A Short-Time Approach for Fatigue Life Evaluation of AISI 347 Steel for Nuclear Power Energy Applications

Abstract

AISI 347 austenitic steel is, as an example, used in nuclear energy piping systems. Piping filled with superheated steam or cooled water is particularly exposed to high stresses, whereupon local material properties in the pipes can change significantly, especially in the case of additional corrosive influences, leading to aging of the material. In the absence of appropriate information, such local material property variations are currently covered rather blanketly by safety factors set during the design of those components. An increase in qualified information could improve the assessment of the condition of such aged components. As part of the collaborative project “Microstructure-based assessment of the maximum service life of core materials and components subjected to corrosion and fatigue (MiBaLeB)”, the short-time procedure, StrainLife, was developed and validated by several fatigue tests. With this procedure, a complete S–N curve of a material can be determined on the basis of three fatigue tests only, which reduces the effort compared to a conventional approach significantly and is thus ideal for assessing the condition of aged material, where the material is often rare, and a cost-effective answer is often very needed. The procedure described is not just limited to traditional parameters, such as stress and strain, considered in destructive testing but rather extends into parameters derived from non-destructive testing, which may allow further insight into what may be happening within a material’s microstructure. To evaluate the non-destructive quantities measured within the StrainLife procedure and to correlate them with the aging process in a material, several fatigue tests were performed on unnotched and notched specimens under cyclic loading at room and elevated temperatures, as well as under various media conditions, such as distilled water and reactor pressure vessel boiling water (BWR) conditions.

1. Introduction

As part of the ongoing collaborative project “MiBaLeb” the fatigue behavior of the metastable austenitic steel AISI 347 (1.4550) is being characterized under various boundary conditions relevant to nuclear power engineering. AISI 347 steel is used in pipes, volume control, reactor water purification, feed water system and other components of German nuclear power plants (e.g., in current light–water reactors (LWR) plants) in a temperature range up to 350 °C.

Non-destructive testing (NDT) techniques offer great potential for detecting the microstructural degradation that precedes micro crack formation. However, this potential has not been sufficiently explored when it comes to the fundamentals of quantitatively characterizing the material behavior under static, cyclic, as well as any combinations of thermal and corrosive loading using NDT, which might provide an insight into the interactions between an NDT-based measurement signal and a material′s microstructure. The MiBaLeb project claims to determine valuable results in this regard and to provide those results for future application. The aging-related degradation processes affect the plant components individually, resulting from both static and dynamic loading, and this in combination with thermo-mechanical loads. In piping systems of light water reactors, the temperature range from 200 to 330 °C is of particular interest, which is mainly based on the run-up and run-down processes of a power plant system [1].

Evidence that the degradation level D (traditionally expressed as “damage”) can be defined within certain limits (0 ≤ D ≤ 1) is currently provided by structural mechanical analyses. For this purpose, fatigue analyses distinguish between elastic, simplified elasto-plastic and general elasto-plastic cases. Elasto-plastic finite element analyses of model transients have been proposed as an example to determine the real damage level very precisely [2]. In addition, probabilistic methods are used to describe the uncertainties arising with respect to the loads applied and scatter in the material properties to be expected. These methods are prescribed in the regulations of the respective countries where the plants are operated. In this context, the Gesellschaft für Anlagen und Reaktorsicherheit (GRS) has developed the probabilistic analysis tool PROST (PRObabilistic Structure calculation) for the determination of structural reliability and durability, respectively, to be used for the assessment of piping and vessels [3] This allows crack formation, as well as crack behavior, to be determined in relation to various damage mechanisms. Probabilistic investigations of the leakage and fracture of piping components as a function of the operating time can also be carried out with PROST [4,5,6].

The design of cyclically loaded components is usually based on the service life derived from S–N curves. To generate such curves, many fatigue tests are traditionally required, which makes the process time-consuming and costly. The basics of the traditional service life evaluation principles are provided by the application of damage accumulation hypotheses, such as those according to Palmgren and Miner [7,8]. However, experimental investigations have shown that, especially in the case of additional corrosive (chemical) loading, the assumption of linear accumulation damage being related to stresses and strains applied is only valid to a limited extent. Again, safety factors are the current solution to cover the resulting inaccuracies and uncertainties [9].

Short-time evaluation procedures (STEP) developed in recent years need to be mentioned in this context, including the StrainLife method to be considered here, which can reduce the number of fatigue tests, as well as the time and cost required by more than 90%. Many of those STEP methods are based on generalized Morrow and Basquin equations, e.g., in a combination of strain (SIT) or load (LIT) increase tests and stress or strain-controlled constant amplitude tests (CAT), which enable the evaluation of S–N curves for the materials investigated. The validation of the STEP has already been successfully performed on various light metal alloys and steels in different heat treatment conditions, including the determination of scatter bands for different fracture probabilities [10,11,12,13,14,15].

