Numerical analysis of formability and failure in tube inversion using i-Form

Numerical analysis of formability and failure in tube inversion using i-Form b Numerical analysis of formability and failure in tube inversion using i-Form Autor: José Ignacio Delgado Muñoz Tutor: Gabriel Centeno Báez Maria Beatriz Silva (Instituto Superior Técnico, Lisboa) Departamento de Ingeniería Mecánica y Fabricación Escuela Técnica Superior de Ingeniería Universidad de Sevilla Sevilla, 2019 Trabajo Fin de Máster Máster en Ingeniería Industrial Trabajo Fin de Máster Máster en Ingeniería Industrial Numerical analysis of formability and failure in tube inversion using i-Form Autor: José Ignacio Delgado Muñoz Tutores: Gabriel Centeno Báez Profesor Contratado Doctor Maria Beatriz Silva Profesor Auxiliar Departamento de Ingeniería Mecánica y Fabricación Escuela Técnica Superior de Ingeniería Universidad de Sevilla Sevilla, 2019 Trabajo Fin de Máster: Numerical analysis of formability and failure in tube inversion using i- Form Autor: José Ignacio Delgado Muñoz Tutores: Gabriel Centeno Báez Maria Beatriz Silva El tribunal nombrado para juzgar el Proyecto arriba indicado, compuesto por los siguientes miembros: Presidente: Vocales: Secretario: Acuerdan otorgarle la calificación de: Sevilla, Julio de 2019 El Secretario del Tribunal Agradecimientos A Gabriel, por su ayuda y dedicación. Por su compromiso, sus ganas de hacer las cosas bien y porque sin él no habría sido posible este trabajo. A mi padre, ya que ha estado en todos los malos momentos. Resumen El trabajo se basa en un análisis numérico de tubos de Aluminio AA6063-T6 deformados con punzones de diferentes radios para el proceso de inversión. El fin principal de este trabajo consiste en determinar cuáles son los radios límites para este proceso, aumentando así el estado del arte. El trabajo inicia con un análisis de todo lo investigado hasta ahora. Posteriormente se explica cómo ha sido modelado todo el proceso y cómo se extraen los diferentes resultados en el programa empleado: i-Form. El proceso se modela para que se lleve a cabo con una velocidad constante. Una vez hecho esto, se pasa a exponer todos los resultados obtenidos comparándolos con resultados experimentales calculados en investigaciones anteriores, validando así los datos numéricos. Los resultados que se analizan son la fuerza y las deformaciones y, una vez validados, el daño dúctil y el espacio de la triaxialidad. Todo esto permite definir con bastante seguridad los radios límites para llevar a cabo el proceso de inversión de forma satsifactoria tanto para radios pequeños como para radios grandes. Figure 16. Energy absorbed with displacement (Magrinho et al., On the formability limits of thinwalled tube inversion, 2019) .................................................................................... 17 Figure 17. Schematic representation of the rotation of axis (Cristino et al., A digital image correlation based methodology to characterize formability in tube forming, 2019) ....... 20 Figure 18. Pre-Processor menu for modelling and meshing....................................... 22 Figure 19. Options for the geometry construction. ..................................................... 23 Figure 20. Representation of the cut tube ................................................................. 23 Figure 21. Refinement option for the mesh ............................................................... 24 Figure 22. Refinement menu .................................................................................... 24 Figure 23. Structure needed of the file die2.dat to load in i-Form ............................... 25 Figure 24. Final model of the process ....................................................................... 26 Figure 25. Main controls options .............................................................................. 27 Figure 26. Dies options ............................................................................................ 27 Figure 27. Material Data and Yield Criteria ............................................................... 28 Figure 28. Assigning material options ....................................................................... 29 Figure 29. Damage and Fracture controls................................................................. 29 Figure 30. Initial simulation window .......................................................................... 30 Figure 31. Ready simulation window for the process ................................................. 30 Figure 32. Selection of the different graphics types ................................................... 31 Figure 33. XY Graphics options ............................................................................... 31 Figure 34. How to export the information .................................................................. 32 Figure 35. How to access to the surface strains ........................................................ 32 Figure 36. Surface strains menu .............................................................................. 33 Figure 37. How to extract the ductile damage ........................................................... 33 Figure 38. Ductile damage menu ............................................................................. 34 Figure 39. Effective Strain vs. Stress Triaxiality menu. .............................................. 34 Figure 40. Evolution of the force for R=0.5 mm ......................................................... 38 Figure 41. Evolution of the force for R=20 mm .......................................................... 39 Figure 42. Evolution of the force for R=5 mm ............................................................ 39 Figure 43. Different stages of the successful tube inversion ...................................... 40 Figure 44. . Evolution of the forces ........................................................................... 40 Figure 45. Final state of the tube and location of the element analysed...................... 41 Figure 46. Strain evolution for R=0.5 mm ................................................................. 