A gas-atomized nickel-based superalloy powder was selected for use in an LPBF system. The powder exhibited a nominal particle size range of 15-53 µm. Its chemical composition is detailed in Table 1 and includes essential elements such as Ni, Cr, Nb, Mo, Ti, and Al, with Fe as the balance, in proportions designed to promote phase stability and high-temperature creep resistance.

During the build process using an FS301M LPBF system (Farsoon Technologies, Changsha, China), each powder layer was approximately 40 µm thick, with a hatch spacing of 100 µm, a laser scanning speed of 1,000 mm/s, a laser power of 270 W, and an effective laser spot size of approximately 70 µm. The stripe-pattern scanning strategy was employed with a stripe width of 10 mm, a bidirectional scan order, a zero-contour scanning strategy, and an interlayer rotation of 67°. The build platform, consisting of a 316L stainless steel substrate (301 × 301 × 40 mm³), was preheated to 200 °C, and the ambient environment was maintained under argon with an oxygen concentration below 0.1%. A rectangular plate measuring 45 × 80 × 3 mm³ was printed with a vertical build orientation parallel to the Z-axis. The plate was removed from the substrate upon completion, without any subsequent stress-relief heat treatment, leaving it in an “as-built” condition for analysis.

The experimental geometry and sample coordinates are illustrated in Figure 1. Macroscopic uniaxial tensile testing was performed at room temperature using an MTS Landmark servohydraulic test system (MTS Systems Corporation, Eden Prairie, MN, USA). A dog-bone specimen was machined from the LPBF IN718 with its loading axis parallel to the build direction. A strain rate of 0.00025 s^-1 was applied in the elastic-yield stage, followed by a displacement rate of 0.0067 mm/s after yielding. The Young’s modulus was experimentally determined to be 189.5 GPa. Notably, the yield strength of the as-built IN718 part was 646 MPa from the tensile experiment, whereas the measured macroscopic residual stress reached a maximum of 791 MPa, which will be discussed in the RESULTS AND DISCUSSION section in detail.

HEXRD experiments were performed at the 3W1 beamline of the Beijing Synchrotron Radiation Facility (BSRF). The incident monochromatic X-ray energy was 60.12861 keV (λ = 0.02062 nm). The beam size was defined using slits as 380 × 380 µm². A two-dimensional flat-panel detector (Mercu 4343, iRay Technology, Shanghai, China) with a resolution of 3,072 × 3,060 pixels was positioned 945.885 mm from the sample to record Debye-Scherrer diffraction rings. The transmission geometry was used to map the sample cross-section. The sample was mounted on a high-precision translation stage (TSA series, Beijing Zolix Instruments Co., Ltd., Beijing, China) [Figure 1B]. The loading direction (LD), transverse direction (TD), and normal direction (ND) were defined relative to the sample geometry. Diffraction patterns were collected point-by-point along the transverse direction (X-axis) and build direction (Z-axis). The exposure time was set to 5 s per frame to optimize the signal-to-noise ratio.

To achieve a high-resolution map of the macroscopic residual stress while managing the immense volume of raw scattering data, an equally spaced measurement grid of 35 rows × 24 columns was applied over the 48 mm × 70 mm region of interest, yielding 840 spatial points. This grid resolution provides sufficient spatial density to accurately capture the steep macroscopic stress gradients concentrated along the lateral edges of the sample while remaining within the strict beamline accessibility time constraints. At each point, four crystallographic planes [(111), (200), (220), and (311)] were analyzed along four azimuthal directions (LD-left, LD-right, TD-left, and TD-right) from the Debye-Scherrer ring. Consequently, over 13,000 individual diffraction peaks were separated and fitted to obtain residual stress information, and peaks with R² < 0.96 were tagged as low-quality outliers and discarded. This extensive data-processing workload ensures the high spatial fidelity and statistical robustness of the reported stress distribution.

Two-dimensional diffraction images were caked into segments at azimuthal angles corresponding to the LD (0°/180°) and TD (90°/270°), with an integration range of ±5°^[29,36]. Reflections from planes such as (111), (200), (220), and (311) were identified for strain measurement.

