High cycle fatigue behaviour of Invar 36 alloy fabricated by laser powder bed fusion

ABSTRACT The Invar 36 alloy was additively manufactured by laser powder bed fusion (PBF-LB), and systematical observations and experiments for microstructure, defects, metallography, especially high cycle fatigue behaviour and fractography were conducted. Inadequate laser energy density results in hardly overlapping melting traces, generating numerous defects. Accordingly, the fabricated Invar 36 alloy presents an inferior high cycle fatigue life, as it failures from the rapid aggregation of the defects. In contrast, an adequate laser energy density remarkably enlarges the overlapping between adjacent melting traces. The large molten pools with steady boundaries are beneficially to generate favourable microstructures and low porosity. Consequently, the Invar 36 alloy shows superior high cycle fatigue life, completely generated from small crack propagation, long crack propagation and final fracture stages. Above experimental results and analysis primarily link up the PBF-LB process, microstructures (defects) and high cycle fatigue performance for PBF-LB Invar 36 alloy.


Introduction
Invar 36, a typical Fe-Ni-based binary alloy, mainly composited of face-centred cubic (fcc) austenite, is well notable for its unique low coefficient of thermal expansion (CTE) < 1.5-2.0ppm/°C (under its Curie temperature around 230°C) (Qiu, Adkins, and Attallah 2015), only about 1/10-1/7 compare to the CTE of the austenitic stainless steel.This extremely low CTE is known as the INVAR effect, when the content of the element nickel approaches to 36 wt%, first discovered by Guillaume (Wohlfarth 1975).Besides, it also features stable mechanical properties, e.g. the elastic modulus and the microstructure of the fcc austenite without phase change over a wide range of temperature.Accordingly, Invar 36 alloy is widely used in engineering applications, where high dimensional stability is strictly required.Typical applications include optical mounting, valves in aero-engine and controlling devices in aerospace industry, precision instruments, shadow masks, ring laser gyroscopes, antimagnetic watches, composites moulds (Qiu, Adkins, and Attallah 2015;Zhan et al. 2018;Li et al. 2020;Liu et al. 2021;Tan et al. 2020;Zhan et al. 2019) and LNG (Liquefied Natural Gas) carrier (Chen et al. 2019).
Generally, Invar 36 alloy engineering components are conventionally processed by machining from wrought workpieces.However, the high ductility, owing to the single austenite phase and low heat conductivity, make Invar 36 alloy soft and gummy (Kaladhar, Subbaiah, and Rao 2012), undesirably leading to low production efficiency (Jasthi, Arbegast, and Howard 2009;Mills and Redford 2013;Kim et al. 2005) to especially machine the components with complicated geometry and architectures.Fortunately, considering its good weldability, recently, the rapid development of the additive manufacturing processes, especially the laser powder bed fusion (PBF-LB), provides a quite promising technique to fabricate complex geometry featured Invar 36 alloy components.In detail, Qiu, Adkins, and Attallah ( 2015) first initiated the PBF-LB process for Invar 36 alloy with varied scanning speeds, and high densification, considerable tensile strength and low CTE were experimentally verified.Particularly, the influences of TiAl (Qiu, Liu, and Liu 2020) and pure Ti (Liu et al. 2022) powder particles on the microstructural and mechanical properties in PBF-LB Invar 36 alloy were also explored.Subsequently, Yakout and Elbestawi (2020) also conducted a series of work to establish the correlations and to reveal the underlying mechanisms among the key PBF-LB process parameters, microstructures, defects and macroscopical performances, i.e. density, CTE, tensile strength, residual stress, etc. (Yakout et al. 2017(Yakout et al. , 2020;;Yakout, Elbestawi, and Veldhuis 2018;Yakout, Elbestawi, and Veldhuis 2019).The influences of varied laser power on microstructures, defects and CTE for the PBF-LB Invar 36 alloy were also revealed by Asgari et al. (2018), and Harrison, Todd, and Mumtaz (2017) mainly focused on experimentally identifying the low and even negative CTE owing to the PBF-LB induced residual stress affected by the laser energy density.The density and surface roughness characteristics were also explored by Khanna et al. (2019), and the densified Invar 36 alloy, featured by its microstructures, tensile strength and hardness were reported by Wegener et al. (2020).The PBF-LB process correlated density, hardness, defect evolution, microstructures, particularly, high-temperature mechanical performance up to 600°C were also experimentally established in our previous works (Wei et al. 2020;Yang et al. 2020).Additionally, corrosion and building orientation dependence of the mechanical performance were revealed by Heine et al. (2022), and Rishmawi et al. (2022).mainly focused on the influence of heat treatments and build orientation on the CTE and tensile behaviour.
Despite that a majority of the process, microstructures and macroscopical performance were well experimentally identified in the above-reported literature, the fatigue behaviour and corresponding failure mechanism under cyclic loading are not well explored and understood for PBF-LB Invar 36 alloy, except that only low cycle fatigue tests for samples, fabricated under only one group of PBF-LB process parameters, was conducted by Wegener et al. (2020).The significant enhancement of the high cycle fatigue life via equal-channel angular pressing (ECAP) for the conventionally processed ultra-fine grain Invar 36 alloy (Vinogradov, Hashimoto, and Kopylov 2003) suggests that the microstructures will remarkably influence the fatigue performance.Differently, it is widely verified that the porosity is inevitably presented in the PBF-LB process, and is particularly detrimental to fatigue resistance, specially the high cycle fatigue resistance of the PBF-LB alloys, hence, the high cycle fatigue behaviour of the PBF-LB Invar 36 alloy still requires a comprehensive exploration.Actually, as an important reference, it should be noted that, in view of the PBF-LB manufactured 316L austenite stainless steel, which is exactly mainly composed of the austenite phase identical to that in Invar 36 alloy.And it is firmly identified that the PBF-LB process correlated microstructures and especially the defects, e.g. the lack of fusion pores, unmelted powders, give rise to pronouncedly influence, usually detrimentally, on its fatigue behaviour, weakening its fatigue resistance (Leuders et al. 2014, Cui et al. 2023).For instance, experimental results and fatigue life prediction model both indicate that over-melting and undermelting in the PBF-LB austenite stainless steel 316L could lead to porosity-driven cracking and inferior fatigue resistance under cyclic loading (Zhang et al. 2017a(Zhang et al. , 2017b(Zhang et al. , 2018)).Besides, it is revealed that the low fatigue resistance is mainly owing to large internal defects instead of other secondary factors (Yadollahi et al. 2015;Shrestha et al. 2016;Shrestha, Simsiriwong, and Shamsaei 2019).Otherwise, the fine sub-grained cellular microstructures result in excellent fatigue performance (Elangeswaran et al. 2019;Elangeswaran et al. 2020).It is suggested that the microstructure features exert only a slight influence on the fatigue crack growth and fracture toughness (Suryawanshi, Prashanth, and Ramamurty 2017;Riemer et al. 2014), and the PBF-LB induced columnar grains are minimally misoriented and coarse enough to induce crack deflection (Kumar et al. 2020), compared with those fabricated by binder jet printing (BJP).
Above-reported results and conclusions for austenite stainless steel manufactured by PBF-LB indeed explicitly present a close correlation containing process-microstructure-fatigue performance, and comparably, also strongly suggest that the fatigue behaviour of PBF-LB Invar 36 alloy should be carefully explored.However, the fatigue performances, especially under the high cyclic loading, as more essential to practical engineering applications, have not been explored yet.
Accordingly, motivated by originally addressing high cycle fatigue issues for PBF-LB Invar 36 alloy, the main works in following sections include: (1) As the important basis, the microstructures, defects, density and tensile performance, which are correlated to the different PBF-LB process parameters, are experimentally identified.(2) High cycle fatigue tests are performed.Particularly, the PBF-LB process-related porosity forming mechanism and microstructure features are revealed to understand the correlation between the PBF-LB process and fatigue behaviour, i.e. fatigue life, fracture mode and failure mechanism.(3) Based on the reported models (Kumar and Ramamurty 2020;Romano et al. 2018), a fatigue model, effectively considering the PBF-LB induced microstructure and defect features, is established to predict the high cycle fatigue strength.Eventually, above three aspects primarily link up the PBF-LB process, microstructures (defects) and high cycle fatigue performance.The contribution of this work provides an experimental foundation to assess the structural integrity under cyclic loads for PBF-LB Invar 36 alloy, as it is of great significance to its practical fatigue-sensitive engineering applications such as aerospace industry.

