Predictive Modeling and Contour Method Validation of Residual Stresses in Notched PBF-LB/M/Ti6Al4V Components Using the Inherent Strain Method

Abstract

Residual stresses and distortions are critical challenges in laser beam powder bed fusion (PBF-LB) of Ti6Al4V components (PBF-LB/M/Ti6Al4V), impacting structural integrity and dimensional accuracy. This study assesses the inherent strain method (ISM) as a computationally efficient alternative to full thermomechanical simulations for predicting these effects. By integrating ISM with experimental validation via the contour method, the research provides specific insights into stress distribution patterns in geometries featuring stress concentrators such as notches. Results demonstrate a strong correlation between simulation and experimental data, particularly at the mid-height regions. Quantitatively, the orthotropic ISM successfully predicted the peak residual stress at 1101.4 MPa, showing excellent agreement within a 4% error margin against the experimental maximum of 1144 MPa captured via the contour method. These findings underscore how ISM can be effectively applied to practical engineering components to predict high-stress zones, enabling the design of distortion-compensated parts without the high computational cost of traditional models. Ultimately, this method facilitates more robust process optimization and enhances the quality and reliability of Ti6Al4V components manufactured via PBF-LB.

1. Introduction

Laser beam powder bed fusion (PBF-LB) has emerged as a leading additive manufacturing process for the production of complex, high-value metal components in moderate volumes. Mechanical properties of parts manufactured using PBF-LB are frequently comparable to those of bulk materials that are conventionally manufactured [1,2,3], which has resulted in its pervasive adoption across a variety of industries, such as aerospace, automotive, biomedical, and power generation [4,5,6,7,8]. However, the generation of residual stresses during the manufacturing process presents a substantial challenge in PBF-LB, despite these advantages. Consequently, these residual stresses may induce a reduction in dimensional accuracy, part distortion, and even manufacturing failures [9,10,11].

Thermal manufacturing processes such as PBF-LB are intrinsically characterized by the formation of residual stresses and subsequent distortions, resulting from uneven cooling and contraction driven by rapid temperature gradients. The prediction and mitigation of residual stress-induced distortions are essential for assuring the structural integrity and functional performance of PBF-LB components, as these effects are particularly acute in PBF-LB due to the localized and rapid heating involved [12,13]. The challenge of accurately predicting stress distributions in components with complex geometrical features persists, even though a multitude of studies have investigated micro- and meso-scale stress fields using simulation tools [14,15,16,17].

Various techniques have been developed to estimate residual stress-induced distortions in PBF-LB parts, ranging from experimental methods such as X-ray diffraction (XRD) and neutron diffraction (ND) [9,10,18,19] to advanced numerical modeling [20,21]. For instance, XRD has been employed to assess residual stresses in PBF-LB-fabricated AISI 316L, while ND has been used to achieve good correlation in laser-direct metal printing [18]. Optical and hole-drilling techniques have also been utilized for stress analysis in PBF-LB components, demonstrating the versatility of experimental approaches. Liu et al. [9] examined stress distribution using a thermal-gradient method. Childs et al. [9] investigated the effect of several factors on the volume of molten thin layers in PBF-LB and estimated that melting density rose as the scanning rate increased. Read et al. [22] and Kempen et al. [23] concentrated on optimizing the PBF-LB processing variables for the fabrication of near-completely solid AlSi10Mg components. Vrancken et al. [24] studied the residual stress in PBF-LB titanium compact tension samples manufactured in a variety of construction directions. Siewert et al. [25] and Zhang et al. [26] have conclusively proven that the directional nature of layer-wise deposition, coupled with hatch-rotation and stripe scanning strategies, introduces pronounced structural anisotropy. However, despite these advancements, in situ evaluation of residual stress during the PBF-LB process remains a formidable challenge due to the complexity and dynamic nature of the process.

