Elucidating formability limits in warm incremental sheet forming of AZ61 magnesium alloy using integrated grey relational and data-driven analysis

Abstract

The growing demand for lightweight structural components in the automotive and aerospace sectors has intensified interest in magnesium alloys; however, their limited formability at room temperature continues to restrict widespread industrial adoption. In this study, the formability and surface quality of AZ61 magnesium alloy during warm incremental sheet forming (ISF) are systematically investigated. A straight-groove forming test was conducted using a Taguchi L27 experimental design to evaluate the effects of key process parameters. To simultaneously optimize formability and surface roughness, Grey Relational Analysis (GRA) was employed. Analysis of Variance (ANOVA) of the GRA results revealed that step depth was the most influential parameter, contributing 70.12% to the total variance, followed by spindle speed with a contribution of 12.97%, while tool diameter and feed rate exhibited comparatively minor effects. The optimal parameter combination, 8 mm tool diameter, 700 rpm spindle speed, 0.3 mm step depth, and 500 mm/min feed rate, resulted in enhanced formability and improved surface quality. Forming Limit Diagram (FLD) analysis confirmed an expansion of the safe deformation zone under optimized conditions. Scanning Electron Microscopy (SEM) observations indicated a predominantly ductile fracture mode characterized by deep dimples and fibrous tearing. Furthermore, a Random Forest–based machine learning model corroborated the ANOVA findings, providing additional validation and trend confirmation for the identified parameter influences. The integrated experimental and data-driven framework presented in this work supports the development of intelligent and optimized warm ISF processes for magnesium alloys.

Introduction

The demand for lightweight, high-performance structural materials has increased considerably in recent years, with the automotive, aerospace, and biomedical industries prioritizing energy efficiency, emission reduction, and sustainability. The low density, high specific strength, high damping capacity, and recyclability of magnesium alloys fulfil these requirements^1–5. The AZ series, particularly AZ61, is widely favoured because of its good combination of strength and corrosion resistance, which are suitable for lightweight structural components^1,3,5. However, magnesium alloys cannot be used extensively in industry due to their poor room-temperature formability. The hexagonal close-packed (HCP) crystal structure limits the number of active slip systems, which restricts ductile behaviour at room temperature^6–9. These limitations are addressed by warm forming techniques. Increased temperatures trigger non-basal-slip systems as well as reduce the flow stress; thereby, enhancing ductility and formability^6–9.

Incremental Sheet Forming (ISF) is a die-less, flexible process capable of making complex shapes at reduced tooling costs. ISF enables localized deformation and achieves higher forming limits compared to conventional sheet forming processes using a numerically controlled tool path^10–12. Process parameters that affect formability and surface integrity in ISF include tool diameter, step depth, spindle speed, and feed rate. Previous studies on incremental sheet forming have reported that step depth strongly influences strain localization and fracture initiation, whereas tool diameter significantly affects surface quality and geometric accuracy during forming^12–15.

Warm ISF complicates this further since thermal softening, strain hardening, and frictional heating interact in complex ways, making analytical modeling difficult^16–19. In order to measure deformation caused by such complicated loading, strain-based forming limit diagrams (FLDs) are often employed to evaluate formability^2,16,21. The method of scanning electron microscopy (SEM) is used to correlate fracture mechanisms with the macro-deformation that is realized^6,7,12,16,22. These approaches, though informative, are descriptive and do not have predictive ability for new conditions in the process.

In recent years, multi-objective optimization techniques, including Grey Relational Analysis (GRA) based on grey and fuzzy strategies, have been widely used to optimize performance indices such as formability and surface roughness^11,15,16,18. These statistical approaches rely on predefined mathematical formulations and lack adaptive learning capability for highly nonlinear thermo-mechanical relationships. Machine learning (ML) provides an interesting alternative with the emergence of smart manufacturing and Industry 4.0²³. Specifically, ML algorithms excel at establishing predictive models to capture highly nonlinear thermo-mechanical behaviours²⁴, facilitating rapid process optimization to minimize physical experimentation²⁵, and enabling the implementation of real-time predictive control systems for digitalized manufacturing environments²⁶.

