Fracture in stretch flanging by single point incremental forming

International Journal of Mechanical Sciences 278 (2024) 109438 Available online 29 May 2024 0020-7403/© 2024 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Fracture in stretch flanging by single point incremental forming J.A. L´ opez-Fern´ andez a , * , M. Borrego b , G. Centeno b , C. Vallellano b a Department of Civil, Materials, and Manufacturing Engineering, University of M´ alaga, C/ Dr. Ortiz Ramos s/n 29071, M´ alaga Spain b Department of Mechanical and Manufacturing Engineering, University of Seville, Camino de los Descubrimientos s/n 41092, Sevilla Spain ARTICLE INFO Keywords: Sheet metal Forming Fracture analysis Single point incremental forming Stretch Flanging Stress triaxiality ABSTRACT This study presents a comprehensive investigation of fracture in open-stretched flanges formed using Single Point Incremental Forming (SPIF) on AA2024-T3 aluminium sheets. The work systematically explores the impact of geometric parameters, namely initial width and length, on the occurrence of the two primary failure modes: fracture at the corner and fracture at the edge of the flange. The experimental campaign comprises a series of flanging tests on medium-small radii to examine the deformation process on both the inner and outer surfaces of the flanges. The novelty of the investigation lies in two key aspects. Firstly, it offers, for the first time, an experimental characterisation of the onset of failure at the flange corner, which significantly limits the process’s capabilities, and critically compares it with the failure at the flange edge. It is highlighted that the fracture at the corner is strongly influenced by tool-induced friction and the incremental nature of the SPIF process, resulting in a local formability significantly higher than expected with a quasi-proportional deformation process, as characterised by the conventional Fracture Forming Limit (FFL) evaluated through Nakazima tests. Secondly, a detailed formability analysis of the flange in the average stress triaxiality versus equivalent strain space is conducted, identifying a common region of Mode I fracture for plane stress applicable to both open-stretched flanges by SPIF and conventional Nakazima tests. To this end, an explicit numerical model based on the Barlat-89 anisotropic yield criterion is used to assess the non-proportional strain/stress paths, inherent in the incremental process. A numerical-experimental methodology is employed to predict both edge and corner failures of the flanges within the average stress triaxiality - equivalent strain space. The fracture loci for Mode I have been determined for both the incremental flanging tests and the Nakazima tests. A comparison of both fracture loci reveals a noticeably lower average stress triaxiality and larger equivalent strain in the SPIFed flanges than in the Nakazima tests, consistent with the increase in the apparent formability of the material observed experimentally in the incremental processes with respect to the conventional tests. 1. Introduction Incremental Sheet Forming (ISF) processes have had a significant impact on industrial applications since the 1970s. This technology reduces costs and increases flexibility compared to traditional forming methods that rely on dedicated tooling [79]. In their research review, Duflou et al. [28] highlighted ISF as a versatile tool for mass customisation and flexible production. The Single Point Incremental Forming (SPIF) variant of ISF, introduced in the 1990s, simplified the incremental forming process. Based on Leszak’s [45] earlier patent (1964), SPIF employs a CNC-controlled hemispherical tool to shape the sheet metal without the use of dedicated dies, providing adaptability and cost efficiency, particularly for aerospace and automotive components. The broad application of SPIF has been highlighted in various materials, including aluminium [20], titanium [77], and steel [36]. In addition, SPIF has been demonstrated to be effective in a range of other sheet materials, including polymers such as polycarbonate [69], and superalloys [14]). Moreover, recent studies are investigating the application of SPIF to multi-phase materials, such as fibre-reinforced materials [29] and bimetallic sheets [76]. Recent advances in ISF, particularly Single Point Incremental Forming (SPIF), have demonstrated significant improvements in formability and efficiency for flanging operations [25]. Research by Cui and Gao [24] and Movahedinia et al. [63] has shown promising results in controlling material thinning during circular hole flanging and flaring processes using multistage SPIF strategies. In this context, Cheng et al. [19] proposed negative multi-stage incremental forming as an appropriate method to prevent excessive thinning in straight-wall components * Corresponding author. E-mail address: [email protected] (J.A. L´ opez-Fern´ andez). Contents lists available at ScienceDirect International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci https://doi.org/10.1016/j.ijmecsci.2024.109438 Received 15 March 2024; Received in revised form 28 May 2024; Accepted 28 May 2024 International Journal of Mechanical Sciences 278 (2024) 109438 2 with narrow internal spaces. In a similar vein, ˙ Zaba et al. [90] pointed out that reducing step sizes improves geometric stability whereas increasing it can cause cracking, whereas Formisano et al. [32] proposed toolpaths alternating up and down steps to reduce forming forces and improve formability in single-point incremental forming. Furthermore, the characteristics of the forming tools used in SPIF have been studied extensively. For instance, the use of non-hemispherical tools, as demonstrated in paddle forming [12], has enabled an increase in hole expansion ratios in hole flanging. In addition, Chen et al. [18] and Wen et al. [86] investigated the impact of tool geometries on enhancing formability in hole flanging. A study by Coman et al. [23] has recently demonstrated that the use of TiN-coated tools in SPIF reduces the loads and improves surface roughness. Similarly, Centeno et al. [16] and Makwana et al. [54] conducted comprehensive analyses of the formability of circular flanges and square flanges, respectively, produced by single-stage SPIF utilising conventional forming limit curves and critical fracture