high damage X20). Low and medium damage X20 was subjected to 130 000 hours of in-service operation at an operating temperature of 545°C and pressures of 17 MPa and 19.4 MPa, respectively. High damage X20 was exposed to 543°C and 18.1 MPa at 156 000 hours. Further details regarding the service conditions, chemical composition and microstructural characterisation of the materials are detailed in van Rooyen et al. (2019). In summary, increasing subgrain and Cr₂₃C₆ precipitate sizes, higher Laves phase volume density, higher cavity density and lower Vickers hardness measurements were noted for increasing damage states. These trends were not as obvious between the low and medium damage states, despite differences in service exposure and experimentally measured creep strain rates.

Figure 1.

BS-SEM micrographs of (a) virgin and (b) low, (c) medium and (d) high damage X20 steel showing subgrains, Cr₂₃C₆ and Mo-rich Laves phase precipitates.

Various sources of creep curves from different test types are used for input to the CDM model, as summarised in Table 1 with the test setups summarised in the schematic diagrams in Figure 2. These tests are categorised according to (i) duration from start to rupture and whether (ii) uniform (for use in conventional testing, that is, constant temperature or stress) or (iii) variable temperature or stress fields (with the use of DIC) are applied.

Table 1.

Summary of different test types for measuring creep curve data for virgin and service aged X20.

Naming convention	Accelerated		Medium-term
Duration; Stress range	<10 h; High stress, >200 MPa		24–1000 h; Intermediate stress, 100–150 MPa
Subcategory	Variable temperature (DIC)	Conventional, (ASTM E139, 2011)	Variable stress (DIC)	Conventional, (ASTM E139, 2011)
Methodology	Several curves over constant stress and variable temperature (575–600°C)	Single curve over constant stress and temperature	Several curves over variable stress (100–150 MPa) and constant temperature	Single curve over constant stress and temperature
Creep curve data source	(van Rooyen et al., 2019, 2020)	(van Rooyen et al., 2020)	(van Rooyen et al., 2022)

Figure 2.

Schematic diagram of the experimental setups for the (a) variable-temperature tests using stereo-DIC for strain measurement (van Rooyen et al., 2019, 2020; van Rooyen and Becker, 2018), (b) variable-stress tests using 2D-DIC for strain measurement (van Rooyen et al., 2022) and (c) ASTM standardised, conventional creep tests using extensometry for strain measurement (ASTM E139, 2011; van Rooyen et al., 2020).

As shown in Figure 2(a), variable-temperature accelerated creep testing was performed in a Gleeble thermomechanical simulator that applies a near-parabolic temperature profile (with a maximum of 600°C) longitudinally across a specimen. Strain and temperature data are acquired by stereo-DIC and thermal imaging applied to the speckled and highly emissive (black) surfaces of the specimen, respectively. Further details regarding the setup can be found in van Rooyen et al. (2019), van Rooyen and Becker (2018), and van Rooyen et al. (2020).

Variable-stress medium-term creep tests were conducted at 600°C using a constant-load creep setup (Figure 2(b)) where 2 D-DIC is applied using a single camera through a tube furnace viewport. A waisted specimen design (ASTM E139, 2011) results in non-uniform stress development along the uniaxial loading direction. Further details regarding the setup can be found in van Rooyen et al. (2022).

Conventional testing involved the use of a traditional creep setup as stipulated by ASTM E139 (2011) where strain is measured using a linear variable displacement transducer (LVDT) as indicated in Figure 2(c). Constant temperature and engineering stress is maintained per specimen in these tests. Further details regarding the setup can be found in in van Rooyen et al. (2020).

Model formulation

The creep model used is based on the CDM formulation developed by Oruganti et al. (2011) and was selected on the basis of simplicity in the inclusion of key microstructural parameters (subgrains and precipitates) relevant to the creep strength of X20 (Aghajani et al., 2009; van Rooyen et al., 2019). The model is also adapted to include a cavity nucleation-controlled damage parameter to better model the creep curves of aged (cavity-dense) materials.

Modified model equations

The CDM model employed is represented by the collection of coupled differential equations (1) to (6):

{\dot{ε} = ε}_{0}^{'} \exp \frac{- Q}{R T} \sinh (\frac{σ (1 - H^{*} (1 - D_{S}))}{σ_{r o} (1 - D_{P}) (1 - D_{c})})

(1)

{\dot{σ}}_{r o} = K_{1} (1 - \frac{σ_{r o}}{K_{2}}) \dot{ε}

(2)

{\dot{D}}_{S} = \frac{\dot{ε} {(1 - D_{S})}^{2}}{S_{i}} (K_{S 1} + K_{S 2} e xp (\frac{- Q_{S}}{R T}))

(3)

{\dot{D}}_{P} = \frac{{(1 - D_{P})}^{4}}{P_{i}^{3}} (K_{P} e xp (\frac{- Q_{P}}{R T}))

(4)

{\dot{D}}_{C} = \frac{k_{N}}{ε_{r}} \dot{ε}

(5)

\dot{σ} = σ \dot{ε}

(6)

The fundamental creep law in equation (1) relates the creep strain rate, $\dot{ε}$ , to stress, $σ$ , and temperature, $T$ , by tracking the evolution of the state variables including the reference stress, $σ_{r o}$ , the subgrain, $D_{S}$ , precipitate, $D_{P}$ , and cavity, $D_{C}$ , damage parameters shown in equations (2) to (5), respectively. For modelling constant-load creep curves, the stress rate, $\dot{σ}$ , in equation (6) must be coupled with the preceding model equations.

