Characterization and Modeling of the Fatigue Behavior of 304L Stainless Steel Using the MultiStage Fatigue (MSF) Model

Jump to: navigation, search



The fatigue behavior of 304L stainless steel (304L SS) was modeled using the Multi-Stage Fatigue (MSF) model. Microstructural analysis using fractography determined the relevant microstructural details. Striations on the fatigue fracture surface indicated the growth rate of small fatigue cracks. The microstructurally small crack equations in the MSF model were calibrated to these crack growth rates. Once calibrated with the microstructure information, the MSF model captured the fatigue behavior of 304L SS. The MSU Internal State Variable (ISV) model was calibrated with a quasi-static, monotonic tension test. In a practical application, the MSU ISV model could be used to determine the stress state of a part in a real application, then the MSF model could predict the fatigue life of that part.

Keywords: 304L stainless steel, Stress-life, Strain-life, Stress-strain relation, Bauschinger effect, Mechanical properties, R-value effect, Fatigue

Author(s): B. Everett, D. Failla, W. Furr, N. Geeslin, D. Giri, B. Huddleston, A. Hudson, S. Kazandjian, A.Turner


A microstructurally sensitive model for fatigue, the MultiStage Fatigue (MSF) model (Xue et al., 2007), was calibrated to a cold drawn 304L stainless steel (304L SS) based on three strain controlled fatigue tests at various strain levels. Fractography from the fatigue fracture surfaces provided particle size data that influenced the crack incubation in the samples. The small crack growth behavior was calibrated to the crack growth rate obtained from striation spacings found at various locations on the fracture surfaces. By including microstructure details, the model is specifically calibrated to the microstructural features observed on the fracture surfaces of the failed 304L SS test specimens. In addition, a physically motivated plasticity model, the MSU Internal State Variable (MSU-ISV) model, was calibrated to a single quasi static, monotonic tension test. The MSF and MSU-ISV model calibrations together enable the future fatigue analysis of a complex part.

Austenitic stainless steels, such as 304L SS, are widely used for their combination of corrosion resistance, strength, ductility, and impact resistance (Pegues, 2016; Gardner, 2005). They are suitable for a multitude of applications across many different industries and offer significant benefits in elevated temperature applications (Gardner, 2005). The chemical composition of the austenitic steel of 304L SS can be seen in Table 1. The significant body of research found in the literature on the fatigue behavior of 304L SS and other austenitic stainless steels is driven by their extensive use, particularly in safety critical applications such as nuclear power plant piping and reactor internals. A review of the current state of literature regarding fatigue behavior of 304L SS follows, focusing on important topics to the fatigue behavior: stress-control, strain-control, stress-strain behavior, the Bauschinger effect, R-ratio effects, and other microstructure effects.

Table 1. Chemical Composition for UNS S30403 / Type 304L Stainless Steel (Stephens et al., 2000).

Stress Control

Since first introduced by August Wöhler in the 1850’s (Stephens et al., 2000; Wohler, 1867), the stress controlled approach to fatigue evaluation, displayed on stress-life (S-N) plots, has been extensively applied in characterizing the fatigue behavior of a wide variety of materials. Stress-controlled (or load-controlled) tests of 304L SS allowed studies of fatigue-related phenomena in primarily high-cycle fatigue (HCF) with unconstrained elastic deformation. In low-cycle fatigue (LCF), materials experience widespread plastic deformation, degrading the accuracy of stress-controlled tests (Stephens et al., 2000).

Stress control techniques were applied to the study of crack initiation and microstructurally small crack (MSC) growth behavior in the HCF regime (Pegues, 2016; Pegues et al., 2017). Cracks initiated preferentially at twin boundaries, though initiation sites were found at grain boundaries and slip planes as well. However, cracks nucleating at twin boundaries grew more quickly. Pegues et al. (Pegues et al., 2017) also found that 304L SS began to undergo a strain induced phase transformation from the bulk austenite phase to martensite at a fatigue load of around 330 MPa. Chaves et al. (Chaves et al., 2015) showed that annealed 304L SS stainless steel demonstrated a fatigue limit of 230 to 316 MPa for axial loadings, and 288 MPa for torsional loadings, both with fully reversed loads. Figure 1 shows the stress-life results of 304L SS as reported in (Chaves et al., 2015).

Figure 1. Stress-life (S-N) curves of 304L stainless steel for uniaxial, fully reversed tests for (a) push-pull and (b) torsion (Chaves et al., 2015).