toughness to establish formability limits. Although significant advancements have been made in SPIF, there has been a notable absence of research on open flanging when it is compared to the research on hole-flanging. This gap in the literature is addressed by Voswinckel et al. [82,83], who conducted studies that provided comprehensive insights into the influence of geometric and process parameters on formability and accuracy in open stretch and shrink flanging. Their findings demonstrated a notable enhancement in formability for shrink flanges obtained by SPIF, in comparison to conventional flanging processes. Similar results were reported by Zhang et al. [92] in their experimental and numerical study on deformation mechanics in AA5754-O sheets, and by Pandre et al. [66] in a comparative study of conventional and incremental stretch flanging of DP steel. Additionally, numerical simulations have been crucial in advancing our understanding of ISF processes, especially in predicting stress and strain behaviour. Studies that use finite element (FE) models, such as those by Li et al. [47] and Wen et al. [87] have been essential in exploring the application of various flanging tools for ISF. Continuing this trend, recent research in this field, including the work of Jin et al. [43] and Said et al. [70], has aimed to enhance the accuracy of predicting deformation behaviour through the use of advanced hardening models. Maqbool and Bambach [56] performed an intensive study of the energy contribution from different deformation modes on the geometrical accuracy. An important advance in this field is the use of explicit formulations in FE modelling. The principal advantage of explicit FE models is the implementation of mass scaling and/or time scaling, which enables a reduction in calculation time without a substantial impact on accuracy, as demonstrated by Pepelnjak et al., [67]. This reduction in calculation time has enables a more extensive investigation of process parameters, including step-down size [34] and the development of large-size models for multi-stage incremental sheet forming [1]. A challenging aspect of SPIF is the understanding of its fracture mechanism, which is affected by several factors, including stress triaxiality and the localised incremental deformation process. These factors, which can enhance formability beyond traditional limits, have been subjected to extensive study. Eyckens et al. [31] highlighted that SPIF often involves non-proportional strain paths affecting material formability, setting the stage for deeper investigations into this phenomenon. Studies by Silva et al. [72,73] and Soeiro et al. [75] have investigated the limiting strains and the role of stress triaxiality in SPIF, proposing alternative fracture forming limit diagrams (FFLs) for mode I of fracture mechanics. Similarly, Jeswiet et al. [42] explored alternative FLDs in SPIF while Isik et al. [40] proposed a simplified methodology to determine maximum strains at fracture. Separately, research by Centeno et al. [15] found that traditional FLDs, which are effective in conventional forming, are not efficient in SPIF due to their unique deformation characteristics. This emphasised the need to reconsider traditional approaches. In this regard, Tian et al. [78] proposed an experimental method for determining the forming limit in incremental forming for specific applications. Other researchers, such as Mezher and Ahmed Shakir [61] and Bingqian et al. [13], have employed the advances in artificial intelligence and deep learning, respectively. Traditional ductile fracture criteria focus on the growth and coalescence of voids. These criteria are often divided into microscopic criteria and macroscopic criteria [41]. Concerning the microscopic approach, McClintock [60] and Rice and Tracey [68] studied the growth of cylindrical and spherical voids and their influence on the material failure by fracture. The criteria proposed by Cockcroft and Latham [22] and Clift et al. [21] considered major stress and mean stress, respectively, in the calculation of damage. The criteria of Rice and Tracey [68] include stress triaxiality, whereas other criteria, such as those proposed by Bai and Wierzbicki [5] and Lou et al. [52], consider both void growth and distortion by including shear stress in their damage functions. Modified classical coupled fracture criteria have been used to predict fracture in SPIF. Hirt et al. [37] modified the Gurson-Tvergaard-Needleman (GTN) model [80] to analyse the influence of typical SPIF process parameters. Gatea et al. [33] modified the GTN model to include shear stress effects and predict fracture in SPIF with more accuracy. Malhotra et al. [55] implemented a modification of the GTN coupled model proposed by Xue [89] to determine the damage accumulation in SPIF as compared to conventional press forming. However, according to Guzm´ an et al. [35], the GTN criterion may underestimate the failure in SPIF and must be coupled with plastic strain to provide a precise prediction of failure. A new extension of the GTN model proposed by Wu et al. [88] incorporates volume fraction and shear damage to describe damage accumulation under arbitrary loading conditions. To improve fracture prediction, some researchers have integrated anisotropic yield models into fracture criteria. For example, Li et al. [46] used the GTN model coupled with the Hill48 yield criterion. In a recent study, Dizajyekan et al. [26] analysed the initiation of fracture in Al/Cu laminated sheets using the Xue’s [89] damage criterion and the Hill48 anisotropic yield criterion, demonstrating higher fracture strain levels in SPIF in comparison to stretching tests due to nNar loading. Despite their flexibility, coupled models present significant challenges in practical application due to the large number of material constants and the difficulty in predicting fracture under negative values of stress triaxiality. The implementation of uncoupled fracture criteria is simpler, although they are less adaptable to different ranges of stress triaxiality. Huang et al. [38] and Nguyen et al. [64] were pioneers in using the uncoupled fracture criterion proposed by Oyane et al. [65] to predict fracture in SPIF with consideration of stress triaxiality. Bao and Wierzbicki [7](a) proposed a criterion for predicting failure in various fracture