1 Introduction

Single point incremental forming (SPIF) is an innovative sheet-forming technique economically well suited for low batch production and prototyping. In fact, this process contrary to traditional press forming operations [1] does not employ dedicated set of tooling in order to perform three-dimensional shaping. The blank is held at the periphery in a clamping rig, and a simple spherical-end tool tightened in a spindle of CNC milling machine follows a predefined path to produce the target shape in a series of loops. These features render SPIF a very flexible process. The process has demonstrated numerous applications in the automotive, biomedical and aircraft industries [2, 3]. Though its understanding with growing research has improved in the past years, yet the process is immature to be successfully deployed on industrial scale. The research community is spending a great deal of efforts in order to improve its industrial suitability in terms of applications and sustainability [4,5,6,7,8,9,10,11].

The limit by which a sheet can be formed without fracturing (i.e., formability) has been a matter of concern for every sheet forming process, because this limit has influence on the process window thereby directly affecting the achievable product. The SPIF process offers higher formability than a conventional pressing process [12,13,14]. The shear, bending and local deformation have been reported as the main factors in this regard [14,15,16,17]. Centrino et al. [18] have presented a good piece of work on this subject. Isolating the effect of stretching and bending, they found that the localized nature of deformation significantly contributes in raising the formability in SPIF in comparison to conventional forming. Furthermore, tools with smaller radii (R as defined in Figure 1 (a)) suppress the neck formation and delay the fracture. The advantage of employing smaller tools on the formability has been also recognized by many other authors [19,20,21,22,23,24].

Figure 1
figure 1

The terminology and test geometries: a Fixed angle frustum of cone, and b Varying wall angle frustum of cone

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However, a small-radius tool does not necessarily raise the sheet formability in SPIF. Excessively small tools can push the process from a regular forming state to an irregular one whereby the forming tool plows the sheet thereby squeezing out the material from the tool/sheet interface and causing a premature failure. This state happens when the tool size is smaller than a critical value. In Ref. [25], this critical value has been proposed as a function of sheet thickness, i.e., Rc = βTo where the coefficients β is 2.2. The proposed critical small radius is believed to maximize the formability in SPIF. This critical condition was developed by employing the sheet material of 1 mm thickness and step size of 0.2 mm. As bending affects deformation stability in SPIF [16, 18], and the bending condition alters with the sheet thickness and step size, it is most likely that the formability will vary accordingly thereby affecting the critical small radius consequently. The present study investigates this point in detail with an objective to explore a more generic critical small radius condition as opposed to the specific one proposed in Ref. [25].

To achieve the objective, a set of aluminum and steel sheets with different thicknesses is selected and formed to fracture employing a range of tool radii and two levels of step size. The critical small radius offering the maximum formability is identified in each test. The FEA is conducted to study the strain state and deformation physics under the forming tools, especially small ones. Interestingly, the coefficient in critical radius condition is found to show dependence on the sheet thickness thereby specifying a critical range and further clarifying and correcting the condition reported in the early work [25]. This newly proposed critical small radius condition is relatively generic, applicable under wider forming conditions.

2 Procedures for Experiments and Finite Element Modeling

To obtain a generalized condition of critical small radius, the formability tests were performed on five sheet-metals with different conditions (such as annealed, cold worked and T6-tempered) and having different thicknesses (0.8 mm to 3 mm) (Table 1). One sheet from each type of material sheet (i.e., one thickness only) was subjected to tensile test on a universal testing machine. The test was performed at a cross head speed of 2 mm/min and following the ASTM-E8 standard.

Table 1 Experimental plan
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As listed in Table 1, the tensile area reduction (a formability indicator in SPIF [2]) of these materials varied from 24% (AA2024-T6) to 82% (AA1060-O). This ensured that the possible effect of properties variation on small radius condition was included in order to realize a generic tool-radius condition. The formability in the present study was defined as the maximum wall-angle that a sheet endures without fracturing (i.e., θmax), a measure commonly employed in the SPIF literature. The formability was determined employing the VWACF test, proposed in Ref. [26]. This test makes use of a conical frustum whose wall-angle (θ as defined in Figure 1 (b)) continuously increases with the depth thus inducing corresponding thinning. The sheet finally fractures somewhere between 0° and 90° whenever the thinning limit is surpassed, and the wall-angle corresponding to fracture is regarded as θmax.