Primarily, prior research has concentrated on the comprehension and mitigation of residual stress in PBF-LB materials that are more frequently employed. The distinctive metallurgical characteristics of Ti6Al4V, however, require a customized approach. This study addresses these challenges by employing the inherent strain method (ISM) as a computationally efficient alternative to traditional thermomechanical modeling. While previous studies have validated ISM for predicting residual stresses in simple geometries, its applicability to features with high-stress concentrations, common in practical engineering components, remains to be explored. The ISM reduces computational complexity by focusing on inherent strain rates, thereby balancing accuracy and efficiency [27]. Experimental validation is performed using the contour method to provide high-resolution stress maps of the notched specimens. By combining simulation and experimental approaches, this study offers a robust framework for understanding how geometric irregularities influence the residual stress state in PBF-LB-manufactured Ti6Al4V components. The insights obtained from this investigation will ultimately be used to improve the overall quality and reliability of Ti6Al4V parts manufactured through PBF-LB by optimizing PBF-LB process parameters, designing distortion-compensated components, and developing advanced post-processing techniques.

2. Materials and Methods

This study conducted experimental and computational investigations to analyze the distortions and residual stress in PBF-LB-produced Ti6Al4V parts (Pakistan Institute of Additive Manufacturing (PIAM 3D), Islamabad, Pakistan). Cube specimens with dimensions of 15 × 15 × 15 mm³ were additively manufactured to facilitate this analysis. To introduce geometric irregularities, notches were designed on the upper portion of the cube samples. These notches were specifically designed to amplify stress concentration zones, enabling a more rigorous validation of the predictive modeling approach. Figure 1 illustrates the CAD model with dimensions and the corresponding manufactured Ti6Al4V specimen.

The elemental composition of the Ti6Al4V powders used in this study is provided in

Table 1. Chemical composition of Ti6Al4V.

Element	Titanium	Aluminum	Vanadium	Iron	Oxygen	Carbon	Nitrogen	Hydrogen
Weight (%)	Balance	5.50–6.50	3.50–4.50	≤0.25	≤0.13	≤0.08	≤0.05	≤0.01

. The manufacturing process was carried out using a Farsoon FS421M PBF-LB machine (Pakistan Institute of Additive Manufacturing (PIAM 3D), Islamabad, Pakistan), which operates with a single laser, a laser power of 200 W, and a maximum scan speed of 1000 mm/s. The resulting samples were then subjected to stress and distortion analyses using the contour method and 3D scanning, respectively. To simulate the PBF-LB process, the inherent strain method was applied using Simufact Additive 2022.

2.1. Material and PBF-LB Processing

Spherical Ti6Al4V powders with a particle size of 50 µm, a Poisson’s ratio of 0.26, a density of 4.38 g/cm³, a Young’s modulus of 110 GPa, a tensile strength of 1290 MPa, and a yield strength of 1140 MPa were employed and are consistent with standard PBF-LB processed titanium. The PBF-LB process was performed on the Farsoon FS421M LPBF machine using a strip-wise scanning strategy. To ensure the relevance of the results to practical engineering applications, process parameters were selected to replicate high-density industrial printing conditions, as shown in

Table 2. Process parameters for PBF-LB.

Parameters	Values
Laser power (W)	200
Layer thickness (mm)	0.03
Scan speed (mm/s)	1000
Scan width (mm)	10
Scan overlap (mm)	0.14
Beam spot size (mm)	0.08
Hatch distance (mm)	0.09
Recoater spread speed (mm/s)	120
Hatch strategy	Strip-wise
Material type	Ti6Al4V

. These parameters match established baseline criteria for printing low-porosity titanium components with reproducible mechanical properties [6,23].

2.2. Numerical Simulation and ISM Calibration

The inherent strain method (ISM) was employed to predict the macro-scale residual stresses in the manufactured Ti6Al4V components utilizing Simufact Additive 2022. This method simplifies modeling by focusing solely on inherent strain rates and eliminating the need for detailed thermal and phase transformation analyses. Originally pioneered by Ueda et al. [28] for large-scale welding assemblies and subsequently adapted for powder bed fusion processes by Keller and Ploshikhin [29], the ISM serves as an efficient reduced-order macro-scale model. It bypasses the computationally prohibitive layer-by-layer transient thermal-fluid simulations by applying a localized, pre-calibrated strain field, the inherent strain, directly to the hatched volume. To account for the directional nature of the layer-wise printing process, an orthotropic calibration was performed. As illustrated in Figure 2, the mechanical simulation focused solely on inherent strain magnitudes to calculate stresses and distortions, omitting phase transformation and thermal strains to reduce computational time.

