Multi-Step Tool Paths Development for Reducing Geometric Deviation and Pillow Effect in the Single-Point Incremental Forming

With the advent of the actual industrial revolution, known as Industry 4.0, new manufacturing paradigms were defined, moving companies from mass production to customized batch production [1]. This latter is characterized by greater flexibility, where generic tools and low investment costs are mandatory [2]. In this panorama, incremental sheet forming (ISF) process represents a suitable shaping alternative, due to its intrinsic characteristics. ISF concerns the deformation of blanks of different materials [3] using a generic forming tool, usually hemispherical. The process is performed by moving the tool on a sequence of predetermined paths, which are a function of the desired geometry, incrementally increasing the axial penetration in the workpiece up to the final depth. For this reason, the contact area between the tool and the blank is small, leading to a localized deformation zone, which in turn reduces forming forces [4] and increases the formability of the material [5], defined as the maximum wall angle obtainable without breakage. Even though dedicated machines have been developed for ISF, this process can be performed using common three-axis computer numerical control (CNC) machines [6]. At the same time, several variants of the process, concerning the usage of a counter die or a counter forming tool (two-point incremental forming), employing quickly reconfigurable dies (multi-point incremental forming), implementing heating systems for increasing the ductility of difficult-to-form materials (hot ISF), and applying contactless tools specifically designed for allowing the blank deformation by the application of high-pressure fluids [7], are available. In its simplest variant, defined as single-point incremental forming (SPIF), the process is characterized by the absence of counter dies, and the workpiece geometry is merely modified by contact with the forming tool [8]. Despite the numerous alternatives and the advantages of the ISF process, several questions related to product’s quality remain still open [9]. Even in the case of simple geometry parts, in fact, the dimensional accuracy results are limited, showing deviations from the nominal computer-aided design in the order of millimeters [10]. This can be ascribed to the springback effect generated by the material twisting, related to the SPIF intrinsic uneven tool–blank contact, or occurring at the end of the process when the sheet is removed from the blank holder [11]. As a consequence, geometrical tolerances are affected as well, resulting in bulged surfaces rather than planar. This lack of flatness defined as pillow effect and represented in Fig. 1, results to be deleterious for subsequent assembly process [12] and generates precision issues in further applications, i.e., thin-walled molds for thermoplastics and composite materials [13]. Aimed by investigating the effects of the different parameters of the ISF process, a great number of scholars can be found in the literature. With the intent of studying how the material strength could impact the geometrical deviations of the formed part with respect to the nominal one, a comparison between the ISF of AA1050 and AA6160 aluminum alloys was proposed by Choudhary and Mulay [14], observing higher deviations for low strength alloy (AA1050). In regard to the influence of the shape and dimensions of the tool, the part accuracy and the material formability were studied by Najm and Paniti [15], where the use of a flat-end tool allowed a reduction in the pillow effect and an increase in the precision of 43% with respect to the hemispherical tool. On the contrary, an increased formability was observed when working with large-diameter hemispherical tools. The effects of the tool diameter, in addition to feed, spindle speed, and step-down (defined as the axial incremental penetration), were analyzed by Dai et al. [16], revealing the former as the only parameter influencing the springback. In particular, by reducing the tool diameter, a lower springback was observed. The authors implemented then a multi-step tool path strategy allowing a 60% decrease in the geometric deviation. A multi-pass strategy was applied by Liu et al. [17] as well, where a more homogenous thickness distribution and a reduction in the twisting effect were demonstrated, when an alternation in the direction of movement of the tool was employed. A reduction of thinning, forces, and deviations, in the production of circular variable angle axisymmetric and inner concave square cup parts, was accomplished by Chang et al. [18], by generating the tool path with a quadratic interpolation amongst consecutive points. Furthermore, different tool trajectories were explored by Prasad and Nagarajan [19], concerning the pure spiral or the combination of spiral and helical in different percentages, demonstrating the increase of the helical percentage in maintaining a greater thickness, while decreasing the springback. With the aim of increasing the formability of the material and achieving vertical surfaces (90 deg wall angle) in the production of AA1050 aluminum alloy square cup, three different tool path strategies were proposed by Buffa et al. [20]. These involved two monodirectional approaches, in which material failure was observed at a wall angle of 80 deg, and a multidirectional one that was able to produce vertical walls with improved accuracy due to better material redistribution. In general, a significant influence of multi-step tool path on formability and accuracy, especially when tool compensation methodologies were employed [21,22], was demonstrated [23,24]. However, the problem related to the pillow or bulge effect persists, and the associated bibliography is currently scarce. A deep analysis of this problem was proposed by Al-Ghamdi and Hussain [25], identifying the hardening exponent of the material together with the tool diameter as the most influencing factors. When the former is high, a more uniform deformation was observed with an increase in the height of the pillow, while a high value of the latter imposes a normal compressive stress on the sheet leading to a decrease in bulge. The same conclusion was achieved by Isidore et al. [26] where, in addition, the positive impact of the bottom-flat tool was underlined. The application of a vacuum system was proposed by Murugesan et al. [27] and showed an encouraging influence on pillow reduction. Nevertheless, there is a lack of studies concerning how bulges can be affected by tool path trajectories.

