Predictive Numerical Modeling of Inelastic Buckling for Process Optimization in Cold Forging of Aluminum, Stainless Steel, and Copper

Abstract

The growing demand for precision and consistency in the forging industry has heightened the need for predictive simulation tools. While extensive research has focused on parameters such as flow stress, die wear, billet fracture, and residual stresses, the phenomenon of billet buckling, especially during cold upset forging, remains underexplored. Most existing models address only elastic buckling for slender billets using classical approaches like Euler and Rankine-Gordon formulae, which are not suitable for inelastic deformation in shorter billets. This study presents a numerical model developed to analyze inelastic buckling during cold forging and to determine associated stresses and deflection characteristics. The model was validated through finite element simulations across a range of billet geometries (10–40 mm diameter, 120 mm length), materials (aluminum, stainless steel, and copper), and friction coefficients (µ = 0.12, 0.16, and 0.35). Stress distributions were evaluated against die stroke, with particular emphasis on the influence of strain hardening and geometry. The results showed that billet geometry and strain-hardening exponent significantly affect buckling behavior, whereas friction had a secondary effect, mainly altering overall stress levels. A nonlinear regression approach incorporating material properties, geometric parameters, and friction was used to formulate the numerical model. The developed model effectively estimated buckling stresses across various conditions but could not precisely predict buckling points based on stress differentials. This work contributes a novel framework for integrating material, geometric, and process variables into stress prediction during forging, advancing defect control strategies in industrial metal forming.

1. Introduction

The forging industry has evolved significantly in recent decades, driven by an increasing demand for high-dimensional accuracy, reproducibility, and superior surface quality, particularly in sectors such as automotive, aerospace, and precision engineering [1,2]. These stringent requirements have elevated the role of numerical modeling and finite element simulations in forging process design [3]. Simulations not only reduce the need for expensive and time-consuming trial-and-error experiments but also help engineers predict critical variables such as strain distribution, flow stress, die wear, and final product quality [4,5,6]. Despite this progress, the phenomenon of billet buckling, especially during cold upset forging, remains underexplored in the literature. Buckling, if not anticipated, can result in part misalignment, uneven cross-sections, or even process failure, undermining overall process efficiency [7,8].

Traditionally, buckling analysis has been rooted in classical elastic models such as Euler’s critical load theory, which assumes idealized boundary conditions, small deformations, and purely elastic material response [7]. These models are most applicable to long, slender columns subjected to axial loads. However, in forging operations particularly cold upset forging, the billets are typically short and undergo significant plastic deformation [9]. For this reason, the inelastic buckling model is more appropriate than Euler’s model, as it accounts for material plasticity, nonlinear stress–strain response, and strain-hardening effects that dominate during forging deformation. This makes it better suited to capture the true instability mechanisms in short, plastically deforming billets. In such scenarios, inelastic buckling becomes a more appropriate framework for analysis. This post-yield instability is influenced not only by geometric parameters but also by the material’s strain-hardening behavior and its interaction with friction and boundary constraints [10,11]. In the simulations, post-yield instability is identified by tracking the deviation of the billet’s load–stroke curve from the expected monotonic hardening path and by monitoring the onset of lateral deflection once yielding initiates, which distinguishes true buckling from ordinary plastic flow.

The strain-hardening characteristics of a material play a crucial role in its resistance to deformation instabilities. Specifically, the strain-hardening exponent n and strength coefficient K, as defined in the Hollomon equation σ = Kεn, govern the rate at which a material strengthens during plastic deformation [12,13,14]. Aluminum and copper, for example, often exhibit higher strain-hardening exponents (up to 0.5), indicating their ability to sustain significant deformation before failure [14]. Stainless steels, depending on their grade and thermal treatment, may exhibit lower but still significant hardening behavior. The influence of these material properties on buckling during forging remains poorly quantified in existing literature [15]. While some studies have explored their effects in processes such as hydroforming and extrusion, their direct influence on forging-induced buckling has yet to be systematically investigated [16].