In a fully coupled transient thermomechanical continuum analysis, the total strain increment tensor field ${\mathsf{\epsilon }}_{\mathrm{total}}$ is decomposed additively into its constituent components:

$${\mathsf{\epsilon }}_{\mathrm{total}}={\mathsf{\epsilon }}_{\mathrm{e}}+{\mathsf{\epsilon }}_{\mathrm{p}}+{\mathsf{\epsilon }}_{\mathrm{th}}+{\mathsf{\epsilon }}_{\mathrm{ph}}$$

(1)

where ${\mathsf{\epsilon }}_{\mathrm{e}}$ is the elastic strain, ${\mathsf{\epsilon }}_{\mathrm{p}}$ is the plastic strain, ${\mathsf{\epsilon }}_{\mathrm{th}}$ is the thermal strain driven by severe spatial temperature gradients ($\nabla T$), and ${\mathsf{\epsilon }}_{\mathrm{ph}}$ represents the strain induced by microstructural phase transformations.

The fundamental principle of the ISM relies on isolating the non-elastic, non-transient strains that persist after the component has completely cooled to the ambient baseplate temperature. This net permanent structural strain is defined as the inherent strain tensor, ${\mathsf{\epsilon }}^{\ast }$:

$${\mathsf{\epsilon }}^{\ast }={\mathsf{\epsilon }}_{\mathrm{total}}-{\mathsf{\epsilon }}_{\mathrm{e}}={\mathsf{\epsilon }}_{\mathrm{p}}+{\mathsf{\epsilon }}_{\mathrm{th}}+{\mathsf{\epsilon }}_{\mathrm{ph}}$$

(2)

In a finite element formulation, it is convenient to use Voigt notation, where second-order symmetric tensors are mapped to column vectors. In this form, the linear elastic constitutive relation governing the macro-scale mechanical response is given by

$$\mathsf{\sigma }=\mathbf{D}\left({\mathsf{\epsilon }}_{\mathrm{total}}-{\mathsf{\epsilon }}^{\ast }\right)$$

(3)

where $\mathsf{\sigma }={\left[{\sigma }_{xx},{\sigma }_{yy},{\sigma }_{zz},{\tau }_{yz},{\tau }_{xz},{\tau }_{xy}\right]}^{\mathrm{T}}$ is the residual stress vector, and $\mathbf{D}$ is the $6\times 6$ anisotropic elastic constitutive matrix populated by the material’s temperature-independent elastic parameters. It must be explicitly acknowledged that treating the constitutive tensor $\mathbf{D}$ as temperature-independent introduces a fundamental physical simplification. During actual PBF-L processing, Ti6Al4V experiences severe cyclic thermal histories, where its elastic modulus and yield strength deteriorate significantly at elevated temperatures. Because the macro-scale ISM abstracts these complex transient conditions into an equivalent static strain field applied at room temperature, it does not explicitly capture intermediate high-temperature micro-plasticity, transient stress relaxation, or localized phase transformation kinetics during deposition. While this room-temperature assumption is highly effective and widely accepted for predicting macro-scale residual stresses and cumulative elastic springback upon baseplate release, it can introduce localized boundary errors. Specifically, in regions subjected to severe geometric discontinuities and sharp thermal gradients, the omission of transient, temperature-dependent plastic yielding may lead to minor deviations in peak stress magnitudes compared to full thermomechanical representations. This limitation is the primary trade-off for the orders-of-magnitude reduction in computational time offered by the framework.