Laser powder bed fusion
In the present study, Invar 36 alloy was additively manufactured by PBF-LB (Renishaw AM250 with an ytterbium fibre laser), providing samples for the microstructure observations and mechanical, especially fatigue tests.The gas-atomised Invar 36 powder was provided by AMC powders corporation (Beijing, China).Our previous powder size distribution test suggests that the powder size ranges in 14-53 μm, and the corresponding powder diameters at 10%, 50% and 90% of the cumulative volumes of the particles are 14.4,29.7 and 52.0 μm, respectively (Wei et al. 2020;Yang et al. 2020).The meander scanning strategy was used, as this strategy has high efficiency in forming sheet-type parts (Jia et al. 2021).The printing plane was filled with the circular reciprocating motion trajectory, and each scanning layer produced no additional rotation related to the previous layer.The parameters mainly involve the laser power (P), point distance (PD), exposure time (θ), hatch spacing (h) and layer thickness (t), which directly affect the PBF-LB process.Besides, the scanning speed (v) can be obtained by: n = PD/u.Indeed, reported literature has identified that the laser energy density E n = P • u/PD • h • t is an essential thermodynamic variable, quantifying the laser energy input to the powder bed, to effectively modulate the microstructures and correspondingly macroscopical performances of the PBF-LB alloys (Yakout, Elbestawi, and Veldhuis 2018).Currently, for the Invar 36 alloy, the laser energy density ranging between E n = 86.3-104.2J/mm 3 has been broadly established to obtain the optimal thermal expansion and quasi-static mechanical performances (Yakout, Elbestawi, and Veldhuis 2018;Yakout, Elbestawi, and Veldhuis 2019;Harrison, Todd, and Mumtaz 2017;Wegener et al. 2020;Wei et al. 2020), while the influence of the laser energy density on the high cycle fatigue behaviour has not been explored.Therefore, based on the one-factor-at-a-time (OFAT) method, seven values of scanning speed were devised, while the other parameters were constantly kept as listed in Table 1.Then, a wide range of laser energy density E n = 74.1-185.3J/mm 3 was purposely scheduled to additively manufacture the Invar 36 alloy for subsequent microstructure observations and mechanical tests.