ML models have demonstrated effectiveness in predicting material behaviour, surface quality, and stability of processes in machining, forming, and additive manufacturing. However, they have not been used extensively in the warm incremental sheet forming of magnesium alloys, particularly with experimentally validated forming limit analysis. Current ISF experiments of magnesium alloys mostly involve optimization of experiments, or traditional statistics, and few studies have reported the use of ML for predictive modeling in this domain. While various approaches have been explored individually, the fully integrated application of multi-objective optimization and predictive modeling remains a critical gap in the current literature. To contextualize this, a few recent studies have attempted partial integrations within sheet metal forming. For instance, Danish et al.²⁷, utilized machine learning algorithms to predict surface roughness during incremental sheet forming; however, their framework was limited to room-temperature applications and did not address the complex thermo-mechanical behavior inherent to magnesium alloys. Khot et al. (2026) identified a critical gap in the literature regarding the integration of multi-objective optimization and predictive modeling in sheet metal forming. To bridge this divide, their study proposed a fully integrated framework that combines Grey Relational Analysis (GRA) for multi-objective optimization with robust machine learning predictive modeling. Conversely, Wang et al.²⁸, reviewed applied Grey Relational Analysis (GRA) to optimize multi-response parameters, yet their approach relied exclusively on traditional statistical methods without the predictive robustness of modern ML models. Therefore, present study aims to bridge this gap by proposing a holistic, fully integrated framework: combining GRA-based multi-objective optimization with robust ML predictive modeling, specifically tailored for the warm forming of AZ61 alloys, and strictly validated through physical Forming Limit Diagrams (FLD) and SEM fractography. This represents a critical gap in the development of intelligent digital manufacturing systems. The optimization of AZ31 has been performed by most of the research based on statistical means²⁹, and very few studies have been conducted on higher-strength AZ61.

In order to overcome these shortcomings, in this study, an integrated experimental-ML model of warm ISF of AZ61 magnesium alloy is used. A Taguchi L27 design was used to systematically vary tool diameter, spindle speed, step depth, and feed rate. Grey Relation Optimization determines the best conditions of the process, whereas strain-based FLDs and SEM analysis show the way of deformation and fracture. ML model is developed for trend analysis and validationthe formation responses of process parameters so that the formability could be estimated accurately without repetitive experiments. The key contributions include extensive experimental research on warm ISF of AZ61 and the multi-objective optimization of performance using GRA. Furthermore, the proposed integrated experimental-data framework facilitates smart manufacturing solutions for lightweight metallic components.

Experiment methodology

Material selection and properties

AZ61 magnesium alloy sheets were selected in the present study as the work material due to their low density, high specific strength, and corrosion resistance. AZ61 is considered a promising candidate for lightweight structural applications in the automotive and aerospace industries. Commercially available as-received rolled AZ61 sheets of 1 mm thickness were prepared as specimens measuring 145 × 145 mm for the incremental forming experiments. The initial microstructure is characteristic of commercial AZ61, predominantly consisting of a primary α-Mg matrix interspersed with intermetallic β-phase (Mg₁₇Al₁₂) precipitates distributed along the grain boundaries^30,31. In this alloy system, the major alloying element, aluminum (~ 6.00 wt%), provides significant solid solution strengthening and facilitates the precipitation of the β-phase, which effectively pins grain boundaries during elevated temperature deformation. The minor addition of zinc (~ 1.00 wt%) further enhances the matrix strength and improves general corrosion resistance³². However, similar to most magnesium alloys, AZ61 exhibits low formability at room temperature because of its hexagonal close-packed (HCP) crystal structure, which restricts the number of active slips. To overcome this fundamental limitation, forming must be executed under elevated thermal conditions. High temperatures activate non-basal slip systems, significantly enhancing ductility and preventing premature fracture during plastic deformation.

Experimental setup and forming procedure

All experiments were conducted in a three-axis Computer Numerical Control (CNC) vertical machining centre (VMC). To hold the sheet firmly in the machine and reduce vibration during forming, a specially designed rigid clamp was installed on the machine table as shown in Fig. 1. The fixture was equipped with an integrated induction heating system, utilizing an induction coil positioned directly beneath the AZ61 sheet inside the fixture block. This non-contact induction method ensures rapid and uniform thermal generation across the workpiece. A closed-loop PID temperature controller, utilizing K-type thermocouples positioned strategically at the sheet periphery and near the forming zone, maintained the target forming temperature of 250 °C with a stability and accuracy of ± 5 °C. The forming temperature was selected based on the earlier research works that indicated an increase in ductility and low flow stress of magnesium alloys in the 200–300 °C temperature range³³.

