Ratcheting performance of maraging steel fabricated via laser powder bed fusion: experimental evaluation and numerical prediction Open Access

Maraging steels are steel alloys with 18% of Ni, with low carbon content, based on the Fe-Ni-Co-Mo system, offering an excellent combination of high strength and high toughness for various engineering applications (high-precision moulding, specialised tools, aircraft landing gear, etc). The main features of maraging steels are good machinability, formability and weldability in the solution annealed condition, and high hardenability and strength after a suitable heat treatment. The high cycle fatigue (HCF) behaviour of conventionally manufactured (CM) maraging steels, including environmental effects, is discussed in the review paper of Rohit and Muktinutalapati (2021). The CM maraging steel 300 has been investigated in the past for its low cycle fatigue (LCF) characteristics after annealing (1 h at 820°C) and aging (3 h at 480°C) by Van Swam et al. (1975). Cyclic softening has been observed for both variants. Maraging steel 300 (referred to as MS300 in the sequel) is also an alloy widely used in the additive manufacturing (AM) method of laser powder bed fusion (LPBF) AM method, primarily due to its high strength, combined with its adaptability to the LPBF method. Past published research on the mechanical performance of LPBF maraging steel has been mainly focused on the monotonic (tensile/compressive) and HCF (>10⁴ cycles) behaviour, both for as-built and heat-treated alloys (Bai et al., 2017; Meneghetti et al., 2017, 2019; Branco et al., 2019; Damon et al., 2019; Mooney et al., 2019; Douellou et al., 2020; Tshabalala et al., 2021). The influence of heat treatment on the growth of fatigue cracks is reported by Santos et al. (2019).

On the contrary, the LCF (<10⁴ cycles) and the cyclic elastoplastic performance have attracted substantially less attention, as indicated by the few studies published so far (Branco et al., 2018, 2021; Henry et al., 2020; Mooney et al., 2020). In particular, the works of Branco et al. (2018) and Mooney et al. (2020) have examined the elastoplastic characteristics of this alloy under strain-controlled loading histories. While the Branco et al. (2021) paper has investigated multiaxial fatigue by also considering local cyclic plasticity effects, nevertheless, no published research exists on the response of this AM alloy under stress-controlled cyclic loading histories (Halama et al., 2022a). Thus, the phenomena resulting from such loading histories remain unexplored. This includes ratcheting, which is defined as the progressive accumulation of plastic strain during cycling. Generally, the ratcheting rate depends on the amplitude of stress and the mean stress. Cyclically softening materials usually show a progressive type of ratcheting (Hassan and Kyriakides, 1994). The ratcheting phenomenon has been studied extensively for CM metals, both in terms of experimental characterisation and, most importantly, prediction via constitutive modelling. Advances in the constitutive modelling of uniaxial and multiaxial ratcheting of steel alloys based on the Armstrong–Frederick (AF) (Armstrong and Frederick, 1966) nonlinear kinematic hardening rule include, for example, the work by Zhu et al. (2017), Zhou et al. (2018), Xing et al. (2019), Bemfica and Castro (2021) and Ohno et al. (2021). However, past work on steel alloys, either conventionally or additively manufactured, has been mostly focused on Stainless Steel 316 L (SS316 L) and a few on other steel alloys, i.e. Wang and Liu (2017), Zhang and Xuan (2017), Gordon et al. (2019), Lindström et al. (2020). Therefore, there is a gap in relation to ratcheting of LPBF maraging steel, even though this alloy is a very popular choice for this AM method (mainly due to the advantages offered by the adaptability of this material to the LPBF method and the versatility of its mechanical properties after simple heat treatment plans).

In phenomenological modelling of cyclic plasticity, a significant reduction of the number of material parameters has been achieved by implementing the AF nonlinear kinematic hardening rule into the distributed element model (Chiang, 2008), preserving the correct description of important phenomena of the cyclic plasticity (ratcheting, mean stress relaxation or memorisation). An evolution of the AF model is the component assembling model, which is based on a physical mechanism and was formulated in a small strain range by Deng et al. (2006, 2007) and Zhang et al. (2008). The component assembling model was extended to finite strains (Liu et al., 2010). This approach was considered promising due to the good predictions obtained for the subsequent yield surface evolution without the need for an explicit definition of the yield surface shape using mathematical formulas (Liu et al., 2011). Various mathematical models have also been developed to allow for the description of the evolution of the yield surface shape, for instance, by Feigenbaum Heidi and Dafalias Yannis (2008), Zhang et al. (2016), Welling et al. (2017), Parma et al. (2018), Zhang et al. (2020) and Kaneko (2021).