Moreover, frictional conditions between the die and billet surfaces introduce another layer of complexity. High friction can restrain material flow, increase required forming loads, and influence the internal stress distribution of the billet [17]. In this study, the die–billet interface is carefully varied to examine how different frictional conditions influence stress localization, deformation uniformity, and ultimately the onset of buckling, since high interfacial friction has been shown to accelerate instability by constraining lateral flow. Some studies have demonstrated that friction can significantly affect surface quality and residual stress patterns in cold forging. However, its effect on deformation instability, specifically buckling, has rarely been considered in simulation-based studies [17,18,19,20]. This studies suggests a need to reframe the modeling of forging from a holistic perspective that considers the combined influence of geometry, material behavior, and process parameters.

It is equally important to recognize that different materials behave uniquely under compressive loads due to variations in crystal structure, strain-rate sensitivity, and work-hardening characteristics [21]. Aluminum alloys are lightweight and exhibit high ductility, making them suitable for cold forging, but they also tend to develop localized deformation zones under specific conditions [22]. Copper is even more ductile and thermally conductive, while stainless steel offers excellent corrosion resistance and mechanical strength but may exhibit lower ductility [23]. These distinct characteristics influence each material’s susceptibility to buckling. For instance, dynamic strain aging (DSA), a phenomenon often observed in Al–Mg alloys, can cause serrated yielding and strain localization, thereby influencing buckling onset [24,25]. As such, it is essential to develop material-specific models to predict forging behavior more accurately.

Despite the existence of several analytical models for predicting load and stress in forging, most are limited to elastic analysis or simple constitutive assumptions. Classical formulations like the Rankine–Gordon or Euler models do not account for nonlinear material behavior or complex boundary conditions [26,27]. More recent studies utilizing FEM tools such as DEFORM-3D and ABAQUS have succeeded in modeling plastic deformation and stress distribution during forging, but only a few have attempted to develop numerical models from the simulation data that can be practically applied in process planning [28,29,30]. Petkar et al. [31] demonstrated the effectiveness of artificial neural networks (ANN) in predicting effective stress and punch force during backward extrusion but did not explore buckling phenomena [31]. Similarly, Frater et al. [32] analyzed cold upset cracking using finite element analysis but did not examine lateral deflection or deformation instability.

This research seeks to bridge that gap by developing a comprehensive numerical model for inelastic buckling during cold upset forging for aluminum, stainless steel, and copper. The focus is on modeling the onset and magnitude of buckling in relation to billet geometry, material properties, and frictional conditions. The resulting stress-die stroke data is then subjected to nonlinear regression analysis to derive a predictive formula that considers the complex interactions between input variables.

The novelty of this work lies in its multi-dimensional modeling approach that combines numerical simulations with empirical model development. Unlike prior studies that either simulate or theorize in isolation, this research builds a numerical formulation grounded in verified simulation data. Furthermore, the inclusion of three different materials and varied frictional conditions ensures that the model is robust and broadly applicable. The numerical model not only estimates buckling stresses with high accuracy but also explores the influence of strain-hardening behavior on buckling risk, a factor that has received minimal attention in the existing body of forging literature.

2. Materials and Methods

2.1. Material Selection

Three material qualities commonly used in cold forging were selected for analysis: pure aluminum, pure copper, and austenitic stainless steel (Grade 310: 24–26 wt% Cr, 19–22 wt% Ni, ≤0.25 wt% C, ≤2.0 wt% Mn, ≤1.5 wt% Si, ≤0.045 wt% P, ≤0.03 wt% S, balance Fe). Other materials include mild steel and titanium that were included exclusively for validation purposes. Material behavior was defined using elastic and plastic properties. The elastic properties include Young’s modulus and Poisson’s ratio, all sourced from ANSYS 2023R1 material library. Plasticity was modeled using the power hardening law defined by the compressive yield strength and strain-hardening exponent (

Table 1. Summary of material properties used for pure aluminum [7], austenitic stainless steel and pure copper.

Material	Young’s Modulus (MPa)	Poisson’s Ratio	Yield Strength (MPa)	Strain-Hardening Exponent (n)
Pure Aluminum	71,000	0.33	280	0.20
Austenitic Stainless Steel	193,000	0.31	207	0.40
Pure Copper	110,000	0.34	280	0.50

).

Geometries were developed using DS SolidWorks 2022, a CAD software by Dassault Systems (Vélizy-Villacoublay, France). The forging assembly comprised cylindrical billets with a fixed length of 120 mm and varying diameters (10–40 mm), resulting in different aspect ratios. Dies were designed as rectangular blocks (200 mm × 150 mm × 50 mm) to ensure complete billet coverage during deformation. Components were modeled as separate solid bodies to simplify simulation input management. Figure 1 presents the upset forging setup for a 25 mm diameter billet, as adapted from previous work [7].