The global stiffness matrix equation for static mechanical equilibrium of the finite element assembly is governed by

$$\mathbf{K}\mathbf{u}={\mathbf{F}}^{\ast }$$

(4)

where $\mathbf{K}$ is the global structural stiffness matrix, $\mathbf{u}$ is the nodal displacement vector, and ${\mathbf{F}}^{\ast }$ is the equivalent macro-scale nodal force vector induced solely by the introduction of the inherent strain field. The vector ${\mathbf{F}}^{\ast }$ is computed via a volume integral over the consolidated geometry:

$${\mathbf{F}}^{\ast }={\int }_V{\mathbf{B}}^{\mathrm{T}}\mathbf{D}{\mathsf{\epsilon }}^{\ast }\mathrm{d}V$$

(5)

where $\mathbf{B}$ is the kinematic strain–displacement matrix relating local total strains to nodal displacements (${\mathsf{\epsilon }}_{\mathrm{total}}=\mathbf{B}\mathbf{u}$). Solving Equation (4) for the global displacement vector $\mathbf{u}$, the local residual stress state is explicitly reconstructed via

$$\mathsf{\sigma }=\mathbf{D}\left(\mathbf{B}\mathbf{u}-{\mathsf{\epsilon }}^{\ast }\right)$$

(6)

To accurately reflect the layer-wise deposition and directional laser-scanning strategies characteristic of PBF-LB, an orthotropic calibration was performed [19,25]. Rather than assuming an isotropic contraction, the components of the inherent strain vector were defined relative to the printer’s primary kinematic orientations:

$${\mathsf{\epsilon }}^{\ast }={\left[{\epsilon }_{xx}^{\ast },{\epsilon }_{yy}^{\ast },{\epsilon }_{zz}^{\ast },0,0,0\right]}^{\mathrm{T}}$$

(7)

To ensure scientific reproducibility, the calibration of these orthotropic inherent strain values was conducted using a standardized, double-cantilever benchmark geometry within the Simufact Additive 2022 optimization workspace. This physical benchmark artifact possessed a geometry consisting of a central support block (10 × 10 × 10 mm³) with dual overhanging cantilever arms extending 15 mm on opposing sides. Crucially, to isolate the precise strain fields, this calibration coupon was additively manufactured under the identical, high-density industrial process parameters listed in Table 2.

An iterative inverse optimization routine using a localized Gauss–Newton minimization approach was executed until the root-mean-square error (RMSE) between the experimental and simulated displacement fields fell below a threshold of 1.0%. By correlating numerical distortion profiles with experimental benchmarks within the Simufact environment, the orthotropic calibration yielded the specific values utilized in this study: ${\epsilon }_{xx}^{\ast }=-0.0054$, ${\epsilon }_{yy}^{\ast }=-0.0023$, and ${\epsilon }_{zz}^{\ast }=-0.0300$. The heavily pronounced magnitude of ${\epsilon }_{zz}^{\ast }$ explicitly accounts for the additive, layer-by-layer accumulation of thermal contraction acting along the build direction. The physical and mechanical basis for this orthotropic strain distribution stems directly from the thermomechanical constraints of the PBF-LB/M process. The in-plane components (${\epsilon }_{xx}^{\ast }$ and ${\epsilon }_{yy}^{\ast }$) are driven by individual laser track scan vectors; because a standardized 67⁰-layer hatch rotation strategy was employed, the directional thermal contractions within the horizontal plane are partially homogenized, though they remain slightly distinct due to the primary scanning orientations relative to the rectangular baseplate layout. In contrast, the vertical component (${\epsilon }_{zz}^{\ast }$) exhibits a significantly larger magnitude [30]. This pronounced vertical strain explicitly accounts for the cumulative, layer-by-layer accumulation of plastic thermal contractions acting along the build direction, coupled with the rigid structural constraint imposed by the thick baseplate, which prevents free elastic relaxation until wire EDM release.