modes based on the fracture criterion of Vujovic and Shabaik [84]. This is particularly relevant in the context of SPIF, where the strain paths are not proportional, and the stress triaxiality fluctuates, as demonstrated by Eyckens et al. [30]. Another significant contribution was made by Bao and Wierzbicki [8](b), who introduced the concept of average stress triaxiality. Indeed, an understanding of how stress triaxiality affects material behaviour and failure is crucial for accurate fracture prediction in SPIF. Xue [89] highlighted the influence of the Lode angle on the onset of fracture, leading to further development of the MMC criterion by Bai and Wierzbicki [5,6]. Recently, the MMC criterion has been employed by Kasaei et al. [44] to predict ductile fracture in hole hemming of aluminium and magnesium, respectively, and by Seyyedi et al. [71], to compare the formability of conventional versus incremental hole flanging of aluminium sheets. Another fracture criterion that considers the Lode parameter, Lou et al. [52], was used by Zhang et al. [94] in conjunction with the Yld2000–2d criterion to accurately predict fracture in aluminium alloys. In order to account for material anisotropy and low stress triaxiality, Mirnia and Shamsari [62] implemented the criterion proposed by Bai and Wierzbicki [5] in conjunction with the Hill48 plasticity model to predict the fracture of various SPIF geometries addressing nonlinear loading conditions. Basak and Panda [10,11] estimated failure limits in SPIF by integrating anisotropy yield criteria Yld2000–2d into the model of Bao and Wierzbicki [8](b). Recently, Li et al. [48] have used the J.A. L´ opez-Fern´ andez et al. International Journal of Mechanical Sciences 278 (2024) 109438 3 anisotropic hybrid approach introduced by Lian et al. [49] to predict fracture in conventional hole flanging. The authors demonstrated the importance of considering anisotropy for an accurate prediction of edge fracture. Although some fracture models have demonstrated satisfactory results, the nature of SPIF makes it difficult to assess the limits of the fracture. In this regard, stress-based FLCs have offered more reliable predictions of material failure in SPIF than traditional strain-based methods, as they take into account the unique loading paths and contact conditions of SPIF. Some studies have shown that the average stress triaxiality levels in SPIF are lower than those in conventional processes. This has been found to affect the growth of voids, which is the primary damage mechanism in SPIF. Martins et al. [59] used McClintock’s damage criterion and Atkins [4] function for the assessment of fracture in order to assess the importance of stress triaxiality reduction and the dominance of void growth damage mechanism in SPIF. Mirnia and Shamsari [62] and Martínez-Donaire et al. [57] also adopted this approach, which further explored the triaxiality space to predict failure in SPIF. In this scientific context, this study aims to investigate the fracture mechanisms in open-stretch flanges formed by SPIF, exploring the role of non-proportional loading and stress triaxiality in material failure. A systematic analysis combining experimental and numerical approaches is conducted, which is pioneer in the examination of fracture at corners in flanges with medium-small radii. The experimental campaign assesses fracture for different flange widths and lengths, analysing the evolution of the principal strains on the inner and outer surface of the sheet. The results indicate significant straining at both the corner and edge of the flanges, showing values well above the conventional FFL obtained with Nakazima tests. An explicit finite element analysis is used to evaluate the non-proportional strain and stress paths inherent to the incremental process and their influence on failure modes. An experimentalnumerical methodology has been employed to identify the Mode I fracture limit in the average stress triaxiality and equivalent strain space for the first time in both the stretch flanging tests by SPIF and the conventional Nakazima tests. A comparison of the fracture loci reveals a notable discrepancy in the average stress triaxiality and equivalent strain between the SPIF and Nakazima tests. This is consistent with the observed increase in the apparent formability of the material in incremental processes with respect to the conventional method. After the brief review of the current state of the art provided in this section, the paper is organized as follows. Section 2 outlines the material characterisation process, including the determination of the FFL and its representation in the field of stress triaxiality and equivalent strain. This is achieved through two methods: (1) an analytical approach based on the Barlat and Lian [9] plasticity model, which is widely used in sheet metal forming, and (2) via finite element analysis. Section 3 presents the experimental campaign, whereas Section 4 outlines the characteristics of the FE model used for SPIF of flanges. Section 5 analyses the two modes of failure observed in flanges, evaluates formability via principal strains, and critically compares fracture loci from SPIF and Nakazima tests. Finally, Section 6 presents the main contributions of the present study. 2. Material characterization This section presents the characterization of the AA2024-T3 aluminium sheets used in this study. In Section 2.1, the tensile properties of the AA2024-T3 aluminium sheet were obtained in three sheet directions. The stress-strain behaviour in the rolling direction was modelled assuming Swift’s law. Additionally, material anisotropy was characterised by obtaining Lankford coefficients. Section 2.2 analyses the formability for AA2024-T3, determined through Nakazima tests. The strain evolution of these tests was evaluated using digital image correlation for necking assessment. In this regard, necking prior to the onset of fracture was not observed. Finally, the Fracture Forming Limit is presented for strain ratios ranging from uniaxial stress to equi-biaxial strain. In Section 2.3 the formability of AA2024-T3 aluminium sheets is performed with a particular focus on stress triaxiality and equivalent strain. This process has been carried out from two different approaches. On the one hand side, the fracture forming limit is obtained following an