1 Introduction

Cracks established during component manufacture or service may subsequently propagate during high temperature duty as a consequence of the combined effects of cyclic and creep loading. The operating conditions responsible for crack growth at high temperatures are diverse, ranging from predominantly cyclic to mainly steady loading with infrequent off-load transients. The fatigue component of crack extension may be a consequence of high frequency (time independent) or low frequency (time dependent) transients, applied in either stress or strain control. Moreover, cyclic loading can be linear-elastic, elastic–plastic or highly plastic. The creep component of cracking may be the result of primary (directly applied) and/or secondary (self-equilibrating) loading. As a consequence of the very diverse conditions which can be responsible for crack propagation at high temperatures, the assessment of structural integrity is currently covered by more than one approach, each with its own material property input data requirements (e.g. Table 1). These are reviewed in the following paper.

Table 1 Data requirements for the assessment of crack growth at high temperatures
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An important consideration in the characterisation of creep-fatigue crack growth rates and the assessment of high temperature structures is the size of the crack relative to the size of the associated cyclic plastic zone at the surface of the component. In the following review, long cracks are those whose size exceeds r p, and short cracks are those whose size is less than r p.

2 Long Crack Growth

2.1 General

Cyclic crack growth is conventionally considered in terms of three regimes (Fig. 1). These are: (i) a low-ΔK regime close to the fatigue crack growth threshold, ΔK th, (ii) a mid-ΔK regime in which propagation rates are modelled by a power law (Eq. 1), and (iii) a high-ΔK regime in which K max approaches K c (and/or σ ref,max approaches R m).

$${{{\text{d}}a} \mathord{\left/ {\vphantom {{{\text{d}}a} {{\text{d}}N}}} \right. \kern-0pt} {{\text{d}}N}} = A\left( {T,\upsilon ,t_{\text{h}} } \right) \cdot \left( {\Delta K} \right)^{m}$$
(1)

In Eq. (1), A(T, υ, t h) and m are material constants dependent on temperature, environment, frequency (below a limiting value) and hold time (above the insignificant creep temperature). ΔK eff may substitute for ΔK.

Fig. 1
figure 1

Cyclic crack growth rate regimes [2]

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At low-ΔK levels close to ΔK th, the magnitude of da/dN is very sensitive to small increases in ΔK and dependent on the same factors which influence ΔK th, these being: material, microstructure and yield strength, temperature, environment and load ratio (R). Propagation rates in the mid-ΔK regime are less sensitive to microstructure and mean stress (R) effects. In the high-ΔK regime, da/dN becomes increasingly sensitive to the level of ΔK and, in particular K max (and/or σ ref,max) as K c (and/or plastic collapse) is approached [3]. Depending on the deformation and fracture characteristics of the material, high-ΔK crack growth rates can be strongly influenced by size and geometry. In these circumstances, a simple LEFM defined ΔK is not the most effective correlating parameter and alternative energy based cyclic loading parameters such as ΔJ or ΔK eq are employed in Eq. (1) [1, 4]. In addition to the factors already mentioned, da/dN in the high-ΔK regime is strongly dependent on microstructure, mean stress, temperature, environment and frequency (strain rate).

At elevated temperatures, the da/dN(ΔK) diagram may be alternatively split into two crack growth regimes (Fig. 1). In the low strain fatigue crack growth (LSFCG) regime, stress/strain transients result in linear elastic loading cycles and low to mid-ΔK crack growth rates for which ΔK (or ΔK eff) still provides the most appropriate correlating parameter. Stress/strain transients responsible for cyclic loading involving a degree of general yield (in particular in tension) are referred to as high strain fatigue crack growth (HSFCG) cycles. HSFCG rates are due to apparent ΔKs and are influenced by whether deformation is stress or strain controlled, in particular in the material’s creep regime. The overlap shown between the LSFCG and HSFCG regimes in Fig. 1 is due to the fact that higher ΔKs can be generated under linear elastic conditions with strain controlled loading because of shakedown into compression.

Fatigue crack growth rates are increasingly sensitive to frequency with increasing temperature.

2.2 High Frequency

At loading frequencies above those for which time dependent effects are influential (i.e. at elevated temperatures where oxidation is not responsible for significant crack tip oxide-wedging and/or creep damage enhancement), the factors affecting da/dN(ΔK) behaviour are summarised in the previous section. For such conditions, the effect of temperature may be quantified in the mid-ΔK regime by rewriting Eq. (1) to give:

$${{{\text{d}}a} \mathord{\left/ {\vphantom {{{\text{d}}a} {{\text{d}}N}}} \right. \kern-0pt} {{\text{d}}N}} = A_{20} \left( {{{E_{T} } \mathord{\left/ {\vphantom {{E_{T} } {E_{20} }}} \right. \kern-0pt} {E_{20} }}} \right)^{m} \cdot \left( {\Delta K} \right)^{m}$$
(2)

where A 20 is equal to A(T,υ) at ambient temperature in Eq. (1). The effectiveness of this approximation is shown in Fig. 2 for a number of high temperature materials [5]. It is clear that even at relatively high frequencies, there is a temperature (dependent on material) above which da/dN becomes increasingly influenced by time dependent thermally activated processes. As for Eq. (1), ΔK eff may substitute for ΔK.