The characteristic creep constant, $ε_{0}^{'}$ , is a scaling parameter which affects the modelled minimum creep rate and depends on the initial dislocation density, subgrain and precipitate volume fraction (Semba et al., 2008) as well as matrix solute diffusivity of Kadoya et al. (1997). This constant is associated with the Arrhenius relation $ε_{0}^{'} \exp \frac{- Q}{R T}$ where $R$ is the universal gas constant and $Q$ is the creep activation energy. The reference stress, $σ_{r o}$ , is a measure of the strengthening mechanisms offered by dislocation-particle interactions and, in many CDM formulations, is treated as a constant related to the Orowan particle bypass stress.

Primary creep is represented by an evolution of $σ_{r o}$ in equation (2) which phenomenologically models the evolution of dislocation structures to be in equilibrium with the applied stress until a steady value of $K_{2}$ is reached at a rate of $K_{1}$ . This differs from other CDM models that represent the primary hardening through evolution of the normalised kinematic backstress, $H$ , as the stress redistributes between the soft steel matrix and hard precipitates (Ion et al., 1986). In the Oruganti model, the maximum value of normalised kinematic backstress, $H^{*}$ , is fixed at a constant value that depends on the initial subgrain, $S_{i}$ , and high-angle block, $B_{i}$ , widths (Oruganti et al., 2011; Xiao et al., 2019):

H^{*} = 0.26 (1 - ν) \frac{λ_{M 23 C 6}^{4 / 3} θ^{1 / 3}}{{b^{1 / 3} S}_{i}} + \frac{K ″}{\sqrt{B_{i}}}

(7)

Here, $ν$ is the Poisson’s ratio, $λ_{M 23 C 6}$ is the average spacing of M₂₃C₆ precipitates, $θ$ is the average subgrain misorientation in radians, $b$ is the Burgers vector magnitude (=2.48 × 10⁻¹⁰ m for body-centred phases Callister and Rethwisch, 2014), and $K ″$ is a constant.

Damage parameters

The tertiary regime is modelled through the damage parameters and their evolutionary equations. The subgrain coarsening damage parameter is given as,

D_{S} = 1 - \frac{S_{i}}{S}

(8)

where

S

is the instantaneous subgrain short width. Equation (3) gives the rate equation for

D_{S}

where

K_{S 1}

and

K_{S 2}

are the strain and thermally activated components, respectively, of subgrain coarsening and

Q_{s}

is the associated activation energy. The weakening effect of the migration of subgrain boundaries is modelled through the decrease of the backstress parameter

H^{*}

by multiplication of

{(1 - D}_{S})

in equation (1).

The damage term related to precipitate coarsening is a function of the initial particle diameter $P_{i}$ and instantaneous particle diameter $P$ ,

D_{P} = 1 - \frac{P_{i}}{P}

(9)

The Ostwald ripening law based on volume diffusion control (Yadav et al., 2016) leads to the rate equation of $D_{P}$ in equation (4) where $Q_{P}$ and $K_{P}$ refer to the particle growth activation energy and pre-exponent constant, respectively. Previous research (Aghajani et al., 2009; van Rooyen et al., 2019) underlined the importance of M₂₃C₆ carbides in maintaining the creep strength of X20 by hindering subgrain boundary migration, highlighting the need to include the relevant coarsening parameters in the damage modelling of such materials.

Besides microstructural deterioration parameters, physical damage in the form of creep cavitation has long been reported for X20 particularly at low stresses (Parker and Siefert, 2018; Trück et al., 1991), leading to consumed life estimation of in-service components based on cavity density measurements from replica metallography (van Zyl et al., 2005). Based on this line of thinking, a creep-constrained cavity nucleation-controlled damage parameter $D_{C}$ is added to the Oruganti model in equation (1). Dyson (2000) proposed a linear relationship between $D_{C}$ and strain, resulting in the rate equation (5) where $ε_{r}$ is the failure strain and $k_{N}$ is a parameter approximately related to the area of cavitated boundaries before failure and has a maximum value of 1/3 (McLean and Dyson, 2000). The relationship of $D_{C}$ with cavity densities of $N_{c}$ and diameters $d_{c}$ is given as (Dyson and McLean, 1998; Wu and Sandström, 1995):

D_{C} = \frac{π {d_{c}}^{2} N_{c}}{4}

(10)

In reality, both void growth and cavitation occur (as noted previously for aged X20 in van Rooyen et al., 2022), requiring combined creep-constrained cavity growth and growth-constrained cavity nucleation damage equations that prove to be complex when defining for high stresses within the transgranular fracture regime (Dyson and McLean, 1998; Lin et al., 2005; Perrin and Hayhurst, 1996). Furthermore, estimates of $D_{C}$ from Equation (10) are traceable through BS-SEM micrographs from which $d_{c}$ and $N_{c}$ can be measured. For the sake of simplicity, equations (5) and (10) are therefore employed to simulate cavity damage.

