Keywords

1 Introduction

Conventional sheet metal forming processes require large batch sizes because these processes require large energy costs and very high investment in equipment and tooling. Single Point Incremental Forming (SPIF) is a relatively new metal forming process with a high potential economic payoff for rapid prototyping applications suitable for flexible and small quantity production, fulfilling this gap in metal forming processes. The processes differ as SPIF, two-point incremental forming (TPIF), double-sided incremental forming (DSIF), and hybrid incremental forming procedures based on the forming method. SPIF, also known as negative incremental forming, deforms the sheet with a single tool without a die. In contrast, the TPIF method uses a partial or full die to regulate the deformation of the blank. The SPIF process has several key limitations including geometric inaccuracy, sheet thinning, and forming constraints such as wall angle, step depth, tool diameter, temperature, etc. To increase geometric accuracy in understanding the many causes of faults in the SPIF process. The SPIF method may suffer from geometric inaccuracies due to spring back, unwanted plastic deformations, and non-uniform thickness distribution. Single-stage and multistage forming processes use forming will occur in a single-stage moment of a tool. Multistage point incremental forming (MSPIF) is a viable alternative than single-stage SPIF. As it reduces sheet thinning and results in more uniform thickness distributions. Geometrical imperfection, also known as stepped features, formed during the formation phase in the MSPIF process due to intermediary steps. With modeling and experimental data, studies of the development of stepped features at the formed parts are explained by Nirala et al. [1]. Material characterization, forming-limit curve, fracture forming-limit curve (FFLC) determination, numerical modeling, and experimentation, especially strain path and fracture strain evaluation, are multistage processes. The comparison of numerical modeling and experimental shows that the multistage forming sequence significantly impacts the strain sites’ position in the major strain space. Strain routes are linear in the initial stage and significantly non-linear in the later phases of recognized forms by Skjoedt et al. [2]. A new specimen geometry offers to reduce the great amount of experimental work to determine the forming limits, concluding that formability is significantly higher in incremental forming than in conventional sheet forming. The process is very flexible and economical due to the low tool costs discussed by Tisza [3]. As Hirt and Bambach [4] depicted in Two Point Incremental Forming (TPIF), also termed positive incremental forming, the forces act on two points. Firstly, the interaction point between the tool and the sheet, and secondly, at the blank–die interface. Although the setup in TPIF is more complex than in SPIF, greater accuracy is obtained by reducing the spring back due to the presence of a partial or full die. Peng et al. [5] explains the Double-sided incremental sheet forming, two tools are present, i.e., one above the clamped sheet and the other below the sheet to support the blank and provide backup forces for better and controlled movement. DSIF provides extra support to the blank like TPIF but without the involvement of any die. Maqbool and Bambach [6] prove the beneficial influence of the dominating deformation modes on geometrical precision leftover moments by numerical simulations. A decrease in energy dissipation inside the bending deformation mode results in fewer residual moments by increasing pitch and lowering tool diameter. The diminished contribution of the bending deformation mode and lower residual moments, rising pitch, and reduced tool diameter increase geometrical precision. Increased geometrical precision results in a reduced contribution of the bending deformation mode and smaller residual moments. Carette et al. [7] refers to the inaccuracy that may lead to the difference in the formed part from that of the ideal part, such as obtaining fillet instead of a sharp corner, inappropriate height, non-uniform thickness throughout the part, obtaining a smaller size than the actual requirements, etc. Many ways suggest improving geometrical accuracy. They were studied by altering physical process parameters like wall angle, step size, speed rate, feed rate, and lubrication or by varying tool paths. Najafabady and Ghaei [8] explain that Ti-6Al-4V sheets deform into three elementary shapes cone-shaped solid, variable wall angle cone-shaped solid, and pointed solid. It also determines the impact of various method parameters on dimensional accuracy, surface quality, and work hardening. The results discovered that the roughness of the inner surface wherever the tool meets the workpiece is larger than the roughness of the outer surface of a workpiece. The micro-hardness check findings conjointly discovered that the hardness varies from the flange to the vertex of the workpiece. SPIF process indicates additional plastic deformation, and therefore work hardening is observed. Thyssen et al. [9] focus on robot-based incremental sheet metal forming. Experimentally identifies the effect of various parameters. The test with decided parameters demonstrated a functional link between process parameters such as forming velocity, the temperature of the forming zone, the step depth, and the wall angle can be defined as the largest actuating variables of the forming process and measured part deviation. The robot controller has designed an online compensating method based on this functional relationship. The compensatory technique reveals a favorable trend in the dimensional accuracy obtained. Ren et al. [10] discuss a constraint agglomerative hierarchical clustering algorithm approach, divided into three parts: determining critical control points to address geometric complexity, simplified simulation models for anticipating spring back offline, and in-situ tool path change during forming. Experiment results demonstrate that the approach delivers an efficient and resilient solution for many geometries at a low setup cost. Akrichi et al. [11] investigated that deep learning as a valuable tool for geometric accuracy prediction in single point incremental forming. Otherwise, deep learning outperforms shallow learning in performance prediction. Furthermore, compared to the stack, the deep belief network model outperforms it in predicting roundness and position deviation. Raju and Sathiya [12] use a hybrid optimization technique combining Taguchi grey relational analysis (TGRA) and response surface methodology (RSM). Combining TGRA and RSM determines an optimal combination of input process parameters, such as the number of sheets, the tool diameter, the feed rate, the spindle speed, and the vertical step depth. The optimization yielded promising findings confirmed by a confirmatory experiment, suggesting that the process may improve and that the technique is reliable. Mohanty et al. [13] explain that the sheet thickness decreases as the wall angle increases and theoretically becomes zero at the maximum wall angle. MSPIF can be used to obtain a large wall angle by redistributing the material. Several types of research have attempted to find an optimal number of stages to achieve the maximum possible wall angles with satisfactory geometric accuracy. In addition, the actual tool path also significantly influences the successful forming of sheet metal with maximum achievable wall angles. Gonzalez et al. [14] discuss the geometries of truncated cones created using two multistage approaches and compared them to the same geometry created using a single-stage method. Digital image correlation assesses each geometric accuracy and thickness distribution. The results show that multistage forming considerably impacts the geometric correctness of the treated sheets compared to single-stage forming. Kumar et al. [15] considered the tool path is a contour along which the forming tool moves to achieve the object’s targeted shape. The tool moves layer by layer from one contour to subsequent contour. The toolpath is a critical parameter in the ISF process since it specifies the component’s dimensional accuracy. The right selection of the tool path improves the formability, surface finish, and dimensional accuracy, as well as the processing time of the resulting part. Blaga and Oleksik [16] aim to find the best forming strategy by adjusting the press position of the punch and the path it takes to form a truncated cone using single point incremental forming. As selected, three different paths to create a truncated cone for the cone. After each vertical press, the punch covers a circular path with a constant step in the first and second variations. The differences show that the following circular trajectory can begin at the same point as the previous press point or shift at an angle of 90 degrees from the previous press point. The punch in the final variation follows a spatial spiral trajectory. Thibaud [17] developed a single point incremental sheet forming process and a numerical toolbox to carry out fully parametric simulations using finite element software.