2.3. Experimental Setup

The main objective of this study was to evaluate and analyze residual stresses and distortions post-manufacturing. Ti6Al4V cubic samples were fabricated on a PBF-LB machine to analyze the process-induced stresses and deformations. To validate the predictive capability of the orthotropic ISM framework, the internal residual stress fields were experimentally mapped using the contour method, a destructive stress-measurement technique based on solid mechanics principles originally formalized by Prime [31]. The notched PBF-LB component was sectioned along its vertical mid-plane using a high-precision wire EDM machine (FANUC RoboCut) (Pakistan Institute of Additive Manufacturing (PIAM 3D), Islamabad, Pakistan). To guarantee an ultra-clean, symmetric cut and minimize thermal artifacts or localized cutting-induced stresses, a zinc-coated brass wire with a diameter of 0.25 mm was operated under a skim-cut setting. Crucially, to prevent any premature structural shifting or clamping forces on the wire caused by bulk elastic relaxation during sectioning, the specimen was securely constrained using a customized, symmetric dual-clamp fixture. This arrangement ensured that both halves of the component remained rigidly fixed to the EDM machine’s worktable throughout the entire cutting path. Following the sectioning process, the resulting deformed topologies of the twin released surfaces were digitized using a high-precision structured-light 3D optical scanner (ATOS Core, GOM GmbH, Islamabad, Pakistan). The scanner was calibrated to operate with a fine spatial resolution (lateral point spacing) of approximately 45 µm and achieved an out-of-plane measurement precision of ±2 µm. The contour method was chosen for its high spatial resolution and ability to measure complex stress distributions, making it an ideal complement to the ISM, as depicted in Figure 3.

The contour method is an empirical approach for measuring residual stress. Figure 4 reveals the use of the wire EDM operation for slicing the PBF-LB-manufactured sample. The residual stress within the sample causes the sliced surface to distort.

Figure 5 outlines the fundamental principle of the contour approach. The stress analyzed is normal to the sliced surface. A 3D scanner is used to measure the profile of the relaxed surface of the sliced halves. The data are then smoothed, averaged, and processed. The inverse of the observed pattern is applied as displacement boundary conditions in a 3D FEM of the sliced surface, followed by an FEA to evaluate the sample’s stress distribution.

3. Results and Discussion

3.1. Experimental Stress and Distortion Mapping

The contour method was employed to analyze residual stresses along the build orientation of Ti6Al4V cubic samples manufactured via PBF-LB. Quantitatively, the component exhibited a total peak-to-valley geometric deviation profile ranging from a maximum positive displacement of +0.02 mm to a minimum negative displacement of −0.07 mm (as shown in Figure 6 and Figure 7d). For a compact component envelope of 15 × 15 × 15 mm³, this cumulative dimensional variation of 0.09 mm represents a distinct structural distortion. This deformation corresponds to approximately three times the nominal powder layer thickness (30 µm), indicating significant elastic springback driven by the relaxation of internal macro-residual stress fields during baseplate detachment. Frameable within industrial manufacturing tolerances, this measurable deformation underscores the inadequacy of relying on purely qualitative assessments and highlights the technical necessity of employing predictive numerical frameworks like the ISM to anticipate macro-scale distortions prior to manufacturing.

The residual stress map obtained from the contour method reveals a maximum stress of 1144 MPa, localized primarily at the mid-height of the specimen and near the edges, as depicted in Figure 7b. The high-resolution map confirms that the presence of the notches, acting as geometric stress concentrators, significantly alters the stress field compared to a standard solid cube. Notably, line 2 (as per Figure 8a) exhibits higher tensile residual stresses compared to other locations, which is a characteristic result of the constrained thermal contraction during the cooling phase of the PBF-LB process. Figure 7a highlights areas of higher residual stress, particularly near the midpoints of lines 1 and 2 (as indicated in Figure 8a). The total surface deformations are depicted in Figure 8d.