Model calibration

Model calibration involves determining the fixed baseline parameters of the modified Oruganti CDM model which are summarised in Table 2. Details regarding the methods used to determine the values are summarised in Appendix 1. Parameters established from microstructural data ( $H^{*}$ , $S_{i}$ and $P_{i}$ ) and literature ( $Q, Q_{S}, Q_{P}$ and $k_{N}$ ) of virgin X20 are fixed, whereas constants relating to the damage evolution ( $ε_{0}^{'}$ , $K_{1}$ , $K_{2}$ , $K_{S 1}$ , $K_{S 2}$ and $K_{P}$ ) are obtained by means of numerical optimisation in fitting (least squares) experimental data of virgin X20 to the model. Initial parameters and search bounds listed in Table 2 provide the optimisation parameters used. An initial value of $σ_{o}$ was set to 2 MPa as recommended by Oruganti et al. (2011) for 9–12% Cr ferritics.

Table 2.

Summary of initial estimates, bounds and optimised values of baseline parameters.

Material parameter and units	Initial estimates	Search bounds	Established parameters	Source for initial estimates
Fixed parameters
$H$ *	–	–	0.30	Equation (7)
$Q$ , kJ/mol	–	–	300	(Yin and Faulkner, 2006; Hore and Ghosh, 2011)
$Q_{S}$ , kJ/mol	–	–	285	(Nitta et al., 2002)
$Q_{P}$ , kJ/mol	–	–	225	(Straub, 1995), Figure 8a, Equation (13)
$k_{N}$	–	–	0.1	(Dyson, 2000)
$S_{i}$ , m	–	–	0.42 × 10⁻⁶	Current work
$P_{i}$ , m	–	–	152 × 10⁻⁹	Current work
Optimised parameters
$ε_{0}^{'}$ , s⁻¹	1.3 × 10⁷	1 × 10⁵–1 × 10⁹	7.59 × 10⁶	(Murchú et al., 2017; Straub, 1995)Figure 8(c)
$K_{1}$ , Pa	6.5 × 10⁹	0.1 × 10⁹–10 × 10⁹	3.72 × 10⁹	∼1/10th of the shear modulus at 600°C (EPRI, 2006)
$K_{2}$ , Pa	13.6 × 10⁶	6 × 10⁶–15 × 10⁶	12.7 × 10⁶	(Murchú et al., 2017; Straub, 1995)Figure 8(c)
$K_{S 1}$ , m/s	140 < $σ$ < 150 MPa:2.6 × 10⁻⁶(Adjusted) $σ$ > 200 Mpa:1.27 × 10⁻⁸	1 × 10⁻⁸–1 × 10⁻⁵	2.58 × 10⁻⁶	(Aghajani et al., 2009; Oruganti et al., 2011) Figure 8(b)
$K_{S 2}$ , m/s	3.1 × 10¹¹	1 × 10¹⁰–1 × 10¹¹	5.35 × 10¹⁰	(Aghajani et al., 2009; Oruganti et al., 2011) Figure 8(b)
$K_{P}$ , m³/s	7.6 × 10⁻¹⁵	1 × 10⁻¹⁵–1 × 10⁻¹⁴	4.81 × 10⁻¹⁵	(Straub, 1995) Figure 8(a), Equation (13)

Figure 3(a) compares the medium-term experimental data (symbols) and the corresponding CDM model response (solid line) for the DIC-measured creep curves of the variable stress test at several stresses. Data and model predictions for the 140 MPa conventional creep test are also included. Good agreement is noted between the model predictions and the DIC experimental data, including the conventional creep curve at 140 MPa. This demonstrates the ability of the current model to simulate medium-term (<1000 h duration) creep curves.

Figure 3.

Comparison of the predictions of the CDM model and the experimental data of virgin X20 from (a) medium-term variable stress data including a conventional test at 140 MPa, (b) accelerated, conventional test data with adjusted model parameters and (c) accelerated, variable temperature test data with corresponding (d) creep rates (using the same legend in (c)).

At higher stresses and shorter durations typical of accelerated tests, the model underestimates the time to rupture, as shown in Figure 3(b) for a conventional creep test at 250 MPa and 600°C. This is likely due to the change in the dislocation climb mechanism from general to local at higher stresses (Christopher and Choudhary, 2019), or due to the stress dependence of the evolution of internal structures. Straub (1995) and Qin et al. (2003) demonstrated that steady-state subgrain size is inversely proportional to stress. Under the assumption of the stress-dependent subgrain growth mechanism, an updated $K_{S 1}$ was optimised from the conventional creep curve at 250 MPa and 600°C and given in Table 3. The adjusted model results in an improved fit, as shown in Figure 3(b).

Table 3.

Summary of ranges for estimated initial and optimised damage parameters.