42 Figure 47. Location of the element analysed and final position of the tube ................. 42 Figure 48. Strain evolution for R=20 mm .................................................................. 43 Figure 49. Strain evolution for R=5 mm .................................................................... 44 Figure 50. Final state of the tube for each small inversion radius ............................... 45 Figure 51. Example of mode of failure for R=3 mm ................................................... 45 Figure 52.Final state of the tube for large inversion radius ......................................... 46 Figure 53. Moment in which the tube starts to invert ................................................. 47 Figure 54. Elements analysed .................................................................................. 48 Figure 55. Strain paths ............................................................................................ 49 Figure 56. FLD for R=10 mm ................................................................................... 49 Figure 57. Space of effective strain vs stress-triaxiality.............................................. 50 Figure 58. Limit cases ............................................................................................. 51 List of tables Table 1. Critical ductile damage (Cristino et al., A digital image correlation based methodology to characterize formability in tube forming, 2019) .......................................................... 21 Table 2. Strain values for different displacements of the upper die............................. 44 Table 3. Displacement of the upper die at the beginning of the inversion and the ductile damage for each inversion radius .......................................................................................... 47 Table 4. Summary of the main results for each case in Element 596. ........................ 50 Numerical analysis of formability and failure in tube inversion using i-Form 1 Introduction Over the years, there have been done a lot of researches about sheet metal forming processes and are present in everyday life things. For example, thanks to sheet metal forming are produced parts of cars, whose worldwide production is around 100 million cars per year (D. Cooper). However, due to different requirements in different industries such as aerospace constructions, tubes researches are getting importance in the forming field. Nowadays, tones of tubes are produced each year. Tubes can be used both for domestic and industrial fields. The increase in the use of aluminium tubes is basically due to these factors: - Reduced installation costs - Minimization of system leaks - No corrosion - Cheaper and longer service life than copper and stainless steel - Lighter weight All these advantages have made the interest in tubes has increased during the last years. One of the fields where thin-walled tubes are being used is the lightweight structures which are basically designed to carry a given load. This principle is mainly used in the aerospace and automotive sectors, using the highest specific strength materials, in cranes, scaffolds or bridges (J. Jeswiet et al., 2008). Another field where these tubes are used is in the frame structures. It can be distinguished between welded tubes, used mainly in bicycle frames, garden chairs or ski sticks and seamless tubes, which offer better mechanical properties, used in a smaller market because the increase in the cost (J. Jeswiet et al., 2008). Figure 1. Frame structure Numerical analysis of formability and failure in tube inversion using i-Form 2 Finally, this kind of tubes are also present in compressed air systems, suction pipes and clear air pipes, cooling pipes, pipes for agricultural and forestry engineering… (Dietzel Hydraulik). All this has led to the appearance of new processes like the incremental tube forming, which is a combination of an incremental tube spinning and a continuous tube bending process (C. Becker et al., 2014), invented in order to reduce the springback (when a metal is deformed, due its characteristics, it will have elastic and plastic deformation. Then, when the metal piece is removed from the tool only the plastic deformation will remain). (AutoForm: Forming Reality, s.f.) Another innovative process is the friction-spinning, which allows the extension of existing forming limits and the production of more complex geometries compared with the conventional spinning processes (Friction-spinning - Grain structure modification and the impact on stress/strain behaviour, 2018). State of the Art This chapter will review the state of the art on tube forming, focusing on the main theme of the work, inversion. In order to do this, a series of basic concepts will be defined first. Due to the manufacturing process used for tubes, it is correct to use the hypothesis existing for sheet metal forming to the tube forming. There are two main manufacturing processes:  Seamless tubes: The tube is first a cylindrical ingot, which is heated and passed through a cylindrical die and then the hole is made through a penetrator.  Welded tubes: The process starts with a metal sheet, which is bend to shape the tube. Then, the joins of the sheets are welded, closing the tube. (Mariño, 2015) Keeping this in mind, it is logical to use the hypothesis of sheet metal forming, due to the second manufacturing method for tubes is starting from a sheet.  