Introduction

Additive manufacturing (AM)1,2,3,4,5,6, particularly laser powder-bed fusion (L-PBF)7,8,9, has emerged as a promising manufacturing technology for fabricating complex geometries directly from digital models in a layer-by-layer manner. It offers advantages such as reduced material waste, improved energy efficiency, enhanced design flexibility, and rapid prototyping10. Meanwhile, the AM process involves rapid localized melting and solidification, resulting in microstructural features, such as fine cellular-dendritic microstructures, elemental segregation, and residual stresses, that differ substantially from those in conventional casting or wrought counterparts11,12,13,14,15. Although AM printing without post-processing treatment is desirable, many AM-printed materials still require post-printing treatments, such as stress relief, hot isostatic pressing (HIP), and heat treatment, to achieve mechanical performance comparable to wrought materials16,17. The inherent variability introduced by machine-specific parameters, processing conditions, and feedstock composition further complicates property prediction and repeatability, making qualification and certification potentially much more complicated18,19.

On the other hand, rigorous qualification and certification protocols are essential for AM components deployment20,21,22,23, particularly in aerospace and other safety-critical industries where performance reliability and statistical confidence are paramount. Conventional qualification strategies, such as “3-σ” probability of 0.9987 with 95% confidence, are traditionally used for aerospace structures24, typically requiring a large number of mechanical tests to establish property distributions and ensure a specified minimum value, thereby incurring substantial cost, time, and material usage. These demands are further exacerbated by the intrinsic variability of AM processes, such as machine-specific parameters, complex thermal histories, and feedstock inconsistencies22,23. Accelerating the qualification of AM parts thus remains a major challenge for widespread industrial adoption.

Many computational models and digital twins approaches have been developed to reduce the experimental burden of qualification, which could be potentially adopted to the qualification in additive manufacturing25,26,27. Among them, ICME-based Accelerated Insertion of Materials (AIM) approach, implemented by Olson and QuesTek28,29,30,31,32, has been proven to be an efficient approach for qualification. ICME provides a physics-based understanding of the process–structure–property chain, enabling the high-throughput calculations and predictions of material response under varied compositions and processing conditions33,34. ICME-based AIM approaches have thus been employed to incorporate uncertainty quantification (UQ) and enable rapid estimation of minimum property values with reduced experimental datasets for conventional materials qualification30,31,32. With yield strength as an example, Harlow demonstrated that combining Monte Carlo simulations with small-scale experimental datasets can accurately calibrate the predicted yield strength distribution, enabling high-confidence estimation of lower-bound strength values28,32,35,36,37,38. A statistical calibration strategy was thus developed using physics-based modeling to define the shape of the property probability density function (PDF), wherein small sets of randomly selected experimental data were used to adjust the simulated PDF via linear transformation, thereby accounting for residual epistemic uncertainty. The linear transformation includes a shift to align the mean and a rotation to match the standard deviation with the experimental data35. Remarkably, the study showed that incorporating as few as 15 experimental measurements into the statistical calibration yielded an accurate prediction of the 1% minimum yield strength, defined here as the yield strength corresponding to the 1% lower-tail cumulative probability level, within ±1 ksi (6.89 MPa), which is comparable to the typical tensile test uncertainty35. Therefore, this ICME-based uncertainty quantification and rapid qualification methodology can be applied to AM-printed alloys, which is the goal of this study.

In this work, we adopted the ICME-based rapid qualification approach, with an ICME modeling framework, to quantify uncertainty and enable qualification of AM-printed materials. Using AM-printed nickel superalloy ATI 718Plus® as a case study39,40,41,42, we constructed an ICME framework that links post-processing heat treatments to microstructural evolution and mechanical property prediction. As outlined in the system design chart in Fig. 128,31,43, the post-printing treatments include stress relief, HIP, solution heat treatment (SHT), and two-step aging39,44. Key microstructural features include the face-centered cubic (FCC) γ matrix, FCC-L12 γ′ precipitates45, grain size, and grain boundary particles, such as primary γ′, δ phase, carbides, and oxides formed during printing and SHT42,46,47. The property of interest in this study is the yield strength, which serves as a demonstration case for uncertainty quantification and rapid qualification.

Fig. 1: System design chart for AM-printed 718Plus.
figure 1

Each box represents a component in the process–structure–property chain, and each connecting line represents the corresponding computational model. Uncertainty quantification and rapid qualification are performed through this integrated ICME workflow.

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In the system design chart, each connection (line) represents a physical relationship, and the corresponding numerical or physics-based models are annotated in Fig. 1. The ICME framework integrates all these models into a modeling pipeline to enable automated simulation, thus predicting the yield strength based on specified processing and composition inputs. This framework supports high-throughput virtual testing of large numbers of compositions and heat treatment conditions, from which the full PDF of yield strength can be analyzed, including estimation of the 1% minimum value that is required for qualification, as illustrated in Fig. 2. According to the study by Harlow32,35, the simulated PDF can be further calibrated to experimental data via linear transformation, enabling accurate prediction of the 1% minimum strength.