analytical-experimental procedure, using the Barlat-89 yield criterion for modelling plastic behaviour. Secondly, the numerical evolution of the stress triaxiality and average stress triaxiality corresponding to three representative Nakazima tests is presented, demonstrating a good agreement between the numerical and analytical FFL. 2.1. Tensile properties The tensile properties of a 1.2 mm thickness AA2024-T3 aluminium sheet were obtained in three sheet directions: rolling (0◦), transverse (90◦), and diagonal (45◦), using a universal INSTRON™ 1196 testing machine following ASTM E8/E8M–09 [2] standard. A Swift’s law, represented by the following equation, σ =742.36(0.025 + ε P)0.235,(1) was assumed to model the stress-strain behaviour in the rolling direction. The yield stress ( σ Y ), ultimate tensile strength ( σ UTS ) and modulus of elasticity (E) were determined using the force acquisition system of the testing machine and an Epsilon® axial extensometer, which was removed prior to fracture to avoid damage. A Poisson ratio ( ν ) of 0.33 was assumed. Furthermore, Lankford coefficients (r 0 , r 45 and r 90 ) were calculated according to standard ASTM E517–00 [3], thereby demonstrating the material anisotropy. These results, presented in Table 1, highlight the need to consider anisotropy in both experimental analysis and numerical modelling of the stretch-flanging process conducted in this study. 2.2. Formability limits within the forming limit diagram The formability limits for AA2024-T3 were determined through Nakazima tests, conducted on an Erichsen® model 142–20 universal testing machine, using a 100 mm diameter hemispherical punch in accordance with the ISO 12,004–2:2008 [39] standard. To ensure minimal friction, a tribological layer of Vaseline-PTFE-Vaseline was used between the punch and the specimens. Five different specimen geometries were utilised to cover a range of strain states from uniaxial to equibiaxial, following the procedures presented in L´ opez-Fern´ andez et al. [51] (Fig. 1). The strain evolution during these tests was captured using the 3D Digital Image Correlation (DIC) system Aramis® version 6.3 with 1.3 megapixels cameras, following the protocol for frame rate and facet size described by L´ opez-Fern´ andez et al. [50]. Despite employing both the temporal methodology by Martínez-Donaire et al. [58] and the standard ISO 12,004–2:2008 for necking evaluation, no necking was observed or predicted before fracture. This is in line with previous observations for similar materials carried out by Vallellano et al. [81]. Furthermore, the strain paths recorded by DIC showed a near-constant strain ratio β up to fracture. Given the absence of conventional necking, the FFL for AA2024-T3 was obtained in accordance with the approach proposed by L´ opez-Fern´ andez et al. [51], which assumes a constant strain ratio from Table 1 Tensile and anisotropy properties of AA2024-T3 sheet for longitudinal, transversal, and rolling direction. Direction σ Y (MPa) σ UTS (MPa) E (GPa) r θ 0◦336 445 69.4 0.76 45◦306 427 67.1 0.95 90◦319 443 68.2 0.54 Average 320 438 68.2 rθ=0.80 J.A. L´ opez-Fern´ andez et al. International Journal of Mechanical Sciences 278 (2024) 109438 4 the last point captured using the DIC system up to fracture. The thickness strain at fracture ( ε 3f ) was determined by evaluating the average thickness along the fracture zone (t f ), using a NICON® SMZ800 microscope and KAPPA® Image Base Metreo software, and applying volume constancy as performed by Centeno et al. [16] and Silva et al. [74]. This approach resulted in the major and minor strains at fracture, ε 1f and ε 2f , depicted in Fig. 1(a), which represent the FFL. 2.3. Formability considering stress triaxiality Formability in traditional sheet metal forming is often evaluated within the FLD, which is represented by the Fracture Forming Limit (FFL) when there is absence of necking. However, the FFL, derived from Nakazima tests involving quasi proportional strain paths, may not be fully applicable for SPIF processes, as far as the strain paths are nonproportional. This study aims to analyse the formability of flanges formed by SPIF, with a particular focus on stress triaxiality η and equivalent strain ε eq , as presented in a previous study by L´ opez- Fern´ andez et al. [50]. Obtaining the expression of η and ε eq requires selecting a plasticity model that is suitable for modelling the plastic behaviour of the material. In this study, the Barlat and Lian [9] yield criterion has been selected (see appendix A). The material constants of the Barlat-89 model, which are related to the anisotropy properties of the material, have been calculated from the Lankford coefficients r 0 and r 90 . Furthermore, the Barlat’s exponent m may vary depending on the material. Previous studies of Logan and Hosford [85] recommend a value of m =8 for metals with FCC microstructure in non-quadratic yield criteria. It should be noted that although the Barlat-89 plasticity model may present some limitations in accurately capturing the behaviour under biaxial stress states, particularly for aluminium alloys with a high degree of anisotropy, it has been able to reproduce reasonable well the experimental behaviour of the AA2024-T3 sheet in stretch and shrink flanging processes by SPIF; see L´ opez-Fern´ andez et al. [50] and (2023), respectively. In order to obtain the fracture locus in the η − ε eq space for the different tests performed in section 1.1, it is necessary to obtain the expressions of ε eq and η from the equations of the plasticity criterion, following the procedure outlined in Appendix A.2. Fig. 2a depicts the Barlat 89 plasticity criterion in the first quadrant corresponding to the AA2024-T3 material, assuming m =6. In the Nakazima tests, it was assumed that the strain paths were proportional, i.e. β was constant. Whereas β is function of α and material constants, it is not possible to obtain an explicit equation α =f(β). However, as depicted in Fig. 2b, which shows the relationship between α and β obtained using Eq. A.8, the stress ratio α was calculated from the strain ratio β numerically. In Fig. 1. (a) Forming limit diagram of AA2024-T3 sheet represented within the principal strains space. (b) Specimens used to obtain different strain ratios β after performing the Nakazima tests. Fig. 2. (a) Representation of the