The ratcheting performance of LPBF MS300, to the best of our knowledge, has not been reported before in the open literature. Thus, our study is providing a new insight around this LPBF alloy, which enjoys wide use in the AM community. This paper offers an insight into the uniaxial ratcheting performance of LPBF maraging steel (MS300 in particular) by reporting the results and findings from an experimental stress-controlled test campaign conducted on AM (LPBF-fabricated) MS300 and on a set of CM MS300 test coupons (fabricated from a round bar of forged MS300 received in the annealed state). It is also noted that the anisotropic behaviour of the AM or CM material is not within the scope of this work, as only uniaxial cyclic tests were performed. The ratcheting rate is strongly dependent on the mean stress and the stress amplitude with a nonlinear tendency. It is challenging for models to capture (represent) well the effects resulting from different combinations of applied mean stress and stress amplitude. This is a motivation of our new work, to find a model which is capable to describe the mean stress and the amplitude stress effect in the ratcheting analysis correctly. In order to predict the ratcheting accumulation for the various loading histories, an improvement of the constitutive cyclic plasticity model of Abdel-Karim and Ohno (2000) has been developed, by adding a memory surface (Halama et al., 2015), calibrated and implemented for the AM MS300 test data obtained from the present study. Especially for the as-built material, the incremental tests with a small change of mean stress or stress amplitude were experimentally investigated and subsequently simulated by the finite element method to validate the abilities of proposed cyclic plasticity model. The ratcheting parameter of Abdel-Karim and Ohno model is newly dependent on the memory surface size and two useful functions are evaluated.

Test coupons were fabricated from AM and CM MS300. An EOS EOSINT M280 LPBF system was employed for the fabrication of the AM test coupons, with the manufacturing process controlled by the EOS “MS1 Performance 2.0” set of predefined parameters (prescribing a 40 μm layer thickness). An EOS gas-atomised MS1 powder feedstock from the past study of Mooney et al. (2019) was used, which has been characterised via energy dispersive x-ray (EDX) analysis. The CM test coupons were machined from a 0.75-inch diameter forged round bar solution annealed NiMark 300 maraging steel which complied with AMS6514H standard (SAE, 2012). The CM coupons’ axis aligned with the rolling direction. The chemical composition of both AM (powder) and CM materials is provided in Table 1.

The cyclic tests were performed with the use of cylindrical diameter MS300 test coupons complying with the dimensional requirements of the ASTM E606 standard (ASTM International, 2021) (drawing shown in Figure 1(a)). Three sets of MS300 test coupons were used for the cyclic test campaign:

• (1)

  Three coupons fabricated via AM at 90° relative to the build plate plane (along the z axis), as illustrated in Figure 1(b), denoted as AM90–1, AM90–2 and AM90-3,

• (2)

  Two coupons fabricated via AM at 0° relative to the build plate plane (along x axis), denoted as AM0-1 and AM0-2 (also illustrated in Figure 1(b)), and

• (3)

  Three coupons fabricated via CM, denoted as CM-1, CM-2 and CM-3.

A combination of post-processing processes was applied, both in terms of:

• (1)

  Heat treatment, namely age hardening at 490°C for 6 h followed by air cooling (which is the heat treatment plan recommended by the AM equipment manufacturer) and

• (2)

  Surface finishing, obtaining the ASTM E606 prescribed surface roughness of less than 0.2 μm (SAE, 2012).

The AM90 and CM set of MS300 coupons were heat treated and surface finished, while the AM0 were kept at their as-built condition. Table 2 summarises the postprocessing combinations for the AM and CM coupons (heat treatment and surface finishing).

The density and microstructure of the AM material used in the present study (AM90 and AM0 MS300 test coupons), both for in its as-built and heat treated (aged) condition, have been previously measured and characterised for the purposes of a study by Mooney et al. (2019). As such, the reader is being referred to the Mooney et al. work (Mooney et al., 2019) for more details on the microstructural aspects, as well as on the tensile behaviour of the alloy, since the objective and sole focus of the present work is to characterise, analyse and predict ratcheting performance of AM and CM MS300.

The mechanical testing plan comprised of a series of incremental (stepwise) cyclic stress-controlled tests at various combinations of ascending and descending stress amplitudes and mean stresses. This plan is detailed in Table 3 for the various types of test coupons (AM and CM). The objective of this complex campaign was to allow examining the influence of a variety of stress amplitudes and mean stresses on ratcheting of MS300, while keeping low the number of coupons required for that purpose.