A thorough literature analysis reveals several limitations in the single-stage SPIF process in terms of forming ability, thickness distribution. The product can be prepared in better way with the help of multistage forming technique. It is necessary to understand the many reasons for such flaws in the SPIF process and to enhance geometric precision. The geometrical inaccuracy, undesirable plastic deformations, and non-uniform thickness distribution can cause geometric inaccuracies in the process. Multistage forming is an alternative to the limitations of single-stage SPIF since it decreases sheet thinning and produces a more uniform thickness distribution. The main goal is to use FEA simulation to investigate the effect of various multistage forming process parameters such as the number of stages (n), angle interval between successive stages (Da), and tool diameter (D) on section thickness (STH), Equivalent Plastic Strain (PEEQ), and Spring back effect.

2 Methodology

The research activity begins with selecting materials and investigating various process parameters, the first phase of the research development. Figure 1 depicts a project plan. As described above in the introduction chapter in the application section, titanium material has many uses in the biomedical industry. Because of its excellent strength-to-weight ratio, titanium has a wide range of services in the aerospace and marine sectors. Due to its superior corrosion resistance to aluminum and steel is employed in orthopedic and dental applications like prosthetic limbs or knees. Furthermore, titanium has good biocompatibility and corrosion resistance because the oxidation of the outer layer in the presence of oxygen produces a highly stable oxide known as Titanium dioxide (TiO2). The oxidized titanium layer protects the surface or structure of implants against environmental conditions such as fatigue, stress, etc. Titanium’s complete integration with the human body and its durable and flexible nature make it an excellent biocompatible metal referred by Kumar et al. [18]. Various researchers have recently studied process factors such as step depth, tool path, spindle speed, etc. Furthermore, creating the sheet in several phases introduces other characteristics and the ones listed above. Thus, it is necessary to investigate these parameters. Hence, the parameters chosen for the current research activity are the number of intermediate stages, angle intervals between successive stages, and tool diameter.