Therefore, the present article aims to individuate reliable SPIF multi-step tool path strategies capable of producing high-accuracy parts, without the need for further post-process and tool compensation operations. Oriented to the pillow effect and wall deviation minimization, by providing a uniform material thickness distribution, the influence of multi-tool passages is investigated. In consideration of this, four new multi-step tool path strategies, intended to improve product accuracy, in terms of geometric wall deviation and pillow effect reduction, were developed and analyzed. The different trajectories were tested through an extensive experimental campaign consisting of the ISF of frustum cones made of AA1050 H24 aluminum alloy. The resulting geometries were then acquired and compared with the nominal one to evaluate the geometric accuracy achievable from each proposed methodology. Moreover, forming forces, deformation field, and sheet thickness were measured. A single-step trajectory, concerning the same process parameters, was performed, as well, for comparison. Although some limitations related to the employed experimental parameters were present, the research outcomes are reliable. The reason of this will be pointed out in a dedicated section. The analyses of the incrementally formed parts allowed the identification of a peculiar path methodology, consisting of a first roughing step, in which a variable wall angle from the top to the bottom of the cone is described, followed by a finishing step, treading at the desired wall angle. This strategy yields high pillow precision and homogeneity of the thickness distribution, identifying it as the most suitable solution for the production of high-quality parts.

To evaluate the effects of the tool path trajectories on the dimensional accuracy of the final part, an extensive experimental campaign consisting of incremental sheet-forming operations was performed. The tests were executed on an MC 60 evolution three-axis CNC machine, equipped with a GE Fanuc Series 18i-M numerical control. Aimed to analyze the mechanics of the ISF process, several scholars employed simple geometries, such as truncated conicals or pyramidal prisms. Therefore, in this work, a truncated cone geometry, characterized by a major diameter D of 125 mm, a height H of 50 mm, and an internal wall angle α of 60 deg, was selected. Figure 2 reports a sketch of the analyzed geometry where the radius r connecting the conical and the bottom surfaces, equal to the radius of 10 mm of the employed hemispherical tool, together with the adopted reference system, are reported as well.

The starting workpieces were squared-shaped blanks made of AA1050 H24 aluminum alloy, with an edge length of 210 mm and a thickness of 1 mm. The chemical composition and mechanical properties of the alloy are reported in Table 1, where ultimate tensile stress (UTS) and yield stress (YS) are shown as well.

For detecting the deformation field by visioplasticity, a circle grid (5 mm in diameter) was reported on the blanks before the ISF process (Fig. 3).

For supporting the metal sheet and acquiring the working forces during the incremental forming operations, a dedicated fixture was realized (Fig. 4). This consisted of a lower plate where four threaded bars were screwed. At the top of the bars, the clamping system of the blank could be vertically regulated by means of nuts. The clamping system involved bottom and top blank holders having a square-shaped cavity with a 150 mm edge length. The sheet could be positioned directly on the bottom holder, or by interposing a support. This latter consisted of an aluminum plate with a thickness of 4 mm, where a rounded circular cavity with a diameter of 132 mm was realized for reducing the upper blank bending. The lower plate was fastened to a load cell that in turn was secured on the machine working table.