The uniaxial cyclic loading tests were realised using a 100 kN/1,000 Nm LabControl electro-servo-hydraulic biaxial testing machine at VSB-Technical University of Ostrava. All tests were performed under room temperature in a load (force) controlled regime. The frequency of harmonic loading was 0.1 Hz. A low loading frequency was selected not to give rise to rate-dependent effects on the material. The LabNet.Workbench software was used for test setup, real-time force control and axial strain measurement. An EPSILON 3442 extensometer with a gauge length of 10 mm was used for axial strain measurement. Axial strain was captured also by Mercury RT optical system based on Digital Image Correlation (DIC) method containing two charge-coupled device (CCD) cameras with a 2.3-megapixel resolution. A probe of 10 mm length placed in the testing part of the coupon was considered for axial strain evaluation.

In this study, the ratcheting rate $εr$ is considered as the minimal value of plastic strain increment in the peaks of a hysteresis loop obtained in a block of loading (Hassan and Kyriakides, 1994), as shown schematically in Figure 2 (where $σa$ being the stress amplitude, and $σm$ the mean stress applied to the material).

The experimental investigation of heat-treated CM and AM MS300 has been performed under a stress ratio (⁠$R=σmin/σmax)$ ranging between −0.56 and −0.41. In particular, the first loading case employed corresponds to constant stress amplitude and constant mean stress for one AM90 and one CM test coupon, namely 1,300 and 500 MPa for AM90-1, and 1,333 and 513 MPa for CM-1, correspondingly. All other tests, for the remaining AM90, CM and AM0 test coupons, were incremental cyclic tests, with constant mean stress and increasing stress amplitude and vice versa. These two types of incremental tests were devised to examine the effect of mean stress and stress amplitude, for the various postprocessing combinations considered in this study. The incremental tests on as-built coupons were realised within the stress ratio interval from $R=$−0.5 to $R=−0.14.$ The full test plan is detailed in Table 3.

The CM MS300 was tested in the HT state only, due to the significant difference in the yield strength and the ultimate strength of AM0 and CM test coupons. As reported for SS316 L (Halama et al., 2019), the as-built material’s state may offer enhanced ratcheting performance in relation to the CM material, under the same loading conditions. As such, the AM90 and CM test coupons’ test plan was chosen to be the same to enable a comparison between these two sets of coupons (even though the two materials are not interchangeable for engineering applications purposes).

In terms of the test protocol/sequence between the different sets of coupons, the AM coupons (heat treated, AM90 and as-built, AM0) were tested at first in an interactive mode. The change of amplitude/mean value for subsequent loading block was set by the experimenter manually after the stabilisation of the strain response. Then, the same loading block definition was applied on the CM test coupons. By this way, the influence of mean stress and stress amplitude on the ratcheting rate could be assessed, and the data are further used for modelling and prediction purposes (as described in the sequel).

A summary of the achieved maximum ratcheting strain and cycles to failure (LCF) for each test coupon tested is found in Table 4. Focusing on the coupons tested under the same loading cases (CM and AM90), it is observed that the ratcheting strain at failure is slightly higher for the AM90 test coupons, namely AM90-1 and AM90-3 vs CM-1 and CM-3 test coupons, which was +33.6 and +8.8% correspondingly. An exception is noted for the case of AM90-2 in comparison to CM-2, where a −126% difference was record. The LCF life values of CM MS300 are substantially higher than those of AM MS300, ranging between +147 and +492%, for the CM versus the AM90 test coupons.

On the other hand, it is noticed that the AM0 (as-built/non-surface finished) coupons exhibited high ratcheting strain values (∼10.30%), recording a modest LCF life (92 and 372 cycles until failure). Due to the different load cases applied, a comparison between AM0 and AM90 or CM material is neither possible nor desired, as this escapes the objective of this work.

The performance of the AM material has been found to be inferior to that of the CM material. This is contrary to the findings of Halama et al. (2019) for another AM steel alloy (SS316 L). Nevertheless, the overall ratcheting performance of the AM material (both as-built and heat treated) may be acceptable for engineering applications where complex geometry parts (i.e. not otherwise manufactured via other methods rather than AM) might be required to operate at such high stress levels.

The experimental results for the heat-treated AM and CM MS300 are presented and discussed in detail in Section 3.1, followed by the results for the as-built AM MS300 in Section 3.2.

The results from the constant stress amplitude – constant means stress loading case for the AM90 and CM test coupons – are presented and discussed in Section 3.1.1. Moreover, the incremental (stepwise) test results, for the mean stress variation are presented in Section 3.1.2 and that from the variation of stress amplitude in Section 3.1.3.

A cyclic test under a stress amplitude of 1,300 MPa and a mean stress of 500 MPa was performed for the AM90 material (and at 1,333 MPa stress amplitude, 513 MPa mean stress for the CM material) to demonstrate the ratcheting behaviour under the constant amplitude as well as the mean stress effect on ratcheting. The hysteresis loops obtained during the initial (first) cycles are illustrated in Figure 3(a) and (b). Slightly higher micro-plasticity is visible for the AM material than in the case of the CM material, as per the differing size observed for the stress–strain hysteresis loops. This is also evidenced by the accumulation of axial peak strain with the number of cycles, which is represented in the graph of Figure 3(c), with the CM material exhibiting a substantially higher plastic strain accumulation capacity at a lower ratcheting rate.