In addition to conventional mechanical stress–strain measurements, STEP for determining S–N curves can also take advantage of NDT-based techniques and parameters, which may rely on temperature and electrical resistance, but also on magnetism or ultrasound. These methods may, therefore, provide a potential for detecting and tracing the damage evolution in a non-invasive way. Electrochemical parameters based on the corrosion potential [16,17,18,19] or the corrosion current density [18,20] can also be used for damage characterization in corrosion fatigue tests. Microstructural changes, such as the formation of slip bands or the crack initiation/propagation on the material′s surface, can be correlated with these measured variables [20]. This has been successfully shown for different materials allowing the progression of corrosion fatigue damage to be described [21,22].

The STEP mentioned above was extended in “MiBaLeb” to include aspects, such as strain control, thermo-mechanical loading and corrosion, which are important aspects for the design of piping systems in nuclear facilities, and this in view of allowing a transition between component behavior as well as material properties and vice versa.

2. Materials and Methods

Mechanical stress–strain measurements within cyclically loaded total strain-controlled constant amplitude (CAT) and strain-increase tests (SIT) in air, as well as under distilled water media conditions, were performed using an extensometer with an initial length of L₀ = 89 mm, mounted on the specimen shafts and measuring the deformation, allowing the resulting total strain amplitude ε_a,t to be determined during the fatigue test. Since this extensometer setup measures deformation along the minimum cross-section and the transitions of the specimen′s shafts, the calibration to the strain values within the minimum cross-section is essential. A second extensometer was, therefore, applied in the minimum cross-section, allowing a correlated total strain amplitude ε_a,t,cor to be used as the controlled value. All fatigue tests were performed with a strain ration of R = −1 and a constant strain rate of ε_a,t,cor = 4 × 10⁻³/s using triangular strain–time functions.

Additionally, a special in situ cell for fatigue tests performed in distilled water was developed, allowing additional measurement techniques to be included (Figure 1a). It should be noted that in tests with AISI 347 steel, distilled water does not cause any corrosion effects on the material. However, comprehensive, well-adhering oxide layers are formed, which are locally broken up due to surface damage, e.g., by micro cracks, followed by macro cracking along the further process. In this context, the electrochemical open-circuit potential (E_OCP) can be used to describe these effects occurring at the material′s surface on the specimen, and thus enables further fatigue-related damage mechanisms to be separated. For this purpose, a pump-driven water circuit is used to circulate the water through the cell, and a thermostat continuously cools the medium to a temperature of 22 °C. For electrochemical measurement, a reference electrode (silver chloride) and a counter electrode (graphite) are used in the cell, and the fatigue specimen serves as the working electrode.

Fatigue tests at elevated temperatures up to 300 °C were carried out using a custom-built furnace for the servo hydraulic testing machine and a high-temperature extensometer (Figure 1b).

Furthermore, fatigue tests were performed under reactor pressure vessel boiling water (BWR) conditions at 70 bar and 240 °C in a specially designed autoclave (Figure 1c). The high-temperature water used in the fatigue tests was provided by a water treatment plant and had a conductivity of about 0.055 µS/cm. The impurities due to chloride ions were generally below 5 µg/kg, which is in accordance with the VGB guidelines [23] for the quality of reactor water in BWR plants in normal operation according to the guidelines. The oxygen content in the fluid was set to 400 µg/kg (0.4 ppm), which is the upper limit for BWR plants and is mainly used in international laboratory studies to simulate BWR conditions.

To measure the change in electrical resistance, a high constant current (I = 8 A) was applied to the specimens using a DC potential probe. During loading, the change in voltage and the thereon-dependent change in electrical resistance along the specimen was measured via wires mounted at the transitions between the gauge length and the shafts using adapted copper clamps and connected to a voltage measurement card with a maximum voltage range of ±500 mV. The change in electrical resistance depends, on the one hand, on the change in specimen geometry (which can be neglected for a load ratio of R = −1). The deciding factor, however, is the change in the specific electrical resistance, from which direct conclusions can be drawn about the change in the defect density of the materials.

During the fatigue tests, a magnetic field sensor was used to record changes in the tangential magnetic field strength due to the progressively forming ferromagnetic phase fractions, which is generated based on the deformation-induced α′-martensite formation of the AISI 347, as well as on microstructural changes due to dislocation reactions, as well as crack initiation and propagation processes at different load levels. ε-martensite could also be formed, but this is not detectable due to its paramagnetic properties. The probe consists of a Hall sensor encapsulated with epoxy resin in a PTFE housing. For magnetization, a constant direct current was applied to the specimen. Magnetic measurements up to elevated temperatures of about 200 °C can be realized without affecting the sensor functionality (see Figure 1a).