Plastic behaviour: To characterize the differences between the processes it is necessary to understand the plastic behaviour of the tube. This explanation is based on the hypothesis of an Numerical analysis of formability and failure in tube inversion using i-Form 3 isotropic material and it is explained in the situation of a metal sheet, extrapolated to the tube. In the vast majority of the processes, stresses which deform the sheet are those which are in the same plane as the sheet, whereas the stress in the direction of the thickness is usually zero or really small compared with the others. Because of this, it is a good simplification suppose that the sheet is deformed under general plane stress. This allows simplify the expressions of the plastic behaviour of the sheet. (Análisis de los Procesos de Conformado de Chapa) Finally, it is supposed that elastic strains (dεe) are insignificant in comparison with the plastic strains (dεp). dε= dεe+ dεp≈ dεp Figure 2. Stress and strain of a sheet in plane stress (Análisis de los Procesos de Conformado de Chapa)  Basic relations of plasticity: Being axis 1 and 2 the principal strain and stress directions in the sheet plane and the axis 3, the principal direction by the thickness and supposing a general plane stress state, stress and strains existing in the sheet are: σ1; σ2; σ3 = 0 dε1; dε2; dε3 = - (dε1 + dε2) The main direction 1 is the one in which stress is higher (σ1> σ2). Strains dε1 and dε2 can be measured, but to calculate dε3, it is used the conservation of volume expression: dε1 + dε2 + dε3 = 0 Now it can be defined two variables which are important in the following sections. These variables are α and β: Numerical analysis of formability and failure in tube inversion using i-Form 4 2 1    2 1 d d    Where β represents the deformation path. There is also a relation between both variables, which can be demonstrated through the Von Mises yield criterion. This is also useful to determine the triaxiality coefficient. This criterion is expressed as follows and, considering that it is under conditions of plane stress: 2 2 2 2 2 2 1 2 1 3 2 3 1 1[( ) ( ) ( ) ] [1 ] 2Y                    Where σY represent the yield strength. 21 2     21 2      Fracture mechanics: Fracture mechanics is the study of how cracks start, grow and stop. It is also an important tool for improving the mechanical performance of materials and components, applying the theories of elasticity and plasticity. There are three different ways of crack propagation: 1. Mode I: Opening mode: tensile stress normal to the plane of the crack, it tends to open the crack. 2. Mode II: Sliding mode: shear stress acting parallel to the plane of the crack and perpendicular to the crack front, it tends to slide one crack face with respect to the other 3. Mode III: Tearing mode: shear stress acting parallel to the plane of the crack and parallel to the crack front. Figure 3. Different ways of crack propagation (T.L. Anderson) Numerical analysis of formability and failure in tube inversion using i-Form 5 A cracked body can be loaded in any of these modes or a combination of them.  Deformation paths in plane stress: The strains are usually represented in a figure during different moments of the forming process. This figure has two axes, which corresponds with the major and minor strains of the plane. This diagram allows to obtain information about how the sheet (or tube) is deforming: Figure 4. Deformation paths in plane stress (Análisis de los Procesos de Conformado de Chapa) In this schematic representation, the main direction 1 is the one in which the strain (ε1) and stress (σ1) are higher, therefore the first point in the diagram is A, where both strains are the same (ε1= ε2). (Análisis de los Procesos de Conformado de Chapa) Each point in the FLC corresponds with a mechanism of deformation: - OA: Equibiaxial expansion - Between OA and OB there is a mechanism called biaxial tension. - OB: Plane strain. - OC: Uniaxial tension. - OD: Pure shear. - OE: Uniaxial compression.  Forming limit curve and fracture forming limit: The concept of forming limit curve (FLC) was introduced by Keeler (1964) and Goodwin (1968) and represents the first safety criterion for formability processes. Numerical analysis of formability and failure in tube inversion using i-Form 12 The experiments were done with tubes of 100 mm length and the dies used had inversion radii of 3.2, 4, 6 and 10 mm. The results were that the inversion was successful for 4 and 6 mm dies, with the 3.2 mm die, the tube inverted but the bottom extreme of the tube crashed with the tube and for the 10 mm die, the tube expanded, provoking a crack. Figure 11. Deformed inversion specimens for 10, 6 and 4 mm inversion radii (S.R. Reid, 1993) Later, P.A.R. Rosa et al., 2003, carried out a deeper research about the external inversion process with the objective of augmenting the knowledge about this process and issues related with friction and deformation modes. It was done by means of a numerical and experimental investigation. The objective in this work was to characterize the deformation modes associated with the formability limits and the influence of the lubrication, performed on Aluminium Al6060 tube of 70 mm length. The internal radius of the tube was 18 mm in all cases and the thickness varied from 0.5 to 2 mm. Moreover, the inversion radius of the die includes radii between 2 and 10 mm. In this article is analysed the critical instability load (which was fixed in 47.2 kN) by the axial compression of tubes, the external inversion of the tube, the load-displacement figure where the three phases of the successfully inversion process were distinguished: axial compression stage, transient invert-forming and steady-state invert forming. Numerical analysis of formability and failure in tube inversion using i-Form 13 Figure 12. External inversion process (G. Centeno et al., On The Utilization of Circle Grid Analysis in Thin-walled Forming of Tubes, 2017) The ductile damage criterion used was the Cockcroft-Latham and it was observed that for an inversion radius of 5 mm (rcd/r0=0.278) the accumulated damage was higher than the critical value and no cracks appeared. However, for inversion radius of 10 mm (rcd/r0=0.556), cracks were observed. That is why the first forming limits mentioned are buckling for rcd/r0 low ratios (around 0.21) and cracking around the circumference for rcd/r0 high values (around 0.42). It was determined that the lubrication is very important, promoted also due to the air that remains inside the tube and enclosed by the dies. Another conclusion reached was that using the ductile damage to study the appearance of cracks in this process was not recommendable, due to cracks were controlled by the decrease of thickness in the tube. Also during the year 2003 G.S. Sekhon et al. carried out an analysis of the inversion process. In