Three different functions - Gaussian, Pseudo-Voigt, and Pearson VII - were used for peak-fitting and comparison. The Gaussian function is often used for its simplicity, but can be insufficient if tails are pronounced. The Pearson VII introduces an additional shape parameter that can further refine the representation of elongated tails, potentially capturing distortions that simpler functions might miss. The Pseudo-Voigt function is a linear combination of Gaussian and Lorentzian components, and is widely used for its computational efficiency and its ability to model peak tails.

The Gaussian function has the form:

(1) $$ I(\theta)=I_{0} \text { exp }\left[-\frac{\left(\theta-\theta_{0}\right)^{2}}{2 \sigma^{2}}\right] \\ $$

where I₀ is the peak height, θ₀ is the peak center, and σ is a width parameter^[37]. The Pseudo-Voigt is an amalgam of Gaussian and Lorentzian components:

(2) $$ I(\theta)=\eta I_{G}(\theta)+(1-\eta) I_{L}(\theta) \\ $$

where η is the Gaussian-Lorentzian mixing parameter, I_G represents the Gaussian component, and I_L represents the Lorentzian component^[37]. The inclusion of a Lorentzian component improves the accuracy for broadened peaks. The Pseudo-Voigt function is a common choice for fitting diffraction peaks due to its flexibility and efficiency. The Pearson VII is defined by:

(3) $$ I(\theta)=I_{0}\left(1+\frac{K^{2}\left(\theta-\theta_{0}\right)^{2}}{M}\right)^{-M} \\ $$

where K defines the profile width, and the shape parameter M refines the rate of the peak tails ^[37].

Once the peak centers were extracted, the interplanar spacing d was determined using Bragg’s law:

(4) $$ d^{h k l}=\frac{\lambda}{2 \sin \theta^{h k l}} \\ $$

where superscript hkl denotes the specific crystal plane, λ is the X-ray wavelength, and θ is the Bragg angle^[29]. Deviations from a stress-free reference spacing d₀ allowed for the calculation of residual strain ε^[29]:

(5) $$ \epsilon^{h k l}=\frac{d^{h k l}-d_{0}^{h k l}}{d_{0}^{h k l}} \\ $$

The residual stress was then computed by applying generalized Hooke’s law. The specific X-ray elastic constants (XEC) for individual crystallographic planes used in the stress calculations are listed in Table 2 ^[38]. The three principal directions - LD, TD, and ND - were defined relative to the geometry of the plate indicated in Figure 1. The residual stresses in LD, TD, and ND are as follows:

(6) $$ \sigma_{L D}^{h k l}=A\left[B \epsilon_{L D}^{h k l}+v^{h k l}\left(\epsilon_{T D}^{h k l}+\epsilon_{N D}^{h k l}\right)\right] \\ $$

(7) $$ \sigma_{T D}^{h k l}=A\left[B \epsilon_{T D}^{h k l}+v^{h k l}\left(\epsilon_{L D}^{h k l}+\epsilon_{N D}^{h k l}\right)\right] \\ $$

(8) $$ \sigma_{N D}^{h k l}=A\left[B \epsilon_{N D}^{h k l}+v^{h k l}\left(\epsilon_{L D}^{h k l}+\epsilon_{T D}^{h k l}\right)\right] \\ $$

where the superscript hkl represents a specific crystal plane. Here, $$ A=\frac{E^{h k l}}{\left(1+v^{h k l}\right)\left(1-2 v^{h k l}\right)} \\ $$, B = 1 - ν^hkl, E^hkl is the corresponding Young's modulus, and v^hkl is the Poisson ratio^[29]. Assuming $$ \sigma_{N D}^{h k l}=0 \\ $$, the residual strain $$ \epsilon_{N D}^{h k l} \\ $$ in the ND can be obtained as follows^[29]:

(9) $$ \epsilon_{N D}^{h k l}=-\frac{v^{h k l}\left(\epsilon_{L D}^{h k l}+\epsilon_{T D}^{h k l}\right)}{1-v^{h k l}} \\ $$