Microstructural investigations of the metastable austenitic steel AISI 347 (1.4550) carried out along the minimum cross-section of the bar material showed significant microstructural variations in the individual cross-sectional areas. After etching with V2A etchant and subsequent optical microscopy examinations, as well as crystallographic analyses using electron backscatter diffraction (EBSD) measurements (Figure 2a) on the unetched sample, a square region consisting of an inhomogeneous multiphase structure of coarser-grained austenite and deformation-induced α′-martensite with mechanical twins was detected in the center of the cross-section. This microstructure indicates low forming temperatures during the manufacturing process of the semi-finished product (bar material), which leads to the occurrence of TWIP (TWinning Induced Plasticity) and TRIP (TRansformation Induced Plasticity) mechanisms. Apart from the center of the cross-section, an increasingly distorted, fine-grained austenitic microstructure is visible, indicating heat input from the surface during the manufacturing process of the bar material.

The detection of the ferromagnetic portions by means of a Feritescope was able to confirm the previously described metallographic finding of α’-martensite. This ferromagnetic content increases from the cross-section edge to the center and reaches a maximum of 0.8 vol.-%. δ′-ferrite was not detected in any of the metallographic investigations carried out. In addition, micro hardness measurements were performed with a Fisherscope HM 2000 and macro hardness measurements with a conventional hardness tester (Figure 2b). The Vickers micro hardness profile with a testing force of 0.98 N (HV0.1) shows a cup-shaped profile from approx. 340 HV0.1 at the sample edge to approx. 180 HV0.1 in the center. In comparison, macro hardness measurements were performed with a testing force of 98.07 N (HV10). In comparison to the micro hardness distribution, this curve shows an approximately narrower U-shaped progression. Here, the macro hardness is approx. 190 HV10 in the edge area and approx. 145 HV10 in the center. The gradient, which can be seen in both the micro and macro hardness measurements, can be related to the finer-grained microstructure in the edge region. The earlier hardness increase in the macro hardness measurements is attributed to the additional increase in α′-martensite formation due to the 100 times larger testing force.

In this project, specimens for two material conditions representing the initial condition and the aged condition are considered. Testing of the aged specimens served to qualify the NDT methods used to characterize the damage evolution during the tests. For aging, the specimens in the initial condition are set to an artificially defined aged condition by means of a practice-oriented thermo-mechanical loading. The parameters for the setting of the artificially aged condition were selected based on existing practice-relevant applications in nuclear technology [24,25]. These parameters for aging in air include a mechanical loading with a total strain amplitude of 0.3 × 10⁻², a strain rate of 4 × 10⁻³/s, and a thermal superposition with a temperature of 240 °C, which represents the mean value of the temperature range from 150 to 325 °C.

To provide a data base required for the aging process with regard to the determination of the load cycles required to pre-damage, several fatigue specimens were loaded with the thermal and mechanical parameters up to a load drop L of 25%. Using statistical methods, the achieved lifetimes were evaluated, allowing load cycles (up to the failure criterion) to be calculated for different probabilities of survival. At a survival probability of 95%, the obtained load cycle for L25% is 31,450 cycles. Here, the aged condition was defined by 0.5 × L25%. Thus, the mechanical pre-damage results in 15,725 load cycles. The parameters examined experimentally and statistically ensured that the material AISI 347 is already transferred by the thermo-mechanical loading to a material condition that deviates significantly from the initial condition without already exhibiting macroscopically detectable damage characteristics.

3. Results

3.1. Comparison of Strain Increase Tests Performed under Laboratory Conditions in Air and in Distilled Water at Ambient Temperatures

Figure 3 shows results of the metastable austenitic AISI 347 steel in the initial condition obtained from two SITs performed (a) in air and (b) in distilled water with the slopes of the correlated total strain amplitude ε_a,t,cor (yellow), the stress amplitude σ_a (green), the changes in tangential magnetic field strength ΔM_tang (blue), as well as the temperature ΔT (black) and the open circuit potential E_OCP (red).

The SITs performed within the scope of this research start with a total strain amplitude ε_a,t,cor of 0.05 × 10⁻², and after a step length of Δt = 1800 s each, the total strain amplitude is increased by Δε_a,t,cor = 0.05 × 10⁻² until specimen failure. In Figure 3a, σ_a also increases stepwise up to ε_a,t,cor = 0.30 × 10⁻², whereupon from ε_a,t,cor = 0.35 × 10⁻² onwards a hardening of the material takes place, which can be seen in a superimposed continuous increase of σ_a within the step. Just before specimen failure, σ_a decreases, indicating a drop-down in stiffness due to advanced macro crack propagation processes. ΔM_tang also exhibits a continuous decrease up to ε_a,t,cor = 0.35 × 10⁻², which flattens thereafter and can be explained by the phase transformation from austenite to α’-martensite. Another change in the ΔM_tang progression is evident from ε_a,t,cor = 0.60 × 10⁻², which is again related to macro crack propagation.

