Hidden mechanical joints of sheets to rods by axial sheet bending and unbending

Mechanical joints of sheets to rods by plastic deformation produced at room temperature away from the rod ends have been investigated in recent years but new developments are still needed. The process hereby presented relies in the axial bending of a sheet with a pre-drilled hole of smaller diameter than that of the rod, until the diameter of the sheet becomes equal to the initial diameter of the rod. Then, a rod is placed inside this larger hole diameter of the sheet and the sheet is unbent to its original shape, with its material being radially injected to a rectangular cross-section slot previously machined in the rod, thus producing a mechanical interlocking between the two geometries. The major process parameters are identified and their influence in the deformation mechanics is analysed by means of finite element modelling and experimentation. The experiments were carried out in unit cells while the numerical modelling analysed sheets of different lengths and thicknesses, so that to allow a proper understanding of the influence of different specifications on the new joining by forming process. Destructive performance tests were performed in the sheet-rod connections and comparisons with previous joining techniques were made to confirm the success of the new mechanical joint.


Introduction
In the search for mechanical joints without the use of fasteners or rivets, the choice was for many years limited to those that produce a force-fit joint resulting from the residual normal stresses that are developed at the contact interfaces due to the elastic-plastic deformation of the sheets and rods caused by shrinkage of the sheets during cooling [1].
Along recent years, joining solutions have been presented that are limited to the production of sheet-rod connections located at the rods ends by conventional boss forming [2] and incremental boss forming [3], which produce two annular flanges for both supporting and fixing the sheet to the rod. More recently, a cross-rolling process was employed to produce sheet-rod connections away from the rod ends [4], through the expansion of the rod against the sheet by a combined axial and rotational movement of two pairs of V-grooved rolls which form two opposite annular flanges that lock the rod to the sheet.
In the previous approaches, the primary plastic deformation occurred in the rods instead of the sheets, but a new joining by forming process was recently developed in which the sheet is locally indented so that material is injected into a slot previously machined on the rod and a form-fit joint is produced at room temperature away from the rod ends [5]. As for boss forming [6], this process of indentation and injection can also be applied to both rods and tubes and in the case of rods, the axial compression applied by the punches will produce the indention and injection of the sheet material which will fill a circumferential slot machined in the rod (refer to Fig. 1a) without a radial deformation of the rod, as it observed for tubes under the same joining process [7]. The performance of joining by indentation and injection is limited to the remaining resistant cross-section after indentation, which is required to inject sufficient sheet material into the rod slot and produce a sound mechanical joint. It was verified that failure occurred by fracture along a very narrow zone of the remaining cross-section designated as the shear resistance length and that the process is limited to relatively large thicknesses.
Upon these considerations, a new joining process is hereby proposed to produce joints of similar and dissimilar materials without protrusions or indentations on the sheet, and with an increased resistant cross-section while being able to produce hidden mechanical joints between rods and sheets of a wider range of thicknesses. The process relies on bending a sheet with a hole whose initial diameter is lower than the diameter of the slot machined in the rod. As the punch moves downwards, the sheet hole starts to expand similar to the initial and intermediate stages of hole flanging [8,9], until a point where its diameter is equal to the external diameter of the rod to which the sheet will be joined. Afterwards, the rod is placed into the sheet hole with the bottom of the slot being coincident with the upper surface of the bottom die. Then, the upper die moves downwards and unbends the sheet back to its original position, forcing the sheet material to be displaced towards the slot cavity, thus producing a mechanical joint between the sheet and rod that remains hidden inside the rod slot cavity.
Selected test cases having different initial hole diameters were tested in smaller test cells, to understand the mechanics of deformation and force-displacement requirements, as well as to define the conditions that provide the maximum pull-out destructive forces for the new mechanical joints. Finite element simulations were employed to further understand the influence of different operating parameters in the overall process, mainly the influence of different configurations of sheets and slots on the feasibility of the process.

Mechanical characterization of the materials
The research on the new sheet-rod connections made use of aluminium AA5754-H111 sheets with a thickness t 0 of 5 mm and aluminium AW 6082 rods with a diameter d 0 of 32 mm. Those materials and their respective flow curves were retrieved from a previous work of the authors [5] to which a comparison is to be made when discussing the performance of the new joining process.
The material flow curves encompassed the mechanical characterization of the materials by means of tensile tests in accordance with the ASTM standard E8/E8M-16 [10] and stack compression tests, where the specimens were prepared by pilling-up three circular discs with 15 mm diameter and 5 mm thickness that were also machined out from the supplied geometries [11]. The reason for the stack compression tests was due to the need of obtaining the stress-strain response of the materials for values of strain that are larger than those obtained in tensile tests. The hydraulic testing machine Instron SATEC 1200 kN was utilized to obtain the values from which resulted the average flow curves depicted Fig. 1 Schematic representation of joining of sheets to rods away from the rod ends by a indentation and injection, and by b The newly proposed joining process composed of an axial sheet bending stage (left) and unbending stage (right) in Fig. 2, with the entire set of tests being carried out at room temperature with a crosshead speed of 10 mm/min.