Porosity and microstructure characterisation
Micro defects, such as pores, lack-of fusion due to unmelted powder and keyholes induced by over melting, always will be inevitably introduced in the PBF-LB Invar 36 alloy (Yakout, Elbestawi, and Veldhuis 2019;Asgari et al. 2018).Especially, the pores and lack-of fusion are the most challenging and detrimental factors to the fatigue performance (Yadollahi and Shamsaei 2017).Accordingly, density measurements, reflecting the porosity feature, were conducted on the 10.0 × 10.0 × 10.0 mm 3 cubes with three repetitive groups, by using Archimedes principle-based instrument (BR-120N, BEYONGTEST Corporation).Besides, Xray computed tomography (CT) scanning was conducted by an apparatus self-assembled in our laboratory (Zhu et al. 2021) to statistically characterise the size, morphology and distribution of the internal micro defects, which are of great importance to the analysis of the fatigue performances.Cylindric samples with a total volume of ϕ3.0 mm × 5.0 mm were extracted from the 10.0 × 10.0 × 10.0 mm 3 cubes, and 3000 rectangle images were captured by rotating the samples 360°in steps of 0.12°with the voxel size of 5.0 × 5.0 × 5.0 μm 3 .The obtained data were reconstructed, visualised, segmented and quantitatively analysed by using image processing software Avizo 9.4.0 (FEI SAS 2015).
Furthermore, microstructure and texture development were analysed by using optical microscope (OM) and electron backscatter diffraction (EBSD) (Quanta 250 FEG).The samples were mounted into the epoxy resins and were ground up to 5000 grit SiC abrasive paper, followed by a mirror-polishing with 0.05 μm colloidal silica suspension.For OM observation, the pretreatment of etching was performed with a solution containing 40% HCl, 25% HNO 3 and 35% H 2 O, and the etching time was 50 s.For EBSD characterisation, a 45 s polishing and etching was conducted in a solution containing 5% perchloric acid and 95% alcohol with 12 V input voltage.Then, the crystal structure analysis was performed by using a 5.0 μm step size.Besides, the top surface morphology, exactly containing the final laser melting patterns of the PBF-LB process were analysed by using scanning electron microscopy (SEM) in a Quanta 250 FEG microscope.The fracture morphology after the fatigue tests was also observed by SEM.

Quasi-static tensile tests
Tensile tests under room temperature were conducted to provide the basis to the subsequent high cycle fatigue tests.Here, 3.5 × 13.0 × 130.0 mm 3 plate-shaped tensile samples with the gauge length of 50.0 mm as shown in Figure 1(a) were tested using a MTS-Criterion 400 under the loading speed of 0.5 mm/min until fracture, according to the Standard ASTM-E8/E8M-16a (ASTM International 2016).The loading force was recorded by the build-in sensors, and the strain data was measured by the extensometer with a 50.0 mm gauge length and a strain range of 50%.

High cycle fatigue tests
As shown in Figure 1(a), the initial plate-shaped samples with 3.5 × 13.0 × 135.0 mm 3 were fabricated for the unnotched fatigue tests, and the building direction was perpendicular to the loading direction.The initial samples were then machined by the electric discharge machining (EDM) wire cutting into the geometry as specified by the Standard ASTM-E466 (ASTM International 2007).Their surface roughness was reached to be R a = 0.8 by polishing to remove the near-surface adhesive powder particles and residual stress generated by the wire cutting.Afterwards, force-controlled high cycle fatigue tests were conducted under a sinusoidal loading (R = −1) on a servo-hydraulic testing system (Instron 1000) at a frequency of 20.0 Hz.Through the temperature detection of the gauge section in the pre-testing, it was identified that this frequency of 20.0 Hz would not affect the testing results.The specific stress amplitude, to ensure within the high cycle fatigue regime, was dynamically adjusted according to the real-time experimental results, in order to ensure at least five stress amplitudes in each group under the condition that N = 10 6 was considered as the ultimate fatigue life.

Fatigue crack propagation tests
Fatigue crack propagation tests were conducted according to the Standard ASTM-E647-15 (ASTM International 2015), and the crack propagation direction was perpendicular to the building direction as illustrated in Figure 1 (b).The samples featured a thickness of 3 mm, a width (w) of 50 mm and the V-shaped notch had a length of 12.0 mm.The sample surfaces were polished to eliminate the factors which may affect the direction and behaviour of crack propagation.The combination of drilling and wire cutting was used to ensure the position and machining accuracy of the circular holes.The initial fatigue crack was pre-loaded according to continuously increased stress intensity factor amplitude ΔK (Suryawanshi, Prashanth, and Ramamurty 2017;Riemer et al. 2014), and the initial crack length reached to be 8.0-10.0mm (a/w = 0.16-0.2,where a is the crack length).The threshold value of the stress intensity factor ΔK th was obtained simultaneously.For fatigue crack propagation assessment, the sinusoidal loading was adopted by using a servo-hydraulic testing system (MTS Landmark).The experimental frequency was 10.0 Hz, and the stress ratio was R = 0.1.The stress intensity factor amplitude ΔK increases with the crack length enlargement under the condition of keeping the loading unchanged.In addition, the flexibility method was used to measure the fatigue crack length in the crack propagation by a 12.0 mm regulated COD gauge.abruptly, suggesting a poor quality in the PBF-LB Invar 36 alloy.Therefore, Figure 2 is divided into the nonoptimal and optimal regimes by v = 900 mm/s or E v = 82.3J/ mm 3 .In the optimal regime, generally, the enlargement of the laser energy density is beneficial to enhance the relative density raised up to 99%.The over enlargement up to E v = 185.3J/mm 3 slightly reduces the relative density, in accordance with the literature-reported pattern (Wei et al. 2020).Overall, it should be noted that the observations in relative density revealed by Figure 2 suggest that E v = 74.1,105.8 and 185.3 J/mm 3 are three representatives to additively manufacture the Invar 36 alloy exactly over a wide range of laser energy density.Thus, only the samples fabricated at these three laser energy densities were used to conduct the subsequent microstructures observations and mechanical tests, and the results are discussed in the following sections.