Fig. 2: Schematic workflow for ICME-based uncertainty quantification and rapid qualification.
figure 2

Composition and process variations are propagated through ICME models to predict corresponding variations in microstructure and yield strength. The resulting property variations are represented as probability distributions used for qualification.

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Results

ICME models calibration with experiments

As illustrated in Fig. 1, the ICME framework includes microstructural models for simulating microstructural evolution and a property model for yield strength prediction. The microstructural models are CALculation of PHAse Diagram (CALPHAD)-based thermodynamics and kinetics models, including the commercial software Thermo-Calc48 and QuesTek’s in-house software PrecipiCalc®49,50, along with the databases TTNi7 and MobNi751. The yield strength model is based upon prior studies that include the contributions from multiple strengthening mechanisms for polycrystalline nickel-based superalloys52. The model predictions were compared with experiments to calibrate and validate these ICME models and databases.

As the first step, thermodynamic equilibrium simulations were conducted to evaluate phase stability and the evolution of grain boundary particles in AM 718Plus® alloy at various conditions via step equilibrium calculations using Thermo-Calc and the TTNi7 database, as illustrated in Fig. 3. The equilibrium model predicts a microstructure consisting of an FCC γ matrix, strengthening FCC-L12 γ’ precipitates, and other potential second-phase particles such as δ phase, carbides, and oxides.

Fig. 3: Equilibrium phase prediction for the 718Plus alloy.
figure 3

Thermo-Calc calculations using the TTNi-7 database show the equilibrium phases and their phase fractions as a function of temperature. Arrows mark the temperatures corresponding to each heat-treatment process. The two dashed vertical lines indicate the subsolvus solution treatment at 970 °C and the supersolvus solution treatment at 1010 °C.

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On the experimental side, five batches of feedstock powder with slight compositional variation, as shown in Table 1, were printed through laser powder bed fusion (L-PBF), heat treated, and machined for tensile tests by Honeywell, as detailed in related publications40,41,53,54. Specifically, the baseline post-build heat treatment sequence includes stress relief at 1065 °C for 1.5 h, HIP at 1160 °C and 103 MPa (15 ksi) for 4 h, followed by the traditional process for ATI 718Plus®, which is a sub-solvus solution heat treatment at 970 °C for 2 h, and two-step aging for 8 h each at 788 °C and 704 °C. This process was applied to Batch #1 and Batch #2. However, this sub-solvus SHT retained a significant fraction of grain boundary particles, including δ phase and various carbides41, as predicted by equilibrium calculations in Fig. 3.

Table 1 Composition of 718Plus powder (in wt%)
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To reduce grain boundary particles, a revised post-printing route was developed by increasing the SHT temperature to 1010 °C, above the solvus temperature of the FCC-L12 γ’ phase, while keeping the other steps unchanged. This heat treatment was referred to as super-solvus heat treatment for Batches #3, #4, and #5. As predicted by the equilibrium calculation in Fig. 3, a substantial reduction in grain-boundary particles is expected. The remaining grain boundary particles from these sub-solvus and super-solvus SHT are compared in Table 2, showing the dissolution of γ’ and δ particles during the super-solvus SHT. This thermodynamic equilibrium model and database were validated by experimental characterization using scanning electron microscopy (SEM) and energy-dispersive X-ray spectroscopy (EDS) of AM-printed samples after solution heat treatment, which confirmed the expected grain-boundary phases41.

Table 2 phase fraction of grain boundary particles after solution heat treatment
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The second step is to simulate precipitation evolution. Second phase FCC-L12 γ′ precipitate is the major strengthening mechanism for ATI 718Plus® alloy39, and it nucleates and grows during the post-printing heat treatment steps, such as controlled cooling and two-step aging. Simulations were performed using QuesTek’s in-house PrecipiCalc® software49, which simulates the precipitation kinetics, including nucleation, growth, coarsening, and dissolution of precipitates, along with the associated database TTNi7 and MobNi7. In the precipitation modeling, it is assumed that grain boundary particles remain stable, and the model only tracks the precipitate evolution inside the grain.

The simulated evolution of γ′ precipitated during SHT cooling, and the two-step aging process (788 °C and 704 °C for 8 h each) is shown in Fig. 4. The modeling output includes time-resolved estimates of mean particle size, phase fraction, and precipitate composition. Secondary γ′ refers to the precipitate nucleated during the cooling stage, distinguishing it from primary γ′ formed at grain boundaries during printing and SHT. Tertiary γ′ was observed in negligible quantities during the final cooling of two-step aging. Due to their minimal contribution and lack of clear experimental evidence, tertiary precipitates were excluded from further analysis. The experimentally measured γ′ precipitate size at the end of aging is 14.8 ± 3 nm41, and it was used to calibrate the interfacial energy between the γ′ precipitate and γ matrix \({\sigma }_{\gamma /{\gamma }^{{\prime} }}\), resulting in a calibrated value of \({\sigma }_{\gamma /{\gamma }^{{\prime} }}=23\,\mathrm{mJ}/{{\rm{m}}}^{2}\).