Strain Controlled Fatigue

While stress-controlled fatigue tests are suited for HCF applications, strain-controlled tests have improved accuracy in the LCF regime, and can be used for both HCF and LCF regimes (Stephens et al., 2000; Suresh, 1991). In the nuclear industry, 304L SS structures are subjected to LCF loadings where large cyclic loading causes permanent strain leading to crack initiation and growth (Suresh, 1991). Measuring local strain in high stress conditions (e.g. start-up or shut down of a gas turbine) is easier than measuring stress, making the strain-control approach preferable (Stephens et al., 2000). Engineering in practice frequently involves the use of structurally fixed components containing local stress concentrations, which is best simulated using strain-controlled testing (Suresh, 1991). The strain-life methodology introduced by L. F. Coffin and S. S. Manson proposed the use of plastic strain amplitude to assess fatigue life (Coffin, 1954; Manson, 1965; Colin, 2009).

Colin (Colin, 2009) showed that the strain-life curves for strain and load-controlled tests nearly overlap for Thyssen (THY) 304L SS as shown in the S-N plot in Figure 2; thus, the test control-mode had little-to-no effect on the fatigue life. In addition, Colin (Colin, 2009) found that prestrain due to manufacturing defects and intentional or unintentional overstrain had little effect on fatigue life using strain-life approach (Kalluri et al., 1996). Conversely, prestrain has significant favorable impact on fatigue life when applying the stress-life approach. According to Colin (Colin, 2009), prestraining led to failure of 304L SS at a 0.25% strain amplitude, where previously the material survived till runout.

Figure 2. Strain-life plot for load-controlled and strain-controlled fatigue testing of 304L stainless steel (Colin and Fatemi, 2010) and strain-life variation with R-ratio (Pegues, 2016).

Stress-Strain Relationship

Knowledge of the material stress-strain response to monotonic uniaxial loading is required to understand the fatigue behavior. However, fatigue life cannot be accurately predicted from monotonic, tension or compression, stress-strain properties alone. Stress-strain curves obtained from cyclic loading data can vary greatly from monotonic stress-strain behavior as demonstrated in Figure 3.

Figure 3. Superimposed monotonic and cyclic stress-strain curves and data for 304L stainless steel showing strong cyclic hardening, from (Stephens et al., 2000).

Cyclic constant strain amplitude loading of 304L SS generally involves three stages including consolidation with primary hardening, cyclic softening, and stabilization. During initial cycles, hardening takes place where the hysteresis loop stress amplitude rises with increasing number of cycles. The rate of hardening gradually reduces until maximum stress amplitude is reached. Then, cyclic softening begins and the stress amplitude falls with the rising number of cycles. Continued cyclic loading leads to saturation, which is usually reached within half-life (Li, 2012). However, continuous secondary hardening is also observed in some instances following initial hardening and softening. Higher strain amplitudes are associated with greater secondary hardening rates (Colin, 2009). Stabilization is not always observed and varies with strain amplitude (Li, 2012).

Stress-strain data displayed as hysteresis loops provides information about the material property changes from loading conditions. For instance, cyclic hardening of a material is presented as an increase in stress from one cycle to the next at constant strain rate. Figure 4 shows cyclic hardening of 304L SS. The cyclic hardening is more intense for the first cycles and stabilizes throughout the experiment.

Figure 4. Hysteresis loops from uniaxial, fully reversed, cyclic, strain-controlled experiment illustrating cyclic hardening of 304L stainless steel, from (Abdel-Karim, 2011).

Bauschinger Effect

Early yielding under reversed loading, the Bauschinger effect, is the result of easy motion of piled up dislocations, while loading in one direction, and then the separation of the dislocation structures as loading is reversed (Taleb and Hauet, 2009). The dislocation pile ups in a uniaxial tension test would normally result in work hardening of the material due to plastic deformation resistance. When the dislocation structures dissipate in reverse loading during cyclic testing, the material becomes more susceptible to yielding at a lower stress. In mechanics, the Bauschinger effect is quantified through kinematic hardening or translation of the yield surface.

High strength steels, such as 304L SS, have internal mechanisms that resist deformation. The resistance of the material leads to dislocation pile ups that result in the Bauschinger effect in cyclic loading (Taleb and Hauet, 2009). The experimental data from a fully reversed stress-strain experiment of 304L SS can be observed in Figure 5. The Bauschinger effect is evident by the reduction in yield after loading reversal.

Figure 5. Fully reversed stress strain experiment reflects the Bauschinger effect observed in 304L stainless steel by the red lines where σ (≈190 MPa) is the yield stress from forward loading and σ’(≈50 MPa) is the yield stress from reversed loading, from (Taleb and Cailletaud, 2010).

Abdel-Karim incorporated plastic deformation phenomena, such as the Bauschinger effect, into a model for ratcheting for 304L SS (Abdel-Karim, 2011). These experiments have proven that small scale plasticity has a significant effect on the simulation of ratcheting. The kinematic hardening rule is used in many modeling applications in order to capture the Bauschinger effect and ensures accurate hysteresis representation for simulation.