Thus, the residual stress in the LD and TD can be calculated using^[29]:

(10) $$ \sigma_{L D}^{h k l}=\frac{E^{h k l}}{1-\left(v^{h k l}\right)^{2}}\left(\epsilon_{L D}^{h k l}+v^{hkl} \epsilon_{T D}^{h k l}\right) \\ $$

(11) $$ \sigma_{T D}^{h k l}=\frac{E^{h k l}}{1-\left(v^{h k l}\right)^{2}}\left(\epsilon_{T D}^{h k l}+ v^{hkl} \epsilon_{L D}^{h k l}\right) \\ $$

Uncertainty analysis followed standard propagation of errors from the fitted peak centers. The covariance matrix from the least-squares fitting procedure provided the variance in θ₀, denoted δθ₀. This variance influences the final strain uncertainty u_ϵ, and consequently the stress uncertainty u_σ.

The residual strain measurement uncertainty can be given as:

(12) $$ \begin{aligned} \quad u_{\epsilon} &= \left[\left(\frac{\cos \theta_{0}}{\sin \theta} \delta \theta_{0}\right)^{2}+\left(\frac{\cos \theta \sin \theta_{0}}{\sin ^{2} \theta} \delta \theta\right)^{2}\right]^{\frac{1}{2}}\\ &\approx \frac{1}{\tan \theta_{0}}\left[\left(\delta \theta_{0}\right)^{2}+(\delta \theta)^{2}\right]^{\frac{1}{2}}\end{aligned} $$

where δθ₀ and δθ are the fitting uncertainties associated with θ₀ and θ, respectively^[29]. Thus, according to Equation (12), the residual strain uncertainty for LD, TD, and ND can be calculated as $$ u_{\epsilon}^{L D} \\ $$, $$ u_{\epsilon}^{T D} \\ $$ and $$ u_{\epsilon}^{N D} \\ $$. Then, the residual stress uncertainty is calculated as^[29]:

(13) $$ u_{\sigma}^{L D}=\left[\left(\frac{E}{1+v} u_{\epsilon}^{L D}\right)^{2}+\left(\frac{v E}{(1+v)(1-2 v)}\right)^{2}\left(\left(u_{\epsilon}^{L D}\right)^{2}+\left(u_{\epsilon}^{T D}\right)^{2}+\left(u_{\epsilon}^{N D}\right)^{2}\right)\right]^{\frac{1}{2}}\\ $$

(14) $$ u_{\sigma}^{\mathrm{TD}}=\left[\left(\frac{E}{1+v} u_{\epsilon}^{T D}\right)^{2}+\left(\frac{v E}{(1+v)(1-2 v)}\right)^{2}\left(\left(u_{\epsilon}^{L D}\right)^{2}+\left(u_{\epsilon}^{T D}\right)^{2}+\left(u_{\epsilon}^{N D}\right)^{2}\right)\right]^{\frac{1}{2}} $$

The three peak-fitting models were used to investigate differences among reconstruction methods in the resultant residual strain and stress field.

This study used two methods to obtain the stress-free d₀. The first method calculates the average diffraction plane spacing across the entire plate region:

(15) $$ d_{0}=\frac{1}{N} \Sigma_{i}^{N} d_{i} $$

where d_i is the plane spacing at i^th measured position and N is the number of measured points. The d₀ error is derived from the individual d_i errors via error propagation.

Given that the total stress across the entire sample is nearly zero in both the LD and TD directions, in the second method, d₀ is calculated by optimizing its value to minimize the integrated residual stress in both the LD and TD directions, as follows:

(16) $$ f\left(d_{0}\right)=\frac{1}{N}\left[\sum_{i}^{N} \sigma^{L D}+\sum_{i}^{N} \sigma^{T D}\right] \\ $$

(17) $$ d_{0}^{\star}=\operatorname{argmin}_{d_{0} \in R^{+}} f\left(d_{0}\right) \\ $$

where f(d₀) is the optimization objective function, and the objective is to minimize the integrated residual stress in both LD and TD. The d₀ that minimizes the overall net residual stress is believed to correspond to the stress-free spacing criterion. This d₀ is optimized by a selective stochastic iteration algorithm^[39]. The algorithm first starts from the initial point d⁰, and this point is then stochastically shifted to another value:

(18) $$ d_{0}^{k+1}=d_{0}^{k}+\Delta d_{0} \\ $$

where k+1 denotes the next iteration point. The Δd₀ is the step length, and Δd₀ ~ N(μ_d; σ_d). The trial value $$ d_{0}^{k+1} \\ $$ is accepted with probability P = 1 if $$ f\left(d_{0}^{k+1}\right)< f\left(d_{0}^{k}\right) $$, otherwise $$ P = \exp \left(-\frac{f\left(d_{0}^{k+1}\right)-f\left(d_{0}^{k}\right)}{T}\right) \\ $$, where T is the temperature factor governing the probability of accepting unfavorable trial solutions^[39]. The algorithm terminated when no further reduction in f(d₀) was achieved within the tolerance steps, N_tol or when the maximum number of iterations, N_iter was reached. Although two methods were used, the resulting d₀ was identical for both methods.

Lattice strain measurements obtained through X-ray diffraction generally contain contributions from macro, intergranular, and intragranular stresses (i.e., type I as well as type II/III stresses). However, the macro stress, as the main focus of this work, was calculated by weighting the multiple reflections with proper factors^[40]:

(19) $$ \bar{\epsilon}=\frac{\sum_{h k l} \alpha_{h k l} \epsilon_{h k l}}{\sum_{h k l} \alpha_{h k l}} \\ $$

where α_hkl is the weighting factor for the corresponding crystal plane hkl, calculated as follows^[40]:

(20) $$ \alpha_{h k l}=T_{h k l} m_{h k l} \frac{E_{h k l}}{E} $$

where T_hkl is the texture index, taken as 1 because the entire diffraction rings were observed^[22], m is the multiplicity of the reflection plane, E_hkl is the diffraction elastic modulus (DEM), and E is the macroscopic Young’s modulus of IN718^[40]. The E value is 189.5 GPa, as determined from a macroscopic tensile experiment.

To elucidate the local microstructural features and provide physical evidence of crystallographic anisotropy, scanning electron microscopy (SEM), energy-dispersive spectroscopy (EDS), and electron backscatter diffraction (EBSD) analyses were conducted on the YZ cross-section of the as-built LPBF IN718 sample, parallel to the build direction. For SEM observation, the sample surface was ground and polished using standard metallographic procedures, and then etched with Kalling’s reagent. The surface morphology was characterized using a scanning electron microscope (AMBER GMH, TESCAN Co., Ltd., Brno, Czech Republic) equipped with a C-Swift EBSD detector (Oxford Instruments, High Wycombe, UK). The elemental distribution was analyzed using an Aztec EDS system (Oxford Instruments, High Wycombe, UK). Both SEM and EDS analyses were performed at an acceleration voltage of 15 kV and an emission current of 3 nA. For EBSD characterization, the mechanically polished samples were further electropolished in a methanol-sulfuric acid solution to remove residual surface deformation. EBSD mapping was performed using the same TESCAN SEM platform at an accelerating voltage of 15 kV and an emission current of 3 nA. During the EBSD data acquisition, the sample was tilted at 70°, and the working distance was maintained between 13 and 15 mm.

The microstructural morphology and crystallographic orientation of the YZ cross-section were systematically characterized. Figure 2A shows the typical SEM microstructural features of the as-built IN718. Due to the extremely rapid cooling conditions inherent in the LPBF process, substantial thermal undercooling is induced. Together with constitutional supercooling caused by solute redistribution during solidification, this thermal undercooling collectively breaks the stable equilibrium of the solid-liquid interface. Driven by directional heat flow, dynamic melt-pool behavior, and layer-by-layer thermal cycling, crystals preferentially grow epitaxially in the direction opposite to the heat flux, ultimately forming a highly directional dendritic structure. As evidenced by the localized EDS elemental mapping in Figure 2B, pronounced dendritic segregation occurs, with Nb enrichment in the interdendritic regions.