Fig. 1
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Plan for execution

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The Abaqus explicit program creates a square-shaped blank with 222 × 222 * 0.5 mm dimension. The blank creates the 2-D deformable model. The tool was modeled as a rigid analytical body with a hemispherical end because the diameter of the tool was one of the criteria. The diameter tools for each experiment were 8mm, 10mm, and 12mm. Furthermore, the size and form of the blank remained consistent. The tool and blank geometry, as shown in Fig. 2.

Fig. 2
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a Geometry of tool, b Dimensions for sheet

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Material characteristics of Ti Gr 2 (commercially pure titanium) by Yoganjaneyulu and Sathiys [19] were acquired from numerous research articles and entered into the Abaqus explicit software’s property manager. Table 1 displays the mechanical qualities. A scaling factor was also applied to shorten the simulation duration without affecting the results.

Table 1 Mechanical properties of titanium grade 2 [19]
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A spiral tool path with a down technique runs for the numerical study to shape the sheet into a truncated cone shape. The step depth is usually between 0.5 and 1 mm. Furthermore, the shorter the step depth, the greater the precision, and the better the outcomes explained by Sornsuwit et al. [20]. As a result, MATLAB software by Yeshiwas [21] operates to build the tool path with 0.5 mm step depth. The total time required to produce the sheet framed using a feed rate of 40 mm/sec. A total of 18001 steps operate to obtain the desired results.

Furthermore, the distance traveled in each step time was computed using the equation. Again, the total time necessary for each step was determined using total length and feed rate. The total length (L), feed rate (Fr), and total duration (T) are necessary for the forming process calculated using the total length to feed rate ratio. Also, step time is the ratio of total time to complete steps. Interaction attributes such as touch and friction assign the sheet tool interfaces. The coulomb’s friction law establishes the sheet blank and the forming tool with a lubricant with a friction coefficient of 0.045. The sheet has meshed with linear quadrilateral components of the S4R type with a mesh size of 0.8 mm. Sketch the interaction view between the blank and the tool. Later in a multistage procedure, importing the distorted sheet from the previous step will further deform it. The results of the previously produced sheet were added to the load manager using the preset field option. The results achieved in the previous stage were considered the beginning condition for the following stage. As a result, the sheet that deforms in the following step already possesses the circumstances achieved in the previous stage.

A total of sixteen experimental tests were required to simulate using Abaqus explicit. Table 2 shows the number of experiments.

Table 2 Design matrix for the experiments
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3 Result and Discussions

On commercially pure Titanium (Ti Gr 2) of 222 × 222 × 0.5 mm dimension, check the influence of simulation analysis on forming parameters (number of steps, angle intervals, and tool diameter) on the sheet thickness, spring back, and equivalent plastic strain. The forming process utilizes a hemispherical tool. MATLAB creates a spiral tool path to shape the sheet into a cup shape. Abaqus creates a 2-D deformable model of Ti Gr 2 sheet. The tool recognizes a rigid analytical body with a diameter of 10 mm. The sheet was held together by four edges and meshed with linear quadrilateral elements of the S4R type with a mesh size of 0.8mm. Furthermore, the coulomb’s friction law applies between the sheet blank and the forming tool, with a lubricant with a friction coefficient of 0.045.

Figures 3a, b, and c show the simulation results of a cup made at 45° with a 10 mm tool diameter from a 0.5 mm thick Ti Gr 2 sheet When compared to two-stage, three-stage, and four-stage forming, the sheet created in three stages has a more consistent sheet thickness compared to two other. Furthermore, progressive and homogeneous deformation reflects the rise in intermediate phases.

Fig. 3
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Simulation for a 2 Stage incremental forming. b 3 Stage incremental forming. c 4 Stage incremental forming

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According to Table 3, the highest sheet thickness of 0.300949 mm after deformation reflects in experiment 10. The sheet deforms in three stages with an angle interval of 5° between each stage and a tool diameter of 12 mm. When deforming with the same angle interval is impossible, the sheet can operate using a variety of angle intervals. A similar simulation for the experiment in which deforming in the same angle interval was impossible. The first stage had a 5° interval for such studies, and the final stage to complete with a predetermined angle interval. However, in experiment 1(2 Stage, Da = 5°, and TD = 10mm), the greatest equivalent plastic strain of 2.22942 and a minor inaccuracy at the bottom corner of the produced cup were attained. Furthermore, the influence of various responses (Section thickness, Equivalent plastic strain (e), and spring back effect) is depicted in the following subsections using various graphs and tables.