For each performed ISF test, the three components of the forming force were acquired during the whole process. To do this, a Kistler 9257BA load cell, with a range of ±5 kN for the x and y components and from −5 kN to +10 kN for the z one, was employed. The load cell was cabled to a Kistler Type 5233A control unit connected, in turn, to the Hottinger Brüel & Kjær^®catman data acquisition software. By means of this latter, a dedicated interface was designed permitting the recording of three component signals with a frequency of 200 Hz.

The geometrical accuracy achievable from five different tool path strategies was investigated and described in detail in the section “The Proposed Tool Path Strategies.” For each strategy adopted, the ISF process was performed with and without the application of the support, for a total number of ten experiments. No spindle speed was assigned to the tool that was free to rotate. All the tests were conducted in lubricated conditions (grease lubricant), by keeping a constant feed rate of 600 mm/s, and a step-down Δz of 1 mm. A hemispherical tool with a diameter of 20 mm (r = 10 mm), made of AISI 1045 steel, was employed in all the ISF tests. The values of the process parameters kept constant along the totality of the test are in accordance with the ones available in the SPIF literature [12,25], and their selection was oriented to an increased accuracy and a uniform thickness distribution. Since, as reported by Dai et al. [16], feed rate and step down have a negligible effect on the material springback and accuracy, for reducing the processing time by maintaining the reliability of the CNC machine, the feed rate was set to half of the maximum machine speed, while a step-down value equal to the sheet thickness was selected. The tool diameter value was selected to decrease the pillow defect, since it was demonstrated that high values have beneficial effect in this sense [25]. The choice of working in lubricated conditions with an unlocked spindle was applied to avoid blank scratches.

The numerical control (NC) programs were developed by the cad/cam interface of autodesk fusion 360^® software, according to the selected geometry, tool path mode, and tool diameter.

To measure the geometrical accuracy, at the end of each ISF operation, the formed frustum cone was first cleaned with acetone and then its geometry was acquired using the Hexagon^® RA-8525-7 scanning system. This system consists of a scanner (model AS1), coupled to a seven-axis arm, permitting to acquire the scanned geometry in the form of a cloud of points. The collection of the points was devoted to the dedicated software polyworks inspector^®, from which the visual representation of the acquired geometry was obtained. This latter was then managed by the polyworks modeler^® environment, allowing the conversion of the cloud of points in a tri-dimensional mesh, the mesh fixing (e.g., by hole filling or by the removal of fake points), and to export it in an STL file. The geometry of the generated STL file was then compared with the nominal one using the gom inspect^® software. After the alignment of the geometries, this software is capable of showing and measuring the deviations between them providing in addition their distribution.

Once the deformed grid was acquired, the component was cut along the diametric section to allow the measurement of the thickness distribution.

As previously reported, the selection of a determined tool path heavily influences, amongst other properties such as mechanical strength and roughness, the geometrical accuracy of the incrementally formed product. This is attributable to the fact that each tool trajectory is associated with a certain material displacement, emphasizing determined deformation mechanisms, and causing a specific work-hardening of the material. Consequently, the related springback effect leads to a deviation of the geometry from the nominal one reducing accuracy.

For investigating the influence of the tool path on the dimensional accuracy of the conical wall and on the pillow effect, different strategies were developed as reported in Fig. 6. Moving from the first proposed strategy to the last one, material springback, the correlated accuracy, and thickness distribution, result to be influenced by different deformation mechanisms. These are briefly discussed for each methodology aimed to individuate the mechanism combination ensuring the best quality.

The tool trajectory defined as ModeS (Fig. 6(b)) is a widely spread single-step strategy concerning the so-called helical tool path. For this reason, ModeS was selected as a reference to be compared with the ones subsequently described. According to this strategy, the tool starts to move following a 3D circular path with a fixed axial displacement Δz, called step down, up to reaching a predetermined axial depth (entering the phase in Fig. 6(a)). At this point, the tool starts to cover a series of complete circumferences in the x–y plane whose diameters depend on α and Δz (penetration phase in Fig. 6(a)). These movements are repeated until the final depth is reached. It was demonstrated [29] that the deformation mechanism induced by this strategy is a bending under tension state. In this condition, a membranal stretching, causing a thickness reduction, combined with a through-the-thickness compression is experienced by the blank. At the removal of the tool, the material springback causes a recovery of wall tension and bottom compression, promoting pillow formation and conical wall internal deflection.