Damage parameter/sensor	Original damage class	Microstructural/cavity density/hardness data			Range of initial estimates	Optimised parameters
Damage parameter/sensor	Original damage class	Feature	Size range	Source	Range of initial estimates	A	MT
$D_{S} = 1 - \frac{S_{i}}{S}$	Virgin	$S_{i}$ (nm)	409–423	(Marx, 2019)	–	–
	Low	$S$ (nm)	520–560	(van Rooyen et al., 2022; van Rooyen et al., 2019)	*0.18–0.26	$\approx$ 0	0–0.06
	Medium		410–622		*0.18–0.34	0.10	0.17–0.23
	High		710–790		*0.40–0.48	0.39	0.21–0.35
$D_{P} = 1 - \frac{P_{i}}{P}$	Virgin	$P_{i}$ (nm)	136–168	(Marx, 2019)	–	–
	Low	$P$ (nm)	173–191	(van Rooyen et al., 2022; van Rooyen et al., 2019)	*0.02–0.28	0.05	0.02–0.06
	Medium		161–183		*0.02–0.25	$\approx$ 0	0.01–0.02
	High		194–198		*0.13–0.31	$\approx$ 0	0.12–0.17
$D_{C} = \frac{π {d_{c}}^{2} N_{c}}{4}$	Virgin	–	–	–	–	–
	Low	$d_{c}$ (µm), $N_{c}$ (cavities/mm²)	1.2–3.4,71–167	(Marx, 2019)	*0–0.001	N/A
	Medium		1.0–2.4, 146–250	(van Rooyen et al., 2022; van Rooyen et al., 2019)	*0.0001–0.001	N/A
	High		0.9–2.2, 60–960	(van Rooyen et al., 2022; van Rooyen et al., 2019)	*0–0.004	N/A
Hardness	Virgin	$h_{o}$ (HV)	244 ± 4	(van Rooyen et al., 2022; van Rooyen et al., 2019)	N/A
	Low	$h_{e}$ (HV)	211 ± 4
	Medium		238 ± 4
	High		202 ± 4

A: accelerated tests; MT: medium-term tests. Lower bounds indicated by * are used as initialisers for the optimisation routine of subgrain and precipitate parameters and for constants set for cavitation damage.

To demonstrate the ability of the adjusted model to simulate creep data at higher stresses and at different temperatures, the adjusted model was applied to accelerated variable-temperature creep curves measured using DIC from a single specimen. As indicated in the microstructural analysis performed in a previous work (van Rooyen et al., 2020), particle coarsening was ignored for these short duration tests ( ${\dot{D}}_{P} = 0$ ). The comparison of the model predictions and experimental creep strain and creep rate curves is shown in Figure 3(c) and (d), respectively. A satisfactory match (within 5% for the secondary regime) is apparent between the model and experimental strain, barring the primary regions and the tertiary region at 600°C. Differences within the tertiary regime are still minor considering the typical scatter of factor 2 encountered within rupture life measurements (Yin and Faulkner, 2006). As seen in Figure 3(d), the secondary creep regime is reasonably captured by the adjusted CDM model.

Damage parameter extraction

The premise of the paper lies in establishing a relationship between the aging level of ex-service X20 and initial-state (prior to testing) damage parameters used in the CDM model. This can be achieved by applying the calibrated model and fitting initial values of selected damage parameters ( $D_{S}, D_{P}, D_{C}$ ) to DIC-measured creep curves of low, medium and high damage X20. The method considers the baseline parameters (Table 2) to remain constant (within a restricted stress range) whilst initial damage parameters are considered to vary between ex-service states due to the evolution of certain microstructural factors (van Rooyen et al., 2022).

5.1 Sensitivity study

The sensitivity of the strain with respect to the damage parameters must be explored to establish the uniqueness of the solutions and the correlations between the parameters. The influence of a percentage change (Δ $D_{x}$ , where x represents S, P or C) on creep curves of different durations is shown in Figure 4. Note that Δ $D_{x}$ = 0% refers to the creep curves corresponding to the undamaged or virgin X20. Figure 4(a) to (c) show creep curves typical of accelerated testing (such as in Figure 3(b) and (c)), medium-term testing (Figure 3(a)) and long-term tests encountered in the work of Aghajani et al. (2009), respectively. It is clear that the effect of $D_{S}$ is unique in all three test types, whereas $D_{C}$ and $D_{P}$ are difficult to distinguish in the shorter-term tests (Figure 4(a) and (b)) due to the long-term evolutionary nature of these parameters. It is therefore necessary to select a combination of two parameters that can be uniquely determined through a minimisation of a suitable objective function.

Figure 4.

Sensitivity of creep curves to subgrain, precipitate and cavity damage parameters for (a) accelerated, (b) medium-term and (c) long-term tests. Note: Δ $D_{x}$ = 0% corresponds to virgin (undamaged) material.

Although not presented here, several objective functions were considered, including a creep stage-based weighting function as suggested by Christopher and Choudhary (2018). It was found that a smooth and low stiffness evolution is available through a normalised version of equation (14) where $F_{nor} = (\frac{1}{n_{p}}) F_{\min}$ . In an approach similar to Roux and Bouchard (2015), a numerical framework is used whereby a simulated creep curve is generated using nominal damage parameter values typical of ex-service material ( $D_{S} = 0.20, D_{P} = 0.05, D_{C} = 0.02$ ) and compared using $F_{nor}$ to creep curves generated from damage parameters varying within 10% of the nominal values. Because three parameters are involved, minimisation involves a 3 D function that is represented by the three cross-sectional response surfaces shown in Figure 5. A steeper gradient is observed along one diagonal of these surfaces with a more gradual slope towards a global minimum perpendicular this diagonal, as indicated by the presence of ellipses in Figure 5(a) and (b). In agreement with Figure 4, this test shows that optimising for either $D_{S} - D_{P}$ or $D_{S} - D_{C}$ parameter sets is possible for medium-term testing. Figure 5c indicates that the $D_{P} - D_{C}$ set does not possess a unique solution for these medium-term tests due to the correlation between the parameters along a diagonal with an approximate 1:1 gradient.