The geometry of the test used herein study has been shown in Figure 1 (b). In order to ensure plane-strain condition during forming, the diameters of the frustums (especially the top and bottom diameters) were kept large following the guidelines reported in Ref. [27]. All kinds of tests, including 55° cones and formability cones, were performed keeping the operating parameters fixed as: feed rate: 3000 mm/min; spindle speed: 0 rev/min and step size: 0.3 mm. In order to examine the effect of step size on material plowing/flow mentioned earlier and hence on the small radius condition, some additional tests were performed employing AA10160-H24 as the test material. The step size in these tests was increased from 0.3 mm to 0.7 mm. In order to provide statistical means to the data, the tests were repeated twice.

The ABAQUS computing package was used for finite element modeling. The frustum of cone with the fixed angle of 55° was simulated using explicit modeling technique because this modeling method reduces computational time and provides results with reasonable accuracy [28]. The commercial Al (1 mm thick) obeying the following hardening law was used as the modeling material:

$$\bar{\sigma }{\text{ = }}\bar{\varepsilon }^{{{\text{0 }}{\text{.2}}}}$$
(1)

where \(\tilde{\sigma }\) and \(\stackrel{-}{\varepsilon }\) are the effective stress and effective strain, respectively. Both the sheet and forming tool were discretized using C3D8R brick elements (0.5 mm × 0.5 mm × 1 mm). The nature of contact at the tool/sheet interface was set as hard, i.e., the tool was not allowed to penetrate into the sheet. The friction coefficient at the contacting interface, which was acquired from the pin-on-disc test, was defined as 0.05. The simulations were done employing three different normalized tool radii 'R, i.e., 1.1 (lowest), 2.2 and 3.9 (highest).

3 Experimental and Numerical Simulation Results

Figure 2(a–d) presents the formability (i.e., θmax) as a function of normalized tool-radius 'R for four different sheet materials. The normalized radius was used because it determines the mechanisms (bending and shear) that affect the failure [18] and hence serves as a meaningful formability determinant. Contrary to one thickness in [25], the formability is shown for sheets of various thicknesses. As can be observed, formability in each of the sheet materials increases to the maximum value followed by a gradual decrease as the normalized radius 'R decreases. Thus, the formability follows an inverse V-type trend over the range of 'R. There is one maximum value of θmax in each formability curve and this occurs against a specific tool-radius which can be termed as critical small radius and can be denoted as 'Rc.

Figure 2
figure 2

Formability curves of various at varying thickness: a 1060-O, b 2024-O, c 1060-H24, and d 2024-T6: avg dev in the results ranged from 0.03% to 0.12%

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To the right of formability diagram/curve, the formability trend is in agreement with the previous studies [19,20,21,22,23,24] which attribute this trend to increase in the size of deformation zone and damage when the tool-radius increases. To the left of formability curve, however, the trend is uncommon and similar to what observed in Ref. [25] for 1 mm thick sheets. This peculiar trend can be reasoned to corresponding increase in material plowing and damage with a decrease in the tool-radius [25].

This is to observe from Figure 2(a–d) that the formability in SPIF (i.e., θmax) in agreement with the literature [19, 20] shows dependence on the sheet thickness. However, interestingly, there are some examples in which a thin sheet offers higher formability than a thick sheet, despite forming was performed under the same conditions and using a constant tool-radius 'R. This trend is more pronounced in annealed sheets: say in 2024-O against 'R of 2 and in 1060-O against 'R of 1.1. This might be attributed to a reason that thin sheets during SPIF experience greater elastic bending and thus reduced bending strains than the thick sheets. Besides affecting the formability, sheet thickness also influences the maximum achievable value of θmax (i.e., ‘θmax) over the investigated range of tool-radius 'R. For example, the value of ‘θmax of AA2024-T6, increases from 52° to 53.5° as the thickness of sheet increases from 1 mm to 2 mm. A similar result is shown by AA10160-H24. However, for annealed materials (such as AA1060-O and AA2024-O), an increase in the sheet thickness does not significantly affect the value of ‘θmax. In addition to operating parameters and material thickness, the formability also depends on the material properties [13]. Therefore, the above variation in ‘θmax with sheet thickness might have appeared due to corresponding variation in the properties.

