The company has a group of cooperation teams engaged in the clear hollow plastic balls industry for many years, with dedication, innovation spirit and service awareness, and has established a sound quality control and management system to ensure product quality.
4.1
Description of the proposed forging process and methodBased on the results discussed in the previous section, the authors decided to modify the forging process for hollow balls shown in Fig. 1. The modification involved using an additional tool ensuring better control of deformation of the billet. The tool was a plastic insert, i.e., an additional core made of a flexible material. A schematic design of the modified forging process for balls is shown in Fig. 10. The insert was assigned the material properties of low-melting alloy TBC12 (BiPb25Sn12Cd12) [28, 29]. This material was chosen for two reasons: its plastometric similarity to technically pure lead and low melting point (below the boiling point of water). As a result, such insert will have the required deformation resistance, and following the forging process, the deformed core can easily be removed from the forging and used again. The TBC12 alloy also has the desired casting characteristics [30], which ensures trouble-free execution of inserts with the required shape and dimensions.
Fig. 10Schematic design of the modified forging process for hollow balls using a plastic insert: (a) beginning of the process, (b) end of the process
Full size image
The first and intuitive choice with respect to the shape of a plastic insert was a tube with a constant cross section, as shown in Fig. 11a. The choice of that shape was also motivated by a literature review in which such solutions were proposed, e.g., [23, 24]. Nevertheless, a detailed numerical analysis (which is discussed later on in the paper) showed that it was not the best choice. Therefore, the authors began to search for such insert geometry that would ensure optimal forging conditions and the achievement of the set objective and at the same time would be easy to make. These searches took account of the results of earlier studies, in particular with respect to the pattern of material flow during the forming of a hollow ball. The end result of these searches is the insert with the geometry shown in Fig. 11b.
Fig. 11Shape and defined dimensions of a plastic insert: (a) with a constant cross section, (b) with a variable cross section (description in the text)
Full size image
Hence, the next stage of the theoretical and experimental study concerns the determination of the effect of the plastic insert dimensions on the conditions of forming hollow balls. The aim of this stage of the study is to obtain a correctly shaped ball using the same billet as in the previously analysed forging process. As shown in Figs. 5 and 8, it was not possible to produce correctly shaped balls from such billet if the process was carried out without the insert. It was decided that the analysis would be made for the cases in which a hollow ball was formed from the billet with the initial dimensions of d0 = 27 mm and h0 = 30 mm.
A theoretical analysis of the investigated process was performed using Deform-2D/3D in the module for axisymmetric cases. The only difference between the model shown in Fig. 3 and the geometric model of the modified forging process lies in that the latter contains an additional discrete object representing the plastic insert, as shown in Fig. 11. Similarly to the billet, the insert was modelled as a rigid plastic material, and the material model of alloy BiPb25Sn12Cd12 was described by the constitutive equation:
$$ {\sigma}_p=\mathrm{109,5}{\varphi}^{0.042}\exp \left(-0.36\varphi \right){\dot{\varphi}}^{0.16}\exp \left(-0.22\dot{\varphi}\bullet 2.6\varphi \right), $$
(2)
where σp is the flow stress, φ is the effective strain and \( \dot{\varphi} \) is the strain rate. This model was determined in previous studies based on the frictionless upsetting of cylindrical specimens with initial dimensions of 20 mm × 30 mm.
The conditions of contact between the billet (aluminium alloy) and the plastic insert (low melting alloy) were described by a Coulomb friction model and a coefficient of friction μ = 0.09. In the simulation, it was assumed that the plastic insert was a condition-imposing object (the so-called master object). The value of the coefficient of friction was determined in a ring upsetting test performed on anvils that were made of the same material as the billet in the forging process for balls.
The experimental tests of the modified process were carried out in the same laboratory conditions as those applied in the tests of the forging process without the use of an additional plastic insert (see Subsection 3.1).