Barlat-89 criterion and the isotropic Von-Misses plasticity criterion calculated for AA2024-T3 in the space of principal stresses. (b) Relationship between the stress ratio α and strain ratio β obtained using the Barlat 89 plasticity criterion. J.A. L´ opez-Fern´ andez et al. International Journal of Mechanical Sciences 278 (2024) 109438 5 this regard, to obtain the equivalent strains at fracture (see Appendix A.2), α was obtained numerically, whereas the values of the major strain at fracture ε 1f were obtained from Nakazima tests. Fig. 3 shows the fracture points resulting from the Nakazima tests obtained using the expressions of ε eq and η obtained in Appendix A.2. For the FFL points, it was assumed that η was constant in each strain path, and α was obtained using β values at fracture. In addition to the FFL obtained from experimental results and analytical expressions, the η evolution was obtained using the numerical FE model. Fig 2 depicts the numerical β evolution for three Nakazima tests. The parameters of the numerical model are detailed in Section 4. As the FE model does not predict the onset of fracture, the ε eq at fracture was calibrated from experimental values. Regarding stress triaxiality evolution, it can be observed that this remains almost constant throughout the three tests presented in Fig. 3, namely uniaxial, plane-strain and equibiaxial. However, slight deviations can be seen due to the initial contact of the punch in the region where ε eq is very low. Furthermore, the stress triaxiality at fracture η f obtained using the equations from Barlat-89 criterion are almost identical to those obtained using FE. This validates the FE model and the Barlat-89 criterion for predicting the evolution of stress triaxiality under proportional loading. As stated in Section 1, the studies conducted by Mirnia and Shamsari [62] and Martínez-Donaire et al. [57] have highlighted the importance of the strain path in the fracture locus. Some fracture criteria, such as Bai and Wierzbicki [5] and Lou et al. [52], also consider stress triaxiality to predict the fracture locus in the η − ε eq space for processes with proportional loading. However, in non-proportional processes, Bao and Wierzbicki [8] used average stress triaxiality η instead. The following equation, η =1 ε eq ∫ ε eq 0 σ m σ d ε eq,(2) can be employed to determine η for a given equivalent strain ε eq . In subsequent sections, the formability of flanges performed by SPIF will be evaluated using η to consider non-proportionality, as proposed by Martínez-Donaire et al. [57]. Fig. 3 shows that η in Nakazima tests is very similar to η . Concretely, the fracture values of η for the tests represented are almost identical in both the FFL points obtained experimentally and the numerical model. In Section 5, the FFL represented in Fig. 3 will be used to discuss the formability of flanges performed by SPIF in the η − ε eq space. 3. Stretch flanging tests This section presents a comprehensive description of the stretch flanging experiments. It includes an overview of the experimental setup and the preparation of the blanks. Furthermore, this section outlines the process parameters and describes the procedure for performing principal strain measurements. A series of stretch flanging experiments were carried out using an EMCO VMC200 three-axis milling machine within the setup depicted in Fig. 4. The setup consisted of a lower die with a principal radius R, and a sheet holder designed to mitigate bulging effects. Along the main radius, the edge was smoothed with a fillet radius of 3 mm. Additionally, a protective layer of PTFE was used to minimise damage derived from the contact between the sheet and the die. The arrangement of the components is illustrated in the schematic view depicted in Fig. 5, which shows a cross-section of the experimental setup. The forming tool is made of stainless steel and has hemispherical shape. It follows circular arcs that change its direction to avoid twisting, followed by an incremental z-axis movement at the end of each pass. An additional gap of 0.2 mm was set between the die and tool to prevent the sheet from being pressed against the die. Throughout the process, the tool remains always in contact with the flange, especially at the end of each pass. To provide lubrication, Castrol Iloform® oil was applied via an air-oil cooling system as shown in Fig. 4. The blanks were prepared from aluminium sheets that were electroetched using a circle grid pattern on both sheet surfaces. The grid was marked on the surface opposite to the contact with the tool, as well as on the opposite surface in order to measure the strains at the corner of the flange in the bending zone. A strain analysis of the flanges was performed using Circle Grid Analysis (CGA) with ARGUS® software. It is important to note that CGA has limitations in measuring strains at the sheet edges. Given that the distance between adjacent points in the grid pattern is 1 mm, the strains at the sheet edges reported in the experimental contours correspond to strains 1 mm from the edge. Therefore, edge strains and fracture strains were further evaluated through thickness measurements using a microscope. As recently reported by Chang et al. [17] in an investigation of conventional stretch flanging, the degree of roughness on the edge of the blank sheets was found to be a significant factor in the onset of the fracture, particularly in the fracture at the corner. To minimise Fig. 3. Fracture forming limit of AA2024-T3 sheet represented within the average stress triaxiality versus equivalent strain diagram. The figure includes the evolution of the stress triaxiality and the average stress triaxiality for three representative Nakazima tests. Fig. 4. Experimental setup for stretch flanging by SPIF within the EMCO CNC machine. The setup comprises the forming tool, dies, cooling system, and a Kistler dynamometer. J.A. L´ opez-Fern´ andez et al. International Journal of Mechanical Sciences 278 (2024) 109438 6 experimental scatter associated with edge roughness, the test samples were milled and successively polished, finishing with # 1200 sandpaper. The process parameters and initial geometry of the specimens are listed in Table 2. These include the main radius of the flange R, the initial width w 0 , initial length l 0 , spindle speed S, tool diameter Φ tool , feed rate F and step down Δz. The process parameters were constant, and different values of w 0 and l 0 were set to obtain the process windows. The duration of each test was a function of w 0 and l 0 , with a range of 3 min for narrow-short flanges to 20 min for wide-large flanges. 