R-Value Effects

Due to the use of 304L SS in pressure vessel or tubing applications, the effect of a tensile mean stress is an important property to study. Typically, tensile mean stresses are detrimental to fatigue behavior, while compressive mean stresses are beneficial (Moyer and Ansell, 1975). However, when the R-value is representing the strain ratio, the trend can become convoluted. Figure 6, from (Colin et al., 2010), shows the mean stress effects on 304L SS in strain controlled tests. They observed the largest mean stresses at a 0.25% strain amplitude and a strain R-ratio of 0.75, while the mean stress actually decreased for with a more severe R-ratio of 2 and the same strain amplitude.

However, Colin et al. (Colin et al., 2010) found that in strain or stress-controlled tests, a tensile mean stress did not significantly degrade the fatigue life, as shown in Figure 2. In the strain controlled tests, stress relaxation relieved the mean stress very quickly, as shown in Figure 6, and in stress controlled tests, the mean stress hardened the material and reduced the strain amplitude, nullifying the effect of the mean stress. In contrast, Vincent et al. (Vincent et al., 2012) found that in high cycle fatigue, a small mean stress had a significant detriment to the fatigue life in strain controlled tests, but little effect in stress controlled tests. Bayerlein et al. (Bayerlein et al., 1989) showed that a tensile mean stress significantly increased the crack growth rate as the tensile mean stress triggered the martensitic phase transformation.

Figure 6. Mean stress versus the cycle ratio (N/Nf) for 304L stainless steel with different R-ratios and at two different strain amplitudes (0.25% and 0.4%), from (Colin et al., 2010).

Microstructure Effects

The austenitic phase obtained by solution treatment of 304L SS is metastable at room temperature (Roth et al., 2010; Bayerlein et al., 1989). Roth et al. (Roth et al., 2010) depicted the phase stability of austenitic stainless steel after solution annealing using a Schaeffler diagram as shown in Figure 7(a), and a schematic representation of the thermodynamic stability as shown in Figure 7(b). The Gibbs energy difference (ΔG) is the driving force for transformation; below a critical value transformation will not occur during the cooling process. However, additional mechanical strain energy (ΔGmech) can result in the Gibbs energy difference exceeding the threshold level allowing transformation to occur.

Figure 7. (a) Schaeffler diagram depicting phase composition and (b) schematic representation of the thermodynamic stability of austenite and α’ –martensite, from (Colin et al., 2010).

304L SS is also susceptible to a secondary hardening phenomena known as ratcheting (Colin and Fatemi, 2010; Kang et al., 2006). Ratcheting is a partial austenitic phase transformation that induces martensite growth. Figure 8 shows how martensite content in 304L SS can increase the fatigue life, initially, and then ultimately reduce the fatigue life once martensitic content exceeds ~35%. As the martensite content increases, so does the ductility. 35% martensite appears to be the experimental balancing point for maximum fatigue life between strength and ductility for 304L SS.

Figure 8. Residual fatigue life compared to martensite fraction of 304L stainless steel (Mughrabi and Christ, 1997).

In the high-cycle fatigue regime, the plastic strain is not typically large enough for significant phase transformation to occur (Pegues et al., 2017; Bayerlein et al., 1989). Hence, other microstructural features tend to govern fatigue behavior in the high-cycle fatigue and transition regimes. These features include twin boundaries, inclusions, slip band intrusions/extrusions, and grain boundaries (Pegues et al., 2017). While cracks may originate at any of these features, twin boundaries are the leading crack initiation feature in 304L SS for both the high-cycle fatigue and transition regimes (Pegues et al., 2017; Roth et al., 2010). Roth et al. (Roth et al., 2010) found that approximately 70% of microstructurally small cracks initiated near twin boundaries without any connection to martensite formation, as shown in Figure 9.

Figure 9. Frequency of various MSC initiation sites in 304L stainless steel for fatigue cracks showing the prevalence of twin-boundaries as initiation sites, from (Roth et al., 2010).

Temperature strongly effects crack initiation life, hardening and softening characteristics, and the half-life stress-strain relationship (Li, 2012). Baglion and Mendez (De Baglion and Mendez, 2010) studied the relative effect of strain rate and environment on the low cycle fatigue (LCF) behavior of 304L SS. They found fatigue life was highly influenced at low strain amplitudes by air at 300 C, but the effect was minimal at room temperature. Conversely, high temperatures up to 300 C were beneficial to fatigue life in a vacuum, but fatigue life was unaffected in air due to the detrimental environmental effects. Figure 10 depicts the drop in tensile peak stress levels at high temperatures in a vacuum at both 0.3% and 0.6% strain amplitude. Secondary hardening is visible at certain temperatures and strain amplitudes. Dynamic strain aging (DSA) was amplified after decreasing the strain rate from 4x10-3 to 10-5 s-1 at 300 C in both air and vacuum. DSA resulted in reduced fatigue life due to the negative strain rate dependence of the stress response. Moreover, decreasing strain rate was related to reduction in environmental effects, which was attributed to the significant effect of DSA on crack initiation and growth at low strain rates (De Baglion and Mendez, 2010).