To further evaluate the crystallographic texture and grain boundary characteristics, EBSD analysis was performed over a 1,000 × 812 μm² area with a step size of 0.8 μm. Figure 3 presents the inverse pole figure (IPF) orientation and grain-boundary (GB) maps for the YZ cross-section. As shown in Figure 3A, the grain morphology of the as-built plate is predominantly characterized by the epitaxial growth of elongated columnar grains aligned parallel to the build direction (Z-axis). The GBs map [Figure 3B] quantitatively reveals that high-angle grain boundaries (HAGBs, > 15°) account for 79.7%, while low-angle grain boundaries (LAGBs, 5-15°) account for 20.3%. The abundant LAGBs represent typical sub-grain structures and dislocation tangles induced by the repeated thermal cycling and intense thermal gradients inherent to the LPBF process, while the HAGBs outline the distinct boundaries of the columnar grains^[2]. Together, columnar grain structures and the complex boundary network provide microstructural evidence of crystallographic anisotropy and may contribute to the asymmetric diffraction peak profiles discussed below.

The reliability of residual stress mapping relies heavily on the quality of peak profile analysis. Figure 4 provides a comprehensive evaluation of the signal and fitting performance. Figure 4A shows that the Debye-Scherrer rings are continuous, indicating good grain statistics. Figure 4B shows the corresponding integrated diffraction profile. Interestingly, the diffraction peaks exhibit slight asymmetry. The asymmetry observed in the diffraction peaks is a direct consequence of the complex thermomechanical history during AM, which leads to variations in elemental distribution, phase formation, dendritic and columnar segregation^[9], and internal stresses within the material, as discussed in^[9,15]. Figure 4C explicitly compares the fitting results for a (111) peak. The experimental data (black dots) exhibit a distinct asymmetry and heavy tail. The Gaussian fit (green) fails to capture this tail. In contrast, both the Pseudo-Voigt (blue) and Pearson VII (red) functions provide better agreement with the experimental data. Figure 4D further confirms this trend visually through a heatmap, where the red blocks indicate that the Pseudo-Voigt function consistently achieves higher goodness-of-fit (R²) than Gaussian and Pearson VII across the dataset.

The spatial distribution of the raw diffraction angle 2θ provides fundamental insight into the state of the material lattice. Figure 5 maps the variation in the diffraction angle 2θ across the sample dimensions. Since d ∝ 1/sinθ, these maps represent the raw lattice distortions. The maps show a distinct spatial pattern where the 2θ values at the lateral edges differ considerably from the center. This center-to-edge variation in lattice spacing is the physical origin of the residual stress fields calculated later. The heterogeneity in 2θ confirms that the material has undergone thermal contraction and nonuniform plastic deformation during the LPBF process, validating the existence of a complex stress state.

To quantify the impact of fitting function, we examine the specific error values. Table 3 presents the average and maximum residual strain errors. The three fitting functions show similar values for both the average and maximum errors. The maximum residual strain error is below 82.6 με. The residual stress error values in Table 4 are similar. The maximum residual stress error is below 15.3 MPa. These results indicate that statistical uncertainty in determining the peak center is very low.

Tables 5 and 6 summarize the average and maximum absolute differences of residual strain and stress, respectively. Compared with the measurement errors in residual strain and stress, these differences among fitting functions are typically smaller. Given that the total stress range is on the order of hundreds of MPa, the differences among the fitting functions are considered negligible. Difference maps obtained using the various fitting functions are also shown in Figures 6-9. A random “salt-and-pepper” noise pattern is observed, confirming that the choice of fitting function does not introduce artificial stress concentrations.

The crystallographic dependence of residual strain is systematically analyzed in Figure 10, which reveals distinct differences in strain magnitude between planes. The (200) plane maps display the greatest color contrast (from deep red to deep blue), indicating a much larger strain dynamic range than the others. This is physically consistent with the elastic anisotropy data in Table 2. The (200) plane is the most compliant (E₂₀₀ is about 160 GPa), deforming markedly under stress, whereas the (220) plane is the stiffest (E₂₂₀ is about 226 GPa), exhibiting smaller strains^[10]. The distribution of the corresponding residual stress in Figure 11 follows the same trend.