The incremental (stepwise) cyclic test explored the influence of the stress amplitude on the ratcheting performance of the heat treated MS300 (both AM and CM). The stress amplitude was incrementally increased from 1,200 to 1,450 MPa in six steps, maintaining the mean stress constant at 500 MPa. The stress–strain hysteresis loops obtained for the CM-2 test coupon are shown in Figure 4(a). Unfortunately, the axial extensometer slid over the surface for the AM90-2 coupon, thus the hysteresis loop’ data were compromised, and these results cannot be reported. The ratcheting curves (axial strain versus cycles) for both materials are presented in Figure 4(b). It is noted that in the case of AM90-2 test (impacted by the extensometer mishap), the axial strain data were measured via the DIC method. It is observed that the critical value of axial strain is higher for the CM material than that for the AM material. Of note is, however, that both materials had similar ratcheting rate accumulation until the failure of the AM material, with the AM material exhibiting only slightly lower rate (as observed in Figure 4(b)).

The varying value of the ratcheting rate $εr$ as a function of stress amplitude $σa$ can be approximated via an empirical model devised for that purpose. The simple model is described by Equation (1).

where $εr$ is the ratcheting rate, $σa$ is the stress amplitude, $εr0$ is a parameter controlling magnitude, $σ∞a$ is a parameter controlling the asymptote and $ca$ is a parameter controlling curvature.

Equation (1) was calibrated with the use of the AM90-2 experimental data resulting in the parameters $εr0=$ 2.196E−08, $σ∞a=1461MPa$ and $ca=2.42$⁠. The experimental versus predicted data are illustrated in Figure 5.

As non-zero mean stress is the cause of ratcheting, cyclic testing to investigate the mean stress influence on ratcheting behaviour follows, for both sets of heat-treated AM and CM MS300, namely the AM90-3 and CM-3 test coupons. In this cyclic test, the mean stress was incrementally increased from 400 MPa to 550 MPa in four steps, keeping constant the stress amplitude at 1400 MPa. The obtained stress–strain hysteresis loops results are presented in Figure 6(a) and (b).

A high ratcheting strain rate for the AM test coupon is noticeable from the first block of loading in the hysteresis loops (Figure 6(a) and (b)), as well as from ratcheting curves (Figure 6(c)). It is noticed that the critical value of accumulated axial strain is comparable for AM and CM test coupons. The CM material, on the other hand, exhibited steady state ratcheting across many cycles, before the final section of high ratcheting rate (immediately before failure), as observed in Figure 6(c).

Similarly, the empirical model described by Equation (1) has been adapted and applied for the case of mean stress variation. This is described mathematically by Equation (2).

where $εr$ is the ratcheting rate, $σm$ is the mean stress, $σ∞m$ is the parameter controlling the asymptote, $cm$ is the parameter controlling curvature.

Equation (2) was calibrated with the use of the experimental data for AM90-3 resulting in the parameters $εr0=$ 6.578E−06, $σ∞m=655 MPa$ and $cm=1.84$⁠. The experimental versus predicted data are illustrated in Figure 7.

The AM test coupons’ experiments were performed for as-built (non-heat treated, non-surface finished) AM MS300 fabricated at 0° build orientation (AM0). Similarly, as for the heat-treated material, the influence of mean stress and stress amplitude on ratcheting performance was investigated via incremental tests, with the test results from the variation of mean stress presented in Section 3.2.1 and that from the variation of stress amplitude in Section 3.2.2. It is also noted that there is no comparison between the AM and the CM coupons, as the latter have been surface-finished.

The evaluation of the mean stress effect on ratcheting was performed on the coupon AM0-1 by applying a stress amplitude of 700 MPa through an incremental (stepwise) test starting from mean stress of 300 MPa. In each loading block, the mean stress was increased by 25 MPa and then retained constant for the subsequent 20 cycles. Substantial plastic strain has been accumulated in the initial loading cycle, as observed in the axial stress–axial strain results of Figure 8(a). A transient behaviour appears in the subsequent cycles, leading to a stabilisation of the ratcheting rate after 20 cycles, as illustrated in the ratcheting curve of Figure 8(b). It should be noted that peak strains from the last two loading blocks are not included in the ratcheting curve, because the corresponding maximal stress was not controlled sufficiently due to the necking of the test coupon. For the last loading block only one cycle was completed. The dependency of strain rate on mean stress is almost linear up to 425 MPa and then it exhibits a very fast pace of increase. Equation (2) empirical model with parameters $σ∞m=541 MPa$ and $cm=2.1$ matches well the experimental data, as shown in Figure 9.