Fig. 2
figure 2

Effect of temperature on high frequency cyclic crack growth rate [5]

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2.3 Low Frequency

At lower frequencies, oxidation and creep interaction effects become increasingly more influential at high temperatures (e.g. Fig. 3). In stress control, below a limiting frequency, crack growth rates may be regarded as being dominated by time dependent crack growth mechanisms (e.g. creep crack growth in Fig. 4). The high temperature behaviour shown in Fig. 4 is typical for many engineering alloys subject to stress controlled cycling in the mid-ΔK regime.

Fig. 3
figure 3

Effect of frequency on high temperature LSFCG rates in 1CrMoV steel [6]

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Fig. 4
figure 4

Effect of frequency on high temperature LSFCG rates in various alloys [5]

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In contrast, crack growth rates in the low-ΔK regime can reduce and ΔK th values increase with decreasing frequency due to oxide induced crack closure (Fig. 3).

At high temperatures, the high growth rates associated with the high-ΔK or HSFCG regimes (Fig. 1) can be generated either as a consequence of relatively high magnitude cyclic loading applied remotely to a long crack (e.g. [1]), or (more usually at an initial stage of thermal fatigue crack development) as high strain transients applied locally to a small crack contained in a cyclic plastic strain field (e.g. [7], see Sect. 3). In the former case, cyclic crack growth rate behaviour is modelled using a modified form of Eq. (1), e.g.

$${{{\text{d}}a} \mathord{\left/ {\vphantom {{{\text{d}}a} {{\text{d}}N}}} \right. \kern-0pt} {{\text{d}}N}} = A\left( {T,\upsilon ,t_{\text{h}} } \right) \cdot \left( {\Delta K_{\text{eq}}^{{}} } \right)^{m}$$
(3)

where ΔK eq is ΔK eff for purely elastic loading. Acknowledgement of the dependence of A on hold time reflects the dependence of this parameter on associated oxidation as well as creep damage [1].

The high temperature crack growth properties required for the defect assessment of components subject to fatigue cycles involving hold (steady operating) periods may be derived from pure fatigue and pure creep crack growth rate data in a construction of the form given in Fig. 4 when the loading is directly applied (i.e. load controlled). In such circumstances, the effective frequency may be simply determined from a knowledge of the total cycle time (i.e. transient + hold time). Alternatively, creep-fatigue crack growth behaviour is analytically modelled on the basis of fatigue and creep crack growth rate characteristics for the material, i.e.

$$\left( {\text{d}a/\text{d}N} \right)_{\text{total}} = \, \left( {\text{d}a/\text{d}N} \right)_{\text{F}} + \, \left( {\text{d}a/\text{d}N} \right)_{\text{C}}$$
(4)

In Eq. (4), (da/dN)F is given by Eq. (3), where A(T, υ, t h) may be influenced by creep and oxidation damage through its dependence on frequency and prior hold time [1] and (da/dN)C is given by Eq. (5).

$$\left( {{{{\text{d}}a} \mathord{\left/ {\vphantom {{{\text{d}}a} {\text{d} N}}} \right. \kern-0pt} {\text{d} N}}} \right)_{\text{C}} = {{\left( {\int_{0}^{{t_{\text{h}} }} {D\left( {\varepsilon_{\text{r}} } \right)} \cdot \left( {C*} \right)^{\gamma } \cdot {\text{d}}t} \right)} \mathord{\left/ {\vphantom {{\left( {\int_{0}^{{t_{\text{h}} }} {D\left( {\varepsilon_{\text{r}} } \right)} \cdot \left( {C*} \right)^{\gamma } \cdot {\text{d}}t} \right)} \upsilon }} \right. \kern-0pt} \upsilon }$$
(5)

Any enhancement of the total growth rate per cycle due to a creep-fatigue-oxidation interaction is covered by A(T, υ, t h) which is determined experimentally (e.g. Fig. 5).