As observable from Figure 2(a–d), the critical value of normalized radius 'Rc offering the maximum formability ‘θmax consistent with Ref. [25] is about 2.2 (with a slight variation) for 1 mm thick sheet metals. However, its value increases as the sheet thickness increases, e.g., from 2.2 to 3.3 as the thickness of AA2024-T6 increases from 0.9 mm to 2 mm. Overall, 'Rc ranges from 2.2 to 3.3 or alternatively Rc ranges from 2.2To to 3.3To. This finding, as expected, reveals that the coefficient β of the threshold radius condition in SPIF is not fixed rather its value is dependent on the sheet thickness. The reason of this interesting finding can be explained by the bend severity (or bending strain) experienced by the sheet during SPIF. Theoretically, the bend severity and bending strain are determined by the parameter R/To. However, during forming with a constant tool-radius, the thick sheet experiences lower elastic bending than the thin sheet thus correspondingly enduring higher bending strains and material plowing leading to early failure. Therefore, a thicker sheet realizes the maximum formability with a tool of slightly larger radius because the bending strains and plowing reduce accordingly thereby overcoming the irregular state of forming. This is to notice from Figure 2 that the 1.84 mm thick AA1060-H24 sheet has a higher value of 'Rc (3.3) than the 1.84 mm thick AA1060-O sheet (2.7). This means that besides thickness the condition of the material also affects the value of 'Rc when thicker sheets are processed in SPIF.

Figure 3 presents the formability curves for two different step sizes (i.e., 0.3 mm and 0.7 mm). As can be seen, a variation in the step size does not cause any visible influence on the shape of formability curve. More important, there is no noticeable change in the value of 'Rc due to increase in the step size which follows that the step size is not a very influential parameter for 'Rc, at least in the investigated range. Therefore, the following small radius condition to maximize the formability in SPIF can be proposed:

Figure 3
figure 3

Formability curves at varying step size

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$${^{\prime}}{R}_{c}=\beta.$$
(2)

Alternatively, Eq. (2) can be written as:

$${R}_{c}=\beta {T}_{o},$$
(3)

where the coefficient β varies from 2.2 to 3.3.

Figure 4 presents the distribution of in-plane X- and Y-strains on the inner surface, as defined in the figure, for three different settings of 'R: X is the direction of tool travel and Y is the transverse direction. From the strain values, it is evident that the sheet in the X-direction experiences compression which decreases (in terms of magnitude) as the value of 'R increases from 1.1 to 3.9. In the Y-direction, the sheet encounters compression only when processing is done with very small tool (i.e., for 'R < 2.2). Moreover, this is to mention that the compression prevails only on the inner surface while the outer surface for any of the opted tools (i.e., for ‘R ranging from 1.1 to 3.9) was observed to experience tension due to stretching. Further, the nature of strain on the inner surface gradually shifts from compression to tension as the value of 'R increases from 1.1 onwards. Considering the reduced formability and in-plane compression observed for ‘Rc < 2.2, it can be said that the in-plane biaxial compression is not conducive for improving formability in SPIF.

Figure 4
figure 4

Strain state of sheet in front of tool

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In fact, this condition on one hand causes material plowing and on the other hand induces intensive contact pressure (i.e., from − 6×107 Pa to − 3.27×107 Pa as ‘R increases from 1.1 to 3.9, see Figure 5). Consequently, material plowing, in terms of height of squeezed material as depicted in Figure 6(a), increases with each increment of forming loop due to corresponding increase in compression at the tool/sheet interface: see X and Y stress profiles at two different levels of part depth (i.e., 5 mm and 20 mm) in Figure 6(b) and (c). This scenario subsequently causes premature failure and formability reduction. Therefore, such a condition should be avoided in SPIF by choosing a suitable tool-radius.