For the SIT performed in distilled water shown in Figure 3b, σ_a also increases stepwise up to ε_a,t,cor = 0.35 × 10⁻², after which there is also a superposition with continuous increases in σ_a resulting from the strain amplitude-dependent hardening process. At ε_a,t,cor = 0.95·10⁻² specimen failure occurs, indicated by a decrease in the ΔM_tang signal and a prior decrease in stiffness, which in turn is indicated by a decrease in σ_a. ΔM_tang decreases in a step-like manner as ε_a,t,cor increases until ε_a,t,cor = 0.20 × 10⁻². The range of 0.40 × 10⁻² < ε_a,t,cor < 0.70 × 10⁻² indicates a deformation-induced phase transformation from austenite to α’-martensite, since there is a relative decrease within the steps.

3.2. Comparison, Initial Condition and Aged

SITs and CATs performed in distilled water were used to compare the initial condition with the aged condition of AISI 347 defined previously in Chapter 2. Figure 4a (same as Figure 3b) shows results from an SIT for a specimen under fatigue loading for AISI 347 in the initial condition showing the development over the time for the correlated total strain amplitude ε_a,t,cor (yellow), the stress amplitude σ_a (green), the open-circuit potential E_OCP (red) and the magnetic field strength ΔM_tang (blue), respectively, all up to specimen failure at ε_a,t,cor = 0.95 × 10⁻². Here, σ_a increases stepwise up to ε_a,t,cor = 0.35 × 10⁻². At ε_a,t,cor = 0.40 × 10⁻² onwards, a large increase in σ_a can be seen until immediately before specimen failure. Afterward, there is a large decrease in stiffness at ε_a,t,cor = 0.95 × 10⁻², which is accompanied by a decrease in σ_a. From ε_a,t,cor = 0.40 × 10⁻² onwards, a stepwise decrease of the E_OCP signal can be seen inverse to σ_a, which decreases more significantly from ε_a,t,cor = 0.85 × 10⁻² onwards. ΔM_tang also decreases stepwise with increased ε_a,t,cor, although the decrease intensifies from ε_a,t,cor = 0.35 × 10⁻². In the range 0.40 × 10⁻² < ε_a,t,cor < 0.70 × 10⁻², a strong increase occurs within the load steps, which can probably be explained by the martensitic phase transformation. The specimen failure at ε_a,t,cor = 0.95 × 10⁻² is reliable, indicated by a sudden decrease of the ΔM_tang signal.

Figure 4b shows the SIT for the aged specimen. Comparatively, it can be observed that the aged specimen fails four load steps below the one in the initial condition (Figure 4a) at ε_a,t,cor = 0.95 × 10⁻². The slopes of σ_a, ΔM_tang and E_OCP are comparable to the initial condition up to a step of ε_a,t,cor = 0.25 × 10⁻². Thereafter, the slopes are significantly different from the E_OCP. The E_OCP signal of the aged specimen decreases considerably in the range 0.25 × 10⁻² < ε_a,t,cor ≤ 0.40 × 10⁻² within the steps. In the following three steps, the E_OCP signal remains approx. constant in the step and again decreases continuously from ε_a,t,cor = 0.60 × 10⁻² until specimen failure.

The increase of σ_a in the first three load steps of the two SITs is due to a pure elastic behavior, which is reflected by the linear stress–strain relationship. From ε_a,t,cor = 0.40 × 10⁻² for the specimen in the initial condition or from ε_a,t,cor = 0.45 × 10⁻² for the aged specimen, the relative increase in σ_a can be explained by cyclic hardening effects typical for that material under cyclic loading. The decrease in σ_a in the last load step is due to progressive macro crack propagation in both the initial and aged conditions.

Prior the fatigue tests, the specimens were electrochemically stabilized in a load-free period at the beginning of the test, such that there were no active surfaces in the medium-material interface zone, as can be read from the constant E_OCP slope. The subsequent stepwise decrease in relation to the change of ε_a,t,cor or σ_a can be explained by activated surfaces exposed as a result of slip band movements and the associated intrusions and extrusions, which leads to a reduced E_OCP signal in each step. By means of a decreasing strain hardening rate (visible from the σ_a signal), E_OCP continues to decrease in the step, which can be related to a stable and following unstable crack propagation processes with continuously new uncovered active surfaces. The constant range at the beginning of the experiment of the aged specimen is shorter and based on the aged condition of the material. The relatively significant changes at the step transition can probably be explained by pre-existing micro cracks in the material leading to anodic metal dissolution, whereupon the E_OCP is shifted to a cathodic direction. The E_OCP signal stabilizes after macro crack initiation until it transitions to the previously mentioned stable and then to the unstable crack propagation at ε_a,t,cor = 0.55 × 10⁻² [26].