For the testing of coupon AM0-2, the stress amplitude was increased from 600 to 900 MPa in 12 steps with an increment of 25 MPa and by keeping the mean stress constant at 300 MPa, see Figure 10(a) stress–strain hysteresis loops curves. A stabilisation of the strain response required approximately 100 cycles to be achieved. After completing 140 cycles, 20 cycles for each increment of load were realised, except for the last loading block, which was performed until the occurrence of failure (Figure 10(a)). Plastic shakedown was observed for stress amplitude 600 and 625 MPa, as evidenced from the ratcheting curve shown in Figure 10(b). Following that, the dependency of ratcheting rate on the stress amplitude is observed to be almost linear up to 825 MPa, after which a very fast growth is experienced (Figure 10(b)). Similarly, to the other cases examined, the ratcheting rate dependence from strain amplitude can be described by Equation (1) empirical model using the parameters $σ∞a=930 MPa$ and $ca=2.1$⁠, with the obtained results presented in Figure 11.

This section describes the mathematical model employed for the prediction of ratcheting response of the material of various AM and CM test coupons. Since the (AM and CM) material reveals a negligible sensitivity to strain rate, the time-independent theory of plasticity is considered for numerical modelling and simulation. The concept of internal variables with the application of differential equations, which is a standard in phenomenological modelling, was applied using a single yield surface (Lemaitre and Chaboche, 1990). The constitutive model used is the Modified Abdel-Karim and Ohno kinematic hardening rule with Memory surface (MAKOM), which was developed earlier for railway materials applications (Halama et al., 2015), is applied here for AM MS300 for the first time. The two proposed variants of the cyclic plasticity model were implemented in ANSYS using the USERMAT.F subroutine according to the Halama et al. paper (Halama et al., 2015).

All experimental loading cases are uniaxial in this study; however, the constitutive equations will be written in their multiaxial (tensorial) format. The pure kinematic hardening model is based on the isotropic yield condition according to von Mises (since the anisotropic behaviour of the material is not within the scope of this work), i.e.

where the deviatoric parts of stress tensor $σ$ and back-stress $α$ are denoted as $s$ and $a$⁠, respectively. The contraction operator “:” is defined using Einstein’s summing rule, i.e. $aij=b:c=bijcij$⁠. The back-stress $α$ defines the position of the yield surface with its fixed size of $σY$⁠. The total strain tensor $ε$ is composed of the elastic $εe$⁠, and the plastic $εp$ part as specified in the additive rule:

The stress tensor can be obtained from the Hooke’s law following the elastic strain:

where $D$ is the elastic stiffness 4-th order tensor. The symbol: means the contraction, which can be defined by Einstein’s summing rule as $aij=B:c=Bijklckl$⁠. The plastic strain increment calculation is based on associative plasticity, thus the normality rule is used in this form:

where the accumulated plastic strain increment $dp$ is stated as an equivalent plastic strain increment, thus:

The superposition of several back-stress parts

is considered according to Chaboche et al. (1979), where $M$ is the number of back-stresses having their own kinematic hardening evolution equation of the AF type (Armstrong and Frederick, 1966). Eight back-stress parts are used in this study (⁠$M=8$⁠).

The MAKOM model is based on the Abdel-Karim and Ohno kinematic hardening rule (Abdel-Karim and Ohno, 2000) with the evolution equation for the $i$-th back-stress part described by:

where $Ci$⁠, $γi$ are basic material parameters and $μi$ is the ratcheting parameter, which is considered to be the same for all back stress parts in this study, i.e. $μi=μ$⁠.

Cyclic hardening of the material is described by Marquis evolution rule:

where $φ∞$ and $ωφ$ are material parameters.

The memory surface is introduced in the model in the principal stress space (Jiang and Sehitoglu, 1996) considering the following scalar function:

where R_M is the radius of memory surface. Its evolution equation is given by

where $cM$ influences the rate of contraction of the memory surface. No contraction is considered in this study for simplicity.

We consider two variants of the MAKOM model in this paper with emphasis on the ratcheting modelling. The first variant is based on the work of Halama et al. (2015). The material parameter influencing the ratcheting rate $μ$ is given by:

In the last term of Equation (14), there are four material parameters $μ01$⁠, $μ∞1$⁠, $ω1$ and $ω2$⁠. The other two parameters are dependent on the memory surface size $RM$ based on an exponential dependency:

where $A0$⁠, $B0$⁠, $A∞$ and $B∞$ are material parameters. The parameters $A0$ and $A∞$ are unitless, while the units for $B0$ and $B∞$ are MPa^-1. Note that Equation (15) is based on the exponential function, with this version named MAKOM–EXPONENTIAL.