3.2. Predictive Accuracy of Orthotropic ISM

The numerical simulation using orthotropic ISM predicted a maximum residual stress of 1101.4 MPa, which closely aligns with the experimental results (contour method reached a peak value of 1144 MPa near the notched boundaries), with a deviation of less than 4% from the experimental peak, as depicted in Figure 7c. The proximity of the peak stresses to the nominal yield strength of 1140 MPa prompts a crucial mechanical evaluation regarding localized micro-plastic yielding. As-built PBF-LB Ti6Al4V components are characterized by a highly non-equilibrium microstructural state consisting of ultra-fine, hierarchical acicular α’ martensite within prior-β grains, a direct consequence of ultra-high thermal cooling rates. Consequently, while the material can structurally sustain near-yield elastic stress fields, localized micro-plastic yielding inevitably develops at severe stress concentrators, such as the notch roots investigated here, during the intense thermal contraction cycles of the layer-wise printing process. Furthermore, because the contour method acts as a destructive evaluation technique, the gradual release of bulk internal stresses during wire EDM sectioning creates a moving stress concentration at the cutting tip. If this transient stress concentration induces minor localized plastic deformation or micro-yielding along the cut path, the linear elastic formulation utilized during the boundary-value finite element reconstruction will treat this non-elastic deformation as additional elastic springback. This baseline behavior can lead to a slight mathematical overestimation of the peak value, explaining why the experimental elastic reconstruction (1144 MPa) slightly exceeds the nominal yield strength and sits higher than the statically bounded numerical ISM prediction (1101.4 MPa). The stress distribution profiles (Figure 8) demonstrate a strong correlation, especially along the midsection of evaluation lines (red lines, Figure 8a). The simulated stress distribution map closely resembles the experimental results from the contour method (Figure 7b), with slight discrepancies observed in the upper regions of the specimen. These variations may be attributed to localized thermal effects during the end-of-printing phase, which the ISM does not fully capture. While the ISM effectively captures the global strain field through orthotropic calibration, it does not account for the localized reduction in heat conduction near the end of part fabrication, which may lead to minor stress relaxation in the final layers that the static model does not fully replicate.

3.2.1. Parametric Influence of Notch Geometry on Residual Stress Fields

To leverage the computational efficiency of the validated ISM framework, a numerical parametric study was conducted to evaluate how notch size, sharpness, and shape profile govern local residual stress concentrations. Two alternative configurations were simulated using the identical process conditions and calibrated orthotropic strains detailed in Table 2: a sharp V-notch (root radius halved to r = 0.25 mm) and a smooth U-notch (semi-circular geometry with a radius of r = 1.0 mm at an equivalent depth). The numerical extractions along the notch root plane reveal a distinct geometric sensitivity. The sharp V-notch profile drives a higher localized constraint, shifting the peak tensile residual stress upward to 1173 MPa. This localized escalation represents a 6.5% increase over the baseline design, suggesting that sharper manufacturing radii accelerate local micro-plastic yielding. Conversely, the rounded U-notch profile mitigates local thermal-contraction constraints, distributing the stress fields more uniformly across the geometric boundary and lowering the peak residual stress to 969 MPa (a 12% reduction relative to the baseline). This rapid comparative evaluation demonstrates that the calibrated ISM framework operates as an effective, low-overhead design tool for optimizing geometric features and mitigating cracking risks in complex PBF-LB/M/Ti6Al4V components prior to printing.

3.2.2. Supplementary Error Analysis and Physical Limitations of the Macro Framework

An evaluation of the stress distributions near the upper boundary of the specimen reveals a localized deviation between the numerical ISM predictions and the experimental contour method profiles. This minor spatial mismatch serves as an indicator of the inherent physical trade-offs associated with reduced-order macro-solvers. Mechanically, the final residual stress state of additively manufactured PBF-LB/M/Ti6Al4V is heavily influenced by transient temperature fields and solid-state phase transformations. The ultra-rapid cooling rates trigger a non-equilibrium β → α’ martensitic transformation. This metallurgical transition is accompanied by a localized volumetric expansion and transformation-induced plasticity, both of which act to partially relax the accumulating thermal tensile stresses. Because the macro-scale ISM translates complex, cyclically transient thermal histories into a static, equivalent inherent strain tensor applied at room temperature, it does not explicitly calculate these intermediate, time-dependent phase change expansion steps or the localized boundary variations in heat dissipation experienced by the final deposited top layers. Consequently, the omission of these microstructural kinetics is expressed mathematically as a minor localized overestimation of the stress state in the upper regions compared to the physical contour measurements [32]. Identifying these underlying mechanisms provides a clear boundary context for the model while reinforcing that the framework maintains high engineering accuracy for bulk macro-stress patterns at a highly accelerated computational speed.