Table 3 Responses data for various parameters
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3.1 Section Thickness Distribution of Multistage Incremental Forming

A detailed analysis was complete by Li et al. [22] to study the influence of the number of forming stages and the angle interval between the consecutive stages on section thickness (STH). The MSPIF process uses the DC06 sheet as material and finds that increasing the number of stages resulted in a more uniform thickness distribution. However, as the number of steps increases, the minimum sheet thickness does. Furthermore, when the angle interval (Da) grows, the thickness initially increases but decreases. The maximum uniform sheet thickness is achievable with a 10° angle separation between successive steps. The minimum section thickness was 0.262133 mm for experiment 8 (3 stages, Da = 5°, and TD = 8 mm), whereas the highest thickness was 0.300949 mm for experiment 10 (3 stages, Da = 5°, and TD = 12 mm). Figure 4a depicts the influence of angle interval and tool diameter on section thickness in a two-stage forming process. However, experiment 1 (2 stage, Da = 5°, and TD = 10mm) exhibits more consistent wall thickness, but it also shows early thinning of the sheet compared to the other three experiments. Experiment 3 (2 stages, Da = 15°, and TD = 10mm) reveals an immediate increase and decrease in thickness at the bottom. The increase and decrease could be due to the previously formed part and large angle interval. Performing repetitions for experiments 4, (2 stages, Da = 10°, and TD = 8mm) and 6, (2 stages, Da = 10°, and TD = 12mm) can be found with the same angle interval but different tool diameter. However, the change was smooth. As can be seen, the influence of tool diameter was minor or non-existent compared to the effect of angle interval. Figure 4b shows the outcome of three stages of shaping and adjusting the angle interval and tool diameter. As a result of an experiment with a Da of 5°, the sheet’s final thickness is more uniform and maximum. However, as the Da between subsequent phases rises, the homogeneity declines.

Fig. 4
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Forming in a Two-stage, b Three-stage & c four stages with varying angle intervals and tool diameter

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Furthermore, among 5° Da trials, a sheet created with an 8mm tool diameter had a somewhat better thickness distribution than a sheet formed with a 12mm tool diameter. When the Da grows, so does the unevenness of the thickness distribution. Figure 4c depicts the influence of angle interval and tool diameter on the section thickness created in four stages. Experiment with Da of 10° and 12 mm tool diameters to see if you can get a more consistent thickness at the start. On the other hand, the trial with 15° Da and 10 mm tool diameter exhibits higher consistency towards the bottom of the cup than the other experiments.

The influence of the number of intermediate phases on section thickness reflects in Fig. 5. As seen in graph 4, stage formation exhibits a delay in thinning, with the most negligible thickness measured at around 70 mm from the fixed edge (clamp). However, as the number of steps grows, the unevenness increases, as the sheet generated with two stages exhibits a smooth thickness distribution. The formation of the previously formed cup would cause unevenness (formed in the previous stage). Furthermore, three steps forming demonstrate the maximum sheet thickness in the produced section. Similarly, when the number of steps grows, the bending of the sheet at the clamps increases.

Fig. 5
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Effect of the number of intermediate stages on section thickness

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Figure 5 depicts the combined influence of the number of steps and angle interval on sheet thickness. It clearly shows that the minimum sheet thickness value increases initially as the number of steps grows but decreases after the threshold. Initially, the sheet is 0.5 mm thick and is reduced by 39.81 to 47.57%, from 0.300949 to 0.262133 mm.

3.2 Spring Back of Multistage Incremental Forming

Another crucial impact noticed in the formed portion after the forming process is Spring back. It is an inaccuracy or deviation detected in the produced part due to the material’s yield strength. The displayed graph in this section depicts the influence of the number of steps and tool diameter on the spring back. The impact of angle interval and tool diameter on a sheet created with a 5° angle interval reflects in Fig. 6a. The inaccuracy in the created cup raises as the number of steps increases, as the spring back effect incorporates each iteration.

Fig. 6.
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The effect of the number of stages and tool diameter a with 5° angle interval of spring back b with 10° angle interval of spring back c with 15° angle interval of spring back

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Furthermore, the sheet created with four steps has a higher cup height than necessary due to sidewall stretching during the forming process. The mistake overlap in the former top section also has a mistake overlap. Furthermore, the 12 mm tool diameter cup exhibits less stretching than the sheet formed with a 10 mm tool diameter. The component created in the second step had higher accuracy among the plotted experiments than the part formed in the first stage. However, the pillow effect reflects in all four experiments. The sheets created at 10° angle intervals with varied numbers of stages and tool diameter yielded a similar outcome. However, the stretching phenomenon found in the 3 and 4 stages forming was relatively more than in the two stages. Figure 6b depicts the influence of the number of intermediate stages and tool diameter on the spring back effect with a constant angle interval of 10°. The extension of the wall rises as the number of phases increases.