The second variant of MAKOM model is based on the empirical model described by Equations (1) and (2), used in the analysis of the ratcheting data of MS300. This variant of will be named MAKOM–ASYMPTOTIC. Consistently with the empirical model equations Equations (1) and (2), applied to capture ratcheting rate, and with Equation (14), the ratcheting parameters are given by:

where $σ∞$⁠, $M0,M∞,c0$ and $c∞$ are material parameters. The $M0$⁠, $M∞,c0$ and $c∞$ parameters are unitless.

It is important to note that for both variants of MAKOM model, the value of $μ$ has to be inside the range 0 ≤ $μ$ ≤ 1. That is why the rule $μ$ ≤ 1 must be applied within the implementation of the constitutive theory (Halama et al., 2022a, b). In other words, when the value of $μ$ calculated by means of Equation (14) is higher than 1, then the Chaboche et al. model (Chaboche et al., 1979) is considered.

All experiments for CM material and both variants of AM MS300 (heat-treated and as-built) were simulated via the finite element method (FEM) in ANSYS 2020R1. The constitutive equations were implemented using the algorithm proposed by Kobayashi and Ohno (2002). The FEM results (model predictions) are directly compared with the experimental data obtained from the tests described in Section 3 of the paper. Ratcheting predictions for CM, heat treated AM (AM90) and as-built AM MS300 are presented and discussed in detail in Sections 4.3.1, 4.3.2 and 4.3.3 correspondingly.

In the case of the (heat treated) CM MS300, the elastic isotropic material constants are Elasticity modulus E = 180 GPa and Poisson ratio ν = 0.3. The material parameters for the MAKOM model are provided in Table 5. The basic material parameters $Ci$⁠, $γi$ were identified from a saturated hysteresis loop and $φ∞$⁠, $ωφ$ were calibrated to capture the monotonic stress–strain curve and cyclic hardening respectively.

The material parameters influencing ratcheting are listed in Table 6. The calibration of material parameters $A0$⁠, $B0$⁠, $A∞$⁠, $B∞$⁠, $ω1$ and $σ∞$⁠, $c∞$ was performed from a comparison of the experimental ratcheting data of the CM-2 case and its prediction with a constant value of the ratcheting parameter $μ$ = 0.05 (pure Abdel-Karim and Ohno model) as described in Halama et al. (2022b). The parameters $μ01$⁠, $μ∞1$⁠, $ω1$ were calibrated from the CM-3 experiment to describe the initial ratcheting rate more accurately.

The number of cycles in the CM material experimental cases are very high. To allow for a better representation of the differences between predictions by the two variants of the MAKOM model results, the prediction results are presented at the half-life of the test. After that, the responses of both material models were saturated. The first prediction results are presented for the incremental ratcheting test with variation of stress amplitude (CM-2 coupon). As shown in Figure 12(a), both variants of MAKOM model underpredict the ratcheting rate in the first two blocks of loading leading to the same response. After that phase, the MAKOM–EXPONENTIAL model predicts higher accumulation of plastic strain. For a comparison, the response of the original Abdel-Karim and Ohno model is also shown, whereas the value of ratcheting parameter used in the simulation is 0.05.

A good description of ratcheting rate for block loading, especially after stabilisation of the response, is observed in the results of Figure 12(b), where the numerical predictions by two variants of the MAKOM model are presented along with the experimental data of the CM-3 coupon. The MAKOM–ASYMPTOTIC model offers a slightly less conservative prediction than that of the MAKOM–EXPONENTIAL.

The last experimental case for the CM-1 coupon considered in the study is for the constant stress amplitude and mean stress loading case. The ratcheting predictions obtained by both variants of MAKOM model are shown in Figure 12(c). There is a significant underprediction of ratcheting by both variants of the MAKOM model. It is also noted that the MAKOM material model leads to non-conservative predictions for small stress amplitudes in all three experimental cases (see Figure 12(a)–(c)).

For the heat-treated AM MS300 (AM90), the following elastic isotropic material constants were used: E = 185 GPa and ν = 0.3. The material parameters of the MAKOM model, which are the same for both variants, are provided in Table 7. The basic material parameters $Ci$⁠, $γi$ were identified from a saturated hysteresis loop and $φ∞$⁠, $ωφ$ to capture monotonic stress–strain curve and cyclic hardening respectively.

The material parameters influencing ratcheting are stated in Table 8. The calibration of material parameters influencing ratcheting parameter evolution were identified by the same way as in the CM MS300 case, see Section 4.3.1.