To provide a macroscopic view, Figures 12 and 13 present the weighted-average results. Figure 12 shows the weighted macroscopic residual strain, and Figure 13 shows the corresponding macroscopic residual stress. Figure 13 clearly illustrates the “core compression, edge tension” distribution. The LD stress map [Figure 13A] shows substantial tensile concentrations at the vertical edges, which are much more pronounced than in the TD map [Figure 13B]. This anisotropy confirms that the residual stresses are dominated by constraints along the build direction. The literature consistently reports such directional dependencies in AM materials, which are linked to processing parameters and thermal effects^[2].

The line profiles for residual strains and stresses are illustrated in Figures 14 and 15, respectively. Figure 14 presents residual strain variation along the X-axis at four distinct Z-heights within the build: near the plate bottom (10 mm), at the mid-height (45 mm), near the top surface (76 mm), and immediately below the top surface (78 mm) along LD (red), TD (blue), and ND (green). The LD residual strain magnitude is substantially larger than in TD and ND. For LD, the residual strain profiles exhibit a characteristic “U-shaped” tensile distribution, with high tensile strain concentrated at the bottom and at the two vertical edges. By contrast, TD exhibits an inverted “U-shaped” compressive residual strain distribution, with considerable tensile residual strain at the top edge (Z = 78 mm) and the bottom. The overall geometric features of the residual stress field outline are consistent with previous reports^[2,23]. However, the absolute magnitudes of residual stress measured in this as-built sample are higher. In direct thermodynamic comparison with thinner-walled struts or smaller test coupons often analyzed in the literature^[23], the higher magnitudes measured may be attributed to the severe thermal confinement within the solid core of our thick plate. This thick internal structural bulk restrains the horizontal cooling shrinkage of the peripheral layers, thereby driving higher localized macroscopic tensile stresses at the lateral boundaries, as will be discussed in the following section, “Physical mechanisms of stress formation”.

Figure 15 provides quantitative proof of stress evolution. By comparing the distribution at different Z positions from bottom (Z = 10mm) to top (Z = 78 mm), the LD direction experiences a transition from “U-shape” tensile residual stress to compressive residual stress. In the LD profile, residual stress reaches a maximum of +791 MPa at the lateral edges. Accounting for the TD residual stress, the von Mises equivalent stress^[41] at this location is slightly lower than 791 MPa. This measured LD residual stress peak exceeds the macroscopic yield strength of 646 MPa because the latter represents an average value over the bulk material, whereas the local yield strength may be higher. To balance the edges, the core region reaches approximately -355 MPa. The TD residual stress peak is +436 MPa.

To elucidate the physical origin of the measured residual stress fields, it is important to distinguish the static residual stress measured in this study from the dynamic thermomechanical mechanisms reported in the literature. Although our experimental data quantify the final solid-state stress equilibrium, the transient cyclic evolution underlying these characteristics is fundamentally governed by the temperature gradient mechanism (TGM), widely documented in prior LPBF research^[11,42-44]. As inferred from these theoretical models and conceptually illustrated in Figure 16A, during the localized heating and deposition stage, the newly deposited (n+1) layer is heated to a high temperature and attempts to undergo massive thermal expansion (ε^thermal > 0). However, the underlying cooler (n) layers (or the substrate) constrain this free expansion, inducing irreversible compressive deformation within the (n + 1) layer (ε^plastic < 0)^[11]. As shown in Figure 16B, the TGM framework indicates that this constrained thermal expansion induces a downward “push” effect at the edges. Subsequently, during the bulk cooling stage as illustrated in Figure 16C, the hindered thermal shrinkage of this plastically shortened layer forces a dramatic stress transition into severe tension (ε^elastic > 0)^[11]. The extreme boundary tensile stresses and compensating internal compressions, directly captured in our spatial mapping in Figure 16D, provide macroscale validations of the cumulative structural equilibrium dictated by these theoretical layer-by-layer thermal cycling models.