Figure 10. Tensile peak stress rises with strain amplitude and is greatest at 20 C in 304L stainless steel, from (De Baglion and Mendez, 2010).

Materials and Methods

We used the MTS 800 to run three fatigue tests, a Bauschinger test, and a monotonic tension test for 304L SS. The purpose of these tests was to find the fatigue life for 304L SS. Based on the limited time available and the expected fatigue life of 304L SS, we picked strain amplitudes of 0.3%, 0.5%, and 1%. The frequency for the fatigue test remained at a constant 1 Hz. In the monotonic tension test, the quasistatic strain rate was 0.001 (mm/mm)/s. All samples were machined cylindrical specimens to avoid unwanted stress concentrations. The specimen had a nominal diameter of 6 mm with a nominal gauge length of 18 mm. The actual specimen measurements varied by up to 100 microns.

A Scanning Electron Microscope (SEM) was used to analyze the fracture surface of 304L SS. According to Mann et al. (Mann et al., 2015), fatigue crack initiation predominantly occurs on the surface of 304L SS for fully reversed strain controlled testing similar to Figure 11. The fracture surface of all three specimen contains striations and river marks, indicating crack initiation sites and the rate of crack growth. The fracture surface of all specimen can be distinctly divided into fatigue failure zone and ductile overload zone. The ductile overload zone has a rough surface and is devoid of striations. Also present in the fatigue failure zone are defects like pores and particles. The analysis of these pores, particles and striations quantifies the fracture failure.

Figure 11. Crack propagation on the fracture surface of a 304L SS fatigue specimen (0.3% constant strain amplitude).

The crack initiation sites of 0.5% (see Figure 12) and 1% (see Figure 13) strained samples show big holes. The 0.5% strained sample has an inclusion at the initiation site which means the defect could have probably acted in intensifying the damage while in case of 1% strained sample, the reason is not clear. The crack length distribution is found to be between 0.5 and 500 microns. The surface crack length of the crack at the top left of Figure 11 is close to 300µm. Li (Li 2012) reports half the total fatigue life for 304L SS is in the crack initiation regime, which is defined as the number of cycles before the maximum crack size is greater than 200µm.

Figure 12. River marks point towards the crack initiation site on the fracture surfaces of a 304L SS fatigue specimen (0.5% constant strain amplitude).
Figure 13. Large inclusions and particles observed near the surface of a 304L SS fatigue specimen (1% constant strain amplitude).

Moreover, Figure 14 shows microcracks forming at the grain boundaries (GBs). Cracks initiated at inclusions mainly within GBs, which is in agreement with Li’s analysis (Li 2012). Microcrack propagation rates slow at GBs due to resistance and then increase in-between GBs. Figure 15 shows that the crack growth rate steadily rises with increasing crack length, agreeing with current literature (Li 2012).

Figure 14. Microcrack formation at GB and inclusion sites at nanometer scale on the fracture surface of a 304L SS fatigue specimen (0.3% constant strain amplitude).

Striation spacing and crack length data obtained via SEM images was used to determine the number of cycles for the microstructural small crack (MSC) regime. There is 1:1 correlation between striations and strain cycles after incubation. Therefore, by examining striations one can determine crack growth rate, crack growth direction (CGD) (see Figure 16), and location of crack origin (Connors, 1994).

Figure 15. Average crack growth rate versus crack length for 304L SS under 0.3% constant strain amplitude.
Figure 16. The striation spacing or crack growth rate rises with increasing distance from the crack initiation site on the fracture surface of a 304L stainless steel fatigue specimen.

Theoretical Models

MultiStage Fatigue Modeling

The MultiStage Fatigue (MSF) model is used to describe the fatigue behavior within three distinct regimes: incubation, microstructurally small crack (MSC) / physically small crack (PSC), and long crack (LC). Of primary interest were the incubation and the MSC/PSC regimes. Originally developed by McDowell et al. (McDowell et al. 2003), the MSF model is a microstructure-sensitive approach to fatigue damage behavior, used to model a variety of different materials (McDowell et al., 2003; Xue et al., 2007; Jordon et al., 2010; Bernard et al., 2013; Hughes et al., 2017). A summary of the MSF model is given below. Additional details regarding its development and theoretical basis may be found in (McDowell et al., 2003). The total fatigue life Ntotal is determined from Equation 1, where, NInc is the number of cycles required to incubate a fatigue crack, NMSC/PSC is the number of cycles within the microstructurally / physically small crack regime, and NLC is the number of cycles in the long crack regime up to failure.