Density and porosity
As the relative density only provides a preliminary estimation of the sample's quality, the size, shape and distribution of the defects within the Invar 36 alloy were statistically analysed by the reconstruction results from X-ray CT as summarised in Figure 3.To suppress errors from the image noise, only the defects with equivalent diameter d > 15 μm, thrice larger than the pixel size, are visualised (Wu et al. 2015;Hu et al. 2020).Intuitively, the spatial distributions visualised in Figure 3(a,d,g) clearly demonstrate that the adequate laser energy density increased from 74.1, 105.8 to 185.3 J/mm 3 could significantly reduce the defects, as a large number of the elongated pores are randomly distributed in the sample fabricated at Ev = 74.1 J/mm 3 .
Subsequently, the sphericity c = p 1 3 (6V) 2 3 /A, where V and A are the volume and surface area of the defects, is quantified to feature the defects (Larrosa et al. 2018).Sphericity c reflects the shape feature of the defects, as low sphericity quantifies the sharp edges which will cause stress concentration, remarkably detrimental to the fatigue strength.The results in Figure 3(b,e,h) reveal that the sphericity approaches to be low value with respect to the large equivalent diameter d, suggesting that the large defects always have relatively sharp edges, which will induce stress concentration under tensile and fatigue loading.Fortunately, the high laser energy density will lower the equivalent diameter, consequently raising the sphericity, so as to enhance the mechanical performances.Moreover, the defects can be classified into two types according to their equivalent diameter d and sphericity c (Hu et al. 2020;Kumar and Ramamurty 2020).In detail, when d is less than 40 μm, and c is higher than 0.8, these defects are classified as pores.Oppositely, when d is larger than 40 μm, and c is less than 0.8, they are defined as lack-of fusion.Accordingly, the comparison of Figure 3(b,e,h) suggests that the enlargement of the laser energy density remarkably reduces the lackof fusion defects and oppositely raises the proportion of the pores, which will have a slight influence on the macroscopical performance (Zerbst et al. 2021).
The further statistical analysis identifies that the equivalent diameter d of the defects fulfils the law of a Gauss distribution as shown in Figure 3(c,f,i).Accordingly, the large defects ranging between d = 80-150 μm can be found in Figure 3(c).Oppositely, the relatively small defects ranging between d = 15-40 μm are found in Figure 3(f,i).The expected value x c of equivalent diameter d in the Gauss distribution is obtained as x c = 112.5,18.9 and 23.0 μm for the PBF-LB Invar 36 alloy fabricated at E v = 74.1,105.8 and 185.3 J/mm 3 .Above results indicate the quality improvement of the Invar 36 alloy is achieved under the condition that the adequate laser energy density is used in the PBF-LB process.

Microstructures
The parameters used in the PBF-LB process will remarkably affect the microstructures, such as the grain size, orientation and boundary angle, and then influence the macroscopical performances.The OM metallographic analysis of the side surfaces and cross sections is presented in Figure 4. Generally, the side surfaces are decorated by vertical columnar grains growing across the molten pool.The increase of the laser energy density raises the height of the columnar grains, as comparing Figure 4(a-c).The metallographic analysis in Figure 4(d-f) further shows that the high laser energy density generates a large and more stable molten pool, which is more beneficial to the continuous growth of the columnar grains along the building direction.The cross sections in Figure 4(h,i) present equiaxed grains epitaxially growing across the molten paths, while for the samples manufactured at E v = 74.1 J/mm 3 as shown in Figure 4(d,g), a large number of open lack-of fusion pores, owing to unmelting of the powder, both in the side surface and cross sections can be found.
The microstructures were also characterised by EBSD as shown in Figure 5, which also identifies the columnar grains in the samples fabricated at the high laser energy density (see Figure 5b,c).Under the high laser energy density input of E v = 185.3J/mm 3 , along the building direction, a large number of textures toward the <001> pole is prevalent in Figure 5(c), as the high degree of the overlap along the building direction enables the grain to grow epitaxially across the multilayers.In addition, the maximum multiples of the uniform distribution (MUD) values in Figure 6 also demonstrate that increasing the laser energy density from 74.1 to 185.3 J/mm 3 promotes the preferred orientation of texture along the <001> pole.The obvious grain texture both in the side surfaces and cross sections can be observed in Figure 5(f), indicating the superior fusing degree in the adjacent molten pools.The phase composition of PBF-LB Invar 36 alloy is dominated by γ austenite phase with a small amount of α ferrite phase (see Figure 5h,i).This dominated γ austenite phase is also consistent with our previous XRD result (Yang et al. 2020).In addition, the obtained grain sizes are of 5.6, 14.1 and 17.6 μm for the PBF-LB Invar 36 alloy fabricated at E v = 74.1,105.8 and 185.3 J/mm 3 , respectively.

Tensile behaviour
The uniaxial tensile stress-strain curves and results are presented in Figure 7 and summarised in Figure 8, respectively.When the laser energy density is as low as 74.1 J/ mm 3 , the average yield and ultimate strengths are only of 305.0 and 355.0 MPa, coupled with a low elongation of only 5.1%.These inferior mechanical performances are mainly attributed to a large number of defects due to the inadequate laser energy density input.When the laser energy density is adequately increased to be 105.8 and 185.3 J/mm 3 , the average yield and ultimate strengths are significantly enhanced as shown in Figure 8(b), and the elongation is also raised to be as high as 29%, suggesting the considerable tensile performances compared with the literature results (Qiu, Adkins, and Attallah 2015;Yakout, Elbestawi, and Veldhuis 2019;Wei et al. 2020).The relative deviation of the stress in Figure 7(c) for three repetitive results is about 10%, while the scatter in the elongation is relatively remarkable.It is well known that the characteristics and distribution of defects generated from additive manufacturing process are quite random.The reported literature (Salarian, Asgari, and Vlasea 2020;Voisin et al. 2018) experimentally verified that the elongation is strongly influenced by the spatial characteristics of the defects.Accordingly, the scatter in the stress and elongation may be attributed to the randomness of the defect space characteristics.