Figure 5 shows the results of two CATs in the low cycle fatigue (LCF) regime. The CAT for the initial condition was performed at ε_a,t,cor = 1.46 × 10⁻² (Figure 5a) and for the defined aged condition at ε_a,t,cor = 1.00 × 10⁻² (Figure 5b), respectively. Due to the strain hardening potential of the AISI 347 steel in the LCF regime, the total strain amplitudes lead to a constant increase in σ_a. The E_OCP signal shows similar slopes for both conditions. In the case of the aged specimen, the E_OCP signal starts at a higher level and shows a lower increase to the local maximum of the signal slope as it progresses. It increases in the first 25 load cycles, decreases steadily thereafter and becomes more pronounced briefly before specimen failure. ΔM_tang shows a predominantly inverse behavior compared to σ_a, with specimen failure indicated with a significant increase from at about 85% of the lifetime. The ΔM_tang slopes of both conditions can be compared very well with the individual slopes with corresponding total strain amplitudes in the SITs.

3.3. Constant Amplitude Tests Performed under Reactor Pressure Vessel Boiling Water Conditions

To investigate the influence of application-relevant BWR conditions (c.f. Chapter 2), various fatigue tests were carried out in a respective environment. In this context, the course of the stress amplitude and the open-circuit potential were used to characterize the fatigue behavior. Figure 6 provides the results from two CATs performed under BWR conditions with total strain amplitudes of 0.3 × 10⁻² and 0.55 × 10⁻², respectively. Since with the CATs under BWR conditions, only a measurement of the specimen elongation at the specimen shoulders outside the autoclave is possible, a correlation between the elongation measured directly on the cylindrical gauge length of the specimen and the elongation determined at the specimen shoulders must be made on the basis of the fatigue tests in air as it was already described before. For this purpose, a strain measurement has been developed that makes it possible to perform tests after measuring the total strain amplitude at the gauge length of the specimen while observing symmetrical elongation amplitudes related to the measurement of two inductive displacement transducers at the specimen shoulders. After each loading cycle, the resulting strain amplitude is automatically evaluated online by the control system, and, if necessary, the elongation amplitude is increased or reduced in steps to subsequently obtain the desired total strain amplitude. By doing so, correlation curves were determined, which reflect the specimen elongations to be applied to the specimen shoulders in order to achieve constant strain amplitudes of the test length as a function of the number of cycles. These curves must be redetermined for all tests with different parameters via a best-fit correlation.

Within the CAT with ε_a,t = 0.3 × 10⁻² (Figure 6a) it can be seen that the course of the stress amplitude remains almost constant from the beginning up to about 100 cycles and slightly decreases in the following, which is due to a cyclic softening of the material. Subsequently, saturation occurs, followed by a strong softening at specimen failure, indicated by a drop of the stress amplitude. With regard to the E_OCP signal, this remains constant at a value of approx. 0 for the first 1000 cycles, after which there is a continuous increase until specimen failure. As it was already explained before, the signal changes in the open-circuit potential being dedicated to surface defects caused by the fatigue process.

Since the initiation of damage in the high cycle fatigue (HCF) regime almost certainly starts from the surface, the information obtained from this measurement technique is essential for the design of the components used in the field of nuclear energy. The exposure of new surfaces first leads to an increase in the corrosion current, with the open-circuit potential dropping locally. The subsequent re-passivation processes then lead to a respective increase in the open-circuit potential, which can be seen in Figure 6a.

In Figure 6b, the results from a CAT with a total strain amplitude of ε_a,t = 0.55 × 10⁻² are shown. Due to the higher total strain amplitude when compared to the results in Figure 6a, cyclic deformation behavior changes to a continuous cyclic hardening behavior over the entire lifetime indicated by a continuous increase in the stress amplitude. The E_OCP signal also shows a behavior comparable to the CAT with the lower total strain amplitude, changing by about 55% from its initial value to specimen failure.

3.4. StrainLife

StrainLife is a STEP for the fast and cost-effective evaluation of S–N curves, which is considered here to be adapted and validated to nuclear engineering applications. The value of this method is, as mentioned before that only one SIT and two total strain-controlled CATs are required for the calculation of a complete total strain S–N curve. The input variables for the StrainLife method include the measurements of σ_a, ε_a,p, ΔT, ΔR, E_OCP and ΔM_tang, respectively, with their development over the fatigue life to be considered as an additional source of information, as shown for the different experimental results presented before. In the following, the procedure is described for the calculation based on electrical resistance values.

Equation (1) is a relationship by Morrow [27], which is commonly used to describe the stress amplitude (load variable) and to be a function of plastic strain amplitude (material response).

$${\sigma }_a=K^{\prime }\times {({\epsilon }_{a,p})}^{n^{\prime }}$$

(1)

By using the load quantity L and the material response M, Equation (1) can now be described as follows.

$$M={K^{\prime }}_M\times {(L)}^{n_{M^{\prime }}}$$

(2)

Here, the coefficient K′_M and the exponent n_M′ are additionally marked with the index M to express that this formula is used for a quantity other than the mechanical plastic strain amplitude. Those relationships provide the basis to determine S–N curves may be derived from an SIT, where the load quantity could be considered to be a total strain amplitude ε_a,t and the material response electrical resistance ΔR written then as

$$\Delta R={K^{\prime }}_{\Delta R}\times {({\epsilon }_{a,t})}^{n_{\Delta R^{\prime }}}$$

