1. Fatigue behavior of the 304L steel

The metastable austenitic stainless steel 304L exhibits a true fatigue limit of 260 MPa [Fig. 1(a)]. No specimen failed beyond 10⁶ cycles, even though remarkable surface roughening could be observed on the surface of run-out specimens.

FIG. 1.

(a) S–N curve for 304L tested at 90 Hz and (b) cyclic deformation behavior at different stress amplitudes represented by means of the change in resonant frequency.

The cyclic deformation behavior in the HCF and VHCF range was characterized on the basis of the resonant frequency, as described above. From a physical point of view, a decrease of resonant frequency corresponds to an increase of damping and thus results from cyclic softening, while an increase of resonant frequency corresponds to cyclic hardening. At all three stress amplitudes depicted in Fig. 1(b), the material exhibits a cyclic softening (decrease of f _res) followed by cyclic hardening (increase of f _res). By increasing the stress amplitude from 250 to 270 and 280 MPa, the curves move to lower values of f _res. At the same time, the onset of cyclic hardening moves to lower numbers of cycles and indicates a dependency of the hardening process on the cumulated plastic strain during cyclic loading.

Microstructural characterizations were carried out to determine the microstructural changes that govern the cyclic deformation. Figure 2 reveals the formation of protrusions at a very low stress amplitude of 240 MPa in the softening phase followed by an increase in density of slip markings during the hardening phase. However, no significant intrusions were observed by AFM measurements. Additional observations made by confocal microscopy and FIB cross-sectioning confirmed the lack of pronounced intrusions in the surface grains.

FIG. 2.

Evolution of surface topography for 304L at 240 MPa (AFM micrograph and surface profile).

For a better understanding of the plastic deformation mechanisms in the VHCF regime, the evolution of dislocation arrangements and phase composition in some surface grains was investigated by means of ECCI technique. A fatigue test performed at 240 MPa was interrupted at 10⁵ cycles (maximum softening), and a layer of less than 2 μm was removed by vibration polishing followed by electrolytic material removal. This method allows a perfect flat surface without additional artifacts induced by mechanical preparation, which is a mandatory requirement for ECCI measurements. After examination, the fatigue process was resumed and the specimen was tested up to 10⁷ cycles. Subsequently, the sample was prepared using the procedure mentioned above and exactly the same grains were analyzed in the SEM. The results for one grain are presented in Fig. 3.

FIG. 3.

ECCI micrographs after 10⁵ cycles (a) and after 10⁷ cycles (b) for one surface grain from a sample tested at a stress amplitude of 240 MPa without failure.

The micrographs show the formation of wavy-type dislocation arrangements during the softening phase with low dislocation density regions in between, which can be explained by a high amplitude of the plastic strain, corresponding to a strong decrease of the resonant frequency at 10⁵ cycles. During cyclic hardening, the microstructure changes and in the saturation stage the microstructure consists of a high number of narrow slip bands [Fig. 3(b)]. This is in reasonable agreement with the described changes in the morphology of the slip markings at the surface. These slip bands consist mainly of closely packed stacking faults. Martensite nuclei are formed at the intersection points and within the shear bands and impede the glide process. A more comprehensive description of the cyclic deformation mechanism of 304L in the VHCF range containing high resolution transmission electron microscopy (HRTEM) investigations and a detailed discussion can be found in Ref. Reference Grigorescu, Hilgendorff, Zimmermann, Fritzen and Christ19.

2. Fatigue behavior of the 316L steel

The results of fatigue tests carried out on the 316L steel under stress-controlled conditions are depicted in Fig. 4(a). This grade shows a decreasing fatigue strength (i.e., no true fatigue limit) in the VHCF regime with sample failure even above 10⁷ cycles at testing frequency of about 20 kHz. It should be noted that the scatter of results from samples tested at 140 Hz with a larger critically loaded volume is significantly lower than the scatter of samples with a smaller volume tested at 20 kHz. In both cases, the S–N curves approach a horizontal asymptote with no failure occurring up to 10⁹ cycles. While the material reaches a fatigue limit of 230 MPa during ultrasonic fatigue testing (at 20 kHz), the cyclic strength decreases by about 50 MPa when tested at 150 Hz. The frequency dependence of the fatigue life can be explained by the increase of Δε_pl/2 at the lower frequency. Similar observations were made by Nikitin and Besel. ^{Reference Nikitin and Besel20}

FIG. 4.