Fig. 1.

Experiment setup.

Hemispherical forming tools with diameters of 8 mm, 10 mm, and 12 mm were machined from EN8 steel. Following fabrication, the tools were subjected to heat treatment to significantly increase their hardness and wear resistance. This elevated hardness is critical during the warm forming of magnesium; it prevents galling (the adhesion and transfer of the softer AZ61 material onto the tool surface) and ensures the tool maintains its highly polished finish throughout the process. Consequently, this sustained hard and smooth tool interface minimizes interfacial friction and prevents severe surface degradation, scoring, or scratching on the AZ61 workpiece. The computer-Aided Manufacturing (CAM) toolpath was generated and cut in a straight downward spiral configuration to create a straight groove geometry, which is standard in incremental sheet forming experiments^34,35. Prior to forming, the sheet surface was lubricated to avoid friction and tear at surfaces. The motion of the tool was governed in three directions where the vertical step-down was used to determine the amount of deformation incremented during each pass. The forming tool was programmed to traverse a continuous linear path with a total length of 40 mm. Upon reaching the end of each 40 mm stroke, the tool plunges vertically by one specified step depth. This linear traversal and downward plunge cycle is repeated continuously until macroscopic fracture occurs.

Design of experiments and process parameters

The effect of important process parameters on formability and surface quality was studied in a systematic manner with a Taguchi L27 orthogonal array. The Taguchi approach was selected due to its ability to minimize the number of experiments, though it has a high level of statistical strength^36–38. Four process parameters were measured including: Tool diameter (TD), Spindle speed (SS), Step depth (SD) and Feed rate (FR). The process parameters and their levels are summarized in Table 1.

Table 1.

Process Parameters and levels.

Parameter	units	Level 1	Level 2	Level 3
Tool Diameter (TD)	mm	8	10	12
Feed Rate (FR)	mm/min	500	800	1000
Spindle Speed (SS)	rpm	300	500	700
Step Depth (SD)	mm	0.3	0.5	0.7

The operating ranges for these process parameters were selected strictly based on established formability limits for the warm incremental forming of magnesium alloys¹⁶. Literature indicates that step depths exceeding 0.7 mm rapidly induce severe strain localization and premature fracture in HCP metals, thereby dictating our restricted 0.3–0.7 mm range³². Concurrently, spindle speeds up to 700 rpm were chosen to provide optimal localized frictional heating without causing surface galling. Finally, the feed rates and tool diameters were mathematically constrained within ranges proven by previous studies to effectively balance thermal dissipation and prevent excessive sheet thinning during the forming of AZ-series alloys³⁹.

All experiments were run in controlled thermal conditions and the formed specimens left to cool naturally to room temperature, and then characterized further. A total of 27 experiments were conducted based on Taguchi L27 design in order to extend the influences of all the process parameters on formability and surface integrity. Formability and Surface Roughness: To accurately capture the process responses, two distinct measurement techniques were employed.

Measurement of formability and surface roughness

Strain measurement and forming limit diagram

As a measure of formability, a circular grid pattern 2 mm in diameter was marked on the surface of the sheet prior to forming with a laser marking system (Fig. 2). The high-resolution digital microscope was used to measure the major and minor axes of the deformed circles after deformation. Formability (evaluated through major and minor strains) was measured using an Olympus CX 41 optical microscope. This equipment utilizes a high-resolution Universal Infinity-corrected (UIS2) optical system coupled with a built-in transmitted Koehler illuminator (6 V/30 W halogen) to ensure edge clarity of the printed grids. Precise coordinate tracking across the localized deformation zone was achieved using a mechanical X-Y stage with a movement range of 76 mm x 50 mm and a coarse focus stroke of 36.8 mm per rotation, ensuring accurate strain mapping. Principal strains were obtained with the help of standard logarithmic strain relations, and strain-based forming limit diagrams (FLDs) were created to test safe and failure deformation zones. Using FLD analysis, one can easily gain a good understanding of the strain distribution and the behaviour of deformation in incremental forming conditions^21,26,40.

Fig. 2.

Formed sheet with fracture location.