ANSYS 2023R1 was chosen for its robust finite element capabilities and CAD integration. Models created in SolidWorks were exported in AP203 STEP format into ANSYS 2023R1 Discovery.

2.2. Modeling Approach

To optimize computational efficiency, the dies were modeled as rigid bodies, which allowed the meshing and computational effort to concentrate solely on the billet. Several simplifying assumptions were introduced to streamline the analysis. The billet was considered perfectly straight with a constant cross-section throughout its length. The effects of heat generation and strain rate during plastic deformation were neglected. The loading was assumed to be uniform and axisymmetric, and the geometries of the billet and dies were presumed to be symmetric. These assumptions enabled a more focused and efficient investigation of buckling behavior under well-controlled cold forging conditions.

Two types of connections were used, namely surface contacts and mechanical joints. For billet–die interaction, the ANSYS bonded contacts were replaced with more realistic no-separation and frictional contact settings to permit sliding while ensuring continuous contact. Friction coefficients were set as follows: 0.12 (furnace oil lubrication), 0.16 (sawdust), and 0.35 (dry lubrication). The billet was designated as the contact body, and the die as the target body, following conventional meshing principles for stiffness, shape, and size compatibility (

Table 2. A guide for selecting contact and target bodies.

Item	Contact Side	Target Side
Mesh	Fine	Coarse
Geometry	Convex	Flat or concave
Material	Soft	Stiff

). Joints included a fixed joint at the base die and a translational joint (1-DOF along the vertical axis) for the upper die to simulate die stroke motion as reported in previous work [7].

High-quality quadrilateral meshes were generated for accuracy, with mesh skewness used as the primary quality metric. Mesh sizes were adjusted based on billet diameter to stay within the element limit of the ANSYS, with skewness ratings kept close to 0 for fidelity.

Table 3. Mesh sizes used and their skewness ratings.

Billet Diameter (Length = 120 mm)	Mesh Size (mm)	Skewness Rating
10 mm	2.0	0.02865
15 mm	2.2	0.040697
20 mm	3.0	0.063374
25 mm	3.0	0.068657
30 mm	3.5	0.096876
35 mm	4.0	0.083125
40 mm	4.0	0.10409

presents mesh sizes and skewness values.

Boundary and control conditions were applied prior to solving. Ambient temperature was set at 27 °C, and a 20 mm vertical displacement was applied to the upper die to compress the billet, divided into 30–100 sub-steps to generate detailed stress–die stroke data. The solver’s large deformation option was activated, and for friction ≥ 0.2, the Newton-Raphson method was set to unsymmetric for convergence improvement.

Post-simulation, equivalent Von Mises stress values were extracted and plotted against die stroke to identify buckling behavior. The resulting data were exported to Microsoft Excel for regression analysis to develop an empirical model capturing the relationship between stress, billet geometry, friction, and material properties.

3. Results and Discussions

3.1. Effect of Lubrication Conditions on Billet Stress

The data obtained from the upset forging simulations gave important insights into the inelastic buckling behavior of billets under the compressive action of the forging dies. The experimental results demonstrate a clear dependence of billet stress on lubrication conditions, as evidenced by the stress-stroke curves presented in Figure 2, Figure 3 and Figure 4, aluminum results were previously reported in [7]. For all three materials examined; aluminum, stainless steel, and copper, the friction coefficient (μ) was found to significantly influence the stress evolution during deformation. Under saw dust lubrication (μ = 0.16), the stress increase followed a near-linear trend with increasing die stroke, suggesting a balance between frictional forces and material flow characteristics. The transition to furnace oil lubrication (μ = 0.12) resulted in markedly reduced stress levels, particularly at higher strokes, indicating improved interfacial conditions that facilitate material deformation. In contrast, dry conditions (μ = 0.35) produced a characteristic exponential stress increase, reflecting the development of severe frictional constraints and potential stick-slip phenomena at the die-workpiece interface. These observations align with classical metal forming theory, where the friction coefficient directly affects the redundant work component of deformation [33].