this case, the tubes were longer (80, 100 and 100 mm length) and thicker (1.12, 1.68 and 1.3 mm respectively). It was used a finite element software, FORGE2, to compare the experimental results with the computational model. In this occasion, the parameters analysed were the key dimensions of the tube, defined in this work as be, external diameter of the inverted tube, and bi, the diameter of the bottom extreme of the inverted tube, shown in the Figure 13. Inversion process and parameters . It was observed that be and bi increased when the inversion radius increased. Numerical analysis of formability and failure in tube inversion using i-Form 14 Figure 13. Inversion process and parameters (G.S. Sekhon et al., 2003) The most important conclusions reached were that during the inversion process there were two unloads, the energy absorbed was larger for small inversion radius of the die, the friction decreased notably the capacity to absorb energy and finally that there was a friction value which made the inversion impossible. All these advances were achieved for external inversion of the tube. It was in 2004 when P.A. Rosa et al. focused in the internal inversion, due to the external inversion was well established. It was based on the fact that both parameters and the lubrication conditions were well defined, as well as the deformation modes that would probably occur. The objective was to determine the formability diagrams taking into account the major parameters of the process (thickness of the tube and inversion radius of the die) and also study the influence of the frictional conditions. This study was done in a theoretical and a experimental way. Figure 14. Internal inversion process (P.A. Rosa et al., 2004) Numerical analysis of formability and failure in tube inversion using i-Form 15 The tubes used were 70 mm length and the thickness were 1 or 2 mm. The inversion radius varied from 2 to 6 mm and it was done with two different materials: Al6060 (Aged) and Al6060 (Annealed). The results obtained after doing all the experiments are collected in the following forming diagram, which shows the relative load P/Pcr as a funtion of the ratio rcd/r0, with a fixed thickness of 1 mm and a value of h=30 mm (see Figure 14). Figure 15. Formability diagram (P.A. Rosa et al., 2004) As it happened for external inversion, explained in P.A.R. Rosa et al., 2003, low values of rcd/r0 produced buckling, whereas high values provoked wrinkling and in some situations, cracks. It was also appreciate that the lubrication had an important role in the success of the process as well as the parameter h. It has been in the last year when the biggest advances have been achieved in this field, thanks to Magrinho et al., 2019, which has not yet been published since it is under review. In that document are used new methodologies, previously applied for sheet metal forming and, later, in tube expansion. The objective here was to analysed the formability Numerical analysis of formability and failure in tube inversion using i-Form 16 limits by local buckling, necking and fracture during the external tube inversion. For this, it was used the combination of digital image correlation (DIC) for the strains, thickness measurements, force evolution and finite element modelling. For the research AA6063-T6 aluminium tubes were used, with 70 mm length, 18 mm of internal radius and 2 mm of thickness. Thanks to previous investigations performed by the authors for expansion process of the same tubes the mechanical properties (G. Centeno et al., Towards the characterization of fracture in thin-walled tube forming, 2016) and FLD were defined (J.P. Magrinho, M.B. Silva, & and G. Centeno et al., 2019). To analyse the formability limits, it was defined three different inversion radii: 0.5, 5 and 20 mm, in order to capture the entire range and, thus, have all the possibilities of evolution of the process possible previously described. For the R=0.5 mm case, it was proven that the evolution of the strains corresponded at the beginning with 𝛽=-2 (pure compression) and that there were two phases: 1. Compression along the meridional direction and stretching along the circumferential direction. 2. Stretching along the meridional and circumferential direction. For the R=5 mm case, it was checked the evolution in terms of forces that S.T.S Al- Hassani, W.Johnson, & W.T. lowe described in 1972. Moreover, in terms of strains, the evolution follows approximately the 𝛽=-0.5 line (pure tension) there were defined two different phases as well: 1. Compression along the meridional direction and stretching along the circumferential direction until the tube inverts. 2. Steady-state values Finally, for the R=20 mm case, it was also verified what was defined before, failure due to the necking followed by cracks. The strain evolution corresponds also with 𝛽=-0.5 and only one phase was identified: increase in the load and compression along the meridional direction and stretching along the circumferential direction until the fracture of the tube. Therefore, the conclusions reached are that within the inversion process there are three possibilities: - Local buckling for small inversion radii - Successful inversion for intermediate radii - Necking for large inversion radii Numerical analysis of formability and failure in tube inversion using i-Form 17 Moreover, it is shown that in terms of absorbed energy, the biggest value is for the successful inversion, allowing to absorb a lot of energy during large displacements. The smallest value is for small radii, but still allows to absorb relatively high energy in small displacements. Lastly, it is concluded that large inversion radii can be used as a mechanical fuse, absorbing a fixed amount of energy for a specific displacement. Figure 16. Energy absorbed with displacement (Magrinho et al., On the formability limits of thin-walled tube inversion, 2019) Document structure This document follows a structure trying to make itself comprehensive even for those who has never heard anything about tube end forming. That is why, in the first place, there is a state of the art which includes the main concepts of formability, explaining