Figure 5.

Response surfaces of the objective function for medium-term creep tests for (a) D_S–D_P , (b) D_S–D_C and (c) D_P–D_C parameter combinations.

Further consideration is given to the effect of stress and temperature measurement error on the confidence in optimised damage parameter values (determined as the absolute difference between nominal and identified parameters normalised by the search range). The simulated creep curves in this case include a Gaussian distribution of strain noise with a standard deviation of 150 µε which approximately corresponds with the longitudinal temporal noise floor (142 με) calculated for the DIC-measured creep curves (van Rooyen et al., 2022).

It was found that the same percentage change in temperature, $Δ T$ , results in higher error in parameter identification than for corresponding changes in stress, $Δ σ$ . This is likely due to the presence of $T$ in the pre-exponential function of the strain rate equation (1) and emphasises the need for a uniform temperature distribution in the gauge section of the waisted specimen design employed in the variable stress, isothermal tests. Furthermore, for the stress and temperature errors considered, relatively low errors in $D_{S}$ (<10%) compared to $D_{C}$ and $D_{P}$ (<40%) were estimated. This is most likely due to the dependence of these parameters on the tertiary stage for their unique identification (Figure 4) coupled with the sensitivity of the tertiary stage and damage evolution equations to small changes in $T$ and $σ$ (Hore and Ghosh, 2011; Yin and Faulkner, 2006).

For the purpose of this study, the $D_{S} - D_{P}$ parameter set will be optimised based on several considerations: (i) The convergence to a minimum is most likely for $D_{S} - D_{P}$ as evident from Figure 5(a). The use of $D_{S}$ in this set also appears to be more robust to uncertainties in stresses and temperatures. In this work, the temperature is maintained within 2°C in the gauge region and stress uncertainty is attributed to the accuracy of the load cell and position accuracy of the DIC virtual strain gauge relative to the geometry of the specimen. This results in expected damage parameter errors of <1.4% and <3% for $D_{S}$ and $D_{P}$ , respectively. (ii) Both $D_{S}$ and $D_{P}$ have direct microstructural definitions (equations (8) and (9)) whereas $D_{C}$ in equations (5) and (10) depends on cavitation and strain accumulation relations observed during monotonic tensile creep tests.

It should be noted that the optimised $D_{P}$ is mathematically equivalent to a combination of $D_{P}$ and $D_{C}$ on account of the similar responses of accelerated and medium-term creep curves, as shown in Figure 4, as well as their identical location within the master equation (1). Compared to $D_{P}$ , relatively low values for $D_{C}$ are expected due to the nature of equation (5) that limits the upper bound to 0.1 ( $k_{N}$ ). This is bolstered by the observation that the higher cavity densities before testing did not reduce the rupture ductility for the high damage states (∼0.24%) compared to the virgin counterpart with no creep cavities (∼0.20%) (Dyson, 2000; van Rooyen et al., 2022). Although the distinction between $D_{P}$ and $D_{C}$ cannot be made, it is expected that a larger portion of the optimised $D_{P}$ will consist of precipitate contributions.

As indicated by equation (7), $H^{*}$ should evolve with time and condition based on the evolution of dislocation densities, block and subgrain sizes (Christopher and Choudhary, 2019) and could therefore also be considered a damage indicator. However, including $H^{*}$ in the optimisation routine would result in numerical instability due to the product of $H^{*}$ with the subgrain damage parameter $D_{S}$ in Equation (1). The authors therefore chose to fix $H^{*}$ (as calculated in Appendix 1) to stabilise the optimisation routine and reduce numerical complexity. In this manner, the collective effect of subgrain and block size evolution is captured in the $D_{S}$ parameter for the ex-service states.

Subgrain and precipitate damage optimisation

Extraction of $D_{S}$ and $D_{P}$ is undertaken using the same optimisation procedure outlined in Section ‘Model calibration’ and Appendix 1. The high creep curve density with variable stress states offered by DIC provides sufficient input data to the CDM model in the medium-term. In addition, adjusted model fitting is also applied to the accelerated test data with a variable temperature distribution.

Table 3 reports the initial estimates used in the fitting procedure (due to the heterogeneity of the microstructural data, a range of initial estimates are reported), including the estimates of $D_{C}$ that are kept constant. Note that the evolution ( ${\dot{D}}_{C}$ ) from this constant initial value still occurs according to Equation (5) based on the observations of increasing cavity densities across the specimen gauge region (van Rooyen et al., 2022).

The experimental and fitted curves for the ex-service material states are shown in Figure 6 for accelerated, variable temperature creep tests (top row) and medium-term, variable stress creep tests (bottom row). Similarly to the baseline fits in Figure 3, a satisfactory match is observed between the model predictions and medium-term data in the bottom row of Figure 6 whereas relatively poorer fits are noted for the accelerated creep tests (top row) for the same reasons as mentioned in Section ‘Model calibration’. The current formulation of the CDM model only optimises for two parameters (initial values of $D_{S}$ and $D_{P}$ ) as opposed to six in the original version in Oruganti et al. (2011). This places a limit on the ability to exactly match all regions of the curves across vast ranges of stresses, temperatures and ex-service states involved in this work.

Figure 6.