3.2.3. Benchmarking ISM Against Conventional Transient Thermomechanical Solvers

To contextualize the practical and industrial significance of the validated orthotropic ISM framework, a comparative evaluation was conducted against conventional transient thermomechanical finite element modeling (FEM).

Table 3. Computational and capability benchmarking of orthotropic ISM against conventional transient thermomechanical FEM.

Performance Metric	Conventional Transient Thermomechanical FEM	Orthotropic Inherent Strain Method (ISM)
Primary Physical Mechanism	Time-dependent transient thermal dissipation, melt pool fluid dynamics, and temperature-dependent plastic flow.	Static application of calibrated macro-scale thermal contraction tensor (${ϵ}^{\ast }$) applied layer by layer.
Computational Time ($15\times 15\times 15{ \mathrm{mm}}^3$ envelope)	$\sim 36 \mathrm{to} 48 \mathrm{h}$ (dependent on track-by-track mesh refinement).	$\sim 8 \mathrm{to} 12 \mathrm{m}\mathrm{i}\mathrm{n}$ (mesh independent of laser track spot size).
Data Storage/Disk Space Overhead	High ($>150 \mathrm{GB}$ due to micro-step transient increments).	Ultra-low ($<1 \mathrm{GB}$ via static structural increments).
Micro-scale Resolution	Captures localized track-scale melting behavior and microstructural thermal cycles.	Abstracted; cannot resolve localized melt-pool track edge phenomena.
Macro-scale Bulk Stress Fidelity	High accuracy across the entire volume.	High accuracy ($<4\%$ peak stress error verified at the core regions).

details the performance trade-offs, computational overhead, and analytical capabilities characteristic of both numerical pathways.

As shown in Table 3, conventional transient thermomechanical FEM tracks time-dependent micro-scale phenomena and path-dependent thermal variations, but its intensive computational time and massive data storage requirements make it highly restrictive for large-scale industrial components or rapid iterative design optimization loops [33]. In contrast, the orthotropically calibrated ISM framework operates as a reduced-order model. By abstracting transient thermodynamic cycles into a static equivalent strain tensor, it resolves the bulk macro-scale stress concentration fields and sharp geometry-driven gradients with an error margin under 4% while reducing computational execution times by orders of magnitude. This balance of efficiency and accuracy establishes the orthotropic ISM as a highly practical pathway for the routine optimization and distortion compensation of complex PBF-LB/M/Ti6Al4V components.

3.3. Comparative Analysis

Figure 8a shows the corresponding stress profile evaluation lines 1, 2, 3, and 4. The stress distribution profiles along lines 1, 2, 3, and 4, as shown in Figure 8b, exhibit slight fluctuations, with higher residual stresses observed along edges 1 and 2. This variation is likely attributed to the build orientation during PBF-LB printing.

The stress distribution profiles from the simulation and contour methods, as shown in Figure 8c, exhibit reasonable agreement, with minor discrepancies at the edges. Figure 8d compares the stress profiles along lines 1 and 2, showing a strong correlation between the ISM predictions and the contour method results, except for slightly higher numerical stresses at the upper locations.

For lines 3 and 4, as illustrated in Figure 8e, both methods show good correlation in stress distribution along the diagonals, though there are differences at the upper corners. The contour technique indicates lower stresses at the top corners, while the numerical model predicts slightly higher stresses in these regions. Nonetheless, both approaches confirm that the opposing edges of the Ti6Al4V sample exhibit high residual stresses.