the plastic behaviour of the material, the plasticity relations and the forming limit diagram. Furthermore, all the advances about inversion will be described to locate the starting point of this work. Subsequently, it will be described how the software works in each of its modules. Special emphasis will be placed on the procedure followed in order to understand how the model has been generated and how the results have been obtained, so that anyone is able to carry it out. Finally, the results obtained will be exposed in terms of forces, surface strains, ductile damage and triaxiality, followed by the conclusions obtained from the analysis of them. Numerical analysis of formability and failure in tube inversion using i-Form 18 Objectives Once it has been demonstrated the effect of the inversion radius, the main objective of this document is to is to determine the radii that characterise the formability limits for the inversion process through a numerical software. This software is i-Form, which will be explained in the following sections. To achieve this target, it will be analysed the force evolution within the inversion process depending on the inversion radius, as well as the surface strains, ductile damage and triaxiality by the use of a finite element analysis. Taking into account all these results, it can be determined how it is expected that the tube behaves depending on the inversion radius of the die. Numerical analysis of formability and failure in tube inversion using i-Form 19 Numerical model Introduction The aim of this section is to define which is the software employed, how it works to carry out the inversion processes and how it is modeled. The simulation will be carried out for tubes of 70 mm length, with an external radius of 20 mm and a thickness of 2 mm. The material used for the simulations is Aluminium AA6063-T6. There have been performed several experimental researches for this material, allowing to have well defined the FLC and FFL, as indicated in the previous section. i-Form i-Form is a software developed by the Instituto Superior Técnico of Lisboa and the Technical University of Denmark. This program allows to make simulations in 2D of different modelling processes. Analytical framework The finite element analysis is based on the irreducible finite element flow formulation (Cristino et al., A digital image correlation based methodology to characterize formability in tube forming, 2019):   2 0 1 2 r TT u V i i f r V V S S dV K dV T u dS du dS              Where: :  effective stress :  effective strain rate : V  volumetric strain rate :K large positive constant imposing the material incompressibility constraint Numerical analysis of formability and failure in tube inversion using i-Form 20 : i T surface tractions : i u surface velocities : T S target surface : r u relative sliding velocity : f  friction shear stress : f S contact interface between the tube and the die :V volume As it is explained in this article, the results that the program provides are in global coordinates (x, y, z), so it is necessary to rotate the strain tensor into the local coordinate system. To do this, it is required to make two rotations: one, rotating the axes (x, y, z) so they coincide with the axes (𝜃, r, z) and the second, rotating the (𝜃, r, z) axis, making it coincide with axis (t, 𝜃, 𝜙), where axis t is perpendicular to the tube thickness and axis (𝜃, 𝜙) are the in-plane axis, 𝜃 in the circumferential direction and 𝜙 in the meridional one. In order to make this more comprehensive, here can be found the mathematical expression and the graphic representation:   , , 2 , , 2 2 1 , , 1 2 T T T t r z x y z R R R R R R        1 cos sin 0 sin cos 0 0 0 1 R         2 cos 0 sin 0 1 0 sin 0 cos R         Figure 17. Schematic representation of the rotation of axis (Cristino et al., A digital image correlation based methodology to characterize formability in tube forming, 2019) Numerical analysis of formability and failure in tube inversion using i-Form 21 To evaluate the damage, the chosen criterion will be the void-growth damage-based one, proposed by McClintock, and utilized in several articles evaluating the fracture limit within (almost completely) tube expansion, due to the crack in expansion opens by tension (mode I of fracture mechanics) (as in Cristino et al., A digital image correlation based methodology to characterize formability in tube forming, 2018, J.P. Magrinho, M.B. Silva, & and G. Centeno et al., 2019, Cristino et al., Theory of single point incremental forming of tubes, 2019 (under review)). This will allow to make comparisons between the analytical results obtained for inversion and these obtained before. This criterion expresses the damage in function of the effective strain,  , and the stress-triaxiality, / m  and allows determining the critical ductile damage at the onset of failure by necking. Taking the expressions deducted by Cristino et al., Theory of single point incremental forming of tubes, 2019 (under review), it is possible to express the ductile damage in function of the strain ratio 𝛽. The hypothesis done is working under plane stress which translated into equations corresponds with: 0 t   If this restriction is introduced in the equations of effective stress,  , and effective strain increment, d  , it is obtained the following: 2 2 1 1 2 1 33 1 m               21 3 f crit D    Where the term f   corresponds with the meridional strain at the moment of the fracture. Moving back to the research of Cristino et al., 2018, in the next table it is shown the critical values of ductile damage at the onset of failure, comparing the analytical and experimental results obtained for the different geometries. Table 1. Critical ductile damage (Cristino et al., A digital image correlation based methodology to characterize formability in tube forming, 2019) Numerical analysis of formability and failure in tube inversion using i-Form 28 During the modelling, the first die defined was the upper one, so it will be the Die Number 1. A friction value of m=0.05 will be introduced to simulate the effect of the lubricant used during the experimentation. To end, the velocity of the