(a) S–N curves for 316L at 140 Hz (black full symbols) and at 20 kHz (blue empty symbols) and (b) resonant frequency of specimens of 316L tested at different stress amplitudes.

The transient cyclic deformation behavior of the more stable austenitic stainless steel 316L in the HCF and VHCF regime is much less pronounced [Fig. 4(b)]. A measurable cyclic softening at the stress amplitude of 230 MPa followed by cyclic hardening can be observed. The reduction of the applied stress amplitude toward VHCF regime and fatigue limit (e.g., see curve for the stress amplitude of 190 MPa) results in a modification of the cyclic deformation behavior in the sense that cyclic hardening right from the beginning of the test dominates. A change in the trend of the natural frequency of fatigued samples from a decreasing to an increasing frequency beneath a critical stress amplitude was also observed by Straub. ^{Reference Straub21}

Investigation of the surface of run-out specimens (Δσ/2 = 240 MPa, N = 10⁹ cycles, f = 20 kHz) revealed a strong localization of the cyclic plastic deformation in single isolated grains, which is in contrast to the earlier described findings for 304L. Figure 5(a) depicts an FIB cross-section in such a grain showing the formation of minor extrusions and deep intrusions followed by subsequent micro crack initiation. Similar cracks parallel to twin boundaries were observed in many samples; however, basically in all cases, these cracks were arrested within the first grain. It should be noted that this crack arrest could not be attributed to the barrier effect of grain boundaries. Rather, martensite formation within the slip bands was found to be the mechanism responsible for the arrest of the microcracks. ^{Reference Grigorescu, Hilgendorff, Zimmermann, Fritzen and Christ19} An ECCI micrograph taken from a plastically deformed grain of a run-out sample is shown in Fig. 5(b). In comparison to 304L, the saturated microstructure is characterized by a planar dislocation arrangement with very few stacking faults. The martensite formation occurs within the slip bands, and the ECCI method did not provide evidence of a second active slip system involved in the phase transformation. More investigations using TEM technique should be carried out to explain the exact transformation mechanisms for these conditions.

FIG. 5.

(a) Surface topography showing extrusions, intrusions, and microcrack; (b) ECCI micrograph in a surface grain after 10⁸ cycles.

The change in the deformation mechanism can be explained by a higher SFE of 316L. The higher tendency to wavy slip enhances the formation of wider slip bands without promoting the stacking fault formation between these bands as in the 304L steel. The developments of critical intrusions as well as martensite formation seem to be a consequence of this slip behavior.

In the following, the experimental results are used as basis for the model development and the corresponding numerical description of the relevant deformation mechanisms of austenitic stainless steels. By means of systematic parameter variations in the simulations, the previously presented assumptions drawn from the experiments shall be verified.

1. Shear band model

In the simulations, each shear band is directly represented by its slip marking in the 2-D surface plane. Shear band formation occurs once the simulated maximum resolved shear stress τ_MRSS in the most critical slip system within the microstructure exceeds a critical shear stress τ_c for shear band formation. Following the models of Tanaka and Mura ^{Reference Tanaka and Mura12} and Lin et al., ^{Reference Lin13} a shear band is approximated by two closely located parallel layers in the 2-D plane.

In alloys such as austenitic stainless steels, the so-called ‘short range order effect’ has to be taken into account. ^{Reference Gerold and Karnthaler22} This effect is related to a local shear plane softening once an obstacle has been penetrated by dislocations in the same shear plane (for more details, see Ref. Reference Hilgendorff, Grigorescu, Zimmermann, Fritzen and Christ23). It is assumed that the 304L steel due to its more planar slip character (low SFE) and therefore its reduced tendency to easily bypass obstacles by spatial dislocation motion (e.g., cross slip) is subjected to the ‘short range order effect’, whereas 316L with its less planar slip character (higher SFE) is not. The mechanism of ‘short range order effect’ is transferred into the model by assuming that the critical resolved shear stress τ_c for shear band formation is higher than the flow stress τ_F once sliding in an already formed shear band occurs.

