Development of Two-Wrinkled Tubes Using an Electrostatic Structural Analysis

Abstract

The primary aim of this study is to develop an axisymmetric numerical model, employing the finite element approach, to simulate a two-wrinkling tube in T6 aluminum. The method uses an electric potential applied to the tube mesh, which passes through a solid die to induce the wrinkling process, facilitated by contact elements between the tube and the die. A lateral incremental voltage electric potential (0–50 kV), due to an electric coil, and applied axial and compressive displacement (0–12 mm) was considered. The materials’ properties were established as nonlinear, with elastoplastic behavior. The results were analyzed, which allowed the tube deformation with two wrinkles, comparable with previous results.

1. Introduction

1.1. Tube Deformation Processes

Traditional methods of mechanical tube formation encompass a wide variety of processes that have been developed and refined over time to shape metal tubes into the desired geometries. These methods are crucial in sectors such as automotive, aerospace, and construction, where precision and material properties are of paramount importance [1].

Although conventional tube forming methods are well-established and widely employed, they are not without limitations. For instance, these methods can be labor-intensive and may require substantial energy input, particularly in processes such as hot extrusion and rolling. Moreover, achieving complex or high-precision geometries may demand multiple processing stages or specialized equipment [2]. As a result, ongoing research has focused on innovative forming techniques, such as hydroforming and electromagnetic forming, which, combined with traditional methods of obtaining tube joints, offer potential advantages in terms of efficiency and material utilization [3].

Electromagnetic forming (EMF) and mechanical deformation represent two distinct methodologies for tube deformation, each with unique advantages and constraints regarding efficiency and accuracy. EMF, characterized by its rapid, high-energy pulsed electromagnetic forces, enables fast plastic deformation, whereas mechanical deformation relies on slower and more methodical physical forces. Currently, EMF finds significant application in tube end-forming processes, including various techniques such as expansion, bending, bulging, and shrinking, which are employed to modify tube ends for specific applications. These processes are essential for creating joints and connections in piping systems, allowing the integration of tubes into larger assemblies [2].

Understanding traditional methods is crucial for studying innovative tube processing and forming technologies in various industries. Numerical modeling aims to investigate these emerging processes, reduce the need for tooling and prototype fabrication, and ensure increased productivity and efficiency in tube deformation applications for commercial components [4].

1.2. Numerical Study Applied to Electromagnetic Tube Forming

The numerical study of tube wrinkling under electrostatic loads explores how electric fields interact with mechanical deformations to induce or modify wrinkling patterns in cylindrical tubes. Wrinkles in tubes are local buckles or deformations formed under radial compression. Research has shown that applying a radial electric field to dielectric tubes with flexible electrodes can significantly influence their deformation behavior. The electric field interacts with mechanical loads (such as internal pressure and axial forces), changing the stress distribution and potentially triggering instabilities such as wrinkling or buckling. The deformation and stability of these tubes are highly sensitive to the electric displacement field, which is coupled to the mechanical material response [5,6,7].

Numerical solutions and theoretical models were developed to analyze incremental deformations and bifurcation conditions in electrostatic tubes. These models show that the vibration frequency, wave speed, and critical loads for instability depend on the tube geometry, the applied mechanical deformation, and the magnitude of the electrostatic potential. Distortion analysis elucidates the emergence of wrinkling as a distinct phenomenon when the frequency of vibration approaches zero, while the application of an electric field can serve to either reinforce or undermine the stability of the tube, contingent upon the specific configuration employed in the finite element method [5,6,7,8].

While several studies focus on wrinkling due to mechanical loads (such as bending, torsion, or internal pressure), the addition of electrostatic effects introduces a new level of complexity. The interaction between electric fields and mechanical stresses can modify the critical conditions for wrinkling and affect the resulting wrinkle morphology. For example, the radial electric field increases with azimuthal stretch, influencing the stress propagation and potentially the wrinkling pattern [5,8].

Understanding electrostatic wrinkling is crucial for designing soft actuators, sensors, and biomedical devices that require precise control of deformation and stability. Theoretical methodologies and numerical simulations provide a base for predicting and tuning wrinkling behavior in advanced materials and structures [5,6,7,8]. In other applications, pipes may exhibit wrinkling, a phenomenon that can be achieved through welding techniques or deformation processes, depending on the specific application and the required mechanical stability [9,10,11].

In response to the evolving requirements for manufacturing technologies that entail the conjunction of tubular structures through the fabrication of industrial and engineering components, the prevailing state of the art seeks to address the limitations intrinsic to the process while optimizing both the manufacturing parameters and the geometrical configuration of the tubes. Presently, investigations are underway concerning the joining processes of aluminum tubes via plastic deformation, which initially employed experimental methodologies but have recently progressed to incorporate numerical simulations. The application of numerical simulations within forming processes facilitates the examination of plastic deformations, defects, and failures, along with variations in process parameters, thereby circumventing the necessity for physical manufacturing trials [12].