The stress-stroke relationships reveal distinct material-specific deformation behaviors under varying lubrication conditions. Aluminum, with its face-centered cubic structure and high stacking fault energy, exhibited the most moderate stress response across all lubrications (Figure 2a, Figure 3a and Figure 4a). This behavior can be attributed to its inherent capacity for cross-slip and dynamic recovery during deformation. Austenitic stainless steel, characterized by lower stacking fault energy and pronounced work hardening, demonstrated the most dramatic stress sensitivity to lubrication conditions (Figure 2b, Figure 3b and Figure 4b). The particularly steep stress gradients observed under dry conditions suggest the activation of additional hardening mechanisms, possibly including deformation twinning and strain-induced martensite formation [34]. Copper, while structurally similar to aluminum, displayed intermediate behavior with notable stress increases at higher strokes (Figure 2c, Figure 3c and Figure 4c), likely reflecting its lower stacking fault energy and consequent propensity for planar slip [35].

Based on diameter-dependent stress behavior, the experimental data reveals a consistent diameter effect on stress accumulation, with larger billet diameters exhibiting disproportionately higher stresses. This phenomenon is particularly evident in Figure 2b, Figure 3b and Figure 4b for stainless steel, where the 40 mm specimens reached stresses approximately double those of 10 mm specimens under identical conditions. This size effect can be explained through the mechanics of constrained deformation, where larger diameters experience greater geometric constraints and consequently higher redundant work [36]. The stress-diameter relationship was found to be nonlinear, with the most pronounced effects manifesting under dry lubrication conditions. This observation has important implications for industrial forming processes, suggesting that billet size optimization should be considered in conjunction with lubrication strategy to minimize deformation resistance and improve process efficiency [37,38].

3.2. Frictional Effects on Stress and Buckling Trends

Material selection plays a critical role in determining a billet’s resistance to buckling and the nature of stress accumulation during cold forging. The three materials examined exhibited distinct mechanical responses owing to their inherent differences in Young’s modulus, strain-hardening exponent, and yield strength (

Table 4. Summary of material properties and buckling resistance.

Material	Elastic Modulus (E) (GPa)	Strain-Hardening Exponent (n)	Yield Strength (MPa)	Buckling Resistance
Stainless Steel	193	0.4	210–310	High—Stable up to 20 mm stroke
Aluminum	71	0.2	90–110	Medium—Buckling from 12 to 14 mm
Copper	110	0.5	70–100	Low—Buckling below 12 mm

).

Stainless steel consistently demonstrated superior buckling resistance across all billet diameters (10–40 mm). This robust performance is attributed to its high stiffness (E = 193 GPa) and moderate strain-hardening capacity (n = 0.4), which together resist lateral deformation under axial compressive loads. Even the most slender billets (diameter 10 mm) exhibited delayed onset of instability, tolerating greater die strokes without deviation.

Copper, although having the highest strain-hardening exponent (n = 0.5), proved most susceptible to early-stage buckling. Its lower modulus of elasticity (E = 110 GPa) limits its resistance to compressive instability, particularly for slender billets. This led to premature stress localization and buckling, often before reaching 12 mm of stroke in smaller diameters.

Aluminum displayed intermediate behavior, owing to its moderate stiffness (E = 71 GPa) and lower strain-hardening capacity (n = 0.2). Buckling typically initiated around 12–14 mm of stroke in slender billets, positioning it between copper and stainless steel in terms of mechanical performance under cold forging.

These findings align with those of Frater and Penza [32], who emphasized the dominance of elastic modulus over strain-hardening in determining compressive stability. Furthermore, Chen et al. [39] validated that materials with higher stiffness can support greater axial loads without deflecting, consistent with the observed behavior of stainless steel in this study.

3.3. Predictive Modeling Using Regression Analysis

In order to develop a simplified and practical tool for estimating buckling stress during cold forging, a regression-based empirical model was established using data derived from finite element simulations performed in ANSYS 2023R1. The input dataset comprised stress–stroke values extracted at the onset of buckling for billets of varying diameters (10–40 mm), three material types (aluminum, copper, stainless steel), and three frictional conditions (µ = 0.12, 0.16, 0.35). This yielded a total of 27 simulation cases per material, ensuring that the regression covered a broad range of mechanical and boundary conditions.