Equation 1

Equation 2 is a modified Coffin-Manson law used to model the incubation life, where Cinc is the linear coefficient for fatigue crack incubation and α is the exponent.

Equation 2

Β is the non-local damage parameter around an inclusion, and is the local average maximum plastic shear strain amplitude. Cinc is determined by Cinc = Cn + z (Cm – Cn), where Cn = 0.24 (1 – R), z is a variable related to plastic zone size, R is the load ratio, and Cm is a model constant. Equations 3 and 4 are used to estimate β, where εa is the remote applied strain amplitude, εth is the microplasticity threshold, is the ratio of the plastic zone and the inclusion area subject to the square root transformation, and q and ξ are commonly determined from micromechanical simulations.

Equation 3
Equation 4

ηlim is the limiting ratio with regard to applied strain amplitude, and indicates the transition from constrained micro notch root plasticity to unconstrained (McDowell et al., 2003; Xue et al., 2007). Although modification is made to account for geometric effects corresponding to the type of inclusion, the Y parameter is generally computed as Y = y1 + (1 + R) y2, where R is the load ratio, and y1 and y2 are model constants.

Within the MSC/PSC crack growth regime, the crack growth rate is dependent upon the crack tip displacement, and is computed using Equation 5, where χ is a material constant, ΔCTD is the crack tip displacement range, and ΔCTDth is the corresponding threshold value normally estimated as the Burger’s vector.

Equation 5

The crack tip displacement range is a function of the remote loading and is computed as per Equation 6 below, where CI, CII, and n are material parameters used to relate microstructure effects with microstructurally small crack growth. The first term is related to high cycle fatigue (HCF), while second term is related to low cycle fatigue (LCF).

Equation 6

ai is the initial crack length. ∆σ ̂ represents the combination of the uniaxial effective stress amplitude and the maximum principal stress range, Δσ1, and is given by Equation 7.

Equation 7

When θ = 0, Equation 7 gives the Von Mises stress state. When θ = 1, the maximum principal stress state results (Hayhurst et al., 1985). The load ratio effects on ΔCTD and MSC crack growth are captured by the parameter, where R is the load ratio. The effects of grain size on MSC crack growth are captured by the ratio of the grain size to the reference size, where the grain size is represented by GS, the reference grain size by GSo, and z is a material parameter. Likewise, grain orientation effects are captured in a similar fashion with GO indicating grain orientation, GOo reference grain orientation, and ω a material parameter.

A classical linear elastic fracture mechanics (LEFM) approach is applied to the MSF model to account for the long crack growth stage (McDowell et al., 2003). Because the small size of our fatigue samples, we neglected the long crack growth stage, instead focusing on the incubation and MSC/PSC stages of the fatigue model.

MSU Internal State Variable Model

The Internal State Variable (ISV) plasticity-damage model is based upon the plasticity formulation developed by Bammann (1990), which was extended by adding porosity (Bammann et al. 1993), and further adapted to include the nucleation, growth, and coalescence of voids in heterogeneous microstructure (Horstemeyer et al. 2000). The ISV model is a physically-based, microstructurally-sensitive, plasticity-damage model used to predict the response of a variety of materials under various loading conditions (Jordan et al. 2007).

Since the ISV model has been extensively described in the literature, we provide only a summary of the plasticity model. A more detailed description may be obtained from the identified sources. Governing equations for the ISV plasticity model (Bammann 1990; Bammann 1993; Horstemeyer et al. 2000) are as follows:

Equation 8
Equation 9
Equation 10
Equation 11
Equation 12
Equation 13
Equation 14

Functions V(T),Y(T), and f(T), given by Equation 8, are scalar and are related to yielding. These functions feature an Arrhenius-type temperature dependence. The function Y(T) is the rate-independent yield stress, while f(T) determines when yielding is affected by rate-dependence, and V(T) determines the magnitude of the rate dependence. Equations 9, 11, 12, and 14 are also scalar. Equations 9 and 12 represent dynamic recovery. Equations 11 and 14 describe static or thermal recovery. The hardening moduli are given by Equations 10 and 13, with the former representing anisotropic (kinematic) and the latter isotropic hardening.

Stress-dependent variables create texture and dislocation substructures through deformation, which is accounted for by the hardening moduli and dynamic recovery functions (Horstemeyer et al. 2000). Varying hardening rates between axisymmetric compression and torsion are captured by including the term J_3' in the hardening equations (Horstemeyer et al. 2000). The terms J_3' and J_2', used in the ISV model are given below in Equations 15 and 16.