To rigorously isolate and highlight the effect of the top serrated notches on the resulting stress fields, a quantitative numerical baseline comparison was conducted against a standard, unnotched solid cube (15 × 15 × 15 mm³) simulated under identical manufacturing and boundary conditions. The comparative results demonstrate that geometric discontinuities fundamentally distort the classical macro-scale residual stress profile typical of plain prismatic elements. Along the vertical centerline, a standard unnotched baseline specimen exhibits a typical parabolic residual stress profile that smoothly transitions from high tensile stress near the baseplate interface into a balanced, uniform compressive field before decaying back toward zero near the upper free surface. In stark contrast, the introduction of the serrated notch topology forces a sharp stress redistribution, preventing the expected decay and instead generating an aggressive tensile gradient near the top surface boundary as the material attempts to elastically conform around the structural discontinuities. Along the horizontal notch root plane, the baseline unnotched control cube exhibits an essentially flat, uniform internal stress distribution across the bulk section, fluctuating mildly around a mean of 420–450 MPa [34]. However, in the notched component, the stress field undergoes severe localized magnification, with peak normal tensile stresses soaring to 1101.4 MPa in the simulation and 1144 MPa in the experimental validation. This represents a dramatic localized stress magnification of approximately 156% relative to the prismatic baseline. These quantitative differences explicitly demonstrate that the orthotropically calibrated ISM framework remains highly capable of capturing sharp, geometry-induced stress gradients and severe localized concentrations, validating its use as a rapid, high-fidelity engineering pathway for distortion forecasting and process optimization in complex components.

The results from this study align with previous findings on stress concentrations in PBF-LB-manufactured components, particularly around geometric discontinuities [24,35]. The observed stress distributions and magnitudes provide a deeper understanding of residual stresses and their implications for part performance. The findings of this study have significant implications for industrial applications of PBF-LB. The identification of high-stress regions enables targeted optimization of process parameters, reducing the risk of part failure and enhancing component reliability. Due to the errors induced by wire EDM, 3D scanner measurements, and data smoothing, it is exceedingly difficult to obtain a precise distribution map of residual stresses near the distorted surface using the contour approach [36]. While the ISM provides a practical and efficient framework for residual stress prediction, it does not account for thermal and microstructural transformations during the PBF-LB process. Considering the economical behavior and short computational time of numerical simulations based on the ISM, the alignment between the ISM and the contour approach is of great industrial interest.

4. Conclusions

Laser-based powder bed fusion has emerged as a prominent additive manufacturing technique, widely used across high-performance industrial sectors. However, the inherent thermal gradients during processing often lead to significant residual stresses and geometric distortions, which can compromise the quality and structural integrity of the manufactured parts. This study evaluated the effectiveness of an orthotropically calibrated macro-scale inherent strain method (ISM) framework for predicting complex residual stress fields in PBF-LB/M/Ti6Al4V components featuring severe geometric stress concentrators. By systematically comparing numerical predictions with experimental results obtained via the destructive contour method, several key conclusions were reached:

• Reduced-order macro-scale frameworks, when calibrated using directional inherent strain vectors, exhibit high fidelity for capturing severe, geometry-induced stress gradients. The framework successfully mapped localized tensile stress accumulations forced by geometric bottlenecks without requiring the intensive computational resources associated with transient thermomechanical simulations.
• The introduction of severe geometric discontinuities fundamentally alters the classical macro-scale residual stress profiles typical of plain prismatic additive elements. A quantitative comparison against an unnotched baseline control revealed a dramatic stress magnification at the notch roots, highlighting that structural variations disrupt standard thermal contraction paths and necessitate localized numerical forecasting tools during part design.
• The destructive contour method, augmented by specialized bivariate cubic spline smoothing routines, provides an exceptionally viable, high-resolution bulk validation metric for numerical modeling. The close alignment maintained across diverse structural directions confirms the viability of integrating reduced-order ISM models into industrial post-printing optimization and qualification pipelines.

These findings are particularly relevant for industries such as aerospace and biomedical engineering, where the structural integrity of additively manufactured components is critical. To advance the current macro-scale framework, future research will target a concrete characterization and modeling pipeline designed to investigate the intense interactions between localized residual stress gradients and the unique, highly non-equilibrium acicular α’ martensitic microstructure characteristic of as-built PBF-LB/M/Ti6Al4V. High-resolution electron backscatter diffraction and micro-focus X-ray diffraction (XRD) mapping will be conducted directly adjacent to the notch roots. This experimental workflow aims to quantify localized dislocation densities, microstrains, and potential stress-induced α’ martensitic variant selections driven by severe directional thermal gradients and geometric constraints. These metallurgical parameters will be coupled with computational mechanics to develop a stress-assisted phase transformation constitutive model.