4.2
Plastic insert with a constant cross sectionA numerical analysis of the forging process for hollow balls included cases in which the plastic insert had the initial height hC0 from 22 mm to 30 mm and the initial wall thickness gC0 from 2 mm to 6 mm, as shown in Fig. 11a. Given the assumption of constant dimensions of the billet (d0 = 27 mm, g0 = 2 mm, h0 = 30 mm), in all analysed cases, the plastic insert’s outside diameter dC0 was maintained constant at 23 mm. The tested combinations of initial dimensions of the plastic insert are listed in Table 2.
Table 2 FEM-simulated combinations of initial dimensions of the plastic insert with a constant cross section, denoted as in Fig. 11aFull size table
Experimental tests were carried out for selected cases in order to compare the numerical results with real conditions of forming balls. Generally, it can be claimed that the numerical results reflect the real conditions in a relatively accurate way. A comparison of the force parameters (Fig. 12) and the changes in the shape of the forged ball shows a good qualitative and quantitative agreement. Nevertheless, in the cases with a potential risk of buckling, the simulation indicates that the material is able to “heal” this defect, which is not entirely true as evidenced by the final shape of the balls shown in Fig. 13. These balls were obtained in the experimental tests performed using the plastic insert with a thickness gC0 of 2 mm and 5 mm and an initial height hC0 of 26 mm. The problem of interpreting results and comparing experimental and numerical findings is raised and discussed in detail later on in the paper. Now, let us move on to a discussion of the numerical results.
Fig. 12FEM-simulated forming force for selected cases of forming hollow balls using a plastic insert; the numerical results are compared with the FEM results of the forging process performed without an insert and experimental results
Full size image
Fig. 13Final shape of the experimental ball formed from a billet with the initial dimensions d0 = 27 mm, g0 = 2 mm and h0 = 30 mm using a plastic insert with a constant cross section and the dimensions dC0 = 23 mm, hC0 = 26 mm and gC0 of 2 mm (a) and 5 mm (b), respectively
Full size image
Figure 14 shows the changes in the shape of a ball with a plastic insert with the initial dimensions dC0 = 23 mm, gC0 = 2 mm and hC0 = 26 mm. The wall thickness of the insert was identical to that of the billet, while the insert’s height was 87% of the height of the billet. The results demonstrate that the plastic insert with such a small relative wall thickness has an insignificant effect on the forging process conditions. At an early stage of the forming process, both the workpiece and the insert undergo local compression, and as a result, the contact surface between the insert and the workpiece is reduced to a small area that is denoted as detail A in Fig. 14a. With further upsetting of the workpiece and the insert, a growing gap is created between the inside wall of the workpiece and the outer surface of the insert (it is marked in Fig. 14 as detail B and detail D). Although the contact between the insert and the workpiece is increased with time (detail A in Fig. 14d), the lack of the insert’s impact on the central region of the workpiece (detail C in Fig. 14b) causes a collapse of the workpiece wall, which consequently, leads to buckling. Therefore, the set objective is not attained. Hence, the shape of the balls obtained at a later stage of the process should be considered incorrect, because given the previously observed collapse of the workpiece wall shown in Fig. 13a, it can be claimed that the forging process for this case will fail.
Fig. 14Changes in the shape of a workpiece (d0 = 27 mm, g0 = 2 mm, h0 = 30 mm) and a plastic insert (dC0 = 23 mm, gC0 = 2 mm, hC0 = 26 mm); description in the text
Full size image
One could ask what will happen if the thickness of the plastic insert is increased e.g. to gC0 = 5 mm. The answer to this question is given in Fig. 15. In general, the forming conditions have not improved significantly. Previously observed phenomena still occur. At an early stage of the forging process, the workpiece and the insert are compressed, which gives rise to local contact between these objects (detail A in Fig. 15a). Further upsetting results in the formation of a significant gap between the workpiece and the insert (detail B in Fig. 15b). When the insert with a bigger wall thickness gC0 is used, an additional contact area occurs with time; this area is marked as detail D in the figure. This contact, however, is momentary and does not play a significant role in the forging process. An analysis of the data in Fig. 13b reveals that the increase in the insert wall thickness has improved the forging quality; however, the improvement is not considerable enough to regard the quality of the ball as satisfactory. It can therefore be claimed that the assumed objective has not been accomplished in this case either.