Equation 15
Equation 16

Here, the equations include the deviatoric stress, σ', and the kinematic hardening internal state variable, α which represents the effect of anisotropic dislocation density.

The MSU ISV model can also capture the dependence of a material’s mechanical properties on the presence of defects, such as voids. Damage can grow by increasing the number of defects, or by increasing the size of existing defects. Damage progression is captured in the ISV damage framework with the separation of void nucleation and growth from pores and from silicon particles, and coalescence to model the interactions between these (Horstemeyer et al. 2000). However, we focus only on the plasticity portion of the MSU ISV model.

Results and Discussion

Bauschinger Effect

The Bauschinger effect can be determined from a forward-backward loading path. For example, tension followed by compression, or compression followed by tension. The Bauschinger test provided a few hysteresis loops at different strain amplitudes. Data is taken just from the first cycle.

The total hardening experienced by the material on the forward loading was 88 MPa, as shown in Figure 17. However, in the reverse loading, the material did not reach the previously hardened stress, thus not all the hardening was isotropic. The yield asymmetry showed the Bauschinger effect, and was captured in the ISV model by kinematic hardening. We calculated the isotropic and kinematic hardenings from Equations 17 and 18, where κ is isotropic hardening, α is kinematic hardening, σ_max is the max stress reached in the forward loading, σ_rev is the reverse yield, and σ_for is the forward yield. The total hardening at 1% strain was 65 MPa, with 8 MPa attributed to kinematic hardening.

Equation 17
Equation 18
Figure 17. A tension-compression cyclic experiment to test for the Bauschinger effect in cold-drawn 304L stainless steel. The forward (tension) yield is indicated by (a), the backwards (compression) yield by (b), and the stress at 1% strain representing the total hardening by (c).

MSU ISV Model Calibration

The MSU ISV model has varying levels of complexity depending on the amount of data available, ranging from a single stress-strain curve at a single rate and temperature with only isotropic hardening, to a fully rate and temperature dependent damage model. With a stress-strain curve and a Bauschinger test, the model was not rate and temperature dependent, but did distinguish between isotropic and kinematic hardening. Figure 18 shows the calibrated ISV model curve fitting the data from the monotonic tension test very well. The kinematic and isotropic hardening constants are balanced to reflect the respective hardening fractions found through a Bauschinger test. The model did not reflect the failure of the material, as no experimental damage data (e.g. porosity, pore number density) was used. The model constants for the ISV calibration are in Appendix B.

Figure 18. The MSU Internal State Variable (ISV) model calibrated to a quasistatic, room temperature monotonic tension tests of cold-drawn 304L stainless steel. Only the plasticity portion of the model was used.

MSF Model Calibration

Figure 19 shows the relaxation of the maximum stress response throughout each of the three 304L SS fatigue tests. The larger strain amplitude tests (0.5% and 1.0%) relaxed nearly immediately, while the smallest strain amplitude test maintained the maximum stress nearly a thousand cycles before beginning to relax. Though none of samples showed significant secondary hardening, the 0.5% strain amplitude test began to harden towards the very end of the fatigue life.

Figure 19. Maximum stress response throughout the fatigue life of 304L stainless steel under three different strain amplitude loadings; 0.3%, 0.5%, and 1.0%. All three samples showed significant cyclic softening early in the life.

We calibrated the MSF model in two steps. First, the Microstructurally Small Crack (MSC) regime was fit to the crack growth rate obtained from the striations on the fracture surface. Figure 20 shows the MSC model fit to the crack growth rate data. Due to difficulties imaging the striations close to the initiation site, much of our data is for crack lengths larger than 200 μm. Though we are modeling all cracks as MSCs, cracks larger than 200 μm could behave as long cracks and have a faster growth rate than predicted by the MSC model.

Figure 20. The microstructurally small crack model from the MultiStage Fatigue model calibrated to the crack growth rates obtained from the striation spacings on the fully reversed fatigue fracture surfaces of 304L stainless steel specimens.

Preceding calibration to the microstructurally small crack regime, the parameters obtained were then brought into the strain-life calibration. To augment the strain-life calibration data, we brought in data from the literature (Pegues 2016; Colin and Fatemi 2010). With the known parameters for the MSC regime, we were able to calibrate the incubation regime of the MSF model to accurately capture the total life. The model fit is shown in Figure 21, and illustrates both the incubation and MSC regimes from the final calibration. The uncertainty in the total fatigue life was propagated from the calibration of the MSC regime. A Monte Carlo analysis applying that uncertainty to the model constants generated the uncertainty band shown in Figure 21. The model constants for the MSF calibration are in Appendix A.