High cycle fatigue strength
The high cycle fatigue tests for the PBF-LB Invar 36 alloy were originally conducted for the first time, as currently, only the low cycle fatigue tests were reported by Wegener et al. (Wegener et al. 2020).The high cycle S-N experimental data and correspondingly logarithmic results are summarised in Figure 9. Especially, the logarithmic results present the consistence to the Basquin model: lg s = a + b lg N, suggesting that the loading stress and high cycle fatigue life present a logarithmic linear correlation (the parameters a and b are marked in the figure).
In detail, for E v = 74.1 J/mm 3 , the fatigue strength corresponding to N = 10 6 is only about 100.0 MPa, showing the worst fatigue resistance and degree of correlation to the Basquin model as revealed by Figure 9(a,b).These poor fatigue performances are mainly attributed to a large number of defects as introduced in Section   3.1.For E v = 105.8J/mm 3 in Figure 9(c,d), the fatigue strength is substantially enhanced to be 170.0MPa.In Figure 9(e,f), the fatigue strength is as high as 210.0 MPa, when E v = 185.3J/mm 3 , showing the superior high cycle fatigue resistance.Considering the tensile strengths introduced in Section 3.3, the positive correlation between the tensile strength and high cycle fatigue strength is intuitive, as alloys with high strength usually could withstand the high cyclic loading stress (Zhang et al. 2017a).To assess the fatigue fracture mechanism, fractography analysis was conducted as described in the following sections.

Fatigue crack propagation
Above poor microstructures, tensile and fatigue strengths suggest the Invar 36 alloy additively manufactured at E v = 74.1 J/mm 3 show a large number of pores, making it more similar to a porous alloy with quite different fatigue crack propagation behaviour unlike that of bulk alloys.Thus, the fatigue crack propagation experiments were only performed on the samples fabricated at E v = 105.8and 185.3 J/mm 3 , and the experimental results are given in Figure 10, in which three typical stages can be easily divided into.In detail, firstly, in the crack initiation stage, the crack growth rate is lower than 10 −4 mm/ cycle, and the stress intensity factor amplitude ΔK at the crack tip is determined as its threshold value for the fatigue crack growth, ΔK th .That is when ΔK is lower than ΔK th , the fatigue crack could be considered as no propagation.The experimentally measured ΔK th is of 13.86 MPa • m √ and 13.67 MPa • m √ for the samples fabricated at E v = 105.8and 185.3 J/mm 3 , respectively.
Further, in the stable crack propagation stage, along with the increasing of the crack growth rate, the two experimental curves almost coincide with each other.In addition, a remarkable logarithmic linear relationship between the crack growth rate and stress intensity factor amplitude ΔK can be expressed as: da/dN = C • (DK) m , where C and m are the basic parameters to describe the fatigue crack growth.It should be noted that the difference of the microstructures and defects caused by the different process parameters at E v = 105.8and 185.3 J/mm 3 , is not significant enough to affect the fatigue crack propagation, crack break behaviour and ΔK th for the Invar 36 alloy, which is quite similar to the rules reported for other PBF-LB alloys (Becker, Kumar, and Ramamurty 2021).

Porosity formation mechanism
As widely known the porosity development in PBF-LB alloys is significantly related to the melting flow   pools.Thus, in order to further explore the porosity formation and corresponding mechanism, the top surfaces, which exactly contain the melting flow information of the density testing samples, were imaged by SEM as given in Figure 11.The dimensions of the molten pools from these SEM images are estimated as 70, 120 and 240 μm for the samples fabricated at E v = 74.1,105.8 and 185.3 J/mm 3 , respectively.Besides, the illustration of the porosity formation mechanism, attributed to different molten pools, is given in Figure 12.Straightforwardly, under the constant hatch distance h = 90 μm, the different laser energy density will generate diverse molten pools, consequently, will significantly influence the PBF-LB quality, namely, microstructures and defects.
In detail, the inadequate laser energy of E v = 74.3J/ mm 3 generates a small molten pool of only 70 μm.Considering the hatch distance h = 90 μm, the adjacent melting traces could hardly overlap with each other, obviously leaving the interstitial regions with unmelted powders as illustrated in Figure 12(a).Accordingly, the boundaries between the adjacent melting traces are quite unclear as distinctly illustrated in Figure 11(a).Especially in Figure 11(d,g), a large number of open lack of fusion pores and unmelted powers can be clearly observed.This phenomenon is consistent with those described in the reported literature which systematically investigate the effect of laser energy density on the porosity (Yakout et al. 2017;Yakout, Elbestawi, and Veldhuis 2018).These poor microstructures and numerous defects should be responsible for the low fatigue strengths (Figure 9a,b).Furthermore, the dimension of the molten pool is raised to be 120 μm at laser energy density E v = 105.8J/ mm 3 , thus, in Figure 12(b), the adjacent melting traces progress to be slightly overlapped with each other.However, the inadequate overlapping regions and low temperature still induce the irregular boundaries (Figure 11(b)), elongated pores and inclusive unmelted powders located at the boundaries of the molten pools (see Figure 11(e,h)).Fortunately, the further enlargement of the laser energy input up to 185.3 J/mm 3 significantly raises the dimension of the molten pools up to be 240 μm, more than twice of the hatch distance h = 90 μm, remarkably enlarging the overlapping regions and temperature between the adjacent melting traces as in Figure 12(c).These large molten pools are beneficial to sufficiently melt the hatch and to suppress the generation of the defects.Thus, the relatively flat and smooth top surface is observed in Figure 11(c), and steady boundaries are established between the adjacent melting traces.Except for a few of spatters, no other type of defects could be obviously observed (see Figure 11(f,i)), indicating the favourable microstructures and low porosity, which are beneficial to the corresponding superior fatigue strengths (Figure 9(e,f)).
The reported works suggest that the defects detrimentally influence the mechanical performances (Qiu, Adkins, and Attallah 2015;Yakout, Elbestawi, and Veldhuis 2019).Accordingly, the above-discussed defects, exclusively generated in PBF-LB process, also affect the rule of the tensile strength.Generally, the Hall-Petch law: s = s fr + K g / g √ (g: grain size) describes the strength-grain size relationship for metal alloys (Cooper, Brayshaw, and Sherry 2018).Here, this law is also used for the PBF-LB Invar 36 alloy as shown in Figure 13(a), which suggests a poor correlation between the yield strength and obtained grain sizes (Section 3.2).However, as given in Figure 13(b), by virtue of the form of the Hall-Petch law, but replacing the grain size g to the equivalent diameter of the porosity d, the expression: s = s fr + K d / d √ , where d = 40, 50, 120 μm are obtained from the values at 95% confidence in the Gauss distribution (Figure 3(c,f,i)) for the samples fabricated at E v = 74.1,105.8 and 185.3 J/mm 3 , could well correlate the yield strength and porosity size.The comparison of Figure 13(a,b) indicates that the tensile strength of the PBF-LB Invar 36 alloy is not dominated by the microstructures, i.e. grain size, but, instead, is mainly determinated by the porosity size, showing quite different rule compared with the traditionally processed alloys.