Work development
To understand and characterize the new process, experimental unit cells were prepared to validate the results from the numerical simulations. Sheets with a configuration of 100 × 100 mm and rods with a height of 50 mm were utilized for that purpose.
The process parameters that are shown in Fig. 1b comprise the initial diameter of the sheet hole d hi , the initial external diameter d si and the initial thickness t si of the sheet, and the final thickness t sf of the sheet at the boundary region adjacent to the hole. It also comprises the height h and width w of the rod slot, and the diameter d 0 of the rod. Regarding the tooling system, the parameters include the angle of the punch and the inner radius r of the bottom die. To limit the number of tests and allow for an understanding of the working principle, the parameters of the tools remained constant while variations were introduced to the geometries as listed in Table 1.
The experimental preforms and setup is presented in Fig. 3, where the results for the first and final stage are also presented. For simplification, square sheets were utilized in the experiments which as it will be seen did not create any constraints to the joining process and therefore allow to extend it to different sheet geometries.

Numerical simulations
The numerical modelling of the new process of joining by sheet bending and unbending was performed with the finite element computer program i-form [12], which is based on an extension of the finite element flow formulation to include the relaxation of the incompressibility condition of the velocity field by means of a penalty function and the contact between rigid and deformable objects.
To analyse the process, the law of constant friction f = mk , where m is the friction factor and k is the flow shear strength friction, was utilized to model friction. A friction factor m equal to 0.1 was utilized on the contact interfaces between the objects since under this condition, the value of the predicted numerical forces was compatible with the experimental results.
The sheets and rods were modelled with rotational symmetry conditions and assumed to be deformable isotropic objects subjected to axisymmetric loading. A total of approximately 3000 quadrilateral elements were utilized to discretize the cross-section of the sheet, whereas the rods had their cross-sections discretized by a total of approximately 2000 quadrilateral elements. By other hand, the punch and bottom die were discretized by means of linear contact-friction elements and modelled as rigid objects. An example of a finite element model for (a) the first stage of bending of the sheet and (b) unbending of the sheet to produce the sheet-rod connection is presented in Fig. 4.     For a sheet with an external diameter of 100 mm, the hole diameter enlarged 0.3 mm, whereas for a sheet with an external diameter of 200 mm, the hole diameter only enlarged 0.2 mm. The differences in the final external diameter are even more noticeable, with the external diameter of the sheet increasing 1.0 mm, whereas for a sheet with an initial diameter of 200 mm, it only increased 0.4 mm. These important observations can be explained by the fact that for larger external diameters of the sheet, the additional material increases the overall resistance of the sheet and acts like a blank holder, thus reducing the overall bending of the sheet. When the diameters of the sheet are smaller, the bending of the sheet becomes predominant, which generates larger differences between the initial and final dimensions of both the hole and the external diameter of the sheet.
A similar effect is observed for smaller hole diameters that suffer higher levels of deformation, since the difference in diameters to which the hole needs to be expanded to allow the positioning of the rod becomes significant. As it was verified for an initial hole diameter of 25 mm (Fig. 6a), after the cycle of bending and unbending of the sheet, the final hole diameter has a difference of almost 2 mm due to the outward flow of sheet material. For the specifications of the geometries and tools, this diameter presents itself as a limit for the process since lower values of the hole diameter will demand larger bending levels of the sheet that will create difficulties for a successful unbending of the sheet and could result in a high level of elastic recovery of the sheet material that may compromise the integrity of the mechanical joint. The amount of outer material flow is also higher in the case of smaller hole diameters, which creates significant changes in the dimensions of the sheet. Therefore, when two opposite bending moments are verified in the sheet during unbending, the limit for the conditions of the process is found since the dimensional variations of the sheet become relevant. The choice of the angle of the punch and the position of the bottom die during the bending stage is essential to reduce the occurrence of an opposite bending moment for sheets having larger differences between their hole diameter and the diameter of the rod, thus allowing to fill deeper slots which are able to withstand larger destructive pull-out loads as it will be seen in a later section.
As for a hole diameter of 27 mm (Fig. 6b), it can be observed that the radial velocity is smaller than for a hole diameter of 25 mm (Fig. 6a), although a considerable amount of outward flow is still visible, which translates in differences between the initial and final hole diameters of 1.6 mm. Finally, for a hole diameter of 29 mm (Fig. 6c), the inward radial material flow at the hole region becomes more intense and the differences in the initial and final diameters of the hole become smaller than 1 mm.