With the intent of reducing residual bottom compression, ModeMA and ModeMB were developed. The ModeMA (Fig. 6(c)) represents a multi-step tool path strategy in which a first step similar to ModeS is performed up to a cone depth equal to H_C (point C in Fig. 6), followed by a second phase characterized by a lower wall angle β up to the final depth H (from point C to point D). All the passes are carried out in the X–Y plane considering constant step-down (Δz) values. In the CD segment, the step down is reduced according to the tan(β)/tan(α) ratio. During the CD movement, a membranal stretching is experienced with a resulting hardened material. Once the tool reaches the bottom, it moves from point D to point B by following a spiral path with a radial pitch of 1 mm. This redistributes part of the hardened material (deformed in the CD movement) on the bottom, while further reducing its compressive state. Finally, the tool moves from point B to point C following the wall slope α according to circular movements with increasing radius (rise phase). In the BC movement, the final redistribution of the material on the wall is performed. This last phase is characterized by wall membranal compression but, due to the increased strength of the material induced by the previous cold deformation path, the buckling phenomenon results to be mitigated, limiting material distortion and enhancing accuracy.

A light modification of ModeMA is represented by ModeMB (Fig. 6(d)) where, once reached the cone depth H_C, the tool moves toward the cone center (point E), at a constant z quote, and then it axially penetrates up to the final H (point D). Form this position, the tool moves along D–B and B–C segments as described of ModeMA. In terms of the deformation mechanism, the difference of ModeMB is represented by the lack of membranal stretching with respect to the one applied in the CD movement of ModeMA. However, this is compensated by an increased bending under tension in the ED movement.

In ModeMC strategy (Fig. 6(e)) a first step equal to ModeS, presenting the same deformation mechanism, is realized considering a wall angle α_R smaller than α. This step is associable to a roughing operation and it terminates at the final H depth (point F). From this position, the tool enlarges the lower surface to the desired bottom diameter (point B) following a spiral movement and inducing membranal stretching along the bottom. At this point, the tool moves from B to A with circular trajectories with constant Δz, considering the desired wall angle α. This second step, which can be considered as a finishing operation, permits a total redistribution of the material along the conical wall while enforcing membranal stretching due to the enlargement of the wall angle. Also in this case, the buckling phenomenon caused by the membranal compression, results to be limited by the increased strength of the wall material induced by the previous cold work operation.

The last multi-step strategy (ModeMD in Fig. 6(f)) can be seen as a sequence of roughing and finishing operations. ModeMD involves the same deformation mechanisms experienced by ModeMC, except for membranal stretching resulting from the partial working of the bottom. In this case, the roughing process is performed by realizing a cone with a variable wall angle increasing from a value α_R at the top (point A) to the wall angle α at the bottom (point B). The cross section of the resulting lateral surface is the internal curve in Fig. 6(f). The finishing passage is obtained by a series of circular passes at constant α and Δz, from the bottom to the top (point A), as in ModeMC. This strategy permits to achieve the desired dimension of the bottom part of the frustum cone (in the correspondence of the flat bottom), yet at the end of the roughing phase. In this manner, the direct deformation of the bottom is avoided, limiting its membranal compression and the related pillow effect during material springback. Moreover, the finishing path works only on the conical wall, redistributing blank thickness, merely affecting the bottom area.

Moreover, it is worth pointing out that the proposed tool path strategies can affect tool wear evolution and surface roughness [30,31], as well. These aspects were not taken into consideration in the present work, limiting the analysis to achievable part accuracies and thickness distributions. Nevertheless, due to their importance, a preliminary discussion on them is proposed in the section “Limitations of the Study.”

The comparison between the same strategy, applied with and without the usage of support, underlines a slight accuracy improvement when this is present. Considering the squared hole dimensions of the blank holders (150 mm × 150 mm) in the absence of the support, the minimum unconstrained length of the sheet is placed in the correspondence of the square middle, with a cantilever length on the radius of (150−125)/2 = 12.5 mm. The maximum unconstrained length corresponds, instead, to the square diagonal, with a cantilever length of (212−125)/2 = 43.5 mm. In supported conditions, this is reduced to (132−125)/2 = 3.5 mm. Despite the difference in the cantilever length between supported and unsupported conditions, which ranges from a minimum of 9 mm (at square middle) to a maximum of 40 mm (at square diagonal), its effect is merely appreciable at the top of the frustum cone. Moving toward the bottom, in fact, this effect becomes negligible, leading to a marginal influence of the support on the final accuracy.