4. Numerical modelling This section presents a description of the numerical model of stretch flanging by SPIF. The FE model is thoroughly described, including details of the mesh, element formulation, boundary conditions, and integration scheme. Additionally, a comprehensive mesh sensitivity analysis is presented to ascertain the optimal element size, considering both simulation time and data accuracy. This study uses numerical simulations to analyse the impact of stress triaxiality on the SPIF flanging process. Fig. 6a shows the undeformed mesh of the FE model created in Ansys LS-DYNA, a widely used finite element software for dynamic analysis. The computer utilised had 8 CPUs of 3 GHz and 32 Gb of RAM. Fig. 6(b) and Fig. 6(c) show the major strain contour for a deformed flange, evaluated on the outer and on the inner surfaces, respectively. The FE model was developed using an explicit dynamic integration scheme, employing the Berlytschko-Tsay element formulation with stiffness hourglass control based on the SHELL163 four-node element with five integration points through the sheet thickness. The distribution of the integration points allows for the evaluation of stress and strains at a distance from the sheet surface equal to one-tenth of the sheet thickness. In accordance with the recommendations specified in the Ansys User’s Guide, an hourglass control coefficient of 0.1 was utilised. The model has four parts: sheet, die, sheet holder and tool. The tool was meshed using a mesh of 0.25 mm of element size. The forming tool, die and sheet holder are rigid bodies. The tool movement was identical to the actual SPIF process described in Section 2.2, i.e. arcs in the horizontal plane defined by 300 discrete points. The velocity of the tool was increased by a factor of 1000 in comparison to the real process, while mass scaling was defined by a factor of 5. The sheet was modelled using the elastic-plastic model for anisotropic behaviour proposed by Barlat and Lian [9], following the stress-strain curve defined in Eq. (1), and the anisotropy coefficients presented in Table 1. The contact regions between the sheet and the forming tools were modelled using the Coulomb friction model, with a friction coefficient μ obtained through an iterative process. Initially, a friction coefficient μ =0.05 was used, as suggested in the literature for regular lubrication (see, for example, Zhang et al. [93]). Subsequently, this value was adjusted to μ =0.01 following an iterative optimisation process based on achieving the best concordance between numerical simulations and experimental observations in terms of strain distribution in the flange, as detailed in L´ opez-Fern´ andez et al. [50]. The exceptionally low friction coefficient served to illustrate the effectiveness of the local and continuous lubrication system implemented in the experimental setup. In order to ascertain the optimal element size in terms of both simulation time and data accuracy, a comprehensive mesh sensitivity analysis was conducted for a flange with dimensions of w 0 =63 mm and l 0 =25 mm. In this context, the principal strains at the critical point at the corner (see CPC in Fig. 10) were compared with experimental data obtained using CGA. It should be noted that due to the limitations of the CGA analysis discussed in Section 3, the strains were evaluated at a Fig. 5. Schematic representation of the flanging setup, which depicts the different components, and the principal geometry and process parameters. (a) A cross section of the SPIF setup and the forming tool, and (b) a formed stretch flange. Table 2 Selected process parameters in stretch flanging by SPIF. R (mm) w 0 (mm) l 0 (mm) S (rpm) ϕ tool (mm) F (mm/ min) Δ z (mm) 45 36/ 45 / 54 / 63 / 72 20 / 25 / 30 20 12 1000 0.4 Fig. 6. Finite element model for a flange width w 0 =54 mm and a flange length of l 0 =20 mm. The model, represented in (a), included the following elements: the sheet, the die, the sheet holder and the tool. The major strain distribution on the outer surface of the deformed flange is depicted in (b), while the major strain distribution on the inner surface is represented in (c). J.A. L´ opez-Fern´ andez et al. International Journal of Mechanical Sciences 278 (2024) 109438 7 distance of 1 mm from the flange edge. The results presented in Fig. 7 show a high dependence between the element size and the simulation time, with values ranging from approximately ten minutes to more than three days for element size values from 2.5 to 0.3 mm. The sensitivity analysis also shows the convergence of the principal strains. While the principal strains fluctuate around the experimental value for element sizes above 1 mm, a convergence is observed for element sizes below that value. This indicates that reducing the element size below 0.5 mm does not result in significant enhancements in accuracy. Additionally, reducing the element size below 0.3 showed instabilities that required tuning additional parameters, such as mass and time scaling. According to this, an element size of 0.5 mm was selected to perform the numerical analysis. Further details concerning the specific parameters of the FE model can be found in previous research, as detailed in L´ opez-Fern´ andez et al. [50] and (2023). Although the model was validated in terms of principal strains in the aforementioned studies, this paper analyses a mode of failure not analysed before, namely fracture at corner. Consequently, the model will be validated in Section 5.2 with particular attention to the results in the corner region of the flange. 