Comparison of CDM model predictions and experimental creep curves for low, medium and high damage X20 (left to right columns, respectively) during accelerated variable temperature and medium-term variable stress (top to bottom rows, respectively).

The optimised damage parameters obtained from the accelerated and medium-term tests are provided in Table 3. Apart from the $D_{S}$ values for the low damage X20, most damage parameters extracted from the medium-term tests are within or close to the microstructural estimates. Relatively higher $D_{S}$ values are recorded in comparison to $D_{P}$ values. Values of $D_{S} \approx 0$ are obtained for low damage X20, whereas medium and high damage states exceed values of 0.1 and 0.3, respectively. Medium damage X20 shows the lowest $D_{P}$ values of less than 0.02 which is a third of the low damage $D_{P}$ of about 0.06. High damage X20, on the other hand, displays the highest $D_{P}$ values which exceed 0.1.

Discussion

Differences in creep behaviour between service-exposed X20 steels with varying levels of service histories are identified by the DIC technique. Microstructural deterioration is also revealed through subgrain, precipitate, cavity density and hardness measurements. These aspects are combined using a microstructurally sensitive CDM framework to reveal quantitative differences in damage levels between the ex-service states.

Full-field methods in creep damage modelling

DIC based creep strain measurements can extract multiple, spatially resolved creep curves at several stresses using a waisted specimen geometry during medium-term tests, or multiple accelerated creep curves across a varying temperature field generated through a Gleeble thermomechanical machine. Such approaches are ideal when material efficiency is a key requirement, such as in the assessment of the service-retrieved materials from a power plant. Accelerated creep tests demonstrated time savings and the economy of material advantages over conventional longer-term versions. Variable temperature tests require special feedback control from a Gleeble thermal system to maintain a specific temperature profile across the specimen, making adaptation to longer-term, independent testing difficult. On the other hand, the tests conducted within a traditional creep rig modified with integration of DIC may be better suited to measurement of medium-term creep curves.

Both approaches produce creep curve data that are representative of the damage levels of service-exposed X20 steels with varying levels of service histories. Specifically, differences in damage states are revealed through qualitative comparisons of curves in terms of minimum creep rates and failure times (Figure 6). Previous approaches (van Rooyen et al., 2019) to quantifying damage levels used simple conventional creep relations (such as the Bird-Mukherjee-Dorn and Zener-Hollomon threshold stress-modified equations) together with DIC-obtained accelerated creep curves. In this work, the application of accelerated and medium-term data to CDM models is highlighted.

Calibration of the Oruganti et al. (2011) CDM model (modified to include cavitation damage) for virgin X20 benefits from the data richness offered by DIC analysis over non-uniform temperature and stress fields. From a single test, a range of curves in various stages of development can be systematically input into the model to extract baseline constants. For example, primary and secondary regime-dominated curves can be used to extract baseline reference stress (equation (2)) and subgrain (equation (3)) parameters, respectively. Tertiary segments of the curves are useful in identifying baseline constants related to slow-evolving subgrain and precipitate damage relations, although the final acceleration to failure is only captured for the maximum temperature or stress condition. This is shown in Figure 9 when comparing the tertiary regime of the 140 MPa stress curve from a variable stress test to a curve at the same stress from a conventional test. The rapid increase in strain rate in the final stages of the former curve is caused by the rupture of the specimen at the location of maximum stress (viz. the 150 MPa curve in Figure 9).

The medium-term test is successfully simulated by the model whose calibrated parameters are within the ranges predicted from the literature. For accelerated tests, the primary stage is not well described as shown in Figure 6. This disparity can be attributed to the lack of a stress-dependent evolutionary equation for the backstress contribution to the primary creep through $H^{*}$ in equation (1) (Christopher and Choudhary, 2018) or the difficulty of equation (2) in capturing the stress-dependent evolution of primary creep (Oruganti et al., 2011). Nonetheless, the model predicts the remainder of the curves considering the difficulties commonly encountered with CDM modelling at higher stresses (Christopher and Choudhary, 2019).

Comparison with alternative damage indicators

Application of the calibrated CDM model to ex-service X20 creep curves from various test types provides a set of optimised $D_{S}$ and $D_{P}$ parameters. Figure 7 compares these parameters between ex-service states and between test durations. For comparison, two alternative creep damage evaluation methods are included. Firstly, the accumulated damage in the form of a life fraction $\frac{t}{t_{r}}$ (where $t$ is the exposure time and $t_{r}$ is the time to rupture) is related to the cavity density ratio $\frac{N_{c}}{N_{r}}$ using a creep ductility factor $λ_{D}$ according to the van Zyl et al. (2005) creep life model:

\frac{t}{t_{r}} = 1 - {(1 - \frac{N_{c}}{N_{r}})}^{λ_{D}}

(11)

Figure 7.

Optimised D_S and D_P damage parameter sets from (a) accelerated variable temperature and (b) medium-term variable stress compared with hardness and cavity density damage assessments.

The cavity density at failure, $N_{r}$ , is taken as 10 000 cavities/mm² and $λ_{D} = ε_{r} / ({\dot{ε}}_{m} t_{r})$ where $ε_{r}$ and $t_{r}$ is the rupture strain and time and ${\dot{ε}}_{m}$ is the minimum creep rate (Cane, 1982). Values for $λ_{D}$ of 10.9, 5.1 and 8.2 are used for low, medium and high damage, respectively (Cane, 1982). Equation (11) is commonly used for life estimates by the local power utility due to its relatively simple relationship with the common site-investigative damage monitoring technique of cavity density measurement from surface replicas.