In 2018, [13] introduced an innovative technique for the plastic joining of thin-walled tubes through the mechanism of compression instability. A numerical simulation of the entire deformation process involved in plastic joining was conducted. The study successfully demonstrated a novel plastic joining method for thin-walled tubes, characterized the different joint outcomes based on geometric parameters such as the free length-to-radius ratio and bevel angle, and established a comprehensive framework to evaluate formability, highlighting the critical role of wrinkle geometry in joint strength.

For the manufacturing of bent tubes, hydroforming has been investigated through models validated by numerical simulation, aiming to overcome significant limitations in the fabrication of complex bent tubes. This approach improves product quality through in-line monitoring, providing valuable insights into performance, behavior, and process conditions [14]. The results demonstrate proper model fitting and provide guidelines for manufacturing, as they enable the identification of the appropriate set of parameters required to produce defect-free components. In this analysis, it is equally crucial that the findings demonstrate the impact of geometric designs on strain distribution and the ultimate properties of the component [15,16].

Studies on the deformation of the cross-section of rectangular tubes have also focused on the rotary stretching process, based on theoretical analysis, experimental research, and finite element simulation. Future trends in this specific field indicate continuous advances and refinements in understanding and mitigating cross-sectional deformation [17]. Research efforts devoted to the development and simulation of electromagnetic forming of aluminum alloy tubes have also gained great importance in the current literature, since comprehensive multiphysics models, simulated via FEM, are crucial for understanding and optimizing electromagnetic forming of aluminum alloy tubes, by analyzing the interaction of various physical phenomena and process parameters [18].

This study aims to outline the methodology employed in the numerical simulation of the wrinkling behavior of Al 6061-T6 tubes by applying an electrostatic charge. The part numerically modeled in this work was previously obtained in the work of [12] through mechanical forming, and now an innovative methodology is proposed for obtaining it through EMF. The numerical simulation performed aims to obtain the final geometry with one wrinkle on the plastically deformed tube, carried out using Ansys 2025 R2 software. Mathematical modeling enables the optimization of parameters and geometries, allows for the use of a smaller amount of material in manufacturing, and contributes to reducing the occurrence of defects during the forming process [19,20].

1.3. State of the Art: Wrinkled Tube in Metal Sheets by Forming

However, traditional mechanical forming processes present challenges associated with tube forming, particularly the issue of reduced deformation zones at tube ends and frequent asynchronous deformation rates. To overcome these limitations, novel technology referred to as axial compression electromagnetic bulging has been proposed. It was first developed in the 1960s and has excellent potential that is often underestimated. This technique, which employs the axial magnetic force of an auxiliary coil, effectively enhances the plastic deformability and the forming limit of aluminum alloy tubes by altering their stress state. Electromagnetic forming (EMF) is a high-speed manufacturing technique that utilizes electromagnetic forces to shape metallic products. This process is versatile, capable of compressing or expanding hollow tubular parts as well as forming or joining sheet metal components. It is particularly suitable for processing materials with limited formability and has been widely applied in industries such as automotive, aerospace, and electronics [21].

Analytical approaches based on energy balance have been employed to model high-speed deformation in thin tubes, integrating advanced yield criteria and flow stress descriptions. These studies underline the influence of strain rates and shape functions on EMF, while highlighting the simplicity and physical clarity of analytical methods, albeit with limitations due to inherent model simplifications [22].

Recent research has significantly expanded the understanding of electromagnetic forming of tubes, encompassing advancements in forming techniques, the development of novel methods to improve process efficiency, and the expansion of industrial applications. Recent progress in EMF has included tube perforation and crimping, which are essential for many industrial processes, as the high strain rates achievable with EMF extend material formability and enable the production of complex geometries [23]. Magnetic pulse forming has also been employed to simultaneously form and punch aluminum tubes, with studies demonstrating that punch geometry strongly influences hole quality, and that concave punches provide superior results in terms of slug separation and dimensional accuracy [24]. To overcome the limitations of conventional coil designs, solenoid field shapers have been proposed as a flexible and cost-effective solution, reshaping electromagnetic force distribution to allow processing of tubes of various sizes without the need for new coils, thereby reducing both energy consumption and cost by up to 25% [25]. In addition, dual-power supply systems utilizing outer coils have been introduced for small-diameter tube flaring, generating strong attractive electromagnetic forces that facilitate tube expansion and broaden the applicability of EMF [26]. EMF has also been successfully applied to form-fit joining of aluminum tubes and sheets, involving tube expansion and flanging, with resulting joints exhibiting high strength that often exceeds the material yield limit [27]. Efforts to improve material deformation and force loading during tube expansion have led to approaches such as axial compression and the application of concave coils, which minimize wall thinning, mitigate non-uniform deformation, and enhance overall forming performance [28].