First, we consider the prediction results for the loading case of AM90-2, which is composed of loading blocks with constant mean stress of 500 MPa and increasing stress amplitude (as described in Section 3.1.2). A conservative prediction is produced by the MAKOM–EXPONENTIAL model, as presented in Figure 13(a). In particular, there is an over-prediction of ratcheting in all blocks of loading; nevertheless, the steady ratcheting rate is predicted well for the first two loading blocks. The ratcheting predictions of the original Abdel-Karim and Ohno model (which is used for the calibration of the MAKOM) is also shown in Figure 13(a). A prediction with constant value of the ratcheting parameter $μ$ = 0.05 offers the most accurate strain rate for the fourth loading block. In the previous loading blocks, the original Abdel-Karim and Ohno model predicts high ratcheting rates, while for the next block, a low ratcheting rate is obtained. The MAKOM–ASYMPTOTIC variant model predicts higher ratcheting rates than those observed experimentally in first four loading blocks, with an improvement noticed for the last two loading blocks.

The second loading case (coupon AM90-3) simulations show the influence of mean stress on ratcheting (the test with increasing mean stress is described in Section 3.1.3). The axial strain accumulation predicted by both variants of the MAKOM model in first three loading blocks is less than that observed in the, see Figure 13(b). The MAKOM–ASYMPTOTIC model offers a better prediction of ratcheting rate in the last loading block.

The third prediction results are presented for the case with a constant amplitude of 1300 MPa and a mean stress of 500 MPa, corresponding to coupon AM90-1. As presented in Figure 13(c), the ratcheting curve obtained by the MAKOM–EXPONENTIAL mode underpredicts the experimental data. The steady ratcheting rate prediction shows a good agreement in this case. For the AM90 (heat treated) MS300, we focused on the prediction of steady ratcheting by both model variants and not on the transient part, because the features of the MAKOM–ASYMPTOTIC variant, as a new variant, would be interesting to evaluate in this case (since it has not been applied in other materials also). It is noticed that the MAKOM – ASYMPTOTIC variant leads to slightly higher ratcheting rate predictions in steady state than that observed experimentally. However, the ratcheting curve is still underpredicted.

Improving further the ratcheting predictions for heat treated MS300 (CM and AM) has proven to be challenging, since the plastic strain amplitude is very low for all loading blocks. On the other hand, interesting differences have been discovered between the ratcheting rate values resulting from different tests (and coupons) which have the same combination of stress amplitude and mean stress (σ_a, σ_m). If one refers to Table 3 load cases, two such (σ_a, σ_m) combinations come up: (1,300 MPa, 500 MPa) and (1,400 MPa, 500 MPa). The ratcheting rate following the stabilisation of each material’s response has been calculated for this case and it is provided in Table 9. Similarly, Table 9 also lists the ratcheting rates for the CM tests/coupons’ cases.

From Table 9 data, it is observed that the ratcheting rates in AM90-2 case are significantly lower than in AM90-1 and AM90-3 cases under equivalent loading (stress amplitude, mean stress) conditions. This behaviour cannot be captured by the cyclic plasticity model. These differences may be attributed to factors which are not accounted by the modelling approach, such as the concentration of pores, non-homogenous properties and perhaps anisotropy associated with the vertical fabrication of the AM90 coupons. For example, the distance from the build plate and the positioning may affect mechanical properties of the produced part, which in this case is a test coupon. Minor variations in the yield strength or the elasticity modulus may have a considerable role in the strain response of the test coupon when subjected to cyclic stress-controlled loading histories (such as in the case of ratcheting). This hypothesis is further supported by the fact that for the case of CM coupons the differences between the ratcheting rates resulting from equivalent combinations of stress amplitude and mean stress are minimal or less profound than that for the AM90 cases.

Both variants of the MAKOM model were also applied for as-built AM MS300 without heat treatment (AM0). The elastic isotropic material constants were: E = 170 GPa and ν = 0.3. The material parameters of the MAKOM model (two variants) are listed in Tables 10 and 11.

The test on the coupon AM0-1 with a constant stress amplitude of 700 MPa and varying mean stress was used for the calibration of the ratcheting parameters based on the prediction of Abdel-Karim and Ohno model with $μi$ = 0.1. The achieved predictions of stress–strain hysteresis loops and ratcheting evolution are shown in Figures 14(a) and (b) and 15(a), (b) and (c), respectively. It is observed that the predictions of the MAKOM–EXPONENTIAL model matches well the experimental data. While in the case of the MAKOM–ASYMPTOTIC model, the prediction of ratcheting rate is further improved for some loading blocks.