Plastic deformation in a shear band is directly represented by slide deformation within the layers in the relevant fcc slip system. Ideal plastic sliding occurs in a layer once the shear stress in the corresponding slip system exceeds the flow stress τ_F. In accordance with the models of Tanaka and Mura ^{Reference Tanaka and Mura12} and Lin et al. ^{Reference Lin13} during forward loading, the sliding is allowed to appear on one of the layers only and during reverse loading sliding is confined to the other layer. By doing so, an irreversible fraction p of the previous sliding can be considered on the layer which is inactive during the respective loading half cycle (for more details, see Ref. Reference Hilgendorff, Grigorescu, Zimmermann, Fritzen and Christ24). In addition, dislocation hardening is considered in the model by increasing the flow stress τ_F depending on the sliding deformation. The relation is defined in a cyclic hardening law, ^{Reference Hilgendorff, Grigorescu, Zimmermann, Fritzen and Christ23} which depends among other parameters on a hardening parameter α that is determined in simulations.

The model parameters mentioned in this study were mostly referred to experimentally determined data or were estimated on the basis of semi-quantitative evaluations. For the metastable 304L steel, the parameters τ_c = 70 MPa, τ_F = 50 MPa, and p = 2 × 10⁻⁶ and for the stable 316L steel the parameters τ_c = τ_F = 110 MPa (without ‘short range order effect’) and p = 2 × 10⁻⁵ (higher because of the increased SFE) were chosen. The parameter p of 316L was estimated based on the evaluation of the surface roughening of fatigued specimens, and for 304L the value was chosen relatively smaller due to lower SFE. It is assumed that the shear band model incorporates the microstructural changes due to dislocations, stacking-faults, and ε martensite because they all presumably lead to a shear deformation along a shear band. The influence of twin or grain boundary shearing is not considered in the model.

2. Martensitic transformation model

The 304L steel undergoes a deformation-induced martensitic transformation (from γ to α′) in the VHCF regime. The ε martensite is not explicitly considered in the transformation model. The experimental results shown in Sec. II confirm that the deformation-induced α′ martensite is preferentially generated at the intersection of shear bands. In the following, the mechanisms of martensite nucleation and growth are assumed to obey the model of Bogers and Burgers ^{Reference Bogers and Burgers15} and its enhancement by Olson and Cohen. ^{Reference Olson and Cohen16} According to their model, a bcc structure can be generated from fcc structure by two specific shears acting at a shear band intersection. ^{Reference Hilgendorff, Grigorescu, Zimmermann, Fritzen and Christ23} As a criterion for martensitic transformation, it is stated that martensite is emerging in the microstructure, once plastic shear deformation occurs simultaneously in two slip systems that are compatible according to the two Bogers and Burgers shears. One of the shear deformations is known from the previously described shear band model. Due to the restriction to single slip in the shear band model, the sliding value in the second slip system is calculated in a purely analytical computation based on the theory of continuously distributed dislocations (for more details, see Ref. Reference Hilgendorff, Grigorescu, Zimmermann, Fritzen and Christ23). With the knowledge of the sliding deformation in both relevant slip systems, the size of a generated martensite domain can be assessed.

Furthermore, an important aspect of the martensitic transformation is the volume expansion around a generated martensite phase. The shape deformation due to martensitic transformation is assessed by computing the deformation gradient tensor of both relevant Bogers and Burgers shears and deriving one corresponding transformation-induced strain ε_M (for more details, see Ref. Reference Hilgendorff, Grigorescu, Zimmermann, Fritzen and Christ25).

The numerical method adopted is the 2-D boundary element method which was applied to compute stresses and displacements within modeled 2-D microstructures taken from real fatigue samples. Details about the used boundary element method can be found in Ref. Reference Hilgendorff, Grigorescu, Zimmermann, Fritzen and Christ23. It should only be noted that by using the general plane stress condition in the modeled 2-D surface plane the plastic sliding deformation of shear bands could be simulated in the true spatial slip directions of each grain.

1. Cyclic plastic deformation of the metastable austenitic stainless steel 304L

Figures 6(a)–6(c) show a SEM image, the corresponding EBSD map, and the phase map, respectively, representing an examined microstructure on the specimen surface of 304L steel (fully austenitic condition), which was fatigued for 2 × 10⁷ loading cycles below the VHCF strength of Δσ/2 = 240 MPa. Simulation of plastic deformation was only considered within the grains labeled 1, 2, and 3 in Fig. 6(b) (enclosed by the white dashed lines). This restriction intends not only to rule out boundary influences but also to reduce the computational effort. The microstructure was captured by means of the EBSD map and the modeling parameters of 304L were applied. Figures 6(d)–6(f) show the calculated shear stresses τ_MRSS in each case at the maximum external load (240 MPa) after the respective simulated loading cycles N _sim. Modeled shear bands are emphasized by thin lines. The martensitic transformation is appreciable by distinctive shear stress peaks around the generated martensite nuclei [e.g., marked with black arrows in Fig. 6(d)].