Moreover, as shown in Fig. 6c for the exact stage forming (3 stage forming), the sheet formed using a 12 mm tool diameter shows a flat bottom compared to the sheet formed using an 8 mm tool diameter. The more contact area of the tool with a 12 mm diameter would result from less or no pillow effect in experiment 11 (3 stages, Da = 15°, TD = 12 mm), wherein the tool with an 8 mm diameter has a more negligible contact area sheet observed pillow effect. However, the bending effect was more for 8 mm tool diameter.

As shown in Fig. 7, the deviation in the sidewall of the formed cup is negligible with varying angle intervals. However, the height of the formed sheet increases by increasing the angle interval between the stages. In this case, the cup wall increases with angle interval, and the sheet formed at 15° shows the greatest stretching.

Fig. 7
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The effect of the angle interval (Da) on spring back

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3.3 Equivalent Plastic Strain (PEEQ) of Multistage Incremental Forming

The material’s formability generated by the SPIF process reflects in equivalent plastic strain. The equivalent plastic strain reduces as the number of phases increases. Furthermore, increasing the angled gap between successive steps reduces the equivalent plastic strain. Shen et al. [23] performed a simulation analysis on a DC56 sheet with a dimension of 380 × 380 × 1 mm to investigate the effect of the number of forming stages (n) and the angle interval (Da) between the consecutive stages on the equivalent plastic strain and spring back. As the number of forming stages increases, so do the equivalent plastic strain and the maximum total strain.

Furthermore, as the forming processes progress, more uniform deformation is noticed. Similarly, when the angle interval (Da) between steps rises, the equivalent plastic strain and the ultimate total strain Fig. 8a depict the sheet’s two-stage shaping with different angle intervals and tool sizes. Figure 8b shows that when the angle interval increases from 5° to 15°, the PEEQ value decreases. However, the final PEEQ and total maximum strain in the experiment (Da = 5°, TD = 10 mm) and experiment (Da = 10°, TD = 12 mm) were high compared to the other remaining experiment made with two stages. The sheet generated with a 2-stage, 10° angle interval and variable tool diameter (TD = 8 mm and 12 mm) had a more excellent value of PEEQ. The three-stage forming process with changing angle intervals and tool diameters appear in Fig. 8b. The figure shows that the value of PEEQ after the second stage is nearly the same for the experiment (Da = 5°, TD = 8 mm) and experiment (Da = 5°, TD = 12 mm), as well as the experiment (Da = 10°, TD = 10 mm) and experiment (Da = 15°, TD = 8 mm). However, the maximum value of PEEQ appears after the third stage sheet approves with fewer angle gaps between stages, which decreases as the angle interval grows. Similar results appear for the sheet generated in four stages, as shown in Fig. 8c. After each step, the sheet with a low angle interval exhibits a high PEEQ value.

Fig. 8
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a Two-stage b Three-stage & c Four-stage forming with variation in angle intervals and tool diameter

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Figure 9 shows the effect on an equivalent plastic strain by varying the number of stages. Three-stage forming shows the highest equivalent plastic strain (PEEQ) value. However, after the first stage, the sheet formed with the two stages shows the maximum PEEQ value, while for the three stages, the value of PEEQ was the least. Moreover, after every stage, the value of PEEQ decreases by increasing the number of intermediate stages. Thus sheet shows high formability when formed with a minimum or an optimal number of stages and angle intervals.

Fig. 9
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Effect of the number of intermediate stages on equivalent plastic strain (PEEQ)

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4 Conclusions

In the present work, FE Simulation work is carried through by deforming Titanium Gr. 2 sheet at different parameters. Section Thickness (STH), Equivalent plastic strain (PEEQ), and spring back were evaluated to understand the process mechanism. Below is a summary of some of the findings:

  • The angle interval between phases does not influence spring back. However, the corresponding plastic strain and the ultimate total strain decrease when the angle interval rises.

  • The number of stages doesn’t change the thickness of a section. But as the number of stages increases, so does the amount of plastic strain and the spring.

  • Using the right tool diameter and forming in the transverse direction may improve formability.

  • The simulation findings show that section thickness decreases in multistage forming compared to single-stage forming.

  • The core part of the cup has a consistent section thickness and the appropriate angle interval between the stages.

  • With an increasing number of steps, the minimum sheet thickness increases initially but decreases over time. In the beginning, the sheet had a thickness of 0.5 mm, reduced by 39.81 to 47.57%, from 0.30 to 0.26 mm.