Surface roughness measurement

Concurrently, the surface roughness (Ra) of the formed components was measured using a Taylor Hobson Surtronic S-128 contact profilometer. To ensure high-fidelity measurements and reproducibility, the instrument was equipped with a diamond stylus having a tip radius of 5 μm and a vertical measurement resolution of 0.01 μm, with an overall instrumental uncertainty of ± 2%. The measurement was taken at three locations on the formed groove in the direction of the tool travel and the mean roughness (Ra) has been noted. The cutoff length was selected as 0.8 mm in accordance with the International Organization for Standardization (ISO) requirements for characterizing the surface. The assessment of surface roughness is important in the ISF processes because it directly influences the dimensional accuracy, fatigue behaviour and functional performance of the formed components.

Data preparation for optimization and machine learning

To complement the statistical analysis, a machine learning (ML) framework was developed to model the non-linear interactions between process parameters. The experimental dataset, derived from the Taguchi L27 orthogonal array, consisted of four input features (TD, SS, SD, FR) and two target variables (Strain and Surface Roughness). As the dataset utilized for training consists of 27 experimental runs, it is important to note that these data points are not randomly sampled; they are derived from a rigorously structured Taguchi L27 orthogonal array. This Design of Experiments (DOE) approach ensures a balanced, uniform, and comprehensive representation of the entire multi-dimensional parameter space, capturing the maximum physical variance with minimal experimental runs. To strictly mitigate the inherent risk of overfitting associated with small datasets, a robust cross-validation strategy was employed. Specifically, 5-fold cross-validation was utilized during the model training phase. This iterative validation technique ensures that the model learns the underlying thermo-mechanical trends rather than memorizing data noise, thereby securing the statistical validity and predictive reliability of the algorithm within the defined experimental boundaries. The data processing and modeling were implemented using Python (v3.9) and the Scikit-learn library. Prior to modeling, the input features were standardized using a StandardScaler to normalize the data distribution (mean = 0, variance = 1), ensuring that parameters with larger magnitudes (e.g., Spindle Speed) did not bias the learning process.

Given the limited dataset size (N = 27), a Random Forest Regressor (RFR) was selected for its ensemble nature, which reduces the risk of overfitting compared to single decision trees. The dataset was partitioned using a stratified 80/20 split, allocating 21 samples for training and 6 samples for testing. The model was configured with the following hyperparameters to ensure stability Number of estimators (trees) are 100, Criterion set as Squared Error, bootstrap kept as True. This configuration allowed the model to act as a validation tool for the experimental trends rather than a pure “black-box” predictor.

Methodologically, the ML algorithm in this study serves two distinct, sequential roles. Initially, it functions as a validation tool during the training phase, utilizing the aforementioned k-fold cross-validation on the experimental Taguchi dataset to mathematically confirm its ability to accurately map the underlying thermo-mechanical relationships. Once this high internal accuracy is established, the model’s primary operational role transitions to trend analysis. Rather than acting strictly as a discrete point predictor, the validated algorithm is deployed to simulate the continuous design space. This enables the extraction and visualization of highly non-linear interaction trends between the process variables (e.g., tool diameter, spindle speed) and the multi-response outcomes (formability and surface roughness), providing deeper physical insights than traditional discrete statistical methods.

Result and discussion

Grey relation analysis

In the Grey Relational Analysis (GRA) framework, the macroscopic formability of the AZ61 sheets was specifically represented by the sum of the corrected in-plane strains. In Incremental Sheet Forming, utilizing the sum of the corrected major and minor strains provides a highly comprehensive metric for formability, as it mathematically correlates to the maximum localized sheet thinning (thickness strain) prior to fracture. This parameter, extracted directly from the deformed grid measurements, was established as the “larger-the-better” performance characteristic for the optimization model.

Following the experiments, the strain and surface roughness values were recorded for each trial (see Table 2). It was observed that process parameters such as tool diameter, spindle speed and step depth would influence these results substantially. These results are used to build upon further analysis and optimization of the process.

Table 2.

Experiment-wise responses measured.