Fig. 15Changes in the shape of a workpiece (d0 = 27 mm, g0 = 2 mm, h0 = 30 mm) and a plastic insert (dC0 = 23 mm, gC0 = 5 mm, hC0 = 26 mm); description in the text
Full size image
A detailed comparison of the impact of the plastic insert wall thickness on the changes in the outer surface of the ball profile is given in Fig. 16. Before moving into the discussion, it should be briefly explained how to read the plots in the figure. The data in Fig. 16a demonstrate that as the forging process continues, at some point, the workpiece wall profile undergoes a collapse if no plastic insert is used. A detailed analysis of the results allows us to conclude that if such collapse is observed when the process progress is between 45% and 60%, then buckling is bound to occur. Even though the workpiece wall profile in a later stage of the simulation (79% and 100%, as shown in Fig. 16a) is convex, this observation cannot be used as a sole rationale for deciding whether the forging process is stable or not. It can also be assumed that if we track the history of changes in the shape of a ball profile and show that the line of this profile becomes non-convex at any moment (e.g., the derivative of this line assumes a negative sign at the point for the coordinate H/2 = 0 mm), this would mean that the forging conditions are no longer stable and buckling will occur, as observed in the experiment (Fig. 13a). A different measure of the degree of non-convexity of this line is the Δmax parameter (Fig. 16b), i.e., the difference between the furthest point on the profile line (in the direction of the R coordinate) and the midpoint located on the above-mentioned line at the height coordinate H/2 = 0 mm.
Fig. 16Changes in the outer profile of the workpiece (billet dimensions: d0 = 27 mm, g0 = 2 mm, h0 = 30 mm) in the forging process performed without a plastic insert (a) and with a plastic insert (b) when the process progress is 60% (b) versus the plastic insert wall thickness (insert dimensions are constant: dC0 = 23 mm, hC0 = 26 mm); description in the text
Full size image
An analysis of the plot shown in Fig. 16b reveals that a change in the wall thickness of the plastic insert does not make the contour line of the ball profile convex. This means that the use of the plastic insert with a constant cross section does not make it possible to attain the set objective, i.e., ensuring stable conditions of the analysed forging process for hollow balls. This is evidenced by the increase in the Δmax parameter with increasing the plastic insert thickness (Fig. 16b). Obtained values of this parameter are listed in Table 3.
Table 3 Values of the Δmax parameter calculated for the forging cases shown in Fig. 16bFull size table
Analysing the changes in the shape of the workpiece and plastic insert shown in Figs. 17 and 18, one can come to a conclusion that even by changing the plastic insert height, one cannot be sure that the set objective will be accomplished. Moreover, a reduction in height practically eliminates any positive effect of the insert on the forging process, as evidenced by the details B, C and D in Fig. 17 (the letters correspond to the denotations in Figs. 14 and 15). On the other hand, an increase in the wall thickness of the insert causes filling of insert material in the space inside the workpiece, which leads to an unnecessary increase in the forming force (even by two times, see Fig. 12) and obstructs the deformation of the workpiece, enhancing the effect of workpiece wall collapse (detail C in Fig. 17). The end result is a ball with a lower final height hK as well as a potential risk of flash formation (detail F in Fig. 18). Given the possible occurrence of buckling, overlap may occur, too.