(3)

The next step is to calculate the average values of the change in electrical resistance of each load level of the SIT and then plot the total strain amplitude as a function of these values (see Figure 7a). In addition, two CATs are performed, which are also added to the same plot. In the respective example, the change in electrical resistance at the same total strain amplitude in the LIT and CAT may not be identical, which can be attributed to a difference in load-time history between LIT and CAT, resulting in different material conditions with respect to damage accumulated up to the respective point, as well as cyclic strain hardening (Figure 7b). To correct this a ratio, Q, between the total strain amplitude of the CAT and the SIT is calculated according to Equation (4).

$$Q({\epsilon }_{a,t,1/2})=\frac{\Delta R_{1/2}\left(\mathrm{CAT}\right)}{\Delta R_{1/2}\left(\mathrm{SIT}\right)}$$

(4)

Using the calculated corrective values, Q, for both total strain amplitude levels applied in the CATs, a fitting curve for further extrapolation of the data is calculated, where an exponential fit has been chosen. By using this relationship, the stress–strain behavior of the SIT is transferred to the stress–strain behavior for strain-controlled CATs (see Figure 8).

If electrical resistance is considered to be the relevant material response M, then this may need to be differentiated between a predominantly elastic and predominantly plastic strain region when plotting the relationship between the material response M and the load variable L. Therefore, coefficient K′_ΔR and exponent n_ΔR′ have to be differentiated in accordance with the two regions. Equation (3) turns up in two versions, including the exponents n_∆R,el′ and n_∆R,pl′, respectively (see Figure 8). Using n_∆R,el′ and n_∆R,pl′, the fatigue strength exponent b and fatigue ductility exponent c can be calculated using Equations (5) and (6), respectively [27].

$$b=\frac{-{n_{\Delta R,el}}^{\prime }}{5\times n_{{\Delta R,el}^{\prime }}+1}.$$

(5)

$$c=\frac{-1}{5\times n_{{\Delta R,pl}^{\prime }}+1}$$

(6)

The total strain amplitude contains an elastic portion ε_a,el and a plastic portion ε_a,pl being additive, as it is shown in Equation (7).

$${\epsilon }_{a,t}={\epsilon }_{a,el}+{\epsilon }_{a,pl}$$

(7)

In the case of mechanical stress–strain measurements, the elastic and plastic portion vs. the number of cycles to failure can be described mathematically equivalent to Basquin (Equation (8)) [28] and Mansion–Coffin equations (Equation (9)) [29], respectively. An example of such a separation into an elastic and a plastic portion is exemplarily shown in Figure 9a using the strain measurement as an example.

$${\epsilon }_{a,el}=B\times {(2N_f)}^b\to B=\frac{{\epsilon }_{a,el}}{{(2N_f)}^b}$$

(8)

$${\epsilon }_{a,pl}=C\times {(2N_f)}^c\to C=\frac{{\epsilon }_{a,pl}}{{(2N_f)}^c}$$

(9)

If the slopes determined according to Equations (5) and (6) are entered into Equations (7)–(9), Equation (10) is obtained. Since at least always two CATs are performed, the coefficients B and C can be determined via Equation (10).

$${\epsilon }_{a,t}=B\times {(2N_{f1/2})}^{\frac{-n_{\Delta R,e{l}^{\prime }}}{5\times n_{\Delta R,e{l}^{\prime }}+1}}+C\times {(2N_{f1/2})}^{\frac{-1}{5\times n_{\Delta R,p{l}^{\prime }}+1}}$$

(10)

Figure 9a gives the result of the StrainLife calculation based on stress–strain measurements. Aside from the calculated S–N curve, the curves for Basquin (elastic portion) and the Manson–Coffin (plastic portion) relationships are also given. Figure 9b additionally shows the calculated StrainLife curves based on the changes in electrical resistance and the tangential magnetic field strength [30].

3.5. Uncertainties of Crack Position and Progression on Total Strain Controlled Fatigue Results

Fatigue life is defined by a damage criterion considered, which is in many traditional cases, is the complete fracture of the specimen. However, this is just one damage mechanism (and the last) that occurs during the fatigue life of a material. In many cases, the damage criterion is considered to be crack initiation only. This is to give clear recognition of the description of linear and non-linear fracture mechanics approaches being a mechanism different when compared to the degradation processes appearing before crack initiation. Care has to be taken when the crack propagation phase is included in the fatigue life evaluation, specifically when strain is the controlling mode. In that case, the number of cycles can go up to infinity when a crack opening displacement is erroneously taken as a basis to determine a specimen’s strain.

An alternative to crack initiation is to measure load drop when the fatigue test is performed under strain control. This is well feasible with unnotched specimens when generating materials fatigue data. However, the question arises, which crack size can be associated with a defined load drop criterion.