Expt. No.	TD	SS	SD	FR	Strain	Surface roughness
1	8	300	0.3	500	1.086	2.635
2	8	300	0.5	800	0.925	3.118
3	8	300	0.7	1000	0.510	3.315
4	10	500	0.3	500	0.951	2.439
5	10	500	0.5	800	0.729	2.7699
6	10	500	0.7	1000	0.541	3.155
7	12	700	0.3	500	0.980	1.9556
8	12	700	0.5	800	0.643	2.2565
9	12	700	0.7	1000	0.644	3.055
10	10	700	0.3	800	0.843	1.955
11	10	700	0.5	1000	0.712	2.225
12	10	700	0.7	500	0.791	2.925
13	12	300	0.3	800	0.825	2.4875
14	12	300	0.5	1000	0.615	2.8555
15	12	300	0.7	500	0.646	3.1
16	8	500	0.3	800	0.931	2.115
17	8	500	0.5	1000	0.910	2.895
18	8	500	0.7	500	0.810	3.355
19	12	500	0.3	1000	0.873	2.5155
20	12	500	0.5	500	0.653	2.155
21	12	500	0.7	800	0.471	3.155
22	8	700	0.3	1000	1.120	2.1491
23	8	700	0.5	500	1.047	2.873
24	8	700	0.7	800	0.775	3.129
25	10	300	0.3	1000	0.735	2.335
26	10	300	0.5	500	0.782	2.859
27	10	300	0.7	800	0.565	3.226

The measured strain (a proxy of formability) and value of surface-roughness were normalized using the Grey Relational Analysis (GRA). The Grey Relational Coefficient (GRC) and Grey Relational Grade (GRG) values are determined for each experiment with a ‘larger-is-better’ criterion for formability and a ‘smaller-is-better’ criterion for roughness. Table 3 was used to rank the GRG values and provide them. Then Analysis of Variance (ANOVA) was conducted on the GRG data in order to identify which parameters of the process influence forming the most.

Table 3.

GRA calculations and experiment ranks.

Expt. No	Normalized strain	Normalized Ra	GR grade	Rank
1	0.9443	0.7176	0.7694	3
2	0.7662	0.5779	0.6118	14
3	0.1052	0.5270	0.4362	27
4	0.7966	0.7817	0.7034	7
5	0.5022	0.6761	0.5540	18
6	0.1701	0.5681	0.4562	25
7	0.8306	0.9651	0.8408	2
8	0.3621	0.8463	0.6021	15
9	0.3646	0.5948	0.4964	22
10	0.6632	0.9653	0.7663	4
11	0.4752	0.8579	0.6333	9
12	0.5919	0.6309	0.5629	17
13	0.6392	0.7654	0.6307	10
14	0.3131	0.6509	0.5050	21
15	0.3685	0.5827	0.4935	23
16	0.7734	0.9000	0.7608	5
17	0.7481	0.6395	0.6230	12
18	0.6186	0.5170	0.5380	20
19	0.7020	0.7561	0.6493	8
20	0.3799	0.8845	0.6293	11
21	0.0169	0.5681	0.4368	26
22	0.9786	0.8867	0.8872	1
23	0.9041	0.6458	0.7122	6
24	0.5698	0.5749	0.5390	19
25	0.5110	0.8179	0.6193	13
26	0.5795	0.6498	0.5657	16
27	0.2189	0.5496	0.4582	24

The best overall formability and surface finish was produced with the following parameters according to the GRG rankings: tool diameter of 8 mm, spindle speed of 700 rpm, step depth of 0.3 mm, and feed rate of 500 mm/min.

Analysis of variance

The ANOVA results for tool diameter, spindle speed, feed rate, and step depth are presented in Table 4. This structured evaluation of parameter contributions on multi-response outputs follows recently established statistical frameworks, such as those used to optimize deformation machining of AA6061 alloys⁴¹. Here, ANOVA was used to quantify the statistical significance of each parameter in predicting the derived GRG.

Table 4.

Analysis of variance.

Parameters	DOF	Adj SS	Adj MS	F Value	P Value	% Contribution	Influence of parameter
Tool diameter	2	0.01956	0.019564	14.28	0.001	5.05	Significant
Spindle speed	2	0.05019	0.050189	36.63	0.000	12.97	Significant
Step depth	2	0.27137	0.271367	198.08	0.000	70.12	Significant
Feed rate	2	0.01576	0.015758	11.50	0.003	4.07	Significant
Error	18	0.03014	0.001370			7.79
Total	26	0.38702