Fig. 4: Precipitate simulation during cooling and two step aging.
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a Thermal profile as the input for the precipitate simulations. b Predicted evolution of mean particle size for FCC L12 γ’ precipitate. c Predicted evolution of the volume fraction for FCC L12 γ’ precipitate.

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Simulation results indicate that the final microstructure consists of secondary γ′ with a mean particle size of 14.8 nm and a volume fraction of 25.2%. These predictions were validated through atom probe tomography (APT), which showed strong agreement with experimental measurements. The compositions of both the γ′ precipitates and the FCC γ matrix are compared in Table 3 and Fig. 5. The results confirm that the model captures the key features of the precipitation process, providing good validation of the precipitate model and associated database TTNi7 and MobNi7. Although both Nb and Ti are present at low concentrations in the FCC γ matrix, noticeable discrepancies between the simulated and experimental values are observed. These differences come from limitations in the thermodynamic database TTNi7, which underestimates the solute concentration of Nb and Ti in the matrix phase. This observation highlights the need for improved database assessments to capture the partitioning behavior of Nb and Ti more accurately. On the other hand, although both Nb and Ti are influential γ′-forming elements, their matrix concentrations after solution treatment are small, and variations at these levels have a limited effect on γ′ volume fraction and particle size. As a result, the discrepancies between simulated and measured matrix compositions have only a modest impact on the precipitation-strength prediction and do not alter the calibration or the conclusions of the rapid qualification framework.

Fig. 5: Comparison of APT-measured and simulated compositions for the FCC γ matrix and FCC-L12 γ’ precipitate.
figure 5

A zoomed-in subplot highlights elements with low concentrations (<5 at.%). Diamond markers indicate concentration in FCC γ matrix, and solid circular markers indicate concentration in the FCC-L12 γ’ precipitate.

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Table 3 Composition (at.%) comparison between APT measurements and CALPHAD simulation
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With the microstructural features obtained from the CALPHAD-based equilibrium and precipitation simulations, the last step is to predict the yield strength of AM 718Plus using a microstructure-based strength model that incorporates these microstructural features. The total yield strength is calculated by superposing the individual strengthening contributions, including grain-boundary particle strengthening, Hall–Petch grain boundary strengthening, solid-solution strengthening from matrix alloying elements, and γ′ precipitate strengthening, as shown in the system design chart in Fig. 1. In 718Plus, γ′ precipitate strengthening provides the dominant contribution. The model captures the competing dislocation-precipitate interaction mechanisms across the γ′ precipitate size spectrum, including shearing through dislocation weak pair-coupling or strong pair-coupling at small radii, and Orowan looping at larger radii52. Although LPBF builds can exhibit strong columnar textures, the solution heat treatment applied here produces an equiaxed grain structure after recrystallization and grain growth41, reducing texture effects on yield strength. Pores and cracks primarily influence ductility and fatigue behavior rather than 0.2% yield strength for the heat-treated condition examined in this work. Thus, these microstructure features are not considered in the current strength model.

Calibration of the model was performed using experimental tensile data provided by Honeywell. The calibration focused on the anti-phase boundary (APB) energy \({\gamma }_{{APB}}\), which is the boundary energy when a dislocation shears the ordered γ′ precipitate and therefore directly influences the shearing component of the precipitate strengthening term. Tensile data from AM-printed specimens processed through the full thermal cycle were used to calibrate the value of \({\gamma }_{{APB}}\) that provides the best agreement between predicted and measured yield strength. This calibration procedure resulted in an APB energy of \({\gamma }_{{APB}}=254\,\mathrm{mJ}/{{\rm{m}}}^{2}\). With this calibrated value, the model reproduces the measured yield strength of 952 MPa for the super-solvus heat-treatment condition, supporting the model’s reliability for subsequent uncertainty quantification analyses. The APB energy influences both the magnitude of γ′ shearing strength and the transition between shearing and bypass mechanisms, making calibration essential for an accurate deterministic mean yield strength prediction. The calibration of the APB energy serves to minimize structural model bias by aligning the deterministic strength prediction with the measured mean yield strength prior to uncertainty propagation.

High-throughput simulation and rapid qualification

With all the ICME models developed, an automated model pipeline was constructed to sequentially connect the inputs and outputs of each model, following the system design chart in Fig. 1. Specifically, the thermodynamic equilibrium model takes the feedstock powder compositions and temperature as inputs, predicting equilibrium phases after SHT, while the precipitation model simulates the precipitate evolution during subsequent cooling and two-step aging. All these microstructural outcomes serve as inputs for the yield strength model, which provides the final yield strength value as output.