The measurements of the sectioned parts demonstrate how directly working the bottom of the cone (ModeMA, ModeMB, and ModeMC) negatively affects the pillow effect. Moreover, totally working the bottom surface (ModeMA and ModeMB) removes any trace of flatness leading to the formation of an undesired small angle conical surface. This can be due to the rigid motion of the material not directly in contact with the tool, as considered in Ref. [20]. In fact, when the tool penetrates and starts to move radially from the center, the radial (F_r) and vertical components (F_z) of the force are directed outward and downward respectively (Fig. 8). In this condition, at beginning of the spiral movement of the tool, the desired cone depth at the bottom center is reached. However, as the tool increases its radial position, only the localized deformation area maintains the requested axial quote, while the bottom part previously worked is subjected to rigid translation. This effect is enhanced by the fact that the cone bottom results to be the least constrained are, being it the furthest one from the fixture. Moreover, as in the case of ModeMA, all the material of the bottom results to be work-hardened by the previous deformation, further promoting rigid motion. As a consequence, the spiral tool path permits the enlargement of the bottom diameter up to the desired radial quote but, shows a counter effect on bottom depth and planarity.

A partial rigid motion is observable on the bottom of the part also for the ModeMC strategy, where a small angle of the conical surface in the correspondence of the starting point of the spiral movement is noticeable. In this case, the bottom area is not subjected to deformation resulting flat and only shifted at an increased depth.

Referring to the accuracy of the conical wall, it is evident how it results to be lower for ModeS and in the upper part of ModeMA and ModeMB. Small geometric deviations are visible, instead, in the bottom part of ModeMA and ModeMB and on the walls of ModeMC and ModeMD, where a finishing passage was performed, underlining the positive influence of a roughing and finishing sequence.

The pillow values of ModeMD and ModeS are comparable and noticeably lower being 0.46 mm and 0.66 mm. These deviations result to be lower respect to the ones that are typically achievable by SPIF processes, ranging between 2 mm and 4 mm [13]. Moreover, even considering the request of tolerances correlated to the applications of components produced by SPIF, which is 0.5 mm [32], these are respected by the deviations ensured by ModeMD. Even though the highest conical wall accuracy is ensured by ModeMC, the analysis of the distribution of the deviations indicates less dispersed values for the ModeMD strategy, suggesting this latter as the best solution.

Concerning the loads generated during the ISF process, Figs. 9 and 10 report the values of forces' components along the process time for ModeS and ModeMD, respectively. Related to the x-component (F_x) (in Figs. 9(a) and 10(a)) and y-component (F_y) (in Figs. 9(b) and 10(b)), a typical oscillatory behavior [33], due to the sinusoidal nature of the signal, is visible. After an initial transitory, where F_x and F_y rapidly increase due to the increment of the tool–blank contact area, these components show a slightly increasing trend of positive and negative peaks' values during the descendent phase (linear AB movement for ModeS in Fig. 6(b) and the curvelinear one for ModeMD in Fig. 6(f)). Along this trend, fluctuations due to the dynamics of the measurement system are detectable. A quick force rise at the beginning of the SPIF process is observable for z-component F_z (in Figs. 9(c) and 10(c)) as well. After this, as reported in Ref. [33], F_z shows a drop before reaching a constant value that is maintained until the end of the descendent phase. Along this phase, low valleys of F_z, corresponding to the tool penetration phases, are visible. In addition, Fig. 10 underlines that the finishing path of ModeMD is characterized by lower values of force components, since in this stage the tool–blank contact surface is reduced. At the end of this finishing phase, the tool is getting closer to the upper part of the fixture. Therefore, due to the close proximity to the blank holder, the displacement of the blank is limited, causing a final force increase. For all the proposed strategies, a maximum value of around 400 N for both F_x and F_y, and around 600 N for F_z, were recorded. The only exception was found for ModeMB during the penetration from point E to point D (Fig. 6(d)), where a peak value of F_z equal to 1050 N was found. As visible in Figs. 9 and 10, the use of the support only influences the force behavior at the beginning (for ModeS and ModeMD) and at the end (for ModeMD) of the SPIF process. Its influence is instead neglectable in the rest of the deformation process. Again, this is ascribable to the proximity of the localized deformation area of the blank with the upper part of the equipment. As reported in Ref. [33], when the process is performed close to the blank holder, the forming mechanism is similar to a wiper bending, leading to higher force development. This effect is increased by the presence of the support that further limits the blank displacement. On the contrary, at the end of the finishing path, the force components are higher in supportless conditions. This can be explained considering the fact that, due to the higher material freedom in the absence of support, the roughing step results in the conical part having a lower diameter with respect to the supported structure; hence more material needs to be finished, leading to increased forces. Moving away from the upper part, the wiper bending mechanism is gradually transformed in the steady-state incremental forming regime, making the effect of the support presence negligible.