For the description and evaluation of the incident of crack initiation, the transition between a fatigue analysis from the uncracked to the cracked structure where fracture mechanics-based evaluations in the LCF regime of the components considered are essential. Based on the results of the fatigue tests presented here, the specimens were analyzed with respect to the location of crack initiation and the assumption to be made [31]. The changes in the stress amplitudes of the total strain-controlled fatigue tests (e.g., Figure 4a) are related to crack geometries and crack sizes in order to develop an appropriate correlation.

For this purpose, four possible crack geometry configurations of the incipient cracks in the cross-sectional area of the specimen are considered. These possible incipient cracks are a circular crack (fisheye), a full circumferential crack, a chord crack and a lens crack, all schematically shown in Figure 10. The circular crack may initiate at an inclusion, as it could be expected for ideal geometrically unnotched specimens [32]. In comparison, the development of a chord or lens crack, e.g., due to an arbitrary unwanted condition (e.g., blade of a strain gauge transducer), may occur in non-ideal geometrically unnotched specimens. The full circumferential crack is predominantly to be expected on notched specimens.

The crack area A is different for each crack initiation configuration and depends on the crack depth a. A can be calculated according to Equations (11)–(14), which is also visualized in Figure 11. For this case, the radius of the cross-sectional area of the unnotched fatigue specimens is assumed to be r = 5 mm. Figure 11 also contains the area fractions with associated crack depths of individual fatigue specimens with a load drop of 25%, which were tested with a strain amplitude of 0.3 × 10⁻². The individual fractographs are shown in more detail in Figure 12. The fractographs show that the lens crack is a possible idealized crack shape. In addition, the location of the extensometer clip is marked with a green dot in the fractographs. The respective crack areas depend on the stress level and the present number of cycles until specimen failure has occurred.

$$A_{Circular}=\pi \times a^2$$

(11)

$$A_{fullcircumferential}=\pi \times \left[r^2-\left(r-a^2\right)\right]$$

(12)

$$A_{Chord}=\frac{\pi }{2}\times r^2-\left(r-a\right)\times \sqrt{a\times \left(2\times r-a\right)}-r^2\times \arctan \frac{r-a}{\sqrt{a\times \left(2\times r-a\right)}}$$

(13)

$$A_{lens}=2\times \frac{\pi }{2}\times r^2-2\times \left(r-\frac{a}{2}\right)\times \sqrt{a\times \left(r-\frac{a}{2}\right)}-2\times r^2\times \arctan \frac{r-\frac{a}{2}}{\sqrt{a\times \left(r-\frac{a}{2}\right)}}$$

(14)

The numerical analysis tool PROST uses the structure of a solid cylinder for the minimum cross-section of an unnotched specimen with a chord crack so that the crack propagation, as well as the stability of incipient cracks, can be compared with the previously determined measurement results. In the case of pure tensile load (cf. Equation (15)), an analytical approach for stress intensity factors can be used according to [9], which is based on the British Standard 7910. This is valid for relative crack depths (a/2 × r) in the range of 6.25% to 62.5%.

$$\begin{array}{cc}\hfill K=\sqrt{\pi \times a}\times \sigma & \times [0.926-1.771\times {\left(\frac{a}{2\times r}\right)}^1+26.421\times {(\frac{a}{2\times r})}^2-78.481\times {\left(\frac{a}{2\times r}\right)}^3\hfill \\ & +87.911\times \left(\frac{a}{2\times r}{)}^4\right]\hfill \end{array}$$

(15)

A limiting load criterion can be determined from the ratio of the ligament area to the cross-sectional area of the undamaged specimen with the crack area A_Chord. The fracture mechanics determination of the crack propagation is carried out by the fatigue crack propagation law according to [33] for austenitic steels (cf. Equation (16)).

$$\frac{da}{dn}=3.43\times {10}^{-12}\times S\left(R\right)\times (\Delta K^{3.3}+C\left(c_{O_2}\right)\times \sqrt{S\left(R\right)\times T_R}\times {(\Delta K)}^{1.65}$$

(16)

Here ΔK is given in (MPa √m) and crack propagation in (m/cycle). T_R is the time of load increase in (s), _CO2 is the dissolved oxygen concentration in (ppm). C is a function dependent on the dissolved oxygen concentration. The function S(R), which contains the load ratio R, is defined by Equation (17).

$$S\left(R\right)=\{\begin{array}{cc}1& R<0\hfill \\ 1+1.8\times R& 0<R<0.8\hfill \\ -43.35+57.97\times R& R>0.8\hfill \end{array}$$

(17)