5. Results and discussion This section presents the results of the research, providing a comprehensive analysis of stretch flanging by Single Point Incremental Forming. Section 5.1 presents an analysis of the failure modes observed in stretch flanging tests conducted using SPIF, with a specific focus on the impact of geometrical parameters, such as initial width and length, on the failure modes observed in flanges. The results, derived from an extensive experimental campaign, are compared with earlier research highlighting new insights specific to this study. Section 5.2 examines the strain analysis of the flanges, comparing the principal strain behaviours in successful flanges and flanges with fracture at different locations (corner and edge), using both experimental and numerical data. Furthermore, the section includes a validation of the numerical model based on principal strains that compare numerical and experimental critical sections both at the edge and at the corner of the flanges. Finally, in Section 5.3, the fracture in flanges formed by SPIF is analysed within the stress triaxiality space. Stress triaxiality is introduced as an appropriate metric to consider non-proportionality effects and the resulting void growth reduction, effectively capturing the loading history inherent in SPIF. Furthermore, the evolution of critical points for flanges with different widths and lengths is examined, determining the onset of fracture in the stress triaxiality-equivalent strain space using an experimental-numerical approach. Additionally, the fracture locus for Mode I fracture in SPIF is identified, and compared with the fracture limit obtained for Nakazima tests. 5.1. Modes of failure and process window The experimental campaign is an extension of the series of tests previously carried out by L´ opez-Fern´ andez et al. [50]. In this case, the tests were designed to assess the formability of two critical locations of the flange: the corner and the edge. This required the evaluation of strains not only at the outer surface of the flange, as in the previous study, but also on the opposite sheet surface. To this end, the sheet blanks were marked using a pattern of circles on both surfaces. It should be noted that the failure mechanism at the corner in open stretch flanging by SPIF has not yet been subjected to a systematic analysis in the literature. Three tests were conducted for each set of w 0 −l 0 parameters, providing statistical meaning to the experimental study, as illustrated in Table 3. This table effectively captures the effect of different flange sizes and shapes under the specified forming conditions, classifying them as successful flanges (S), flanges with fracture at the corner (FC), and flanges with fracture at the edge (FE). Fig. 8, which shows a flange from each case in Table 3, provides a visual representation of these results. The experimental results indicate the presence of two critical zones of fracture in flanging by SPIF: the corner and the edge of the flange. Notably, a considerable number of successful flanges were obtained, i.e. without fracture. Only one flange exhibited fracture at the edge for the selected w 0 and l 0 values. This observation could be attributed to the selected values for w 0 . It is typical for fracture to occur at the edge in hole flanging, both in conventional forming and in SPIF. However, the geometry of open stretch flanges with small radii (R =45 mm), as the focus of this study, results in reduced principal strains at the edge due to its open geometry. The sides of open flanges, in particular those with small w 0 and large R, come closer during forming, which reduces the maximum strains at the edge and, in some cases, prevents fracture. It can be established that fractures at the corner (FC) and at the edge (FE) are different. This variation is notably influenced by the dimensions w 0 and l 0 . Successful flanges are typically observed in narrower configurations or in wider flanges with small lengths. Conversely, for widths between 54 and 72 mm and lengths greater than 20 mm, fractures are observed. Fig. 9 presents three cases are presented to illustrate the different outcomes observed in the experiments: a successful flange, a flange with fracture at the corner, and a flange with fracture at the edge. The fractures at the edge are attributed to large strains along the edge, which are slightly larger on the outer surface, while the corner fractures are linked to substantial straining at the inner surface of the corner in the bending zone. In previous studies, such as those by Voswinckel et al. [83], the fracture at the corner was identified as a failure mode in SPIF stretch flanging. However, no detailed strain analysis was conducted to elucidate the underlying mechanics. Similarly, research by Dudra and Shah [27] suggested edge fractures as a characteristic failure mode. With regard to the quantitative results, the maximum l 0 /R ratio observed in this study is 0.66, corresponding to the narrowest flanges of w 0 =36 and w 0 =45 mm. The results indicate that flanges of any length may be achievable for these widths, suggesting an infinite theoretical l 0 / R. This occurs because, for small w 0 , flanges behave similarly to straight flanges with minimal strain at the edge, as long as the edge remains almost straight. Additionally, the stretching of the flange in the l 0 Fig. 7. Simulation time and principal strains for different element sizes evaluated at the corner of a flange with dimensions w 0 =63 mm and l 0 =25 mm. The black and grey dashed lines represent the experimental major and minor strains, respectively. Table 3 Process window for R die =45mm and different initial width and length. S (Successful), FC (fracture at corner) and FE (fracture at edge). w 0 [mm] l o [mm] 36 45 54 63 72 20 S S S S S 25 S S FC FC FE 30 S S FC FC FC J.A. L´ opez-Fern´ andez et al. International Journal of Mechanical Sciences 278 (2024) 109438 8 direction in narrow flanges is minimal, which means that fracture at the corner does not occur if the bending radius is sufficiently large. However, it is important to note that such extensively long flanges are very poor in terms of geometrical properties. This comprehensive approach, which analyses both the inner and outer surfaces of the flanges, offers valuable insights into the failure mechanisms, which is helpful in optimising the SPIF process for practical applications. The study has revealed that fracture at the corner is the dominant failure mode in stretch flanging by SPIF for flanges with a medium-size radius. The experimental data indicate that the failure modes are significantly influenced by geometrical parameters, particularly flange width. The data demonstrate a clear pattern whereby narrower flanges or those with shorter lengths tend to be more successful, whereas wider flanges with longer lengths are more prone to fracture. Furthermore, the findings indicate that for certain widths, flanges of any length may be feasible, suggesting potential for further optimisation in SPIF applications. The subsequent section presents a more detailed analysis based on principal strains. 