Sheet-rod connection
After the identification of the final hole diameter and final sheet thickness, it is possible to define the suitable size of the rod slot. For the experimental tests, sheets of 100 × 100 mm were chosen for the unit cells to test the limit conditions of the process for the presented configuration. After the bending stage, the rod is positioned in the hole that has now a diameter coincident with its external diameter. The height h of the rod was kept constant and equal to the sheet thickness after bending, while the width w of the rod slot was chosen according with the initial hole diameter.
It would be expected that the inner diameter of the rod slot should match the final hole diameter, however during the unbending of the sheet and filling of the rod slot cavity, there is a point when the sheet starts to contact with the vertical wall of the slot and is forced to invert its inward material flow to an outward material flow. This effect is mostly observed for sheets with smaller external diameters since the lower amount of sheet material is not able to offer enough resistance to the radial tensile loads developed in the external boundary. To circumvent this situation and allow the determination of the width w of the rod that offers the best destructive performance, as it will be discussed in the next sections, the hole diameter is always 1 mm smaller than the inner diameter of the slot. A comparison between the numerical and experimental results for a width w of the rod slot equal to 1 and 2 mm is presented in Fig. 7a and b, respectively.
As seen, for a sheet with an initial hole diameter of 29 mm and a rod slot with a width of 1 mm (Fig. 7a), the material starts to flow inward until a point where the sheet material may start to contact with the upper outer region of the rod slot and move outwardly. This outward material flow becomes reduced if the initial contact with the rod slot is initiated at the vertical wall region, which is verified for the value of the chosen punch angle (refer to the inserts of the graph of Fig. 8). However, due to the contact with the vertical wall of the rod slot, the final hole diameter of the sheet after the connection is 0.5 mm larger than when the sheet was unbent solely (refer to Fig. 5).
Conversely, for a rod slot with a width of 2 mm and an initial hole diameter of 27 mm (Fig. 7b), the inward radial flow of material is constrained by the deeper slot and in turn, a larger amount of outward material flow is verified, with the final hole diameter being 0.9 mm larger than initially. The same is not verified for sheets with lower thicknesses, as it will be seen in a later section. As for sheets with a larger external diameter as shown in Fig. 7c, the differences between the initial and final diameters are very reduced independent of the width w of the rod slot that is now completely filled by the sheet material. These variations are listed in Table 2.
Therefore, starting from an external diameter of 200 mm that is typically found in most common sheet applications, no constraints to the material flow are presented for the new joining process, since the variations in both the sheet hole diameter and external diameter become minimal as the hole diameter becomes closer to that of the rod and the external diameter becomes larger.

Force-displacement evolution
The experimental and finite element predicted evolution of the force with displacement for the bending and unbending of the sheet is presented in Fig. 8. For the bending stages of both diameters, no relevant differences are seen in the force-displacement evolutions since the principle behind Fig. 7 Comparison between the numerical and experimental results for a width w of the rod slot equal to a 1 and b 2 mm for a sheet with an initial external diameter of 100 mm, and c a numerical comparison between the same widths w of the rod slot equal to 1 mm (left) and 2 mm (right) for a larger sheet with an initial external diameter of 200 mm those deformations is the bending of the sheet made from the same material. The only difference is that for a smaller hole diameter of 27 mm, the amount of displacement to obtain the same final hole diameter coincident with the external diameter of the rod is larger than for a hole diameter of 29 mm. The curves are characterized by a steep increase due to the initial contact with the punch until the yield point of the material where the curves increase more slightly as the sheet is progressively bent along the longitudinal axis.
For the unbending stage and respective joining process, the curves for the different hole diameters are also similar, starting with an initial peak in force as the sheet makes an initial and localized contact with the upper moving tool that forces the sheet material to be radially injected towards the rod cavity. Then, the force reduces until a point where the sheet starts to contact with the vertical and upper wall of the rod slot and the tools become in full contact with the sheet surfaces. The evolution of the joining operation is illustrated in the cross-section details included in the graph of Fig. 8.
The differences in the evolutions of the different hole diameters during the unbending of the sheet are due to the higher level of bending to which the sheet with a lower hole diameter of 27 mm is subjected, which demands a higher force to unbend the sheet back to its original position and is responsible for the first force peak. The second difference is the higher displacement necessary to fill a deeper slot than in comparison with the previous hole diameter of 29 mm.
The numerical and experimental correlations are adequate and the differences are justified by smaller and tolerated errors during manufacturing of the preforms.