Theoretical and numerical studies, including modeling and simulation, have been critical for predicting final geometries and optimizing system design, with field concentrators and dual-coil arrangements reported to enhance forming accuracy and efficiency [29,30]. Despite these advancements, challenges remain in adapting EMF processes to diverse materials and applications, making theoretical investigations essential for providing insights into underlying mechanics and guiding further process improvements. As research continues, EMF demonstrates increasing potential to transform tube forming technologies, offering an efficient and sustainable alternative to conventional methods.

2. Materials and Methods

This work presents a numerical methodology, based on the finite element method, used to obtain the wrinkling of an Al 6061-T6 aluminum tube. The tube dimensions are an external diameter of 42.6 mm, a thickness of 3.2 mm, and a length of 100 mm. Figure 1 represents the mesh and the boundary conditions to be used in the numerical simulation. The analysis of the numerical model uses the same geometric and mechanical properties as the experimental models previously tested by the authors [9,12].

Figure 2 shows a flowchart with all the fundamental steps for this analysis, during the pre-processor and post-processor. In the pre-processor, the chosen finite element was PLANE 223, a 2D element with a coupled field, 8 nodes, and up to 6 degrees of freedom per node, suitable for axisymmetric analyses. The size mesh considered was to have no fewer than two elements in the thickness of the tube, and all the size elements were maintained throughout the entire model. The implemented mesh configuration ensured a minimum of two elements across the tube’s thickness to accurately represent the tension gradient, rendering further refinements unnecessary, as the deformation levels were moderate and remained within acceptable limits throughout the loading process, facilitated by the axisymmetric configuration and the PLANE223 element, thus achieving stable convergence without adaptive riveting.

The assembly’s axisymmetric necessitated the specification of a rotational axis as a boundary condition, alongside the immobilization of the die parts’ lower region, to prevent movement during force application. For electrostatic–structural analysis, the activated degrees of freedom are displacements (Ux, Uy) and electrical potential (VOLT). An applied electric potential compresses in thickness and elongates. An electrostatic–structural analysis is performed to determine the deformed shape in the thickness direction. The contact model integrates a target and a contact surface to define the interaction between bodies; herein, the tube’s target surface is denoted by the TARGE169 element, while the die’s contact surface is characterized by the CONTA172 element, both of which are optimally suited for the analysis due to their proficiency in accommodating rotational and translational displacements, normal and tangential forces, and moments. For the analyzed tribological system, a Coulomb friction coefficient of 0.15 was employed, reflecting the standard value in metal forming applications [31].

An incremental electrostatic and structural analysis was performed. There are different types of electrostatic and structural analysis, such as static, full transient, linear perturbation static, linear perturbation modal, and linear perturbation harmonic. Static and transient analyses can be used to determine the deformation of an electromechanical device under applied voltage incrementally.

The boundary conditions adopted in the simulation were the symmetry axis, die fixation to constrain the displacement, and fixation in the middle tube. A lateral incremental voltage electrical potential (0–50 kV), due to an electric coil contact, and an axial compressive displacement (0–12 mm) were applied in the tube, with the axisymmetric activated option. For the geometry in two dimensions, the assembly parts (tube and die) have symmetry throughout their axial length. For the nonlinear behavior, the yield stress and tangent modulus of the aluminum alloy AA6061-T6 were defined as 280 MPa and 100 MPa, respectively. These values were obtained from standard material property data reported in the literature for this alloy, which indicates a typical yield stress between 270 and 285 MPa and a tangent modulus in the range of 90–110 MPa [32,33]. The adopted values fall within this range and are consistent with the mechanical behavior expected for AA6061-T6 under room temperature forming conditions. The mechanical properties for the aluminum tube were Young’s Modulus and Poisson’s Ratio, which had values of, respectively, 69 GPa and 0.3. The die is steel material, with a Young’s Modulus value of 210 GPa. The yield stress was considered equal to 600 MPa and maintained in the elastoplastic regime. Preliminary simulations confirmed that the inclusion of strain hardening in the die material had no significant effect on the overall deformation of the aluminum tube. This modeling simplification is supported by established literature, which indicates that the deformation and strain hardening of forming tools have a negligible influence on the forming behavior of the workpiece [34,35,36].

The electric relative permittivity (also known as dielectric constant) is considered equal to 1 [37].

3. Results and Discussion

In the post-processor, after solving the numerical simulation, some results were obtained for the wrinkling aluminum tube: the deformed shape, the displacement field, and the von Mises stresses. For this analysis, a personal computer 12th Gen Intel(R) Core (TM) i5 with 16 GB RAM was used for all simulations.