High cycle fatigue failure mechanism
Actually, the high cycle fatigue life of alloys is mainly composed of the lives in the crack initiation and propagation stages (Suresh and Ritchie 1984).Moreover, as defects will be inevitably introduced in the PBF-LB Invar 36 alloy, the identification and discussion of the influence of the defects on the crack initiation are of great significance to understand the fatigue life and failure mechanism.Accordingly, as the fractography, characterised by SEM, shows similar results for the samples fabricated at same E v but loaded at different cyclic stress amplitude, thus, only the representative fractography, containing the crack initiation (region I) and prorogation (region II) regions, for the samples after fatigue failure under certain loading stress is given in Figure 14 for brevity.Under high cyclic loading, the crack initiation stage dominates the entire fatigue life (Cui et al. 2020).Crack initiation is affected by multiple factors, and especially, for PBF-LB alloys, the porosity has pronounced influence on the crack initiation.Specifically, remarkable stress concentration is usually induced at the edges of the pores, correspondingly generating the local plastic deformation.The squeezing of the slip zone, resulted from this local plastic deformation, further leads to the rapid crack initiation (Polak et al. 2017).Furthermore, it should be noticed that different pore characteristics, i.e. number, size, morphology and distribution, give rise to diverse influences on the fatigue crack initiation.Specifically, a large number of pores are prone to stimulate multiple crack initiations, remarkably accelerating the fatigue failure.Besides, rapid crack initiation could also be attributed to the large size of pores, as high stress concentration results in the low resistance to the crack initiation (Kumar and Ramamurty 2020).The poor sphericity, meaning the sharp boundaries around the pores, always triggers the severe stress concentration, also leading to the rapid initiating of the cracks.Additionally, the spatial distribution of the pores, especially closed to surface, is far more detrimental than the internal ones (Edwards and Ramulu 2014;Li et al. 2016), since the short crack propagation distance to the open surface provides a great opportunity for crack initiating.
The above correlations, clarifying the influences of the pores on the crack initiation under high cycle fatigue for PBF-LB Invar 36 alloy, are actually verified in Figure 14.In detail, for the sample fabricated at E v = 74.1 J/mm 3 , the crack propagation process cannot be observed in Figure 14(a), and a certain number of the open lack of fusion pores, featuring the size about l = 200 μm, are easily observed in Figure 14(b).Thus, similar to a porous alloy and unlike bulk alloys, the crack initiation and propagation regions can be hardly distinguished.When suffering the fatigue stress loading, these PBF-LBinduced pores will quickly aggregate, pronouncedly reducing the load-bearing area at the cross section.Consequently, the crack propagates rapidly, leading to the poor fatigue resistance confirmed by Figure 9(a).
In addition, considering the sample fabricated at E v = 105.8J/mm 3 , loaded by s = 193.8MPa, as shown in Figure 14(c,e), the crack initiation and propagation regions can be clearly distinguished.Especially, Figure 14(d) indicates that the crack is initiated from the aggregated partially melted powders sized about l = 70 μm, and located at the subsurface of the sample.These located unmelted powders undesirably enhance the stress concentration, detrimentally giving rise to the premature crack initiating, which significantly shortens the fatigue life in the small crack propagation stage, and hence weakens the total fatigue life as shown in Figure 9(c).
Unlike the previous two samples, of which their cracks are initiated from the large size of either pores or unmelted powders, the crack initiation in Figure 14(f) for the sample fabricated at high laser energy density input E v = 185.3J/mm 3 , loaded by s = 209.5 MPa, originates from an intergranular crack under the stress concentration with the short length about l = 30 μm.This type of crack initiation is identical to that in the traditionally processed bulk alloys.Accordingly, this sample presents superior fatigue life generated from small crack propogation, long crack propogation and final fracture stages.Usually, compared with the easy aggregation of large pores and unmelted powders in the previous two samples under fatigue loading, the relatively small intergranular cracks aggregation will be more difficult, due to the blocking effect of the microstructures (Zhang et al. 2017b).Moreover, it can be straightforwardly understood that the propagation from the relatively small intergranular crack to the large size crack will contribute a considerable proportion of the fatigue life.Therefore, the above mechanisms are responsible for the superior fatigue life for the PBF-LB Invar 36 alloy as given in Figure 9(e).Besides, the crack prorogation and final fracture regions were also characterised by SEM as summarised in Figure 15.Due to the inadequate laser energy input of only E v = 74.1 J/mm 3 , numerous lack of fusion pores induce the brittle fracture mode as shown in Figure 15 (a, d), which further reveal the underlying mechanism  responsible for the low fatigue resistance.The crack propagation in the sample fabricated at E v = 105.8J/ mm 3 mainly includes cleavage fracture and fatigue striations as shown in Figure 15(b).Especially, the proportion of the fatigue striations is in the majority, benefited from the excellent ductility, as confirmed by the large elongation in Figure 8(a).Accordingly, the final fracture is dominated by a large number of the ductile dimples in Figure 15(e).Besides, the crack propagation in the sample fabricated at E v = 185.3J/mm 3 also includes the cleavage fracture and fatigue striation as shown in Figure 15(c).Opposite to that in Figure 15(b), the proportion of the cleavage fracture is in the majority.Besides, the width of each fatigue stripe indeed represents the crack growing distance under one cyclic loading.Obviously, the width of the fatigue stripe is relatively narrow in the sample fabricated at E v = 185.3J/ mm 3 , experimentally verifying its superior fatigue life.Furthermore, it is notable that crack deflections can be observed in Figure 14(e).Indeed, the blocking effect of the hard microstructures to the primary crack propagation leads to the crack deflections.In fact, these crack deflections are beneficial to lower the growth rate of the fatigue crack, and hence to further improve the resistance of crack propagation.
Accordingly, the fracture surface after the final fracture shows only a few of the ductile dimples in Figure 15(f).Indeed, the fatigue striations and the numerous ductile dimples in Figure 15(b,c) suggest the sample fabricated at E v = 105.8J/mm 3 is expected to experience the stable crack propagation and should have an excellent fatigue resistance.However, considering the different crack initiation as discussed from the fractography in Figure 14, the sample fabricated at E v = 185.3J/mm 3 , initiated from the small intergranular crack, still presents the relatively long high cycle fatigue life as confirmed in Figure 9(e).