To assess the variability in mechanical properties due to variations in process parameters and composition, a Monte Carlo high-throughput simulation was performed through this automated model pipeline following the flowchart in Fig. 2. A total of 2000 virtual scenarios were generated by sampling variations in solution temperature, solution time, cooling rate, aging temperatures and durations, grain size, and alloy composition. All input parameters were sampled from normal distributions centered on their nominal values, with standard deviations designed to reflect realistic AM-process variability. The nominal values and standard deviations used for sampling are summarized in Table 4. The thermal profiles for these 2000 heat treatments are illustrated in Fig. 6. These normally distributed parameters provide unbiased sampling of the expected process fluctuations and avoid clustering within any region of the parameter space. Each scenario was evaluated using the developed ICME model pipeline, and the yield strength distribution was generated by running these 2000 scenarios through the automated ICME model pipeline, showing as the blue dashed line in Fig. 7. The total of 2000 Monte Carlo scenarios provides resolution of tail probabilities down to approximately 0.05% (1/2000), which is significantly finer than qualification requirements of 0.3% minimum or 1% minimum.

Fig. 6: Thermal profiles for high-throughput heat-treatment simulations.
figure 6

2000 heat-treatment scenarios are shown, with each line representing an individual thermal profile, including solution heat treatment (SHT), cooling, and two-step aging, used in the high-throughput simulations.

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Fig. 7: Probability distribution of room-temperature 0.2% yield strength from experiments and modeling.
figure 7

Open circles represent experimental tensile test results from Batches 3, 4, and 5. The light-blue dashed line shows the predicted yield strength distribution before statistical calibration, and the red solid line shows the distribution after statistical calibration. Two dark dashed lines indicate the 1% minimum yield strength and the 50% (mean) yield strength.

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Table 4 Nominal values and standard deviations used in the Monte Carlo sampling
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Compared to the 252 experimental tensile tests performed for Batches #3, #4, and #5, the predicted yield strength distribution (blue dashed curve in Fig. 7) was slightly offset, which is acceptable given the complex variability and large uncertainties inherent in the AM processes. This ICME prediction already captures the mean yield strength and the overall shape of the distribution. The remaining differences in distribution width arise from variability sources not explicitly included in the Monte Carlo sampling.

Following Harlow’s theory32,35, a linear transformation can be applied to statistically calibrate the probability density function (PDF). This linear transformation of simulated yield strength \({{\rm{YS}}}_{{\rm{uncal}}}\) into calibrated yield strength \({{\rm{YS}}}_{{\rm{cal}}}\) includes a shift to align the mean values and a rotation to match the standard deviation. The linear transformation was performed following Eq. (1).

$${{\rm{YS}}}_{{\rm{cal}}}={\rm{a}}\times {{\rm{YS}}}_{{\rm{uncal}}}+{\rm{b}}$$
(1)

In this equation, the parameters a and b are defined as following Eqs. (2) and (3).

$${\rm{a}}=\frac{{{\rm{s}}}_{\exp }}{{{\rm{s}}}_{{\rm{uncal}}}}$$
(2)
$${\rm{b}}={\overline{{\rm{YS}}}}_{\exp }-\frac{{{\rm{s}}}_{\exp }}{{{\rm{s}}}_{{\rm{uncal}}}}\times {\overline{{\rm{YS}}}}_{{\rm{uncal}}}$$
(3)

In these equations, \({{\rm{s}}}_{\exp }\) and \({{\rm{s}}}_{{\rm{uncal}}}\) are standard deviations of experimental measurements and simulated (before statistical calibration) yield strength, respectively, while \({\overline{{\rm{YS}}}}_{\exp }\) and \({\overline{{\rm{YS}}}}_{{\rm{uncal}}}\) are the average values.

The model predictions after statistical calibration are shown as the red solid line in Fig. 7. After statistical calibration, simulation results showed excellent agreement with experimental tensile test data obtained from Honeywell. The predicted probability distribution of room-temperature 0.2% yield strength closely matched the experimental distribution, capturing both the mean and lower-bound values. Specifically, the experimental 1% minimum yield strength was 934 MPa, while the model predicted 932 MPa. And the mean values were 952 MPa for simulation and experiment, as noted in Fig. 7. This statistical calibration applies only a linear shift and scale transformation to align the predicted and measured distributions, preserving the physics-based sensitivity to process and composition variations while accounting for unmodeled sources of variability.

It is noted that the model includes two distinct calibration steps. The first calibration determines the interfacial energy in the precipitation model and the APB energy in the microstructure-based strength model to ensure that the deterministic prediction of yield strength is accurate. This step establishes the correct baseline for the physics-based ICME model. After uncertainty propagation through Monte Carlo sampling, where the initial input variations are not necessarily identical to real manufacturing variability, a second statistical calibration is applied to align the predicted distribution width and deviation with the measured statistical variability. This adjustment accounts for additional sources of variation not represented in the original Monte Carlo sampling. These two calibrations serve different purposes and do not duplicate each other. After the two-step calibration, the model parameter APB energy \({\gamma }_{{APB}}\) primarily affects the mean value of the yield strength distribution, whereas the lower-tail percentiles arise from the propagated variation in composition and heat-treatment parameters. Thus, small deviations in \({\gamma }_{{APB}}\) shift the entire distribution but do not significantly affect its width.