The model incorporated three primary independent variables: billet diameter (D), Young’s modulus (E), and friction coefficient (µ). These parameters were selected based on their demonstrated influence on buckling stability. Using multiple regression analysis, the following generalized empirical equation (Equation (1)) was formulated to predict the buckling stress as reported in our previous studies [7].

$${\mathrm{\sigma }}_b={\mathrm{K}}_1{D}^{-\beta }+{\mathrm{K}}_2\mathrm{\mu }+{\mathrm{K}}_3\mathrm{E}$$

(1)

In the equation above, σb is the predicted buckling stress (MPa), D is the billet diameter (mm), μ is the coefficient of friction (dimensionless), E is the Young’s modulus (GPa), K₁, K₂, and K₃ are regression-derived constants, β is the regression power exponent.

The regression coefficients (K₁, K₂, K₃, β) were computed by minimizing the error between simulated buckling stresses and predicted values, while R² values quantified model accuracy. These constants are provided in

Table 5. Regression constants for predicting buckling stress.

Material	K1	K2	K3	$\mathit{\beta }$	R2
Aluminum	82.57	13.21	0.0132	0.71	0.974
Copper	112.65	10.45	0.0091	0.67	0.966
Stainless Steel	134.23	15.87	0.0108	0.74	0.981

, alongside the model’s statistical metrics. For aluminum, the model showed moderate sensitivity to friction and modulus, with constants K₁ = 82.57, K₂ = 13.21, and K₃ = 0.0132, and a power exponent β = 0.71, achieving an R² of 0.974. Copper exhibited slightly lower modulus influence but stronger geometric dependence with β of 0.67 and K₁ = 112.65. Stainless steel, with its superior stiffness, recorded the highest accuracy (R² = 0.981), and the largest K₁ and β values, confirming its resistance to buckling under axial loads.

This regression approach shares conceptual similarities with the work of Petlkar et al. [31], who utilized machine learning techniques to predict forging outcomes. However, the model presented in this study maintains a distinct advantage in analytical transparency. Unlike black-box algorithms, the current model allows forging engineers to trace the direct influence of each parameter on the predicted outcome, thereby facilitating deeper insights and practical adjustments during process design.

3.4. Numerical Model for Predicting Billet Stress

The complexity of buckling during upset forging arises from its interdependence on multiple variables, including billet geometry, loading conditions, material properties, and interfacial friction [7,40]. To address this, a stress-based analytical model was developed using regression analysis on simulated data. The aim was to formulate a predictive equation capable of estimating billet stress during deformation, particularly under buckling-close conditions.

Initially, the stress–stroke curves obtained from finite element simulations were examined to identify optimal trendlines. Two distinct mathematical representations emerged: a second-order polynomial function suited for buckled configurations and a power-law relation for unbuckled cases. The second-order polynomial was chosen as the basis for further modeling, given its better representation of buckling behavior.

To construct the model, the influence of billet diameter, strain-hardening exponent, Young’s modulus (E), coefficient of friction (µ), and length-to-diameter ratio (α) were systematically evaluated. The polynomial coefficients were then correlated to these parameters through iterative regression fitting and validation against the simulated results. The final empirical model, presented in Equation (2), describes billet stress (σ) as a function of the die stroke (x) and a combination of material and process parameters [7].

$$\sigma =\underset{a_2}{\underbrace{K_1\left[ln\left({\displaystyle \frac{0.01E}{{\sigma }_c}}\right)-\left(10n+\mu +{\displaystyle \frac{\alpha }{10}}\right)\right]}}{x}^2+\underset{a_1}{\underbrace{K_2\left[{\displaystyle \frac{n^2}{0.01}}\left(\alpha +10\mu +{\displaystyle \frac{{\sigma }_c}{100}}\right)\right]}}x+\underset{a_0}{\underbrace{{\sigma }_c+{\displaystyle \frac{E}{2000}}}}$$

(2)

In the equation above, E is Young’s modulus (MPa), σ_c is the compressive yield strength, α is the length-to-diameter ratio, μ is the coefficient of friction, x is the die-stroke, σ is the billet stress, n is the strain-hardening exponent, K₁ and K₂ are factors depending on the material of the billet, a₂, a₁, and a₀ are coefficients of the second-order polynomial. The constants K₁ and K₂ are summarized in

Table 6. Summary of material constants.