Fig. 17Changes in the shape of a workpiece (d0 = 27 mm, g0 = 2 mm, h0 = 30 mm) and a plastic insert (dC0 = 23 mm, gC0 = 5 mm, hC0 = 22 mm); description in the text
Full size image
Fig. 18Changes in the shape of a workpiece (d0 = 27 mm, g0 = 2 mm, h0 = 30 mm) and a plastic insert (dC0 = 23 mm, gC0 = 5 mm, hC0 = 30 mm); description in the text
Full size image
4.3
Plastic insert with a variable cross sectionA further analysis of the investigated forging process for hollow balls concerned cases in which a plastic insert with a variable cross section was used (Fig. 11b). This analysis was divided into two stages. First, a comprehensive analysis was performed for a wide range of values of the geometric parameters of the insert. Next, the range of the tested parameters was narrowed to a region in which the forging process is the most effective. This region was selected on the basis of the results obtained in the first part of the study. Referring to the considerations discussed in previous chapters, the following assumptions were made in the first stage of the analysis:
In the analysed forging process, a hollow ball is formed from a billet with the following dimensions (Fig. 10a): d0 = 27 mm, g0 = 2 mm, h0 = 30 mm.
The geometry of the plastic insert (Fig. 11b) is described by constant and variable parameters.
The constant parameters are the total insert height hc1 = 26 mm (referring to the parameter hc0 of the plastic insert with a constant cross section discussed in the previous section of the paper) and the outside diameter dc1 = 23 mm (resulting from the billet’s dimensions).
The variable parameters are (Fig. 11b) the insert thickness gc1 (taken in the range from 2.0 to 6.0 mm), the necking thickness gc2 (its value is taken in the range from 0.5 to 2.5 mm depending on the thickness gc1), the height hc2 (taken in the range from 8.0 to 12.0 mm), and the necking height hc3 (its value is taken in the range from 1.0 to 5.0 mm depending on hc2).
First of all, the conditions of the forging process for hollow balls in question were assessed. To this end, the history of changes in the workpiece outer profile was examined and the effect of using a plastic insert of specific dimensions was investigated. For a quantitative assessment of the results, and thus for establishing a criterion for the attainment of the set objective, two parameters were adopted: the Δmax parameter, which was previously defined in Fig. 16b, and the forging height hK (Fig. 4). The values of these parameters, depending on the dimensions of the plastic insert with a variable cross section (Fig. 11b) are shown in Figs. 19 and 20, respectively.
Fig. 19Values of the Δmax parameter calculated for the forging process (billet dimensions: d0 = 27 mm, g0 = 2 mm, h0 = 27 mm) performed using a plastic insert of varying dimensions, as shown in Fig. 11b; description in the text
Full size image
Fig. 20Forging height hK calculated for the forging process (billet dimensions: d0 = 27 mm, g0 = 2 mm, h0 = 27 mm) performed using a plastic insert of varying dimensions, as shown in Fig. 11b; description in the text
Full size image
It was observed that the relative size of the plastic insert wall necking in its central region significantly affects the ball forming conditions. A reduction in the thickness ratio gC2/gC1 leads to a significant improvement in the forging conditions, as evidenced by the fact that the Δmax parameter can be lower than 0.02 mm. At the same time, however, an analysis of the results given in Fig. 19b demonstrates that this relationship has its extreme values, i.e., it is possible to indicate optimal dimensions of the necking. The same conclusion can be reached by taking into account the other two dimensions, i.e., the heights hC2 and hC1.
However, in order to be able to fully define the optimal dimensions of the plastic insert, one must additionally analyse the other parameter describing the insert, i.e., the expected height of the forging, hC (Fig. 20). It turns out that obtaining a ball with the highest possible height is possible when we use the plastic insert with higher values of the aforementioned thickness ratio gC2/gC1. In addition, the application of a higher value of the necking height hC1 ensures that the difference between results will be lower. The final analysis based on polyoptimization, where the objective function is to obtain the smallest possible value of the parameter Δmax and the largest possible forging height hK in the tested range of the variable parameters, which can be written using the equation:
$$ {\Phi}_{\mathrm{target}}=\min \left[\left(\frac{\Delta_{\mathrm{max}}-{\Delta}_{\max \left(\min \right)}}{\Delta_{\max \left(\max \right)}-{\Delta}_{\max \left(\min \right)}}\right)+\left(\frac{h_{K\left(\max \right)}-{h}_K}{h_{K\left(\max \right)}-{h}_{K\left(\min \right)}}\right)\right], $$
(3)
which indicates that the optimal forging conditions are ensured when the plastic insert has the following dimensions: hC2 = 10 mm, hC3 = 3 mm, gC1 = 5 mm and gC2 = 0.75 mm. Figure 21 shows successive stages of the forging process performed with the use of such insert. The figure also shows the vectors of material flow velocity, which provides a deep insight into the problem.