The statistical calibration of the yield strength distribution uses 252 tensile measurements obtained from three AM print batches provided by Honeywell, with all specimens undergoing the same designed post-printing heat-treatment process, while still containing realistic batch-to-batch and machine-specific variations representative of industrial AM production. These variations are not fully included in the composition and heat-treatment variations in the 2000 Monte Carlo–sampled scenarios. Therefore, the statistical calibration step aligns the predicted strength distribution with the broader experimental variability. It is noted that the microstructure becomes fully equiaxed after heat treatment41, so build-direction anisotropy is not expected to influence the tensile results.

A global sensitivity analysis was conducted using standardized multiple linear regression applied to the 2000 Monte Carlo–sampled scenarios, as shown in Fig. 8. All input parameters were standardized, and the resulting regression coefficients quantify their relative influence on the predicted yield strength. The analysis shows that γ′-controlling variables, including aging temperatures, solution temperature, cooling rate, and the concentrations of γ′-forming elements (Al, Nb, Ti), dominate yield strength variability. Other alloying elements and process parameters exhibit comparatively small contributions within the tested tolerance ranges. These results confirm that the yield strength distribution is primarily governed by factors influencing γ′ precipitation.

Fig. 8: Global sensitivity analysis of predicted yield strength.
figure 8

Standardized multiple linear regression is applied to the 2000 Monte Carlo–sampled scenarios. The standardized regression coefficients quantify the relative influence of each process parameter and alloying element on the final predicted yield strength.

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Because the APT measurements showed noticeable deviations in Nb and Ti concentrations in FCC γ matrix compared with the CALPHAD predictions, as shown in Fig. 5, it is important to clarify how variations in these γ′-forming elements propagate through the precipitation model and influence the predicted yield strength. The γ′-forming elements Nb and Ti influence precipitation strengthening through their effects on the γ′ phase fraction, precipitate chemistry, and size distribution. However, the uncertainty of Nb and Ti content in the FCC γ matrix is minimized through the physics-based model calibration, including the interfacial energy calibration and APB energy calibration.

This ICME-based uncertainty quantification and rapid qualification approach was also applied to Batches #1 and #2, which underwent sub-solvus SHT, with the comparison shown in Fig. 9. A total of 36 tensile test data points were obtained by Honeywell for Batches #1 and #240,41, which were insufficient to statistically determine the 1% minimum value through experiments. Nevertheless, these 36 data points were sufficient to recalibrate the modeling results, as demonstrated by the dark solid line in Fig. 9, which predicts that the 1% minimum yield strength is 896 MPa.

Fig. 9: Comparison of 0.2 percent yield strength distributions for subsolvus and supersolvus heat treatments.
figure 9

Open dark diamond markers represent subsolvus results from Batches 1 and 2, and open circular markers represent supersolvus experimental tensile-test results from Batches 3, 4, and 5. The dark solid line shows the calibrated yield strength distribution for the subsolvus condition, and the red solid line shows the distribution for the supersolvus condition. Two dark dashed vertical lines indicate the 1% minimum yield strength and the 50% (mean) yield strength. The two heat treatments produce similar mean strengths (947 MPa vs 952 MPa) but differ markedly in the 1% minimum values (896 MPa vs 932 MPa).

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A comparison between samples from sub-solvus and super-solvus SHT, with all other steps remaining the same, shows that the mean yield strength values are very close, slightly increasing from 947 MPa to 952 MPa. However, the sub-solvus samples exhibit significantly greater yield strength variation, with a 1% minimum yield strength of 896 MPa, substantially lower than that of the super-solvus heat-treated samples, which is 932 MPa. This difference is primarily attributed to the presence of complex grain boundary particles remaining in the sub-solvus heat-treated samples, which introduce higher uncertainty in the final properties. In contrast, the super-solvus heat treatment dissolves most of the grain boundary particles and results in a cleaner grain boundary microstructure, as demonstrated in Table 2, resulting in more consistent and narrowly distributed yield strength. Therefore, a super-solvus solution heat treatment is preferred for AM-printed ATI 718Plus® alloy.

Discussion

This combined physics-based and statistical framework enables robust estimation of lower-tail strength values with a limited number of experiments. A key question in this ICME-based rapid qualification approach is how many experimental data points are required to calibrate predictive models with sufficient statistical confidence for AM-printed materials. Harlow demonstrated that as few as 15 data points are sufficient when accurate models are employed, while more data points, such as 20, may be necessary depending on the degree to which the ICME model captures the critical materials behavior35.