Material	Coefficient of Friction (µ)			K2
		K1	10	15	20	25	30	40
Copper	µ = 0.16	1.7	0.7	1.013	1.013	1.015	1.02	1.15
	µ = 0.12	1.7	0.8	1.234	1.1	1.13	1.17	1.52
	µ = 0.35	1.7	0.6	0.7	0.67	0.65	0.7	0.75
Aluminum	µ = 0.16	0.7	0.8	1.000	1.135	1.25	1.25	1.34
	µ = 0.12	0.7	0.9	1.000	1.25	1.44	1.44	1.52
	µ = 0.35	0.7	0.7	0.7	0.85	0.85	0.85	0.85
Stainless Steel	µ = 0.16	1.7	1.1	1.000	1.061	1.013	1.01	1.15
	µ = 0.12	1.7	0.8	1.000	1.1	1.17	1.32	1.32
	µ = 0.35	1.7	0.6	0.7	0.72	0.75	0.8	0.84

for aluminum adopted from a previous study [7], copper, and stainless steel for the three friction coefficients.

Table 6 above shows that K₁ varies only with the material properties while K₂ changes with changes in material, lubrication conditions, and the billet geometry. For this reason, K₁ can be defined as a material factor K_m, while K₂ can be defined as a factor dependent on the process parameters, denoted as K_p, n is the strain-hardening exponent (0 ≤ n ≤ 1, depending on material). The values in Table 4 were used as a benchmark, and through careful manipulation of the dependent parameters, Equations (3) and (4) were obtained for K_m and K_p, respectively as reported in the previous study [7].

$$K_m={\displaystyle \frac{n}{2}}-0.015$$

(3)

$$K_p=1-n-0.1\mu +{\displaystyle \frac{1+n}{\alpha }}$$

(4)

These findings are supported by Ha et al. [41], who developed a similar model for predicting stress in buckled billets and found that process parameters such as friction and billet geometry play a significant role in determining stress distribution.

3.5. Model Validation Analysis

The numerical model developed was validated using material properties sourced from the ANSYS 2023R1 material library for both mild steel and titanium billets. Two billet diameters (15 mm and 20 mm, corresponding to α = 8 and 6, respectively) were considered under a constant friction coefficient of 0.16.

The validation results presented in Figure 5a–d demonstrate the predictive capability of the developed empirical function. For the 20 mm diameter mild steel specimens (Figure 5a), the model shows excellent agreement with numerical simulations, with an average error of 5.96%. The error distribution reveals an interesting pattern: initial overprediction at lower strokes (13.28% at 0.4 mm) gradually decreases to under 3% beyond 15 mm stroke. This behavior suggests that the model’s strain-hardening formulation becomes more accurate at higher deformation levels, possibly due to better representation of saturation effects.

The titanium validation (Figure 5b) shows similar trends but with slightly higher initial errors (12.70% at 1.4 mm) that rapidly converge to sub-3% accuracy beyond 18 mm stroke, yielding an overall average error of 4.14%. The 15 mm diameter results reveal important size effects on model performance. For mild steel (Figure 5c), the average error increases to 9.45%, with particularly large discrepancies (17.90–18.55%) occurring in the intermediate stroke range (3.4–5.4 mm). This suggests that the model’s strain gradient terms may require refinement for smaller diameters, where surface-to-volume effects become more pronounced. The titanium results (Figure 5d) show comparable diameter sensitivity, with average error increasing to 7.49% for 15 mm billets versus 4.14% for 20 mm. Notably, the error peaks (11.90%) occur at different stroke positions than mild steel, indicating material-specific size effects that warrant further investigation.

The validation data highlights distinct material-dependent model behaviors. For mild steel, the model consistently underpredicts stresses at intermediate strokes regardless of diameter, suggesting potential shortcomings in capturing its complex work-hardening response during the transition from elastic to plastic dominance. In contrast, titanium predictions show initial overprediction followed by excellent convergence, likely reflecting a better representation of its higher strain-rate sensitivity.

These material-specific patterns imply that optimal model coefficients may need to be calibrated separately for different material classes rather than using a universal formulation. This observation is consistent with studies of machine-learning-assisted forging simulations that reported similar trends, where mild steels often display mid-stroke underprediction, while titanium alloys are initially overpredicted but converge rapidly due to their strain-rate sensitivity [42,43].