The experimental and model prediction results for the second loading case, for AM0-2, representing the stress amplitude effect (σ_m = 300 MPa), are presented again in the form of stress -strain hysteresis loops (Figure 14(c) and (d)), as well as the ratcheting curve (Figure 15(d), (e) and (f)). Based on these data, it is considered that both variants of the MAKOM model are capable to describe the amplitude and mean stress effect well. It should be noted that the hysteresis loops from the last loading block are not shown in Figure 15(c) and (f).

Based on the experience from both numerical studies, it can be summarised that the MAKOM–ASYMPTOTIC variant offers better overall prediction capabilities. The MAKOM–ASYMPTOTIC model is more conservative than the MAKOM–EXPONENTIAL model, and for structural integrity management purposes, it may be a better choice for ratcheting prediction (i.e. provide estimates on when the back-stress reaches the limit value of $σ∞$⁠).

Both the CM and the AM MS300 material tested under cyclic stress-controlled (non-zero mean stress) loading has exhibited steady ratcheting after the stabilisation of the response. The AM test coupons exhibited an LCF life of about one-third that of their CM counterparts. However, as presented in the incremental ratcheting test results evaluating the influence of stress amplitude (under a constant mean stress $σm$ = 500 MPa), the tested AM MS300 has experienced only slightly lower ratcheting rate for stress amplitudes below $σa$ = 1,300 MPa than that of the CM coupon. The (other two) loading cases evaluated the mean stress effect on the ratcheting behaviour of the heat-treated material and offered useful data for constitutive modelling purposes. On the other hand, significant differences have been found in the ratcheting rates of the various coupons tested, which have provided further insights on the materials’ response. For all loading cases, the CM variant offers a longer steady phase of the ratcheting curve. However, the ratcheting curve of AM MS300 (with and without heat treatment) shows still enough number of cycles in contrast to AM SS316 L, where the decay of the ratcheting rate is directly followed by the progressive increase of the plastic strain accumulation (Halama et al., 2022a).

The two incremental cyclic plasticity tests performed on the as-built AM MS300 coupons (not heat treated or surface finished) were focused on the mean stress and stress amplitude effect on ratcheting. Relatively small increments of mean stress and stress amplitude were considered in the experiments. The obtained ratcheting data extends our understanding around the cyclic stress–strain behaviour of the AM MS300, with a focus on stress-controlled loading, beyond what has been published in the past, i.e. by Mooney et al. (2020).

The experimental data sets obtained for AM MS300 were used in FEM simulation, to allow ratcheting predictions. Two variants of a constitutive plasticity model with a memory surface (MAKOM model) have been used for that purpose. The MAKOM model has previously shown the capacity to capture the stress amplitude effect, which is often problematic for classic cyclic plasticity models based on the Armstrong and Frederick kinematic hardening rule. The mean stress and stress amplitude effect is treated by incorporating the Jiang and Sehitoglu memory surface (Jiang and Sehitoglu, 1996). In the Abdel-Karim and Ohno kinematic hardening rule, the ratcheting parameter is dependent on the size of memory surface, whereas in this case (AM MS300), two functions/variants were evaluated: MAKOM–EXPONENTIAL and MAKOM–ASYMPTOTIC. Based on the prediction results obtained for as-built AM MS300, it was found that the MAKOM–ASYMPTOTIC variant can offer a good description of the complex ratcheting behaviour. It is also noted that the proposed model (either MAKOM–EXPONENTIAL or ASYMPTOTIC) requires a trial simulation by the original Abdel-Karim and Ohno model in order to be calibrated properly (Halama et al., 2022a, b).

Both the experimental results on ratcheting performance, as well as the proposed MAKOM model, tailored for this specific AM alloy (MS300), are expected to be valuable for other researchers and engineers working on structural integrity.

In a further study, we will focus on evaluating more complex loading cases to verify the suitability of both variants of the MAKOM model for other AM alloys (i.e. SS316 L and Ti-6Al-4V). Similarly, the test plans will focus on revealing transient effects occurring from the variation of stress amplitude and mean stress, by employing incremental (stepwise testing), as this has proven useful in the case of AM MS300.

This work was supported financially by the Grant Agency of the Czech Republic (GACR), Project no. GA23-04724S, and has been performed in connection with the DMS project reg. no. CZ.02.1.01/0.0/17_049/0008407 financed by Structural Funds of the European Union and with the Specific Research Project (SP2024/37) supported by the Ministry of Education, Youth and Sports of the Czech Republic and the Faculty of Mechanical Engineering VSB-TUO, and the Faculty of Science and Engineering of the University of Limerick. Moreover, the support of the South Eastern Applied Materials (SEAM) Research Centre of the South East Technological University (SETU), formerly Waterford Institute of Technology (WIT), in kindly providing the test coupons is acknowledged.