Figure 11 reports the F_z behavior for all the analyzed strategies when support is applied. Except for ModeMC, where a smaller wall angle is performed along the roughing stage, a good superposition of F_z is detectable for all the other methodologies. This happens until the end of the primary helical tool path phase, underlining the repeatability of the tests from the force point of view. Despite the forces to process the bottom surface for ModeMA and ModeMB (from point D to point B) being the same, the one related to ModeMA results to be shifted due to the intermediated phase from point C to point D.

Table 3 reports the process times for all the tool path strategies applied, and their percentages referred to ModeS. As expected, multi-step modalities present higher times. Moreover, finishing plus roughing paths (ModeMC and ModeMD) result to be the most time consuming, characterized by a duration greater than the double of ModeS. Despite this, a substantial product quality improvement was detected (Table 2) at the expense of almost negligible forces required for performing the finishing step (Fig. 11).

Starting from the deformed grids reported in Fig. 12 it was possible to evaluate the major and minor strains according to Eq. (1) reported in Fig. 13. The material twisting due to the asymmetrical mechanics of the ISF process, especially in the correspondence of the major diameter (lower parts in Fig. 12), is visible. Moreover, the trace related to the rising phase of ModeMB and the excessive pillow effect for ModeMB and ModeMC, are highlighted. As expected, the material in the wall and at the bottom is subjected to a biaxial stretching condition (Fig. 13). The strain results for ModeS, ModeMA, and ModeMB are more concentrated in the upper part of the graph corresponding to a minor strain ε₂ lower than 0.1, indicating a condition closed to the plain strain. On the contrary, ModeMC and ModeMD strains are positioned at higher ε₂ values, causing a more uniform deformation. All the strains on the cone bottom are located at very low values of both ε₁ and ε₂ and closed to the bisector of the first quadrant, revealing, as expected, an almost equi-biaxial stretching. In addition, the qualitative representation of the forming limit diagram (FLD) for Al1050-H24 achieved by Bouziane et al. [34] is reported. In all the performed tests, ruptures of the aluminum sheets were not reached, meaning that FLD curves obtainable for SPIF processes are positioned at higher ε₁ values. This confirms again the increased formability of materials when processed by means of incremental forming operations.

Despite the outcomes of this study allowed to individuate a remarkable multi-step tool path strategy (ModeMD) in terms of part accuracy and thickness distribution, it is worth emphasizing some limitations in the performed experimental campaign.

As previously reported, there were no variations in tool shape, tool radius, step-down size, feed rate value, and only a single material was tested. Concerning the effects of step-down, it has been demonstrated in Ref. [27] that it does not affect the part accuracy. On the contrary, a reduction of the tool diameter leads to an improved accuracy [16], while increasing the amount of positive bulge effect [25]. Considering the influence of the tool shape, a flat-end tool decreases the geometrical deviation of both wall and bottom surfaces with respect to the nominal ones [15]. Moreover, a material showing high strength [14] and low hardening exponent [25] is characterized by a better accuracy.

Taking into account the tool path direction, no alternate trajectories were applied along the deep of the cone (z-direction) and only the helical methodology was employed. Nevertheless, their variations do not affect either accuracy or pillow [17,19].