Figure 5
figure 5

Pressure contours for three levels of normalized tool-radius

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Figure 6
figure 6

Selected results for a sample formed with the tool-radius of 1.1 mm: (a) Progression of squeezed material; (b) X- stress for two depths of sample; and (b) Y- stress for two depths of sample

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4 Discussion

The experimental and analytical studies in the literature have reported that the formability in SPIF generally shows a downward trend when the tool-radius is increased, because the damage (i.e., volume of voids) and hydrostatic tension increase or in other words the hydrostatic compression decreases [21,22,23,24]. The present study, on the contrary, shows that the formability reduces even with the decreasing of tool-radius if the radius is below a critical value of ‘Rc (i.e., left side of formability curve). Analyzing formability in the pressing process, Kweon et al. [29] have reported that compression under the punch reduces damage; however, too large compression could lead severe damage and premature sheet fracturing. Combining the in-plane compression, metal plowing and reduced formability with ‘Rc < 2.2–3.3 dictates that the in-plane compression leads to sever damage in SPIF. Achievement of the highest formability against the critical small radius ‘Rc (radius that does not cause plowing) indicates that a little shear is useful for enhancing the formability in SPIF. This finding is in good agreement with Ref. [15] in which it has been reported that shear improves the formability. The current study clarifies that excessive shear, however, is not conducive for maximizing the formability.

In order to know the effect of using unduly small tools on forming of the shallow parts (i.e., forming at low angles), a 20° cone was formed employing a tool of 1.2 mm radius and 1 mm thick sheet (i.e., R < ’Rc) of AA1060-O. Contrary to steep forming (i.e., 55° angle cone), material squeezing was not so severe. However, the center of blank was observed to bulge out in the form of a pillow, which occurs due to in-plane biaxial compression, bending moment and thickening of material around the forming tool [30]. Due to increase in the pillow height with forming depth, consequently the sheet metal ruptured at the tool/sheet interface (see Figure 7). For AA1060-O, the formability window was found to be between 25° and 76°. This follows that the use of unduly small tools does not only reduce the process window on the higher side but also on the lower side of forming angle.

Figure 7
figure 7

Rupturing of sheet material due to pillowing during forming a shallow part using an unduly small tool ('R = 1.1 mm)

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Summarizing the present findings, the radius of forming tool affects the strain/pressure state in SPIF. Further, the critical small radius (i.e., 'Rc) offering the maximum formability is almost independent of step size. However, the condition of material has a certain effect on this radius. The value of 'Rc increases as the sheet thickness increases (i.e., from 2.2 to 3.3 as the thickness increases from 1 mm to 3 mm). The formability regardless of the forming parameters demonstrates an inverse V-type trend. Moreover, forming with tools corresponding to left side of the V-curve does not narrow the process window only on the upper side but also on the lower side.

5 Conclusions

In the current work, the formability characterization in SPIF was carried out employing small tools. The formability tests were performed employing a range of sheet thicknesses, various materials and step sizes, and a new level of understanding on the critical small radius in SPIF was attained. The following are the important findings:

  1. 1.

    There is a critical radius Rc that maximizes the formability in SPIF. Its value depends on the thickness of sheet such that Rc = βTo where the coefficient β ranges from 2.2 to 3.3 as the thickness increases from 1 mm to 3 mm. Further, its value is independent of step size in the investigated range. This newly proposed condition can facilitate the users to maximize the formability without doing trial and error tests for a wider range of forming conditions.

  2. 2.

    The selection of tool with R < Rc does not only reduce the high limit of formability but also diminishes the low limit thereby narrowing down the overall process window of SPIF. As instance, the formability window was observed to reduce between 25° and 76° when AA1060-O sheet was processed with the tool of radius of 1.2 mm. The higher limit decreases due to undue shearing and stretching, and lower limit reduces due to pillowing of sheet metal.

  3. 3.

    Bi-axial in-plane compression on the inner side of sheet imparts non-conducive condition to maximize the formability. Moreover, this occurs only when the selected radius of forming tool is smaller than the critical one (i.e., for R < Rc).

  4. 4.

    Regardless of the forming parameters, the formability as a function of tool-radius follows an inverse V-type trend. Therefore, V-curve is a characteristic of formability in SPIF.