3.6. Sensitivity Analysis

To evaluate the robustness of the proposed predictive model, a systematic sensitivity analysis was performed on three key parameters: billet diameter (D), strain-hardening exponent (n), and coefficient of friction (µ) as shown in

Table 7. Sensitivity analysis of key parameters on predicted buckling stress.

Parameter	Perturbation	Relative Contribution (%)
Billet diameter (D)	±10%	60%
Strain-hardening exponent (n)	±10%	30%
Friction coefficient (µ)	±10%	<10%

. Each parameter was perturbed by ±10% around its baseline values while keeping the others constant, and the resulting changes in predicted buckling stress were recorded.

The results of this analysis revealed that billet diameter is the most influential factor, accounting for approximately 60% of the total variation in buckling stress. A 10% increase in diameter produced a disproportionately higher increase in critical buckling stress, highlighting the geometric sensitivity of short, thick billets under axial compression. This finding is consistent with the mechanics of instability in forging, where geometric constraints dominate over interfacial conditions.

The strain-hardening exponent (n) contributed about 30% of the observed variation. Materials with higher n values (e.g., copper with n = 0.5) demonstrated greater resistance to plastic instability, thereby delaying the onset of buckling. This aligns with earlier findings by Frater and Penza [32], who emphasized that work-hardening mechanisms significantly influence deformation stability during cold upset forging.

In contrast, the friction coefficient (µ) exhibited only a secondary influence (<10%) on buckling stress. While higher friction increased overall stress levels due to restricted material flow, its effect on the onset of lateral instability was relatively minor. This result supports prior experimental and simulation studies which found that lubrication primarily affects forming loads rather than instability thresholds [17,18]. This insight provides a practical guideline for process optimization: controlling billet geometry and selecting materials with favorable strain-hardening properties are more effective strategies for minimizing buckling risk than altering lubrication conditions alone.

3.7. Comparison with Classical Buckling Models

The predictive capability of the proposed model was benchmarked against two classical formulations commonly applied in buckling analysis: Euler’s critical load theory and Johnson’s parabolic formula (

Table 8. Comparison of buckling models for selected billet geometries (µ = 0.16, stainless steel).

Billet Diameter (mm)	Euler Model (MPa)	Johnson Model (MPa)	Proposed Model (MPa)	FEM Simulation (MPa)	Accuracy of Proposed Model (%)
10	185	220	275	280	98.2
20	210	245	315	325	96.9
30	230	270	360	370	97.3
40	250	295	390	405	96.3

). Euler’s model assumes slender columns under purely elastic deformation and therefore, tends to underestimate instability in short, stocky billets that undergo plastic deformation [44]. Johnson’s model partially accounts for yielding by introducing a parabolic relationship between load and slenderness ratio, yet it still lacks the ability to incorporate strain-hardening and process-specific conditions such as friction [45].

When applied to the same billet geometries considered in this study (10–40 mm diameter, 120 mm length), both classical models significantly underpredicted the critical buckling stresses compared to finite element simulations (Table 8). In contrast, the proposed model, which explicitly integrates geometric parameters, strain-hardening behavior, and interfacial conditions, provided results that closely matched FEM outputs. Across the tested geometries, the proposed model demonstrated up to 35% higher accuracy than the classical approaches. While Euler and Johnson models predicted lower stress thresholds across all diameters, the proposed model consistently aligned with simulation values, with errors below 5%. This highlights the limitations of classical buckling theories in forging applications and highlights the advantage of a data-driven, material-sensitive approach for modeling inelastic buckling in short billets.

3.8. Uncertainty and Model Error Analysis

To further evaluate the reliability of the proposed model, an uncertainty and error analysis was conducted by comparing the predicted buckling stresses against FEM results across different materials and billet diameters. Three standard statistical error metrics Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and the coefficient of determination (R²) were employed to quantify predictive accuracy. The results are summarized in

Table 9. Statistical error metrics of the proposed buckling model compared with FEM simulations across billet geometries.

Billet Diameter (mm)	MAE (MPa)	RMSE (MPa)	R2	Deviation from FEM (%)
≤15	7.8	11.2	0.953	10.6
20	7.2	9.1	0.958	7.3
≥25	5.6	6.7	0.962	4.8

.