Step depth was the most influential factor (F = 198.08, P < 0.001), and thus had statistical significance in the 95% confidence level. Step depth accounted for 70.12% of the total variation, and the minor variations in step depth significantly alter the outcome. The second-most significant factor was spindle speed, accounting for 12.97% of the variance. Its F value and P value were also significant (36.63 and P < 0.001, respectively). Therefore, control of step depth is the determinant of material response and the secondary roles are played by the spindle speed and tool diameter. There was also a significant contribution of tool diameter and feed rate 5.05 and 4.07% respectively with a P value of 0.001 and 0.003 respectively, that is their optimization is relevant even though their contributions are lesser. The GRG response data in Table 5 indicates that the step depth has the most impact, as it has the broadest GRG range of 0.2456. This is followed by spindle speed (0.1056), tool diameter (0.0659) and feed rate (0.0566). An extended GRG range is a sign of a greater performance influence. The table is also indicative of the best settings: Level 1 in tool diameter (GRG = 0.6531), Level 3 in spindle speed (GRG = 0.6711), Level 1 in step depth (GRG = 0.7364), and Level 1 in feed rate (GRG = 0.6461). According to the GRG analysis, the following settings are suggested as the most successful settings: tool diameter 8 mm, spindle speed 700 rpm, step depth 0.3 mm, feed rate 500 mm/min. GRG results also concur with those of the ANOVA and machine learning, once again, showing step depth as the most influential factor. The response table validates step depth, which is the most dominant parameter, as evidence of ANOVA and ML.

Table 5.

Response table.

Parameters	Levels			Δ	Rank
	1	2	3
Tool diameter	0.6531*	0.5910	0.5871	0.0659	3
Spindle speed	0.5655	0.5945	0.6711*	0.1056	2
Step depth	0.7364*	0.6041	0.4908	0.2456	1
Feed rate	0.6461*	0.5955	0.5896	0.0566	4

Overall mean GRG = 0.65

* Optimum level

Δ = Maximum Value Minimum value

Machine learning-based trend analysis

Recent experiments have demonstrated that machine learning has the capability of modelling the nonlinear behaviour in incremental forming, which is complex^42–46. As an example, Mollensiep et al.²⁴ trained neural networks to predict the presence of geometric accuracy^46,47, and others have combined ML with crystal-plasticity models to predict material behaviour^37,48. In this connection, a Random Forest (RF) model was constructed not only as a black-box predictor, but also as a validation tool, which supports the physical trends determined by ANOVA. Since Taguchi L27 dataset is discrete, the RF regression model offers a general mechanism of modeling nonlinear interaction and ranking feature importance without the parametric assumptions of ANOVA. The Random Forest (RF) model was evaluated across a tool diameter range of 10 mm to 12 mm to assess predictive performance and accuracy of the machine learning model were quantitatively evaluated using three standard statistical metrics. The coefficient of determination (R²), root mean square error (RMSE), and mean absolute error (MAE) were calculated using Eqs. (1)– (3), respectively. The coefficient of determination is defined in Eq. (1), while RMSE and MAE are given by Eqs. (2) and (3), respectively.

1

2

3

where represents the total number of experimental observations (dataset size), is the actual experimental Grey Relational Grade (GRG) value for the -th observation, is the corresponding GRG value predicted by the ML model, and is the mean of the actual experimental GRG values.

The Random Forest model yielded a coefficient of determination (R²) of 0.66, with a Root Mean Square Error (RMSE) of 0.081 and a Mean Absolute Error (MAE) of 0.068 on the test data. While an R² of 0.66 typically suggests moderate predictive capability in continuous domains, this result must be interpreted within the context of the experimental design.

The Taguchi L27 array is inherently discrete, utilizing widely spaced parameter levels (e.g., step depths of 0.3, 0.5, and 0.7 mm). Regression models (Fig. 3), which attempt to fit continuous curves through data, naturally face challenges when interpolating between such distinct, non-continuous process settings. Consequently, the primary utility of this ML framework is not precise numerical prediction, but rather feature importance ranking and trend validation.

Fig. 3.

Correlation between experimental GRG and Random Forest estimated trends.

As illustrated in Fig. 4, the Random Forest feature importance analysis independently identified step depth as the dominant factor affecting performance, followed by spindle speed. This ranking is in near-perfect agreement with the ANOVA results, which attributed 70.12% of the variance to step depth and 12.97% to spindle speed. This convergence of “physical statistics” (ANOVA) and “digital statistics” (ML) robustly validates the finding that step depth control is the critical mechanism for stable warm ISF of AZ61, regardless of the modeling approach used.

Fig. 4.

Feature importance obtained from random forest regression.