Figure 3 shows the two-wrinkled aluminum tube before (a) and after deformation (b) (c). Although the simulation was performed using a 2D axisymmetric model, the deformed shape shown in Figure 3 was obtained by revolving the 2D results around the Y-axis to provide a clearer three-dimensional visualization of the tube deformation. Figure 3d presents an experimental result of a wrinkled tube previously obtained by the co-authors [13]. For the experimental test, a hydraulic press was used with an interchangeable die containing an internal cavity for tube insertion. The numerical results demonstrated the two-wrinkled tube effect, where the plastic deformation achieved corresponds to the desired final geometry.

The deformed results demonstrated the two-wrinkled tube effect. The verified plastic wrinkling represents the required final shape. It is also possible to obtain the 3D formation of the two-wrinkling tube, despite the modeling having been carried out in 2D, due to the axisymmetric option. The proposed numerical model of a two-wrinkled tube in this work also allows for the analysis of the effect obtained in the experimental tests.

The results for horizontal and vertical displacements and von Mises stresses are presented in Figure 4, Figure 5 and Figure 6 for an applied voltage equal to 5 kV.

The published work by the co-authors [8], with the experimental methodology applied to this type of tube, allows us to conclude that, after the cold forming process is used, the wrinkled tube in the front-view presented values equal to 6.9 and 6.1 mm, as can be seen in Figure 3d. These values are comparable with those obtained from the numerical methodology proposed in the present work. The lateral displacement achieved for the two-wrinkled tube in the present study has a double value compared to that previously obtained by the same authors [10], where an elastoplastic and incremental numerical simulation was only used.

Figure 5 and Figure 6 represent the nonlinear behavior in the nodal middle thickness of the wrinkling tube for the lateral displacement and von Mises stresses, respectively. Both numerical results were compared for a numerical simulation, but only with the one-wrinkled tube, where the same numerical methodology was applied.

The lateral displacement, at an intermediate node of the tube thickness, increases nonlinearly with the application of the electric potential. This behavior thus represents the local plasticization that has occurred. Comparing the results from the two-wrinkled tube with the one-wrinkled tube deformation, it is noted that the global lateral displacement has the same behavior. However, the lateral displacement of the tube with two wrinkles has an increased displacement alongside the imposed loading condition.

Related to the evolution of equivalent stresses, the linear regime occurs until an electric potential of 2.5 kV is applied, which corresponds to a von Mises stress of 240 MPa. Beyond this electric potential, there is an increase in plastic deformation, maintaining a plastic stress state in the material up to 272 MPa when a two-wrinkled tube is formed. The stress level is lower than that of the one-wrinkled tube formation.

The wrinkled tube, designed according to the proposed methodology, is useful in structural elements such as scaffolding, ladders, and gym equipment, and can be manufactured in various materials, including steel or aluminum.

Figure 7 illustrates the joint sections: A: area devoid of tube distortion, B: outer section of the joint, and C: inner deformation area of the tube.

The images in Figure 7 were obtained from tube deformations for sheet metal joints using the mechanical forming process [12], the same model used in the presented numerical study using the EMF process; see Figure 3. The process describes the connection within a rectangular aluminum panel through the assembly of tubular structures using sheets composed of different materials and configurations, which is the typical manufacturing process.

4. Conclusions

The objective of this work was to simulate the deformation of an aluminum tube numerically. The simulation was performed incrementally, with an electrostatic and structural analysis, using nonlinear material properties, to obtain the wrinkled tube formation.

This numerical methodology allowed the formation of deformation in tubes, like that which occurred in an experimental prototype, comparable to previous work by the authors [12]. A previous numerical investigation by the authors also allowed obtaining plastic deformation in the tube, but that was caused by an imposed internal pressure [9].

The prior research conducted by the co-authors [8] indicates that the dimensions of the cold-formed wrinkled tube range from 6.9 to 6.1 mm, facilitating the integration of thin sheets, and, corroborating the numerical outcomes of this study, the lateral positioning for the two-wrinkle tube identified herein is double that reported previously [10], which relied solely on elastoplastic and incremental numerical simulations.

The linear regime in the evolution of equivalent stresses persists until an applied electric potential of 2.5 kV, equating to a von Mises stress of 240 MPa, after which plastic deformation escalates, maintaining a plastic stress state in the material up to 272 MPa during the formation of a two-wrinkled tube.

The proposed methodology, presented for obtaining two wrinkled tubes, or even one, is significant for practical industrial applications. Future work should investigate a new experimental setup for obtaining wrinkling in tubes, the use of other materials, and replicate the same characteristics in three-dimensional numerical analysis models.