FIG. 6.

(a) SEM image, (b) EBSD map, and (c) phase map of the examined real surface area of 304L steel (Δσ/2 = 240 MPa, number of cycles N _exp = 2 × 10⁷); distributions of simulated shear stresses τ_MRSS at certain simulated loading cycles N _sim (d–f) (304L, loading amplitude Δσ/2 = 240 MPa, cyclic slip irreversibility p _sim, hardening parameter α_sim, and martensite scaling parameter s _α′). ^{Reference Hilgendorff, Grigorescu, Zimmermann, Fritzen and Christ23}

Below each illustration in Figs. 6(d)–6(f), the values for the cyclic slip irreversibility p _sim, the hardening parameter α_sim, and a parameter s _α′ are given. The parameters p _sim and α_sim have the same physical meanings as the original parameters p and α from the shear band model (see Sec. III.A.1), but in the simulations these parameters were increased (and stored to p _sim and α_sim) to satisfy the true cyclic deformation of many experimental cycles within a significantly lower number of simulated cycles due to computational effort (for more details, see Ref. Reference Hilgendorff, Grigorescu, Zimmermann, Fritzen and Christ23). The parameter s _α′ was used to scale the martensitic transformation in the simulation by multiplying the change of martensite domain size with this parameter. In supposing a linear relationship between cycles and irreversibility, the chosen cyclic slip irreversibility p _sim = 0.2 for N _sim = 10 would correspond to N _exp = 10⁶ experimental cycles with p = 2 × 10⁻⁶. Cycles beyond 10⁷ cycles were not considered in simulations.

A comparison of the surface slip markings found in the SEM image [Fig. 6(a)] with the modeled shear band layers [Figs. 6(d)–6(f)] illustrates that shear band formation in designated slip systems was correctly reflected in the simulation model. Also, the sites of generated martensite domains in the simulation are in good agreement with the distribution of the martensite phase presented in Fig. 6(c).

The simulated plastic sliding deformation in the shear band layers led to a decrease of maximum shear stresses inside the relevant grains, while shear stress peaks arose at the grain boundaries where plastic sliding deformation was piled up. Around generated martensite nuclei, distinctive shear stress peaks appeared because of the transformation-induced volume expansion, but also due to the blocking of plastic sliding deformation at martensite domains. With increasing loading cycles, the number of modeled shear bands increased. This was due to the gradual and renewed increase of shear stresses around existing shear bands owing to the decrease of cyclic plastic sliding amplitudes in shear band layers. This decrease is a consequence of the individual cyclic hardening and cyclic slip irreversibility of each shear band and the ongoing blocking of shear bands by martensite domains. ^{Reference Hilgendorff, Grigorescu, Zimmermann, Fritzen and Christ23}

2. Cyclic plastic deformation of the stable austenitic stainless steel

Figures 7(a) and 7(b) show the SEM image and the EBSD map, respectively, representing an examined real surface area of the stable austenitic stainless steel 316L fatigued at the VHCF strength of Δσ/2 = 190 MPa (reduced compared to 304L) up to 2 × 10⁷ loading cycles. Again, the microstructure was modeled based on the EBSD map and the modeling parameters of 316L were applied. The simulation of cyclic plastic deformation was confined to the grains labeled 1, 2, 3, and 4 as indicated in Fig. 7(b) (enclosed by the white dashed lines). Figures 7(c)–7(e) show the calculation results obtained for the simulated shear stresses τ_MRSS after selected cycles N _sim at the maximum external load (here 190 MPa). As shown in Fig. 6, modeled shear bands are emphasized by thin lines. The martensitic transformation model was disabled due to the stable condition of AISI 316 L. The represented number of experimental cycles after 10 simulated cycles can be assessed to be N _exp = 0.5 × 10⁶ by comparing the parameters of 304L and 316L.

FIG. 7.

(a) SEM image and (b) EBSD map of the examined real surface area of 316L steel (Δσ/2 = 190 MPa, number of cycles N _exp = 2 × 10⁷); distributions of simulated shear stresses τ_MRSS at certain simulated loading cycles N _sim (c–e) (316L, loading amplitude Δσ/2 = 190 MPa, cyclic slip irreversibility p _sim, and hardening parameter α_sim).