Destructive tests
The testing of two rod slots of different widths is supported by the need of identifying the mechanical resistance and the destructive mechanisms inherent to the fabricated joints when subjected to pull-out destructive loads. For the purpose of applying these types of loads, an external cylindrical tool with an inner diameter of 36 mm was employed in these tests to compress the rod and produce its separation from the sheet.
The two joints tested failed by shearing as seen by the morphology of the ring surface left at the rod slot (refer to the photographs included in the graph of Fig. 9). The case with a deeper rod slot having a width of 2 mm was able to withstand approximately more 15 kN than the rod slot with a width of 1 mm. The sheet starts to be sheared along the axial cross-section until it reaches the total sheet thickness and gets separated from the ring that is left inside the rod cavity. Both cases were able to withstand a higher force than  the maximum force of approximately 35 kN that was previously observed for the same set of materials when joined by indentation and injection [5]. The reason is due to the remaining cross-section in that joining process being relatively smaller than the original sheet thickness (refer to the comparison of Fig. 1). This limited not only the performance of the mechanical joint but also the applicability to sheets of smaller thicknesses, which is not verified with the new joining by forming process as it will be seen in the following section.

Thin sheets
For thinner sheets, the morphology of the hole region is much more homogeneous which motivates a closer and larger contact region between the sheet and the rod, as shown in Fig. 10.
In contrast, the smaller amount of resistant material in the thinner sheets leads to the development of an opposite moment during the unbending of the sheet. At the same time, as the sheet thickness becomes smaller, a whiplash effect becomes predominant which motivates a larger flow of sheet material to the rod slot, which in turn results in smaller differences between the initial and final hole diameters. For reference, in a sheet having an original hole diameter of 28 mm, the difference between the initial and final hole diameters is 0.8 mm for the original sheet thickness of 5 mm (Fig. 10a), while for a sheet thickness of 3 mm (Fig. 10b) is 0.6 mm and for a thickness of 1.5 mm (Fig. 10c), the differences are even smaller and around 0.5 mm. Therefore, as smaller the thickness of the sheet, the smaller the differences in the hole diameter and the closer the contact between the sheet and rod. For external diameters larger than 200 mm, those differences are even smaller for the different sheet thicknesses, following what was observed in Fig. 7c for a sheet thickness of 5 mm.

Conclusions
The new joining process brings the possibility of producing mechanical joints between rods and sheets of different thicknesses that are able to withstand larger destructive pull-out loads. The process is carried out in two different stages: a bending stage to enlarge the initial hole diameter previously drilled in the sheet until it matches the outer diameter of the rod, and then an unbending stage which consists in unbending the sheet back to its original shape. It was seen that for sheets with smaller external diameters, the sheet material tends to flow outwardly instead of inwardly to the inside of the slot previously machined on the rod. This effect is diminished when sheets of larger external diameters and/or smaller thicknesses are utilized.
A considerable amount of sheet material needs to be displaced to the inside of the rod slot since it was observed that deeper slots give rise to higher pull-out loads and to a better performance of the joint. Nevertheless, for the tool setup utilized in this work, the limit of the rod slot width was found for a sheet hole diameter of 25 mm, where the differences between the initial and final hole diameter after the unbending stage were very significant due to a strong outward material flow produced by the occurrence of an opposite bending moment during the unbending stage. The significant bending demanded to enlarge the hole diameter until it matches the external diameter of the rod and allows its positioning and consecutive joining contributed to the previous effects. Therefore, it can be concluded that to ensure the success of the joining operation, the Fig. 9 Pull-out destructive tests of the new joints with photographs of the specimens after separation differences between the hole diameter of the sheet and the external diameter of the rod should not be so high so that to avoid out-of-plane deformation during the axial sheet unbending.
In conclusion, the process can be applied to industrial size applications and relies on conventional manufacturing techniques, fostering new products and applications of joining by forming of rods to sheet panels.
Funding Open access funding provided by FCT|FCCN (b-on). This work was supported by Fundação para a Ciência e a Tecnologia of Portugal and IDMEC under LAETA-UIDB/50022/2020.

Data availability
The authors confirm that the data and material supporting the findings of this work are available within the article.
Code availability Not applicable.

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Ethical approval The article follows the guidelines of the Committee on Publication Ethics (COPE) and involves no studies on human or animal subjects.

Consent to participate Not applicable. The article involves no studies on humans.
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Competing interests The authors declare no competing interests.
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