High cycle fatigue strength model
The discussion in Section 4.2 suggests that the size of the crack initiation source is critical to the high cycle fatigue life.Especially, compared with those of the samples fabricated at E v = 74.1 and 105.8 J/mm 3 , the crack initiation source in the sample fabricated at E v = 185.3J/mm 3 is small enough and is comparable to the size of the microstructures.Thus, the high cycle fatigue behaviour should consider the influence of the width of the plastic area at the crack tip.Indeed, the small crack propagation has an intrinsic crack length l 0 .When the crack initiation source length is larger than l 0 , the threshold value of the stress intensity factor amplitude DK th keeps constant, while the fatigue strength will be weakened with the increase of the crack initiation source length.When the crack initiation source length is lower than l 0 , the fatigue strength approaches to be constant, while DK th will be lowered with the reduction of the crack initiation source length.Actually, the effective crack length should include the intrinsic crack length l 0 and crack initiation source length l, expressed as (Milne, Ritchie, and Karihaloo 2003): (1) Accordingly, considering the elastic-plastic stress at the crack tip, the fatigue strength s th should be calculated as: When the crack initiation source length l approaches to be zero, the intrinsic crack length l 0 could be estimated as: where the value of DK th is 1/4 of 13.67 MPa• m √ obtained in Section 3.5, considering the different stress ratio R used in the fatigue and crack propagation tests (Maierhofer, Pippan, and Gänser 2014).s f = 210.0MPa is obtained in Figure 9(e), and Y = 1.1 is the shape factor.Subsequently, by substituting above parameters into Equation (3), an intrinsic crack length l 0 = 60 μm is obtained.
In addition, by considering the blocking effect of the microstructures on the crack tip slipping, the BSB (blocked slip band) model was proposed, in which the critical condition of the crack propagation is that the crack tip slipping should break through the resistance of the grain boundaries those misorientation angles between two grains should be larger than 15° (Kumar et al. 2020).The intrinsic crack length l 0 is defined as the width of the plastic zone at the small crack tip, and can be estimated as the half size of the grains where misorientation angles between two grains are larger than 15°.Here, the grain size is obtained about 120 μm from the EBSD results in Figure 16, giving l 0 = 60 μm, which is consistent to that obtained from Equation (3), demonstrating the effectiveness of above model, considering the elastic-plastic stress at the crack tip.Finally, by substituting the crack initiation source length l, the predicted fatigue failure strength is given as: The Kitagawa-Takahashi (K-T) diagram, which illustrates the fatigue failure envelope under cyclic loading expressed by Equation ( 4), is given in Figure 17, in which the experimental data, including the experimentally measured fatigue strength s th in Figure 9 and l = 200, 70 and 30 μm obtained in Section 4.2, are marked for comparison.
In Figure 17, the experimental data show good consistence with the envelope, identifying the effectiveness of the established high cycle fatigue strength model.In detail, when the adequate laser energy density is of E v = 185.3J/mm 3 , the PBF-LB Invar 36 alloy has the small crack initiation source length l = 30 μm, far less than the intrinsic crack length l 0 = 60 μm.Thus, the type of the crack initiation source is related to the intergranular crack, not the defects as verifed in Figure 14(f).The predicted strength is of 180.0 MPa, relatively conservative to the experimental fatigue strength of 210.0 MPa, since the fatigue strength s f used in the fatigue model should obviously be lower than the ideal fatigue strength limitation.This comparison suggests that the Invar 36 alloy fabricated at E v = 185.3J/mm 3 shows good quality, as its defects cannot detrimentally influence its fatigue strength, namely, presenting excellent fatigue resistance.
For the sample fabricated at E v = 105.8J/mm 3 , as observed in Figure 14(d), the crack is initiated from unmelted powder, sized about l = 70 μm, larger than the intrinsic crack length l 0 = 60 μm.Then, the microstructures could not give rise to the blocking effect on the crack propagation.Then, this sample suffers the quick linear elastic crack prorogation, generating relatively low fatigue strength as shown in Figure 17.As illustrated in Figure 12, the different porosity formation mechanism, owing to the disparity in the laser energy density input, leads to the remarkably different crack initiation source and propagation as shown in Figure 14, and consequently different fatigue strengths in Figure 17.This comparison also indicates that the fatigue resistance of the PBF-LB Invar 36 alloy should not be evaluated or predicted just by its relative density.When the adequate densification is ensured, the high laser energy density input is more favourable to ensure the high fatigue resistance for the PBF-LB Invar 36 alloy.For the sample fabricated at E v = 74.1 J/mm 3 , the low relative density, owing to the large number of the pores and unmelted powders, makes it similar to a porous alloy.Accordingly, as confirmed in Figure 17, this sample presents the low fatigue strength, and should not be preferred for engineering application under the cyclic loads.