5.2. Strain analysis and numerical validation Prior to presenting the strain analysis, it is necessary to identify the critical regions of interest in the flange, as far as the two modes of failure appear there. Fig. 10 offers a schematic representation of these critical areas, which include the Critical Section at Corner (CSC), Critical Section at Edge (CSE), Critical Point at Corner (CPC), and Critical Point at Edge (CPE). In fractured flanges, these critical sections and points coincide with the locations of failure. In case that these locations could not be measured using CGA, the measurement was performed in the symmetrical point. For flanges that were successfully formed, the critical areas align with regions exhibiting the highest major strains. In particular, the circle grid pattern marked on the upper surface of the flanges allows for an accurate analysis of strains at the corner, which is of special interest in this study. In this context, Fig. 11 depicts a flange with dimensions w 0 =63 mm and l 0 =20 mm. The strain distribution on both the inner and outer surfaces is depicted. The corner region, illustrated in Fig. 11a, shows significant straining on both sides of the flange, which is a result of the friction in l 0 direction induced by the forming tool. Referring to Fig. 11b, Fig. 8. Selection of flanges of different initial withs and initial lengths formed in the experimental plan representing the process window obtained for stretch flanging by SPIF. Fig. 9. Flanges in different positions representative of the various modes of failure observed in stretch flanging by SPIF. (a-c) Flange of w 0 =63 mm and l 0 =25 mm with fracture at corner. (d-f) Flange of w 0 =72 mm and l 0 =25 mm with fracture at edge. (g-i) Successful flange of w 0 =54 mm and l 0 =20 mm. J.A. L´ opez-Fern´ andez et al. International Journal of Mechanical Sciences 278 (2024) 109438 9 which represents the strain distribution within the FLD, there are strain points considerably above the FFL. This shows the enhancement in formability that is typically achieved in SPIF. This strain distribution and increase in formability at the corner was not previously experimentally analysed in the literature. The strains at the critical point at the corner (CPC), were determined via microscopy performing thickness measurement assuming uniaxial stress due to its free edge condition. The proximity of these strains to the FFL indicates that this flange configuration is on the limit of failure, particularly at the corner, as evidenced by the occurrence of larger l 0 configurations leading to fracture in this region. On the other hand, Fig. 11c presents the strain analysis on the flange outer surface, i.e., the surface with no tool-sheet contact. Here, the strains concentrate in two lobes at the edge, resulting in the critical point at the edge (CPE). Fig. 11d depicts a similar analysis is depicted for the outer surface, with strains closely approaching the FFL. These observations, particularly the proximity of strains to the FFL in Fig. 11b and 11d, indicate that this flange configuration is approaching the failure limits, primarily at the corner. Indeed, for flanges with the same width but longer l 0 , fractures have been observed in this region, as indicated in Table 3. This comprehensive strain analysis in Fig. 11 serves to highlight the criticality of these regions for the determination of formability and failure in flanging by SPIF. Fig. 12 illustrates a flange with dimensions w 0 =63 mm and l 0 =25 mm that has suffered fracture at the corner. The strain map (Fig. 12a) indicates that the corners are the most strained regions, as observed in Fig. 11a. Fig. 12b shows the critical section at the corner (CSC), which has few strain points above the FFL. The fracture strains at the critical point at the corner (CPC) were obtained by measuring the thickness of the fracture lips. The strain analysis on the outer surface (Fig. 12c and 12d) indicates that the flange was far from failing at the edge. The highest strains did not concentrate at the edge due to the prior corner fracture, and they are low compared to the strains observed at the corner. The critical point at the edge (CPE) was identified in the region exhibiting the highest strain within these lobes. Additionally, as shown in the strain distribution of the outer surface represented in Fig. 12d, the strains are slightly below the FFL, suggesting that although the flange fractured at the corner, it was far from experiencing a fracture at the edge. The strain analysis for a flange with w 0 =72 mm and l 0 =25 mm, which fractured at the edge, is presented in Fig. 13. The strain contours in this case are significantly different from those observed in previous cases. The strain levels at the corners are above the FFL, including the CSC. The analysis of the outer surface (Fig. 13c and 13d) suggests that the fracture initiates within the sheet, very close to the edge, and propagates towards it. The strain distribution demonstrates that a multitude of points are above the FFL, even in regions where no fracture occurred. Fig. 14 compiles the fracture points from the three flanges analysed Fig. 10. Picture of a deformed flange with a schematic representation for a successful flange of critical section at corner CSC, critical section at edge CSE, critical point at corner CPC and critical point at edge CPE. Fig. 11. Experimental strain analysis obtained using CGA of a successful flange of w 0 =63 mm and l 0 =20 mm. (a) Major strain distribution on the inner surface; (b) Strain point cloud and evolution along the critical section at corner (CSC) on the inner surface; (c) Major strain distribution on the outer surface; (d) Strain point cloud and evolution along the critical section at edge (CSE) on the outer surface. J.A. L´ opez-Fern´ andez et al. International Journal of Mechanical Sciences 278 (2024) 109438 16 α = σ 2 σ 1 (A.11) d ε eq = σ 1d ε 1+ σ 2d ε 2 σ =(1+ α β) σ 1d ε 1 σ (A.12) Then, using the expression of σ from Eq. 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