The moderate R² of the ML model can be greatly attributed to the discrete nature of the Taguchi experimental design that is orthogonal. Hence, the ML framework is viewed as a decision-support tool, as opposed to a high-accuracy surrogate model. Nevertheless, the given limitation does not negate the advantages of using ML with experimental and statistical results as it makes the confidence in the tendencies of dominance in parameters high and creates the basis of further data-heavy intelligent manufacturing systems, such as digital twins and adaptive process control.

Strain based forming limit diagram (FLD) analysis

Strain-based forming limit diagram (FLD) in Fig. 5 measures the behaviour of AZ61 magnesium alloy in a straight-groove test in warm incremental sheet forming (ISF). Unlike conventional stamping FLDs, the strain-based FLD in ISF represents fracture-controlled limits under localized deformation paths. A total of 27 experiments were conducted with varying tool diameter, spindle speed, step depth, and feed rate based on a Taguchi L27 design. Circle-grid analysis at the location of deformation of each specimen was used to measure the major and minor strains. The strain values were as low as 0.471 in Experiment 21 up to 1.120 in Experiment 22 which shows the influences of the process parameters and their interaction on formability.

Fig. 5.

Strain based forming limit diagram of total 27 experiments.

The FLD indicates that major strains of above 0.9 were obtained in experiment 1, 4, 16, 22 and 23 which is a good formability. These were runs with small step depth (0.3–0.5 mm) and low to medium feed rates (500–800 mm/min). Experiment 22, where the tool had a diameter of 8 mm, the spindle speed was 700 rpm, step depth was 0.3 mm and feed rate of 1000 mm/min, maximum strain attained was the optimal balance between the tool-workpiece contact and the continuous flow of plastic. Other trials 2, 7, 10, 13, 17, 19 employed the same step depths but different feed rates and spindle speeds using intermediate tool diameters (10–12 mm). These conditions provided an equal strain distribution with a bit of resistance to deformation as compared to the high strain group. Lower strains were measured in experiments 3, 6, 9, 14, 21 and 27 and indicate early thinning and possible fracture. They used a greater step depth of 0.7 mm and increased feed rates (800–1000 mm/min), and this led to unstable deformation and minimal time was available to stress relaxation. As an example, experiment 21, where the tool was 12 mm, step depth was 0.7 mm and the feed rate was 800 mm/min, recorded the lowest strain value of 0.464, which means that it was a poor formability. FLD indicates three deformation regimes, primarily controlled by step depth. Lower step depth promotes a stable strain path and delays localization; higher step depth increases thinning rate and promotes early fracture. To quantitatively evaluate the enhancement in formability, the critical plane strain intercept, FLD₀, was extracted directly from the experimental forming limit data. Under the optimized warm forming conditions (250 °C), the maximum safe FLD₀ value for the AZ61 sheet was determined to be approximately 0.67. This represents a massive quantitative improvement when compared to the room-temperature formability of AZ-series magnesium alloys documented in the literature, where typical FLD₀ values are strictly limited to approximately 0.10–0.15^49,50. This quantitative expansion of the safe-forming zone mathematically confirms the efficacy of the elevated temperature and localized incremental deformation in activating non-basal slip mechanisms.

Fracture behaviour

To investigate the fracture mechanism of AZ61 magnesium alloy during warm incremental sheet forming, scanning electron microscopy (SEM) analysis was performed at the fracture location of the specimen corresponding to Experiment 22, which exhibited the highest strain value of 1.12. After forming, the sheet was sectioned at the fracture location, and SEM observations were conducted on the cut cross-section of the fractured region.

The SEM micrograph at 2,500× magnification (Fig. 6) shows the overall fracture morphology characterized by numerous microvoids, irregular cavities, and localized material separation zones, indicating significant plastic deformation prior to fracture. The observed voids have approximate diameters ranging from 3 to 10 μm, with several regions showing clusters of voids suggesting void growth and coalescence during deformation.

Fig. 6.

SEM micrograph of the cross-section at the fracture location of AZ61 sheet formed under optimized warm incremental sheet forming conditions (Experiment 22) showing microvoid formation and ductile fracture morphology at 2,500× magnification.

A higher magnification SEM micrograph (7,500× magnification, Fig. 7) reveals more detailed fracture features including deep dimples, micro-dimple clusters, and tearing ridges. The dimples exhibit approximate sizes of 2–5 μm, which are typical indicators of ductile fracture through the microvoid nucleation–growth–coalescence mechanism.

Fig. 7.