simulation will be 1mm/step. 2.2.2.4 Material Data and Yield Criteria Next thing to do is to define the properties of the material. As it happened for the modelling, there are two options: introducing the material by fulfilling the different gaps in the menu or by loading a database with all the information. In this case, the chosen option was to load a database. In “Material Database”, it has to be selected the “User Defined Aluminium 6xxx” option due to during the experimentation the material was Aluminium 6063-T6. Finally, in the bottom table (“User Defined Material”), it has to be ticked the Tabulated Data option. Later, the data has to be loaded by pushing “>Input Data” and, at last, pushing “Save Material” in the upper-right corner. Figure 27. Material Data and Yield Criteria Numerical analysis of formability and failure in tube inversion using i-Form 29 Once the material has been saved, the next thing is to assign the material to the workpiece. To do this, and remembering that there is only one piece, the material will be assigned by pushing “Assign All”. Figure 28. Assigning material options 2.2.2.5 Damage and Fracture controls It is time now to define the damage model and taking into account the different analysis that will be studied, the selected one is “McClintock – Stress Triaxiality” which will allow to study the critical damaged in the different cases. Figure 29. Damage and Fracture controls Numerical analysis of formability and failure in tube inversion using i-Form 30 Simulation process To carry out the simulation, it will be opened now the I-form 2D program. Once it is opened the windows command screen appears. All the necessary to perform it is to specified the root of the folder containing all the data defined in the pre-processor. It is only necessary that this data is in a folder named Run2d contained in the root of the hard disk (C:\Run2d\...). For instance: Figure 30. Initial simulation window When you press “Enter”, if everything is alright, everything defined previously will appear as “Active”: Figure 31. Ready simulation window for the process Finally, by pressing “Enter” again, the simulation will be carried out. Numerical analysis of formability and failure in tube inversion using i-Form 31 Postprocessor As soon as the simulation has finished, it is possible to analyse a huge quantity of variables, such as surface strains, velocity, force, stress, ductile damage… This project it is focused in analysing the evolution of the force, the surface strain, the ductile damage and the stress triaxiality, therefore, it will be explained how to extract these results. 2.2.4.1 Forces One of the main purposes of this document is to show how the forces vary depending on the tube’s radius. Getting this information is easy. The methodology to follow is: Figure 32. Selection of the different graphics types When the XY Graphics menu is opened there are several options: Figure 33. XY Graphics options Numerical analysis of formability and failure in tube inversion using i-Form 32 In this case, the required information is the step number (later is immediate to change the step into vertical displacement) in the X axis and the vertical load in the Y axis. When those options are marked, it is possible to export an asci file with the data as follows: Figure 34. How to export the information The data will be export in a text file in the working directory with the name “fem.asc”, which can be easily copied to an excel file. 2.2.4.2 Surface Strain During the analytical experimentation, the external surface of the tube was painted in white and later sprayed with black droplets, generating a non-uniform speckle pattern, in order to use de digital image correlation (DIC). In this case, using the numerical model it is more immediate to get these results: Figure 35. How to access to the surface strains Later, the following menu appears: Numerical analysis of formability and failure in tube inversion using i-Form 33 Figure 36. Surface strains menu Here, it can be selected the step number to extract the strain and the desired element. It can be extracted the global strains, local strains and principal strains of it. 2.2.4.3 Ductile damage Another analysis to be done is to establish ductile damage criteria for the tube inversion. For this reason, it is important to know how the damage varies and where it is concentrated. To extract this information, there is a simple option: Figure 37. How to extract the ductile damage Numerical analysis of formability and failure in tube inversion using i-Form 34 Once the menu is opened, the most important option is the first one, to select the step number. Figure 38. Ductile damage menu 2.2.4.4 Effective strain vs stress-triaxiality To analyse this data, the menu Effective Strains vs. Stress Triaxiality has to be selected in the Graphics option. Once it is selected the next menu appears: Figure 39. Effective Strain vs. Stress Triaxiality menu. Numerical analysis of formability and failure in tube inversion using i-Form 35 The fields to fulfil are the Final Step Number and the Tracking Element, which will allow to define the results to extract. As it happened for the forces, it is possible to export an asci file with the data. Numerical analysis of formability and failure in tube inversion using i-Form 37 Results Introduction The aim of this section is showing the different possibilities that a tube has when it comes to external inversion processes. For this reason, results will be shown since a really small radius (R=0.5 mm) to a large radius (R=20 mm), comparing the force evolution, the strain paths and, finally, the ductile damage that the tube is able to withstand in each case. All these results will be also compared with experimental results obtained in previous researches, shown in the document J.P. Magrinho, M.B. Silva, & and G. Centeno et al., 2019, performed on a hydraulic testing machine (Instron SATEC 1200 kN). This will allow to validate the numerical results obtained within this work. Force analysis In the force analysis can be distinguished mainly three different evolutions, depending on the inversion radii. This is the reason why it will be divided