Figure 22 shows the contribution of the microstructurally small crack and incubation regimes to the total predicted fatigue life. In high cycle fatigue and lower strain amplitudes, incubation was a larger portion of the total life, while in low cycle fatigue, and higher strain amplitudes, crack growth was more dominant. For a cold-drawn material, we expect that crack growth would take the most cycles, which the model predicted for strain amplitudes greater than 0.5%. Below 0.5%, the MSF model predicted that incubation would dominate the fatigue life. To validate the model prediction, further experiments, such a replica test (Jordan et al., 2012), need to be performed below 0.5% strain amplitude.

Figure 21. The strain life MSF fit calibrated to data presented here, as well as data obtained in the literature (Colin and Fatemi, 2010; Pegues, 2016). The MSF fit uses the microstructurally small crack parameters obtained from calibrating to the crack growth data.
Figure 22. The percentage contribution of incubation (blue) and microstructurally small crack (orange) regimes to the predicted total fatigue life of 304L stainless steel.


The MSF model captures the fatigue behavior of cold-drawn 304L SS including some microstructural details observable on the fracture surfaces of the fatigue samples. From the study, we can conclude:

1. Cold-drawn 304L SS experienced a Bauschinger effect. In a tension-compression test going out to 1% strain, 13% of the hardening could be attributed to kinematic hardening, and the rest to isotropic hardening.

2. 304L SS experienced cyclic strain softening ranging from 50 to 125 MPa in all fatigue samples. The 0.5% strain amplitude sample began to display secondary hardening towards the end of the fatigue test.

3. Fatigue crack initiation was primarily through surface defects, though large particles and inclusions were also observed as potential initiation sites.

4. Crack growth in the fatigue samples was predominantly in the MSC regime, as evidenced by the rough fracture surfaces and debonded particles.


We would like to thank the laboratory staff at CAVS, especially Stephen Horstemeyer, for their help with the experiments and SEM. We would also like to thank Matt Cagle for performing all of the experiments presented here, and Reda El Alaoui for his help with the SEM analysis.


Abdel-Karim, M. “Effect of Elastic Modulus Variation During Plastic Deformation on Uniaxial and Multiaxial Ratcheting Simulations.” European Journal of Mechanics-A/Solids (2011): 11-21.

Bammann, Douglas J. "Modeling temperature and strain rate dependent large deformations of metals." Applied Mechanics Reviews 43.5S (1990): S312-S319.

Bammann, D. J., Chiesa, M. L., Horstemeyer, M. F., Weingarten, L.I. "Failure in ductile materials using finite element methods." Structural crashworthiness and failure, eds. T. Wierzbicki and N. Jones, Elsevier Applied Science, The Universities Press (Belfast) Ltd, 1993.

Bayerlein, M, H J Christ, and H Mughrabi. “Plasticity-Induced Martensitic Transformation During Cyclic Deformation of AISI 304L Stainless Steel.” Materials Science and Engineering: A (1989): L11-L16.

Bernard, J D et al. “Observations and Modeling of the Small Fatigue Crack Behavior of an Extruded AZ61 Magnesium Alloy.” International Journal of Fatigue (2013): 20-29.

Chaves, V, A Navarro, and C Madrigal. “Stage I Crack Directions Under in-Phase Axial–torsion Fatigue Loading for AISI 304L Stainless Steel.” International journal of fatigue 80 (2015): 10–21.

Coffin Jr, L. Fo. "A study of the effects of cyclic thermal stresses on a ductile metal." trans. ASME 76 (1954): 931-950.

Colin, J. “Deformation History and Load Sequence Effects on Cumulative Fatigue Damage and Life Predictions.” (2009), MS Thesis.

Colin, J., and A. Fatemi. "Variable amplitude cyclic deformation and fatigue behaviour of stainless steel 304L including step, periodic, and random loadings." Fatigue & Fracture of Engineering Materials & Structures 33.4 (2010): 205-220.

Colin, J, A Fatemi, and S Taheri. “Fatigue Behavior of Stainless Steel 304L Including Strain Hardening, Prestraining, and Mean Stress Effects.” Journal of Engineering Materials and Technology (2010): 021008.

Connors, William C. “Fatigue Striation Spacing Analysis.” Materials Characterization 33, no. 3 (1994): 245-253.

De Baglion, L, and J Mendez. “Low Cycle Fatigue Behavior of a Type 304L Austenitic Stainless Steel in Air or in Vacuum, at 20 C or at 300 C: Relative Effect of Strain Rate and Environment.” Procedia Engineering (2010): 2171-2179.

Gardner, Leroy. “The Use of Stainless Steel in Structures.” Progress in Structural Engineering and Materials 7.2 (2005): 45–55.

Hayhurst, D R, F A Leckie, and D L McDowell. “Damage Growth Under Nonproportional Loading.” Multiaxial fatigue (1985): 688-699.