Fig. 21Changes in the shape of a workpiece (d0 = 27 mm, g0 = 2 mm, h0 = 26 mm) in the forging process performed with a plastic insert with the dimensions dC1 = 23 mm, gC1 = 5 mm, gC2 = 0.75 mm, hC1 = 26 mm, hC2 = 10 mm, hC3 = 3 mm and the vectors of material flow; description in the text
Full size image
From an early stage of the forging process, one can observe three regions of contact between the plastic insert and the inner surface of the workpiece wall; these regions are marked in Fig. 21 as details A and B. Despite the occurrence of a significant void between the insert and the workpiece (detail C in Fig. 21), the forging conditions can be considered stable. The stability of the forging process predominantly depends on the way in which the insert is deformed in its centre (detail B in Fig. 21). A collapse that occurs in this area of the insert (detail D in Fig. 21) exerts a strong impact on the workpiece wall, pushing the workpiece wall in the radial direction, as evidenced by the material flow velocity. In effect, the buckling of the workpiece wall is prevented. It can therefore be concluded that the set objective has been attained and the proposed shape of the plastic insert meets the requirements.
The positive results obtained in the first stage of the study encouraged the authors to undertake further analyses. In the second stage, it is assumed that the insert necking dimensions gC1, gC2 and hC3 are constant and their values were calculated based on the results obtained in the first stage of the analysis. In contrast, the heights hC1 and hC2 are variable parameters. Such assumption makes it possible to investigate the impact of all dimensions of the plastic insert with a variable cross section on the forming conditions and, consequently, to determine optimal values of these dimensions.
Figure 22 shows the effect of changing the plastic insert’s heights hC1 and hC2 (for the constant values of the thicknesses gC1 = 5 mm and gC2 = 0.75 mm and the necking height hC1 = 3 mm, according to Fig. 11b) on the value of the parameter Δmax. An increase in the total height hC1 of the insert leads to higher stability of the forging conditions. The value of the height hC1 is very close to the billet height h0 (in this case, it is 30 mm), and the outer surface of the ball profile can basically be considered convex, which means that no buckling will occur. When the height hC1 is at least 28 mm (which is about 90% of the height of the billet), the parameter Δmax is below 0.02 mm; the best result is achieved for the height hC1 = 30 mm when Δmax = 0.003 mm. A comparison of these values with the data listed in Table 3 reveals that the use of the plastic insert with a variable cross section ensures stable forging conditions. However, the dependence of the parameter Δmax on the height hC2 is parabolic, with a clear extreme value. The parameter Δmax reaches the lowest values when the height hC2 is approximately 10 mm.
Fig. 22Parameter Δmax calculated for the forging process (billet dimensions: d0 = 27 mm, g0 = 2 mm, h0 = 27 mm) performed using a plastic insert with variable necking, as shown in Fig. 9b, the constant dimensions being hC3 = 3 mm, gC1 = 5 mm and gC2 = 0.75 mm; description in the text
Full size image
In addition to the parameter Δmax, the workpiece height hK (Fig. 4) was also taken into consideration in the analysis. The results demonstrate that an increase in the height hC1 has a negative effect on the workpiece height hK. The forming conditions resemble those observed in the previous case shown in Fig. 18. Namely, due to the fact that the height of the insert in the workpiece is too high, the material of the insert fills the space above the workpiece, preventing the flow of the workpiece material in the axial direction. The effect of changing the insert dimensions is plotted in Fig. 23, where it is shown that the change in the height hC1 has a more significant effect than the change in the height hC2. By taking into account the variables and analysis results shown in Figs. 22 and 23, in combination with parameter polyoptimization based on the objective function Φtarget in accordance with Equation (3), it is confirmed that the previously determined dimensions of the plastic insert hC1 = 26 mm, hC2 = 10 mm, hC3 = 3 mm, gC1 = 5 mm and gC2 = 0.75 mm, are optimal when a hollow ball is forged from a billet with the dimensions h0 = 30 mm and d0 = 27 mm.