in three sections. Small inversion radius To analyse the effect of small radius in the inversion process, the chosen radius was R=0.5 mm. In these cases, where the radius is almost non-existent, it can be observed that an instability appears, concretely local buckling at the top of the tube. This effect also occurs in the instability process, where the tube is pushed against a plane die in order to determine the maximum force that can be applied to the tube. Numerical analysis of formability and failure in tube inversion using i-Form 44 Figure 49. Strain evolution for R=5 mm This figure depicts the strain path for the successful inversion process. As it happened for the large radius, the trajectory that follows is 𝛽=-0.5, corresponding with Uniaxial tension, with the difference that in this case the absolute values of the strains grow until the inversion is completed in that point. Later, the strains values decrease and remains constant. In the next table, the values of the strains are presented: Table 2. Strain values for different displacements of the upper die Once the three possibilities for inversion has been defined, it is the time to define which are the ranges of radii for a successful inversion. At one extreme, the failure is provoked by local buckling at the top extreme of the tube; at the other, the excessive decrease of the thickness favours the appearance of a crack. After carrying out some simulations more for small radius (from 0.5 mm to 5 mm), the results reached are: Displacement of the upper die 0 -0.00006 0.00011 2.5 -0.02077 0.0419 5 -0.06227 0.13474 7.5 -0.09234 0.19824 10 -0.12198 0.25703 12.5 -0.13226 0.26928 15 -0.12941 0.25005 17.5 -0.11873 0.22508 20 -0.10711 0.2048 22.5 -0.09771 0.19093 25 -0.09131 0.18245 27.5 -0.08741 0.1776 30 -0.08519 0.17485 32.5 -0.08388 0.17311     Numerical analysis of formability and failure in tube inversion using i-Form 45 Figure 50. Final state of the tube for each small inversion radius As it is shown in the table, for inversion radii smaller than R=0.8 mm, there is directly local buckling at the top of the tube, while for radii equal or larger than R=0.8 mm to R=3 mm, there is a partly inversion. In the table are indicated the maximum displacement of the upper die without the appearance of instabilities. For R=3 mm or larger, the potential mode of failure is that the tube may end up bumping against itself, causing an instability in the top of the tube. For example, if the simulation for R=3 mm continues until an upper displacement of 40 mm, the result will be the following: Figure 51. Example of mode of failure for R=3 mm Numerical analysis of formability and failure in tube inversion using i-Form 46 On the other hand, if the radius is larger than 5 mm, the results are these: The table depicts the state of the tube at the end of the simulation and as it can be seen, there is a successful inversion for the range of radius between R=5 and 15 mm. For larger radius, these are so large that the tube cannot invert, but adopt the shape of the die as in the inversion process. Next step is analysing the ductile damage in each case because, although in the simulation the result is satisfactory in terms of final shape, this may not happen in a laboratory. Ductile damage analysis In this section, it will be only analysed the ductile damage for large inversion radii, due to the results obtained in the previous chapters. The critical value for the ductile damage obtained for expansion in previous researches was around 0.15. During the inversion process with a 5 mm radius die, the ductile damage accumulated at the end of process is bigger than that value. However, when the tube begins the inversion process, the ductile damage is lower and the process is carried out successfully. Having this in mind, the critical ductile damage for expansion will be used for inversion as the critical damage until the moment the tube separates from the die. Figure 52.Final state of the tube for large inversion radius Numerical analysis of formability and failure in tube inversion using i-Form 47 Therefore, it will be considered that once the inversion process has started, it is easier for the tube to complete the inversion process rather than starting the necking process. The beginning of the inversion will be considered the moment that the internal face of the tube separates from the die, as it is shown in the figure: Figure 53. Moment in which the tube starts to invert The next table shows the displacement of the upper die at the beginning of the tube inversion and the ductile damage at that moment for different inversion radii: Table 3. Displacement of the upper die at the beginning of the inversion and the ductile damage for each inversion radius As table shows, the ductile damage is lower than the critical value in each case except for R=11 and 12 mm, so it can be concluded that the inversion is successful until an inversion R=10 mm. All the figures showing the ductile damage in the moment corresponding with the beginning of the inversion value are attached in the Annex B – Ductile damage. If the inversion radius is larger than R=10 mm, the ductile damage surpasses the critical value, generating a crack and making the process unsuccessful. Inversion radius Displacement of the upper die at the beginning of inversion (mm) Ductile damage 5 14.5 0.1353 616 0.1368 7 17.5 0.1346 8 19.5 0.1394 920 0.1322 10 21.5 0.1359 11 24.5 0.1504 12 26.5 0.1595 Numerical analysis of formability and failure in tube inversion using i-Form 48 Formability limits The objective of this chapter is to determine the limits for a successful inversion process analysing the ductile damage (previously analysed) and the surface strains. As it was proved in the section 3.4Error! Reference source not found., in terms of ductile damage, it is not known if the process is satisfactory from R=8 mm and larger due to the critical value for the damage is it not clearly define for the inversion process. This is why the strain paths for R=8 mm and larger will be represented in the FLD. In each case, the element analysed will be Element 596. The choice of this element is due to the fact that strains are