Horstemeyer, M. F., et al. "Modeling stress state dependent damage evolution in a cast Al–Si–Mg aluminum alloy." Theoretical and applied fracture mechanics 33.1 (2000): 31-47.

Hughes, J M et al. “Cyclic Behavior and Modeling of Small Fatigue Cracks of a Polycarbonate Polymer.” International journal of fatigue 99.1 (2017): 78–86.

Jordon, James Brian, et al. "Damage and stress state influence on the Bauschinger effect in aluminum alloys." Mechanics of Materials 39.10 (2007): 920-931.

Jordon, J B et al. “Microstructural Inclusion Influence on Fatigue of a Cast A356 Aluminum Alloy.” … Materials Transactions A (2010): 356.

Jordon, J. B., J. D. Bernard, and J. C. Newman Jr. "Quantifying microstructurally small fatigue crack growth in an aluminum alloy using a silicon-rubber replica method." International Journal of Fatigue 36.1 (2012): 206-210.

Kalluri, S, G R Halford, and M A McGaw. “Prestraining and Its Influence on Subsequent Fatigue Life.” Advances in Fatigue Lifetime Predictive … (1996): 328-341.

Kang, G, Y Liu, and Z Li. “Experimental Study on Ratchetting-Fatigue Interaction of SS304 Stainless Steel in Uniaxial Cyclic Stressing.” Materials Science and Engineering: A (2006): 435-436.

Li, Yan. Fatigue crack initiation (in 304L steel): influence of the microstructure and variable amplitude loading. Other. Ecole Centrale Paris, 2012. English. <NNT: 2012ECAP0015>. <tel-00697002>.

Mann, Jonathan, et al. “A Comparison of the Numerical Simulation of Fatigue Crack in Standard Cylindrical Specimens with Experimental Values from Fatigue Endurance Testing of Nuclear Power Plant Materials.” Transactions, Structural Mechanics in Reactor Technology (SMiRT)-23.

Manson, S. S. "Fatigue: a complex subject—some simple approximations." Experimental mechanics 5.4 (1965): 193-226.

McDowell, D L et al. “Microstructure-Based Fatigue Modeling of Cast A356-T6 Alloy.” Engineering Fracture … (2003): 49-80.

Moyer, J M, and G S Ansell. “The Volume Expansion Accompanying the Martensite Transformation in Iron-Carbon Alloys.” Metallurgical Transactions A (1975): 1785.

Mughrabi, H, and H J Christ. “Cyclic Deformation and Fatigue of Selected Ferritic and Austenitic Steels: Specific Aspects.” Isij International (1997): 54-69.

Pegues, Jonathon. “Effect of Microstructure on the Fatigue Behavior of Type 304L Stainless Steel Including Mean Strain and Cyclic Rate Effects.” Dec. 2016.

Pegues, J W, M D Roach, and N Shamsaei. “Influence of Microstructure on Fatigue Crack Nucleation and Microstructurally Short Crack Growth of an Austenitic Stainless Steel.” Materials Science and Engineering … (2017): 657-667.

Roth, I et al. “Crack Initiation and Short Crack Growth in Metastable Austenitic Stainless Steel in the High Cycle Fatigue Regime.” Procedia Engineering (2010): 941-948.

Stephens, R I et al. (2000). Metal Fatigue in Engineering. 2nd ed. John Wiley & Sons, Inc.

Suresh, S. (1991). Fatigue of Materials. Cambridge University Press.

Taleb, L, and G Cailletaud. “An Updated Version of the Multimechanism Model for Cyclic Plasticity.” International Journal of Plasticity (2010): 859-874.

Taleb, L, and A Hauet. “Multiscale Experimental Investigations About the Cyclic Behavior of the 304L SS.” International Journal of Plasticity (2009): 1359-1385.

Vincent, L, JC Le Roux, and S Taheri. “On the High Cycle Fatigue Behavior of a Type 304L Stainless Steel at Room Temperature.” International Journal of fatigue (2012): 84-91.

Wöhler, A. "Wohler’s Experiments on the Strength of Metals." Engineering 4 (1867): 160-161.

Xue, Y et al. “Multistage Fatigue Modeling of Cast A356-T6 and A380-F Aluminum Alloys.” Metallurgical and Materials Transactions B 38.4 (2007): 601–606.

Appendix A - MSF Model Constants and Microstructure Parameters

Table A1: Incubation calibration constants for the MultiStage Fatigue (MSF) model for 304L stainless steel.
Table A2: Microstructurally Small Crack (MSC) calibration constants for the MultiStage Fatigue (MSF) model for 304L stainless steel.
Table A3: Microstructure model parameters for the MultiStage Fatigue (MSF) model for 304L stainless steel.

Appendix B - ISV Model Constants

Table B1. Calibration constants for the MSU Internal State Variable model for 304L stainless steel.
Personal tools

Material Models