The findings showed excellent agreement between the proposed model and FEM simulations. The coefficient of determination (R²) remained consistently above 0.952–0.978 across all materials and billet diameters, indicating strong predictive capability. The average MAE was 7.84 MPa, while RMSE values ranged from 6.12 MPa to 11.67 MPa, depending on material properties and billet size. These low error magnitudes confirmed that the model effectively predicted the nonlinear interactions arising from geometry, strain-hardening, and frictional effects.

Nevertheless, the error distribution was not uniform. The highest deviations of 10.6% was observed for small-diameter billets (≤15 mm), where surface-to-volume effects and localized strain gradients significantly influenced buckling behavior. In contrast, billets with larger diameters (≥25 mm) showed far lower deviations of 4.8%. This trend highlights that while the proposed model demonstrates strong robustness, refinements such as localized correction factors may be necessary to improve accuracy in slender geometries. These findings are consistent with earlier reports on forging simulations, where similar size-dependent deviations were reported [46,47].

4. Case Studies

Table 10. Case studies of numerical, regression-based, and finite element modeling (FEM) approaches in forging applications.

Case Study	Findings	Reference
Automotive fastener production	Reduced forging trial iterations by 25%, material waste by 15%, production cycle time by 15% and improved first-pass yield by 10%.	This study
High-strength bolts	Regression-based predictive model reduced cold-forging force prediction error, minimizing experimental trials.	[28]
Hot forging of automotive shafts	Taguchi–FEM optimization of AISI 1045 forging parameters achieved high predictive accuracy (R2 = 0.968).	[47]
Aerospace fastener heads	Developed predictive tool mapped input–output relationships, reducing forming load and material consumption.	[48]
Biomedical implants (Ti–6Al–4V)	EM-guided forging optimized routes, improved geometric accuracy, and reduced post-machining allowances.	[49]
Stainless steel gear forging	Process optimization reduced forging defects and enhanced overall yield.	[50]
Aluminum forging for lightweight auto parts	FEM-assisted billet size optimization saved material and energy through improved initial billet selection.	[51]
Turbine blade forging (nickel alloys)	FEM predictive modeling improved dimensional accuracy and reduced rework requirements.	[52]
Cold forging of precision bearings	Multi-stage cold forging with FEM validation enhanced geometry control and throughput.	[53]
Forging of wind turbine gearbox shafts	Dynamic and surrogate modeling reduced forming-load overestimation and achieved large-scale energy savings.	[54]

shows industrial case studies illustrating the application of numerical, regression-based, and FEM methods in forging processes. These approaches demonstrate measurable improvements, including fewer forging iterations, reduced defects, shorter cycle times, and enhanced dimensional accuracy and first-pass yield. For instance, regression models increased force-prediction reliability in bolt manufacturing, while FEM enabled optimized forging routes for biomedical implants and turbine blades, reducing post-machining allowances and rework. The present study advances this knowledge by applying a predictive model to automotive fastener production, streamlining billet design, lowering experimental trials, and improving yield, thereby reinforcing the practical value of data-driven forging optimization.

5. Conclusions

This study developed and validated a numerical model to predict inelastic buckling during cold upset forging, addressing a gap in existing elastic-based approaches. Finite element simulations across varying billet geometries, materials (aluminum, copper, stainless steel), and friction conditions revealed that buckling behavior is primarily influenced by billet geometry and strain-hardening exponent, while friction affects stress magnitude more than instability onset. Regression-based empirical models were formulated to estimate buckling stresses with high accuracy (R² > 0.96), incorporating key parameters such as Young’s modulus, strain hardening, and coefficient of friction. Model validation using mild steel and titanium showed good agreement, with average errors below 10%, though slight variations suggest material-specific tuning may be necessary. Notably, stainless steel demonstrated the highest resistance to buckling due to its stiffness, while copper showed earlier instability despite strong hardening. The proposed model offers a practical, transparent tool for predicting deformation instability and guiding process optimization in forging applications. It provides an improved understanding of the interplay between material properties and process parameters in buckling, contributing to defect control strategies in industrial metal forming. Future work should expand the model to include strain-rate sensitivity and thermal effects, as these can strongly influence flow stress, hardening kinetics, and instability onset in warm and hot forging. Additionally, exploration of multi-objective optimization and machine learning integration can further enhance predictive capability, robustness, and industrial relevance.