To address this issue, a trial analysis was performed by randomly drawing N = 5, 15, 30, and 50 data points from the 252 available experimental data points in Batches #3, #4, and #5. For each case N, N data points were randomly selected from the full set of 252 tensile results to perform statistical calibration, and this procedure was repeated 2500 times to generate the distribution for that N. As shown in Fig. 10, 95% of the calibrated strength distributions fall within the shaded regions for each N. As expected, increasing the number of calibration points narrowed the uncertainty bands and brought the predicted distributions into closer agreement with the ground truth. The ground truth is represented by the black line in Fig. 10, which was calibrated using the full dataset.

Fig. 10: Effect of sample size on the spread of the calibrated yield strength distribution.
figure 10

The width of the calibrated yield strength distribution is shown by varying the number of experimental data points used for statistical calibration (N = 5, 15, 30, and 50).

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To quantify the effect of experimental data size, we continuously varied the number N of randomly selected experimental data points from 3 to 250. For each sample size N, 2500 Monte Carlo iterations were performed, similar to those in Fig. 10, and the 0.3% minimum and 1% minimum yield strength values, corresponding to the 0.3% and 1% lower-tail cumulative probability levels, were extracted from each recalibrated distribution. The distributions of the 0.3% minimum and 1% minimum values across the 2500 Monte Carlo iterations were collected, and the 95% probability interval (a centered range containing 95% of the data) was plotted as a function of the selected sample size N in Fig. 11, clearly quantifying the effects of experimental calibration data points on the lower-tail (0.3% minimum and 1% minimum) strength estimation. The results suggest that both the 0.3% minimum and 1% minimum values converge rapidly toward the true values (the green dashed lines in Fig. 11), which are assumed to be the calibrated results based on the full dataset. Another notable observation is that the average predicted values for the 0.3% and 1% minimum strengths are slightly higher than the reference values obtained using all 252 data points. This behavior arises from the statistical difficulty of estimating extreme lower-tail percentiles from small samples, which causes the calibrated distributions to under-represent the true tail region. The deviation decreases rapidly as N increases and becomes very small, less than 1 MPa when N exceeds 10.

Fig. 11: 95% probability intervals for minimum yield strength metrics as a function of sample size.
figure 11

a Probability intervals for the 0.3% minimum yield strength under different calibration sample sizes. b Probability intervals for the 1% minimum yield strength under the same conditions. Error bars mark the N values at which the probability-interval width falls within 1 ksi (6.89 MPa). The green dashed lines indicate the reference values (0.3% minimum and 1% minimum) calibrated using all 252 experimental data points, and the black solid lines represent the average calibrated 0.3% and 1% minimum yield strength across 2500 iterations for each N.

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If an uncertainty tolerance of ±1 ksi (6.89 MPa) is acceptable, which is a typical uncertainty in tensile tests, the analysis suggests that 18 experimental data points are sufficient to determine the 1% minimum, and 21 points suffice for the 0.3% minimum, indicating that the ICME models achieved reasonable predictive accuracy. This is slightly larger than the minimum of 15 data points suggested by Harlow35, reflecting the larger complexity and variability associated with AM-printed materials.

The slightly higher data requirement observed for AM 718Plus relative to the conventional 15-specimen criterion proposed by Harlow can be attributed to the combined influence of process- and microstructure-level variabilities inherent to AM that are not accurately captured by this ICME model. On the process side, melt-pool dynamics, layer-by-layer thermal cycling, and localized fluctuations in energy input produce spatially varying thermal histories across the build. These variations propagate into microstructural heterogeneity, including compositional segregation, differences in γ′ nucleation density and growth kinetics, local variations in precipitate size and volume fraction, grain-structure variability, and fluctuations in defect population. Although post-printing heat treatment reduces the overall anisotropy, the underlying process–microstructure coupling in AM still generates broader performance variability than is typically observed in forged materials, where thermomechanical processing promotes microstructural homogenization. This wider intrinsic spread in yield strength necessitates a slightly larger experimental dataset to estimate lower-tail strength metrics with comparable confidence.

It is also useful to distinguish between structural model error and data-sampling error in this work. The structural error of the microstructure-based strength model is minimized through the calibration of the interfacial energy and APB energy, which ensures that the deterministic prediction of yield strength matches the experimentally measured mean value. Once this calibration is applied, the dominant remaining source of uncertainty arises from the limited number of tensile measurements used to fit the distribution, producing sampling variability in the estimated 1% minimum and 0.3% minimum values. The convergence behavior in Fig. 11 therefore reflects the reduction of sampling error with increasing data, rather than residual structural bias in the mechanistic model.

Interestingly, both the 0.3% and 1% minimum values exhibit similar convergence behavior with increasing data, suggesting that the tail behavior of the strength distribution can be robustly captured using relatively few experimental data points. Moreover, stricter qualification thresholds can be determined using the same experimental dataset. This finding highlights the key advantage of this ICME-based rapid qualification approach, which is the ability to quantify property variability and define confidence limits with minimum experimental effort.