Conclusions
In the present study, the microstructures and mechanical, especially high cycle fatigue performances, coupled with a fatigue strength model were explored for Invar 36 alloy (has the unique feature of low coefficient of thermal expansion) additively manufactured by laser powder bed fusion, aiming at originally revealing the correlation of the PBF-LB process-microstructure(defect)-high cycle fatigue performance.The main conclusions are summarised as: (1) The PBF-LB Invar 36 alloy is mainly composed of γ austenite phase and a low fraction of α ferrite phase.Columnar grains across the molten pools and equiaxed grains epitaxially across the molten paths are clearly observed.The adequate laser energy density (E v = 105.8and 185.3 J/mm 3 ) generates large and stable molten pools, beneficial to the continuous growth of the columnar grains, while a large number of open lack-of fusion pores is generated, owing to the unmelting of the powder caused by an inadequate laser energy density input (E v = 74.1 J/mm 3 ).(2) Visualisation and statistical analysis of X-ray CT results identify that due to the inadequate laser energy density input (E v = 74.1 J/mm 3 ), the large defects always have the relatively low sphericity, i.e. sharp edges, which induce stress concentration, and consequently result in the low tensile and high cycle fatigue strengths.
(3) The variation of the tensile strength of the PBF-LB Invar 36 alloy is not dominated by the microstructures, i.e. grain size, but, instead, is mainly determinated by the porosity equivalent diameter d, expressed by s = s fr + K d / d √ , showing quite different rule compared with the traditionally processed alloys.(4) The inadequate laser energy density (E v = 74.1 J/ mm 3 ) gives rise to the large number of the pores and unmelted powders, making the PBF-LB Invar 36 alloy similar to a porous alloy with low fatigue strength, failure from the rapid aggregation of the defects.In contrast, the adequate laser energy density (E v = 185.3J/mm 3 ) enables the crack initiation source to be the intergranular crack.Then, the defects cannot detrimentally influence the fatigue strength, and it has superior and completed fatigue life generated from the small crack propagation, long crack propagation and final fracture stages.(5) A fatigue model, effectively considering the PBF-LB induced microstructure and defect features, is established and experimentally verified to predict the high cycle fatigue strength.The fatigue resistance of the PBF-LB Invar 36 alloy should not be evaluated or predicted just by its densification.When the adequate densification is ensured, the high laser energy density input is more favourable to ensure the high fatigue resistance for the PBF-LB Invar 36 alloy.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Funding
This

Figure 2
Figure 2 summarises the relative density of the PBF-LB Invar 36 alloy, revealing a process-porosity relationship.When the scanning speed is up to 900-1000 mm/s (Figure 2(a)), corresponding to a laser energy density of 74.1-82.3J/mm 3 (Figure 2(b)), the relative density drops

Figure 1 .
Figure 1.Schematic diagram of samples designed for high cycle fatigue and fatigue crack propagation tests (all dimensions in mm).

Figure 2 .
Figure 2. Relative density versus (a) laser scan speed and (b) laser energy density for the PBF-LB Invar 36 alloy.

Figure 3 .
Figure 3. Visualisation of spatial distribution (a, d and g), sphericity (b, e and h) and Gauss distributions (c, f and i) of the defects obtained from X-ray CT for Invar 36 alloy additively manufactured at E v = 74.1,105.8 and 185.3 J/mm 3 .

Figure 4 .
Figure 4. Metallographic analysis by optical microscope (OM) for side surfaces (a-f) and cross sections (g-i) of Invar 36 alloy additively manufactured at E v = 74.1,105.8 and 185.3 J/mm 3 .

Figure 9 .
Figure 9. High cycle fatigue S-N experimental data (a, c and e) and correspondingly logarithmic results (b, d and f) for Invar 36 alloy additively manufactured at E v = 74.1,105.8 and 185.3 J/mm 3 .Basquin model with parameters a and b is used to describe the experimental data.

Figure 10 .
Figure 10.Fatigue crack propagation curves for Invar 36 alloy fabricated at E v = 105.8and 185.3 J/mm 3 .The crack propagation direction is perpendicular to the building direction.

Figure 11 .
Figure 11.SEM images show various defects and melting boundaries at the top surfaces of the samples fabricated at E v = 74.1,105.8 and 185.3 J/mm 3 .

Figure 12 .
Figure 12.Illustration of porosity formation mechanism attributed to diverse molten pools and overlapping distances basically resulted from input laser energy densities of (a) 74.1,(b) 105.8 and (c) 185.3 J/mm 3 .

Figure 13 .
Figure 13.(a) Poor correlation between yield strength and grain size in Hall-Petch law: s = s fr + K g / g √ , and excellent correlation between yield strength and porosity size referred to the form of Hall-Petch law: s = s fr + K d / d √ .

Figure 16 .
Figure 16.Grain boundaries with their misorientation between two grains are large than 15°obtained by EBSD in (a-c) side surfaces and (d-f) cross sections.

Figure 17 .
Figure 17.Kitagawa-Takahashi (K-T) diagram predicts the fatigue strength established from the failure envelope under high cyclic loading, and experimental data are marked for comparison.

Table 1 .
Process parameters used in the laser powder bed fusion for Invar 36 alloy.
research was supported by Natural Science Foundation of Hunan Province under grant #2021JJ30085, The Science and Technology Innovation Program of Hunan Province under grant #2021RC30306, Open Research Fund of State Key Laboratory of High Performance Complex Manufacturing, Central South University under grant #Kfkt2021-01 and The Fund of State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body under grant #52175012.