The simulation result shown in Fig. 7(c) indicates that after the first simulated loading cycle only one shear band next to the grain boundary between grain no. 1 and 2 was generated [see arrow in Fig. 7(c)]. It confirms the reduced tendency of the stable austenitic alloy to undergo plastic deformation in shear bands. This observation can be referred to the higher values for the critical resolved shear stress τ_c for shear band formation and flow stress τ_F, along with a smaller external loading (VHCF strength). Figures 7(d) and 7(e) indicate a further shear band formation with increasing simulated loading cycles. In this case, the strong accumulation of cyclic plastic deformation in shear bands led to a gradual increase of the shear stress peak at the grain boundary causing the generation of new shear bands. A comparison of the surface slip markings found in the SEM image [Fig. 7(a)] with the simulated shear band layers [Fig. 7(e)] illustrated that shear bands were formed in the correct slip systems.

The high shear stress peak evolving at the grain boundary in Figs. 7(d) and 7(e) indicates a critical site regarding crack initiation, thus leading to the experimentally identified microcracks in 316L (see Fig. 5). By contrast, the martensitic transformation in the metastable austenitic stainless steel 304L led to a stagnating increase of plastic deformation due to the blocking of shear bands by martensite domains. Thus, it seems reasonable to suppose that this barrier to plastic deformation provides an explanation for the higher VHCF strength of AISI 304 compared to AISI 316 L. In the following, a comparison based on the resonant behavior is carried out.

3. Influence of cyclic plastic deformation on the resonant behavior

In simulations, the force displacement hysteresis loops were used to determine the resonant frequency ratio η_res = f _res/f ₀ for each simulated cycle. This was done by applying a hysteretic and viscous damping model (for more details, see Ref. Reference Hilgendorff, Grigorescu, Zimmermann, Fritzen and Christ23). From experiment, the resonant frequency ratio η_res was also evaluated by using the monitored resonant frequencies f _res divided by the additionally determined natural frequencies f ₀ of the individual specimen.

Figure 8 shows a qualitative comparison of resonant frequency ratio η_res from simulation and experiment. The simulated cycles N _sim are shown at the top and the cycles from experiment N _exp at the bottom. The solid black curves represent the results of η_res from experiment for the metastable and the stable austenitic stainless steel (304L and 316L). A decrease of η_res corresponds to cyclic softening and an increase to cyclic hardening. The results from simulations are indicated for 316L and 304L by a dotted and a dashed curve, respectively. The evaluated ratio η_res for the simulation of 316L (see Fig. 7) showed no appreciable change (η_res ≈ 1) owing to low simulated cyclic plastic deformation. Therefore, the value of the critical resolved shear stress was reduced from 110 to 90 MPa, leading to the calculated results for η_res depicted in Fig. 8.

FIG. 8.

Qualitative comparison of resonant frequency ratio η_res from simulations (dashed and dotted curves with scale of simulated cycles at the top) and experiment (solid curves for 304L and 316L with scale of cycles from experiment at the bottom). ^{Reference Hilgendorff, Grigorescu, Zimmermann, Fritzen and Christ23}

The dashed curve in Fig. 8 indicates that the simulation of cyclic plastic deformation of 304L at RT results in a cyclic softening (decrease of η_res) at the beginning and a subsequent cyclic hardening (increase of η_res), which is qualitatively consistent with the experimental results [also shown in Fig. 1(b) in terms of the resonant frequency]. The cyclic softening can be attributed to the continuous formation of new shear bands in combination with the ‘short range order effect’. ^{Reference Hilgendorff, Grigorescu, Zimmermann, Fritzen and Christ23} Once shear bands are reaching a minimum critical distance, further formation is prevented in the simulation. Subsequently, the local cyclic hardening and irreversibility in each shear band and the blocking of shear bands by martensite domains lead to global cyclic hardening.

In case of the simulated stable austenitic stainless steel (but with reduced τ_c and τ_F), the dotted curve in Fig. 8 shows a constant increase of the ratio η_res and thus continuous cyclic hardening. This is in good agreement with the experimental results obtained for 316L. In particular, the reduced tendency to form shear bands and the absence of ‘short range order effect’ and martensitic phase transformation prevent cyclic softening and reduce cyclic hardening. The necessity to increase the tendency to form shear bands in the simulation to visualize the expected moderate cyclic hardening is referred to local plastic deformation in the matrix between shear bands which was not considered.