Additionally, this study does not consider the effect of the applied tool path strategies on tool wear and surface roughness. In this sense, several scholars observed that by increasing feed rate, step down, and blank thickness, individuated as the most influencing factors, the tool wear rises [30,35,36]. On the other hand, tool diameter and step down appear to have significant and opposite effects on the surface roughness. When the former increases and the latter decreases, the surface roughness is reduced [30,31]. However, even when multi-stage SPIF is concerned, there is a lack in the bibliography directly correlating multiple deformation passages with tool wear and roughness. Hence, based on the literature at the time, the effect of the proposed tool path strategies can only be hypothesized. In consideration of the increased process time and material hardening experienced during the first deformation stage, an augmentation of tool wear for multi-stage strategies can be expected. In particular for ModeMC and ModeMD where longer rising paths and higher process times are required. As reported in Ref. [31], the second tool passage (finishing path) in ModeMC and ModeMD should reduce wall waviness, causing a better surface quality.

In this preliminary analysis, each experimental test was performed once, without any repetition. Even if during the helicoidal tool path, the forces show a good repeatability, it is not ensured from a geometrical point of view, possibly undermining the improvements of results achieved by the implementation of ModeMD.

In consideration of the aforementioned gaps, further tests should be carried out, by varying the material and the tool, to evaluate their effects on the validity of the proposed multi-step strategy, by monitoring tool wear and surface roughness, to assess the strategy influence on them, and by adding repetitions for repeatability purposes.

This article presents the analysis of different multi-step tool path strategies, aimed to improve the accuracy of frustum cones manufactured by the SPIF process. In particular, the accuracy was assessed concerning the geometrical precision of the conical wall and the reduction of the pillow effect. Five tool trajectories were investigated: a single step one, two multi-step methodologies regarding the total deformation of the truncated cone bottom, and two multi-step strategies where a sequence of roughing and finishing passes were implemented. An extensive experimental campaign, consisting of SPIF operations were performed on AA1050 H24 alloy blanks on a three-axis CNC machine. The tests were performed with and without a support structure aimed to reduce the sheet bending. Forming forces, realized geometries, strain fields, and part thicknesses were measured. Despite the reported limitations of this study, the analysis of the experimental results leads to the following outcomes:

• The supported employment has negligible effects on the whole accuracy and merely influences the three components of forming loads.

• The total or partial work of the bottom surface (ModeMA, ModeMB, ModeMC) negatively affects the pillow effect, causing an increased deviation from the nominal geometry. This is due to the formation of a conical surface on the bottom as a consequence of the rigid motion of the material.

• The application of a roughing-and-finishing multi-step strategy (ModeMC, ModeMD) allows to achieve a uniform distribution of thickness and a thicker conical wall.

• Employing a roughing step with a variable wall angle that avoids the need of directly working on the bottom surface (ModeMD), allows a substantial reduction of pillow defect.

• The exploited roughing-and-finishing trajectories take a double process time with respect to the single-step approach. Nevertheless, the loads in the finishing step, and in particular the vertical component, are considerably lower than the roughing ones.

Following these considerations, ModeMD has been individuated as the best solution for the selected geometry manufacturing. Concerning the time-consuming nature of a roughing-and-finishing methodology, further studies will be carried out by increasing tool feed to evaluate its effect on part quality, forces, and eventual failures, aimed to reduce process times. Moreover, the effect of the proposed strategies on different materials and of varying process parameters will be supplementary investigated.

There are no conflicts of interest.

No data, models, or code were generated or used for this article.

d =

diameter of the undeformed circle in visio-plasticiy grid, mm

r =

forming tool radius, mm

t =

local sheet thickness, mm

D =

frustum cone top diameter, mm

H =

frustum cone height, mm

d_max =

maximum diameter of the grid circle in deformed conditions, mm

d_min =

minimum diameter of the grid circle in deformed conditions, mm

t_i =

initial sheet thickness, mm

F_r =

radial component of forming force, N

F_x =

component of the forming force along the x-axis, N

F_y =

component of the forming force along the y-axis, N

F_z =

component of the forming force along the z-axis (axial component), N

α =

frustum cone wall angle, mm

Δz =

step down, mm

ε₁ =

major strain

ε₂ =

minor strain