Secondly, the hardness model of Endo et al. (2003) is also included as:

\frac{t}{t_{r}} = \frac{1 - \frac{h_{e}}{h_{o}}}{0.19}

(12)

where

h_{e}

and

h_{o}

are the ex-service and virgin X20 Vickers hardness measurements, respectively as supplied in Table 3. It is necessary to note that the higher the damage parameter, the greater the material’s life exhaustion as indicated with the aid of Robinson’s rule (Cardoso et al., 2015).

Figure 7 demonstrates that comparative damage assessments between ex-service states using optimised damage parameters correspond well with traditional damage sensors of $t / t_{r}$ (using equations (11) and (12), for instance). Interestingly, these latter methods suggest that low and medium damage X20 have similar damage levels. This is clear from the higher hardness values for medium damage traded off with higher cavity densities (Table 3), as well as from the similar Zener-Hollomon parameters determined for these states in van Rooyen et al. (2019).

It is also evident from Figure 7(a) that only one parameter from the $D_{S} - D_{P}$ set can be discerned from the accelerated creep curves, while the other attains an insignificant value. Specifically, low damage X20 demonstrates superior subgrain properties but weaker precipitate contributions compared to its medium damage counterpart. Although the measurements in Table 3 and Figure 1 indicate smaller subgrains for the sampled medium damage X20, the values are close to those of low damage X20 considering the typical scatter presented by heterogeneous microstructural characteristics within these materials.

Nonetheless, the optimised damage parameters agree with the microstructural data previously sampled from these specimens (van Rooyen et al., 2019), which indicated larger grains but denser M₂₃C₆ distributions for medium damage when compared to the low damage state. This would explain why the medium damage material has similar medium-term creep behaviour to the low damage material (Figure 6(e) and (d), respectively) on account of the strengthening benefits offered by precipitates for intermediate stresses (Murchú et al., 2017). Similarly, high damage X20 showed higher $D_{S}$ values corresponding to large, equiaxed subgrains. In this case, however a meaningful value of $D_{P}$ was not obtained due to the lack of tertiary data in these fast-developing curves. High $D_{P}$ values are anticipated as high damage X20 exhibits lower M₂₃C₆ number densities and larger particle diameters compared to other states (van Rooyen et al., 2020). The relatively low values of $D_{P}$ for the lower damage states coupled with the slow development of this parameter (Christopher and Choudhary, 2018), makes it difficult to obtain discernable values from creep curves generated from accelerated tests.

For accelerated and medium-term tests, it is difficult to discern whether optimised $D_{P}$ values are disguised by $D_{C}$ given their similar mathematical implementation in the master CDM equation (1). Given that finite values are measured for the apparent $D_{P}$ values which are in accordance with microstructural evidence of differences in precipitate distributions, however, it is more likely that $D_{P}$ is the outcome of the optimisation routine than $D_{C}$ which is estimated to have near-zero values (Table 3). By contrast, $D_{S}$ parameters are relatively larger and show the same trend in damage as indicated by the commonly employed cavity density method. This is most likely due to the respective sensitivities of the secondary and tertiary creep regimes to $D_{S}$ and $D_{P}$ (Christopher and Choudhary, 2018) coupled with the limitations of the current technique in measuring full curve development into the tertiary stage, as mentioned in Appendix 1 and in van Rooyen et al. (2022).

As the duration of the test increases for medium-term tests, both $D_{S}$ and $D_{P}$ can be discerned with the same conclusions drawn as before, barring higher $D_{P}$ values for high damage X20 as expected from the microstructural analysis in Table 3. Consistent damage parameters are extracted across the different medium-term tests at different stresses as evident from the high damage X20 values. It is useful to understand that the variations in measured damage states for high damage X20 (as seen in the larger differences between the variable stress tests in Figure 7(b)) could be attributed to the extreme microstructural heterogeneity of this material especially when considering the microstructural and cavity density spread in Table 3.

Recommended CDM-based damage philosophy

CDM-based damage extraction is an effective and material efficient damage assessment tool, especially when combined with the high density of creep curves available through DIC measurement. This approach assumes that ex-service material has accumulated damage during operation which is represented by the non-zero, initial state ( $D_{x}$ at $t$ = 0) damage parameters extracted using the Oruganti CDM model. A similar notion was applied in work by Tu et al. (2004) where the accumulated damage for repaired CrMoV weld material is reset to zero while the accumulated damage of the unrepaired parent material is retained.

This work demonstrates the quantitative correlation between the extracted damage parameters and the original classifications of ex-service X20. Hardness methodology show less consistency in terms of the observed creep behaviour. For example, the higher hardness damage observed for low damage X20 compared to medium damage X20 is inconsistent with the longer rupture times observed in medium-term, low damage tests in Figure 6(d). The optimised damage parameters also display less scatter than the cavity model, especially with respect to high damage X20. Furthermore, higher consistency and hence confidence in parameters is noted for the medium-term, variable stress tests than single-specimen, conventional tests. This is attributed to the use of multiple creep curves in the optimisation routine in the latter approach, which allows for more meaningful parameter extraction.