The present ICME framework also has several limitations that should be acknowledged. First, equilibrium phase fractions are obtained from thermodynamic calculations, whereas actual heat-treatment schedules may not provide sufficient time for complete dissolution or reprecipitation of γ′ or other phases. These kinetic limitations can introduce discrepancies between predicted and measured compositions or phase fractions. Second, the accuracy of the microstructural model is constrained by the fidelity of the underlying thermodynamic and kinetic databases. Limitations in these databases are the primary source of the observed discrepancies in Nb and Ti concentrations between the model predictions and the APT measurements. Third, the strength model assumes a well-densified AM component and does not explicitly incorporate defects such as pores or lack-of-fusion regions, which may reduce the yield strength through stress concentration or reduced effective load-bearing area. While these factors are not explicitly represented, the framework includes a statistical calibration step that aligns the predicted strength distribution with experimental measurements. This statistical calibration partially compensates for model-form errors and unmodeled sources of variability, allowing the ICME approach to remain predictive for qualification purposes even when certain microstructural details are approximated or not fully captured. Consequently, as the fidelity of the underlying ICME models improves, the reliance on statistical calibration would diminish, and fewer experimental data points would be required to achieve the same confidence in lower-tail value estimation.

This work establishes a practical framework for uncertainty quantification and rapid qualification of AM 718Plus heat treatments by integrating physics-based ICME models with Monte Carlo high-throughput simulations. The combined approach propagates variability through the process–structure–property chain and, when calibrated statistically with tensile tests, yields accurate predictions of full property distributions, including the 1% minimum yield strength. The analysis indicates that approximately 18 tensile data points are sufficient to bound the 1% minimum yield strength within ±1 ksi, which substantially reduces the experimental burden relative to conventional qualification practices. The methodology is broadly transferable to other AM alloys and components and provides a scalable path to physics-informed, cost-efficient qualification workflows.

Methods

Experimental processing

The experiments were performed by Honeywell and are documented in public literature40,41,53,54. Specifically, 5 batches of ATI 718Plus® feedstock powders were printed via L-PBF, under a previously optimized LPBF process window for ATI 718Plus41 (laser power 195 W, scan speed 1000 mm/s, hatch spacing 0.10 mm, and a 5-mm stripe width), which produces stable melt pools with low porosity. After printing, post-printing heat treatments were performed, followed by tensile tests on the printed 718Plus® samples. The standard heat treatment process for ATI 718Plus® includes a stress relief at 1065 °C for 1.5 h, a HIP at 1160 °C and 103 MPa for 4 h, a sub-solvus solution heat treatment at 970 °C for 2 h, and a two-step aging at 788 °C and 708 °C for 8 h each39,41. This standard heat treatment was applied to Batches #1 and #2. A super-solvus heat treatment process was developed by raising the SHT temperature to 1010 °C for 2 h while all other steps remaining unchanged. This super-solvus heat treatment was applied to Batches #3, #4, and #5. Grain size was measured by SEM, precipitate morphology was characterized using SEM and atom probe tomography (APT). Some additional compositional analysis was performed via EDS41.

ICME framework overview

The integrated ICME framework developed in this study comprises two major components: microstructure evolution modeling and mechanical property prediction. The microstructure models link composition and thermal processing conditions to the evolution of grain boundary phases and γ′ precipitates, while the strength model integrates the predicted microstructure with empirical relationships to estimate yield strength. The overall workflow is shown in Fig. 1.

Thermodynamic simulations were conducted using the commercial CALPHAD software Thermo-Calc48 with the TTNi-7 database to model equilibrium phase fractions at different solution heat treatment temperatures. These calculations informed the evolution of grain boundary particles such as δ phase, primary γ′, and carbides. It is noted that the CALPHAD-based equilibrium model employed in this work is validated and broadly used for Ni-based superalloys across wide temperature ranges. Although experimental characterization for model checking is available only for selected conditions, the thermodynamic database provides consistent predictions at both sub-solvus and super-solvus solution temperatures, which are used in the subsequent precipitation and strength modeling. Precipitation simulations were carried out using QuesTek’s in-house software PrecipiCalc®49, with the TTNi-7 and MobNi751 databases. The model tracks nucleation, growth, coarsening, and dissolution of γ′ precipitates under time-temperature conditions matching the heat treatment cycle. The output includes precipitate size, volume fraction, and composition, as well as matrix chemistry, which is the input for the yield strength model. Interfacial energy between precipitate and matrix was calibrated by matching prediction with APT-measured precipitate size and volume fraction.

Yield strength model

The yield strength \({{\rm{\sigma }}}_{{YS}}\) of the nickel superalloy was evaluated by superposing the major strengthening contributions in polycrystalline nickel-based superalloys52 following Eq. (4).

$${{\rm{\sigma }}}_{{YS}}={\sigma }_{0}+{\sigma }_{{HP}}+{\sigma }_{{SS}}+{\sigma }_{{precipitate}}+\mathop{\sum }\limits_{i}{f}_{{GB},\,i}{\sigma }_{{GB},\,i}$$
(4)

In this yield strength model, each contribution is described as follows.

Base strength \({\sigma }_{0}\) is the intrinsic strength of the nickel matrix, which is taken as 22 MPa.