concentrated in this extreme and where the FLC and FFL were calibrated in previous researches, as explained in 1.1. It will be also represented the Element 586, at the same distance but in the internal side of the tube. Figure 54. Elements analysed In the next figure, it will be represented the strains since R=8 mm until the radius in which the FLC is clearly surpassed. The figure only shows the evolution since the beginning of the process until the moment the tube starts to invert, corresponding with the upper displacement shown in Table 3. Displacement of the upper die at the beginning of the inversion and the ductile damage for each inversion radius. Numerical analysis of formability and failure in tube inversion using i-Form 49 Figure 55. Strain paths Looking in the FLD, it can be appreciated that there is no point beyond the FLC for radii smaller than R=10 mm, which means that the inversion process is carried out successfully. Analysing now what happens with the R=11 mm case, there are some points which are above the FLC, so it is not clearly determined if the process can be carried out successfully or not. Finally, the R=12 mm case have many points above the FLC. On the other hand, the strains in the internal tube side are larger than in the external one, but it happens the same as before: there is no point beyond the FLC for radii smaller than R=10 mm. Figure 56. FLD for R=10 mm Numerical analysis of formability and failure in tube inversion using i-Form 50 Finally, it will be analysed the FLD for R=10 mm, for the elements shown in the Figure 56, to see if the choice of the element analysed affects to the final conclusion. As it is represented, the strains are bigger in the Element 583 and 575, but they are still below the FLC, so the conclusion will be the same. Looking to all these data (ductile damage, strain paths and state of the inversion), it can be concluded that the inversion process is completed successfully until R=10 mm in the cases of R>5 mm. The R=11 mm case is not clear, but taking the results obtained it will probably have a crack. Table 4. Summary of the main results for each case in Element 596. Triaxiality analysis In this chapter, it will be represented the effective strain vs stress-triaxiality for the limit cases in large radii, corresponding with R=9, 10 and 11 mm for the elements previously used: Element 596 (external tube side) and 586 (internal tube side). Figure 57. Space of effective strain vs stress-triaxiality Inversion radius Displacement of the upper die at the beginning of inversion (mm) Ductile damage Fracture 5 14.5 - - 0.1353 No 616 -0.07979 0.17197 0.1368 No 7 17.5 -0.09039 0.20369 0.1346 No 8 19.5 -0.10424 0.24756 0.1394 No 9 20.8 -0.11752 0.26983 0.1322 No 10 21.5 -0.12552 0.28012 0.1359 No 11 24.5 -0.15034 0.33943 0.1504 ? 12 26.5 -0.16691 0.37469 0.1595 Yes     Numerical analysis of formability and failure in tube inversion using i-Form 51 As it happened before, there are no points beyond the FLC for the cases R<10 mm in the Element 596, but there are for R=11 mm. This reinforces the idea that the tube will have a crack for R=11 mm. Moreover, if Element 586 (internal side) is analysed, there are a lot of points beyond the FLC. This leads to the idea that the FLC is defined conservatively and, maybe, if it is recalculated, the R=11 mm case is also below it. Figure 58. Limit cases Numerical analysis of formability and failure in tube inversion using i-Form 53 Conclusions and future work In this work, it has been carried out a numerical analysis using the i-Form software for the inversion process for Aluminium AA6063-T6 tubes. The results obtained have been validated with experimental researches made before for inversion processes with radii 0.5, 5 and 20 mm of the die. Once all the results - divided in four different categories: forces, surface strain, ductile damage and triaxiality - have been analysed and the conclusions reached are: - The assumptions taken from sheet metal forming are valid for tube forming. - The evolution of the forces obtained in i-Form are very close to the experimental ones, making possible to predict this evolution for the different inversion cases. - In terms of surface strains, the results obtained are very good when the inversion takes place. However, for small inversion radii, although the evolution has the same trajectory, the values of the strains differs substantially. - During the inversion process the trajectory of the strains for a point at the lower extreme of the tube follows the 𝛽=-0.5 line, which means Uniaxial tension. This also happened during the expansion process. - It has been demonstrated that the forming limits for the tube are local buckling for small inversion radii and necking for large inversion radii. - The ranges where the different failures take place are: 1. R ≤ 0.8 mm  Local buckling at the top of the tube. 2. 0.8 < R < 3 mm  Inversion, with a limit displacement for the upper die. Otherwise, local buckling will appear. 3. 3 ≤ R ≤ 10 mm  Successful inversion. Potential mode of failure will be that the tube bumps with itself. 4. R ≥ 11 mm  Necking before the inversion process. All the results obtained have been purely numerical except those from R=0.5, 5 and 20 mm inversion radii. In order to validate the forming limits radii, it will be interesting to prove these results in an experimental way. Another possibility is to study the ductile damage for the inversion process due to, as it has been proved, the damage for this process is bigger than the value for the expansion process. This would generate new possibilities in terms of inversion processes. Numerical analysis of formability and failure in tube inversion using i-Form 60 Rinv= 7 mm – Displacement of the upper die: 17.5 mm Rinv= 8 mm – Displacement of the upper die: 19.5 mm Numerical analysis of formability and failure in tube inversion using i-Form 61 Rinv= 9 mm – Displacement of the upper die: 20.8 mm Rinv= 10 mm – Displacement of the upper die: 21.5 mm Numerical analysis of formability and failure in tube inversion using i-Form 62 Rinv= 11 mm – Displacement of the upper die: 24.5 mm Rinv= 12 mm – Displacement of the upper die: 26.5 mm