Fig. 5
figure 5

Long crack cyclic/hold creep-fatigue crack growth test data for 2¼CrMo cast turbine steel at 538/565 °C [1]

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3 Short Crack Growth

The HSFCG rates associated with small cracks contained in local cyclic plastic strain fields (typically a ≤ 5 mm) are most effectively modelled as a function of Δε [7–12] (e.g. Fig. 6), i.e.

$${{{\text{d}}a} \mathord{\left/ {\vphantom {{{\text{d}}a} {{\text{d}}N}}} \right. \kern-0pt} {{\text{d}}N}} = B^{\prime } \cdot a^{b} \cdot \left( {\Delta \varepsilon } \right)^{q} \;{{{\text{d}}a} \mathord{\left/ {\vphantom {{{\text{d}}a} {{\text{d}}N}}} \right. \kern-0pt} {{\text{d}}N}} = B^{\prime } \cdot a^{Q} \cdot \left( {\Delta \varepsilon } \right)^{b}$$
(6a)

although other correlating parameters may be employed [12]. For relatively short cracks contained in high cyclic plastic strain fields, (da/dN)total is effectively modelled using a refinement of Eq. (6a) for a range of power plant steels [7, 10–12], i.e.

$$\left( {{{{\text{d}}a} \mathord{\left/ {\vphantom {{{\text{d}}a} {{\text{d}}N}}} \right. \kern-0pt} {{\text{d}}N}}} \right){}_{\text{total}} = B^{\prime } \cdot a^{Q} \cdot \left( {\Delta \varepsilon } \right)^{b} \cdot \left( {1 - D_{C} } \right)^{ - 2}$$
(6b)

Typically, in Eq. (6), Q = 1 [13]. The effect of hold time on HSFCG rate for a cast 1¼CrMoV steel at 550 °C is shown in Fig. 6. In this example, D C was modelled empirically as a function of hold time (see also [11]), although in a formal assessment it would be determined in terms of ductility exhaustion [14].

Fig. 6
figure 6

Comparison of crack growth rates after 0.5 mm crack extension from notch root in large SENB feature specimen creep-fatigue tests on cast 1¼CrMoV steel at 550 °C [7]

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For advanced 9/11 % Cr martensitic steels, which are more prone to creep-fatigue deformation interactions, D C may be replaced by a microstructural (deformation) condition parameter [15].

It is evident that the material property data required for the defect assessment of high temperature components subject to creep-fatigue loading can be strongly dependent on the specific operating conditions relating to the practical application under consideration. This means that the rigorous creep-fatigue defect assessment of a component can require a significant investment in the determination of appropriate material property data. It is therefore important to demonstrate that both creep and fatigue loading are significant at the critical feature to be assessed.

4 Defect Assessment

A creep-fatigue crack growth assessment is only necessary when both creep and faigue are shown to be significant [3, 14].

As a generality, creep is regarded as being significant if, for the total number of cycles, the sum of the ratios of hold time to the maximum allowable time at the temperature of interest is greater than or equal to unity, i.e.

$$\sum\limits_{j = 1}^{N} {\left[ {{{t_{\text{h}} } \mathord{\left/ {\vphantom {{t_{\text{h}} } {t_{ \hbox{max} } \left( {T_{\text{ref}} } \right)}}} \right. \kern-0pt} {t_{ \hbox{max} } \left( {T_{\text{ref}} } \right)}}} \right]}_{j} \ge 1$$
(7)

Values of t max depend on material crack size and temperature [3]. For materials with ε r ≥ 10 %, t max is taken to be the time required to achieve an accumulated creep strain of 0.2 % at a stress level equal to the reference stress. Alternatively for ε r < 10 %, t max is determined on the basis of an accumulated creep strain of 0.2 ε r.

Fatigue is regarded as being significant if cyclic loading influences the development of creep damage. This is likely if the elastic range exceeds the sum of the steady state creep stress and the stress to cause yield at the other extreme of the cycle [14]. Fatigue is also considered significant if the estimated crack growth due to cyclic loading exceeds 10 % of the calculated creep crack growth.

When both creep and cyclic loading are shown to be significant, the extent of creep-fatigue interaction should be determined [3, 14]. As a generality, the effect of creep damage on fatigue crack growth rates has little influence on the total crack growth per cycle provided the latter includes an explicit calculation of creep crack growth (i.e. Eq. 4). In such circumstances, there is no creep-fatigue interaction and no requirement to enhance creep-fatigue crack growth rates. It is only necessary to consider a creep-fatigue interaction when the effect of cyclic loading on creep is shown to be significant despite fatigue crack growth rate having been estimated to be only a small fraction of the total crack growth rate per cycle. For such conditions, the constants in Eq. (3) should be determined from tests with hold times relevant to the service application being assessed (e.g. Fig. 5).

Similarly, in cases where cracks are propagated by fatigue through material heavily damaged by prior creep, propagation rates are likely to be increased. In these circumstances, a factor should be applied to the fatigue crack growth constant to account for the amount of prior creep damage. This should be determined experimentally [3].

5 Concluding Remarks

The crack growth rate data used for the defect assessment of high temperature structures has been reviewed with reference to examples for a number of power plant steels.

The material property parameters necessary for the defect assessment of such components subject to creep-fatigue loading can be strongly dependent on the specific operating conditions relating to the practical application under consideration. The rigorous creep-fatigue defect assessment of a component can therefore require a significant investment in the determination of appropriate material property data.