Based on these grounds, the following damage assessment philosophy is recommended: Conservative assessments of service-retrieved material required in entry-level integrity evaluation procedures can involve using DIC to measure multiple accelerated creep curves (as shown in the top row of Figure 6) from single specimens and optimising for $D_{S}$ through a curve-fitting routine. This will identify ex-service states with large differences in apparent deterioration, viz. high damage X20 in this work. In this case, $D_{S}$ can be used as an indicator for damage as it usually exceeds 0.1 (as noted in Figure 7). The higher damage parameter value is most often used as the basis of comparison between states (Cardoso et al., 2015; EPRI, 2007). For more subtle differences, in the case of medium and low damage X20, consideration should be given to creep test duration as subgrain strength usually dominates within medium-term creep tests whereas precipitate hardening has longer-term advantages (Murchú et al., 2017). To this end, medium-term testing using variable stress profiles can refine the assessment in terms of both $D_{S}$ and $D_{P}$ . For testing exceeding 1 000 h, consideration should be given to $D_{C}$ , as well as other forms of microstructural deterioration, including precipitation of new phases and solid solution strengthening. Additional modes of damage characterisation should also be considered including cavity density and hardness measurements. This is particularly necessary in the instance when one damage parameter dominates for a particular ex-service material, as demonstrated previously from the accelerated tests. Future work should also consider correlative studies between ex-service X20 states and volumetric cavity density estimates measured using more advanced techniques (than surface replication) such as microtomography.

Although beyond the current scope, the technique proposed in this work can then be used to estimate times to failure of components in a similar manner to the hardness and cavity damage evaluation techniques (Cane, 1982; Cardoso et al., 2015). Operating temperatures and stresses along with the optimised damage parameters can be applied to the CDM model to predict evolution to failure. This has a considerable advantage over wholly empirical methods, as the CDM model is directly tied to microstructural evolution. Relative comparisons of the parameters between ex-service materials can also allow for refinement of traditional damage classification techniques (viz. the cavity-based classification used in Abe et al. (2008) to include more subcategories that reduce the uncertainties in expended life fractions. Demonstrative of this is the apparent disparity in damage verdicts for medium damage X20 when considering cavity density, $D_{P}$ parameter, microstructural and hardness measurements (Figure 7). Results suggest that the medium damage material has a life-exhaustion extent closer to that of low damage material than the high damage counterpart, necessitating an updated subclassification of damage. Identifying possible conservatisms in this way presents great economic and plant availability benefits in the context of power plant maintenance.

Conclusions

The emphasis of this work is damage assessment using the wealth of creep curve data available from DIC strain measurements within a microstructurally relevant CDM framework. This technique is ideal for applications in the power industry due to the preservation of material volume and it introduces a new potential benchmarking technique for future damage evaluation procedures. The key conclusions are:

Experimental techniques that make use of full-field strain measurement of virgin and ex-service X20 power plant steel supply several creep strain curves across various temperatures and stresses that are in various stages of curve development while preserving material economy. This is useful for applications to data heavy CDM model calibration where the accuracy of the parameter extraction process is greatly improved by the large number of creep curves.

The evolution of subgrains, M₂₃C₆ precipitates and cavity distributions during testing result in the adaptation of a CDM model with representative damage parameters and evolutionary equations.

Following calibration of the baseline parameters on virgin X20, optimisation of subgrain and precipitate damage parameters is performed on ex-service X20 using creep curves from medium-term with variable-stress tests and from accelerated with variable-temperature tests. Medium-term tests result in measurable values of these parameters, especially in terms of subgrain damage, which are higher for highly exhausted material. Smaller differences in intermediate exhausted states can also be identified by pairing the optimised damage parameters with traditional forms of creep damage assessment.

Good agreement is demonstrated with alternative evaluation philosophies, demonstrating the potential of the technique for use as a life-assessment tool. Relatively lower damage parameter values and higher hardness values for the medium damage material suggest an expended life fraction closer to that of the low damage counterpart, despite a higher cavity density and original damage categorisation.

Future work could focus on more accurate cavity density damage characterisation using 3 D imaging techniques such as microtomography and applying CDM models to the damage parameter extraction from small specimen testing techniques such as small punch creep tests. The latter can be achieved by combining strain mapping from DIC and integrating the CDM model for iterative stress calculations using finite element methods.

Footnotes

Acknowledgements

Sourcing of piping material is gratefully attributed to the ESKOM Power Plant Engineering Institute (EPPEI) Material Science Specialisation programme. Micrographs were provided by the electron microscopy facilities of the Centre for High Resolution Transmission Electron Microscopy (CRTEM) at the Nelson Mandela University.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Research Foundation of South Africa [grant number TTK13070320379].

ORCID iDs

Melody van Rooyen

Thorsten Hermann Becker

Note

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Creep damage parameter extraction from ex-service 12% Cr steel using digital image correlation computed strain data

Abstract

Keywords

Introduction

Materials and experiments

Model formulation

Modified model equations

Damage parameters

Model calibration

Damage parameter extraction

5.1 Sensitivity study

Subgrain and precipitate damage optimisation

Discussion

Full-field methods in creep damage modelling

Comparison with alternative damage indicators

Recommended CDM-based damage philosophy

Conclusions

Footnotes

Acknowledgements

Declaration of conflicting interests

Funding

ORCID iDs

Note

References