The description of the crack propagation according to [28] is only valid for load ratios R ≥ 0. Investigations on austenitic steels have shown that the application of the correlation for R = 0 is also possible for negative load ratios in fatigue tests in air as an ambient medium [34]. Therefore, the function S(R) for the crack-closure load was extended to include extrapolation to negative R values. The first term of the crack propagation law (cf. Equation (16)) describes the crack propagation in an air environment, while the second describes the medium component. The crack propagation law in an air environment and R = −1 is shown in Figure 13 (blue curve, left axis). The fatigue crack propagation law becomes a straight line in a double logarithmic plot. In the area shaded in blue, the load peak has a linear–elastic stress intensity factor K_max > K_IC. In this area, even linear–elastic crack initiation or even specimen failure can be expected (in the elastic–plastic calculation, initiation can be expected earlier). In order for this behavior to be set in relation to the crack load, ΔK is also plotted as a function of crack depth for different nominal strain amplitudes (green lines). Thus, for a given crack depth a, the corresponding ΔK can be read from the desired load amplitude. Starting from the abscissa value, the crack propagation rate can be determined. This procedure is exemplified in Figure 13 with brown arrows. The process shown in Figure 13 is only a rough approximation since the crack propagation law may not lie exactly within the range considered.

Figure 14 shows the function of load drops for a constant strain amplitude of 0.3 × 10⁻². The diagram also contains the crack sizes determined during the test, which are further used as input variables in the calculation, e.g., using the PROST analysis tool. Horizontal dashed lines indicate the geometrical estimation of the crack depth a. The bars (slightly shifted to avoid overlapping) indicate the possible ranges of possible crack lengths depending on the extensometer clip position. The calculation results are focused on a load drop of 25%.

For total strain amplitudes larger than 1 × 10⁻², it should be noted that the reliability of the present calculations is limited since the Chaboche parameters were adjusted for relatively small total strain amplitudes. Further investigations and results of this work are described in more detail in [31].

From the investigation of crack sizes associated with the fatigue life achieved, an estimation can be proposed that gives a probability of a certain crack size at the defined load drop. The relation is based on the range spanned by two idealized crack shapes related to the experimental findings. The following equation approximates the full span of values for possible crack sizes for an assumed load drop d_L (relative value), based on data of CATs with a total strain amplitude of 0.3 × 10⁻² (Figure 14).

$$a\in \left[4.7\mathrm{mm}\times \sqrt{d_L}\times 4.4\mathrm{mm}+6.4\times d_L\right]$$

(18)

These values of a minimum and maximum envelope of the crack depths measured in the CATs are shown in Figure 15. Considering the possibilities, the experimentally observed cracks are within the range predicted by the numerical simulation, which allows possible corrections in terms of the scatter of fatigue life data with respect to a defined crack initiation criterion to be made.

4. Conclusions

This paper underlines that parameters determined from non-destructive and electrochemical testing can be used as an alternative or even as an improvement to the traditional parameters of load, stress, deformation and strain to determine the fatigue life of metallic materials and components. This has been demonstrated for the metastable AISI 347 (1.4550) steel in an initial and aged condition with a clear view on nuclear engineering. Changes in temperature, electric resistance, tangential magnetic field strength or in open circuit potential are among those NDT-based techniques from where measurements can be derived, which show adequate if not even better performance when compared to mechanical stress–strain hysteresis, being one of the state-of-the-art measurement techniques. Parameters derived from those NDT techniques can possibly be easier accessed when compared to plastic strain measurements, and they may possibly be more sensitive when looking at specific cases, where this article is only providing a limited example and can still not claim to demonstrate the full potential of such an approach lying ahead.

It has been shown for media conditions, such as distilled water, as well as reactor pressure vessel boiling water (BWR), that NDT measurements still hold, and that the NDT-based parameters can even be derived appropriately under such environmental conditions. Since the NDT-based parameters derived can be used to determine S–N curves, even in the context of short-time procedures where only a fraction of effort is required to generate an S–N curve when compared to state-of-the-art, a completely new field of generating S–N data is opened.

Considering components in nuclear engineering often being exposed to strain-controlled loading conditions, at least in the locations where fatigue damage becomes critical, the StrainLife method has been specifically developed and explained based on electrical resistance measurements. It is based on the fact that materials’ fatigue data for the stage until crack initiation can be used and transferred to evaluate and assess the fatigue life of components. In that context, the question of what damage—or better, degradation—is, is still not sufficiently solved. However, there is hope for improvement. The current view is still that cracks, and specifically their size, are a characteristic of damage and a structure’s vulnerability. Numerical simulation has proven that an understanding between loading conditions and degradation criteria (damage) within a material and a structure can be well established and can help to further improve the reliability of materials’ fatigue data generated, specifically with respect to short-time procedures, such as shown with the assessment of crack initiation uncertainties in the case of StrainLife here.

5. Outlook

StrainLife is on the way to becoming an integral part of structural mechanical analysis codes in nuclear engineering, where the tool of PROST, which has been referenced in this paper, is an example. Implementing such methods in tools for integrity assessments does not only ease the provision and application of S–N data within the code but can also help to guide inspection processes of components in the future in a way that the code may be able to predict which parameter should be measured best at which location and how accumulated damage may be assessed best at a macroscopically still-uncracked stage. Evidence for such approaches is currently being tested as part of the “MiBaLeb”-project, of which the second phase is due to be run into the year 2023.