Fig. 23Forging height hK calculated for the forging process (billet dimensions: d0 = 27 mm, g0 = 2 mm, h0 = 27 mm) performed using a plastic insert with variable necking, as shown in Fig. 11b, the constant dimensions being hC3 = 3 mm, gC1 = 5 mm and gC2 = 0.75 mm; description in the text
Full size image
The numerical results were verified by experimental tests involving the formation of a hollow ball from a tube with the same dimensions as in the numerical analysis (i.e., d0 = 27 mm, g0 = mm, h0 = 30 mm). The forging process was performed with the use of the plastic insert (Fig. 11b) with a constant outside diameter dc1 = 23 mm and constant dimensions of the necking hC2 = 10 mm, hC3 = 3 mm and gC3 = 1 mm. The tests were performed using two heights hC1 (28 mm and 30 mm) and two thicknesses gC1 (4 mm and 5 mm). It should be clarified that some of these dimensions slightly differ from the optimal values that were determined in the theoretical analysis. The use of an insert with its thickness gc1 below 1 mm is pointless due to the difficulty of ensuring sufficiently high tolerance. The value of the height hc1 was selected deliberately due to practical reasons.
Examples of hollow balls obtained in the experimental tests are shown in Fig. 24. The top photographs show the view from the side of the top die, while the photographs in the bottom, from the side of the bottom die. All cases considered in the experiments were successful. The balls have the correct outer profile and are free from any shape defects. The experimental height hK of the forged balls (Figs. 4 and 24) shows high agreement with the numerical results.
Fig. 24Examples of hollow balls formed from a tube (d0 = 27 mm, g0 = 2 mm, h0 = 27 mm) in a forging process using a plastic insert (as in Fig. 11b) with the dimensions hC2 = 10 mm, hC3 = 3 mm, gC3 = 1 mm and gC1 and hC1 given in the figure; description in the text
Full size image
Additionally, Fig. 25 shows the axial section of a hollow ball obtained in two analysed cases with two different insert thicknesses gC1. The experimental results show agreement with the theoretical findings. The application of the plastic insert with a variable cross section is effective, as evidenced by detail A in Fig. 25. During the forming process, the insert exerts in the marked area the required load ensuring stable forming conditions, ultimately resulting in the production of a ball with the correct shape and relatively constant wall thickness. In addition, the void marked in Fig. 25 as detail B is identical to that obtained in the numerical analysis and has no negative impact on the final result. The only deviation from the theoretical results occurs in the area marked as detail C. This can be explained by the fact that during the forming process, the insert is slightly displaced towards the bottom die. As a result, the diameter of the hole in the lower area of the ball (i.e., from the side of the bottom die) is larger than the hole in the upper area. This difference ranges from 1 mm to 3 mm approximately, depending on the insert height hC1.
Fig. 25Axial section of a hollow ball formed from a tube (d0 = 27 mm, g0 = 2 mm, h0 = 27 mm) in a forging process performed using a plastic insert (as in Fig. 11b) with the dimensions hC1 = 28 mm, hC2 = 10 mm, hC3 = 3 mm, gC3 = 1 mm and gC1 = 4 mm (a) and 5 mm (b); description in the text
Full size image
For more information clear hollow plastic balls, please get in touch with us!