Grain boundary particle strength \(\mathop{\sum }\limits_{i}{f}_{{GB},{i}}{\sigma }_{{GB},{i}}\) is the strength due to grain boundary particles, including primary \({\gamma }^{{\prime} }\) particles, δ phase, oxides, and carbides. Here, \({f}_{{GB},i}\) and \({\sigma }_{{GB},{i}}\) denote the phase fraction and strength of each grain boundary particle type. The fraction \({f}_{{GB},{i}}\) is obtained from equilibrium thermodynamic calculations at the solution temperature and is generally very small, as shown in Table 2. The corresponding particle strength values \({\sigma }_{{GB},{i}}\) use a constant value of 100 MPa52.

Grain boundary strengthening \({\sigma }_{{HP}}\) is the grain-size contribution and follows the Hall-Petch relationship in Eq. (5).

$${\sigma }_{{HP}}=\frac{{K}_{{HP}}}{\sqrt{{D}_{{gr}}}}$$
(5)

In this equation, \({K}_{{HP}}\) is the Hall-Petch coefficient, and \({D}_{{gr}}\) is the grain size in micrometer (\(\mathrm{\mu m}\)).

Solid solution strengthening \({\sigma }_{{SS}}\) is predicted based on the Labusch model55, which incorporates the effect of lattice misfit and modulus misfit between the solute atom and Ni, shown in Eq. (6).

$${\sigma }_{{SS}}\left({\left\{{x}_{i}\right\}}_{\gamma }\right)=\sqrt{{\sum }_{i}{{k}_{\,i}}^{2}{x}_{i}}$$
(6)

In this equation, \({x}_{i}\) is the mole fraction of element \(i\) and \({k}_{\,i}\) is the coefficient of each element.

Precipitate strengthening\(\,{\sigma }_{{precipitate}}\) from γ‘ is the dominant contribution for 718Plus. The model accounts for three dislocation-precipitate interaction modes: shearing through dislocation weak pair-coupling, shearing through dislocation strong pair-coupling, and Orowan looping. The effective precipitate strengthening is a competition among these three modes, as illustrated in Eq. (7).

$${\sigma }_{{precipitate}}=M\times \min \{{\tau }_{{weak}},{\tau }_{{strong}},{\tau }_{{orowan}}\}$$
(7)

In this equation, M is the Taylor factor for FCC Nickel alloys, with an average value of 3.06.

The three threshold stresses for three interaction modes52 can be predicted following Eqs. (8), (9), and (10).

$${\tau }_{{weak}}=\frac{{\gamma }_{{APB}}}{2b}\left\{\sqrt{\frac{{\gamma }_{APB}d}{2{T}_{L}}}\frac{d}{L}-\frac{\pi }{4}{\left(\frac{d}{L}\right)}^{2}\right\}$$
(8)
$${\tau }_{{strong}}=\frac{2{T}_{L}}{\pi {bL}}\sqrt{\frac{\pi d{\gamma }_{{APB}}}{2{T}_{L}}-1}$$
(9)
$${\tau }_{{Orowan}}=\frac{{Gb}}{L}$$
(10)

Here, \(d\) and \({f}_{{\gamma }^{{\prime} }}\) are the mean diameter and phase fraction of \({\gamma }^{{\prime} }\) precipitates. \(b\) is dislocation burgers vector, \(G\) is the shear modulus of the Ni matrix, and \({\gamma }_{{APB}}\) is the APB energy of γ‘ phase. \(L\) is the distance between precipitates, defined as Eq. (11).

$$L=\sqrt{\frac{8}{3\pi {f}_{{\gamma }^{{\prime} }}}}d-d$$
(11)

\({T}_{L}\) is the line tension of dislocations defined as Eq. (12).

$${T}_{L}=\frac{G{b}^{2}}{4\pi }\left(\frac{1-0.25\nu }{1-\nu }\right)\mathrm{ln}\left(\frac{R}{b}\right)$$
(12)

where \(\nu\) is the Poisson’s ratio and R is the outer cutoff distance, taken as 10 nm.

As can be seen from the above equations, the APB energy \({\gamma }_{{APB}}\) of γ‘ phase strongly influences both shearing mechanisms, and is the most sensitive parameter in the precipitate strengthening model. Physics-based model calibration was performed by matching the predicted yield strength with experimental tensile data, and the calibrated value obtained is \({\gamma }_{{APB}}=254\,\mathrm{mJ}/{{\rm{m}}}^{2}\).

Statistical sampling and data fusion

To evaluate the required number of experiments for qualification, a Monte Carlo sampling strategy was applied. Random subsets of size N from the experimental dataset (total of 252 data points) were drawn for model statistical calibration. This statistical calibration is a linear transformation including a shift to align the mean values and rotation to match the standard deviation. The predicted minimum yield strength values at the 1% and 0.3% probability levels were tracked across 2500 iterations for each sample size N, providing the upper bound, lower bound, and mean values for these 2500 iterations.