Higher magnification SEM micrograph of the cross-section fracture surface of AZ61 magnesium alloy (Experiment 22) showing dimples, void coalescence, and tearing ridges, indicating ductile fracture behaviour at 7,500× magnification.

The SEM micrographs were oriented with respect to the tool travel direction during incremental forming, allowing a clearer interpretation of fracture morphology relative to the deformation path. The alignment of dimples and tearing ridges along the forming direction indicates that fracture occurred after significant localized plastic flow.

The observed fracture morphology can therefore be classified as ductile dimple rupture, which is typical for magnesium alloys deformed under elevated temperature conditions. At the forming temperature of approximately 250 °C, activation of additional non-basal slip systems enhances plastic deformability and delays crack initiation, resulting in extensive microvoid formation before final fracture. Similar ductile fracture characteristics in warm forming of magnesium alloys have also been reported by Bao et al. and Magdum et al.^51,52.

Overall, the SEM fractography confirms that fracture in warm incremental sheet forming of AZ61 magnesium alloy occurs predominantly through a ductile microvoid coalescence mechanism, consistent with the high formability observed in the FLD analysis.

Conclusions

This study presents an integrated experimental and data-driven framework of optimizing warm incremental sheet forming of AZ61 magnesium alloy. At the end of the Taguchi experiments, the Grey Relational Analysis and machine learning validation, the following conclusions were made.

The process window with the highest formability and lower surface roughness was determined by the use of the Grey Relational Analysis. The best parameters are a tool size of 8 mm, spindle speed of 700 rpm, step depth of 0.3 mm and feed rate of 500 mm/min.

ANOVA of the Grade of Grey Relational reveals that the step depth is dominant with 70.12% of the overall performance variance. Spindle speed accounted for 12.97% of the total variance, significantly influencing process stability by controlling the extent of friction-induced thermal softening.

The strain-based FLD reveals that with low step depths (0.3–0.5 mm) a stable strain distribution is achieved and localized necking is delayed. The SEM fractography of the specimens of the highest formability displays ductile fracture characteristics, deep dimples and tearing fibers, which proves that non-basal slip systems were facilitated under the warm forming conditions.

The Random Forest model served as an effective decision-support tool. Its feature-importance analysis was equivalent to the ANOVA, which independently verified the step depth as the most important factor without using parametric assumptions. This validates the reliability of the experimental results in spite of the discrete data. The results, supported by the ANOVA (Table 4) and FLD analysis (Fig. 5), demonstrate that maintaining a step depth between 0.3 mm and 0.5 mm is strictly required to control strain gradients and prevent premature fracture. Concurrently, regulating localized heat input via the spindle speed (optimized at 700 rpm) is essential. Higher spindle speeds generate beneficial frictional heating which, when combined with the external heating system, enhances thermal softening. This effectively lowers the flow stress and sustains the activation of non-basal slip systems, as evidenced by the highly ductile fracture morphology (deep dimples) observed in the SEM fractography (Fig. 6) of the optimally formed specimens. Together, these parameters dictate the successful forming of lightweight AZ61 magnesium alloys. The demonstrated consistency between statistical and ML approaches provides a foundational framework for developing future adaptive control systems and digital twins in the context of Industry 4.0.

From an industrial perspective, the identification of optimal process parameters for warm incremental sheet forming of AZ61 magnesium alloy provides a practical guideline for manufacturing lightweight components with improved surface quality and formability. The dominance of step depth as a controlling parameter offers a direct and implementable strategy for process optimization in industrial ISF setups. Future research may focus on extending the present framework toward real-time process monitoring, adaptive control strategies, and integration with digital twin systems. Additionally, further investigations involving larger datasets and advanced hybrid machine learning models may enhance predictive capability and enable robust intelligent manufacturing applications.

Author contributions

A.R.M. and R.A.M. conceived and designed the study, conducted the experiments, and wrote the main manuscript text. P.C. contributed to experimental planning, process analysis, and interpretation of formability results. P.P. assisted with statistical analysis, Grey Relational Analysis, and optimization methodology. A.K. performed microstructural characterization and SEM analysis and contributed to the discussion of fracture mechanisms. O.O.B. developed and validated the data-driven and machine learning models and assisted with result interpretation. All authors reviewed, revised, and approved the final manuscript.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Data availability

The analyzed data are included within the article.

Declarations

Competing interests

The authors declare no competing interests.