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  • Towards modelling elastic-plastic deformation of a tube-shaped work-piece under axisymmetric load Koos van Putten*, Klaas van der Werff**, Kurt Steinhoff** and Jaap Fontijne***

    *Research Assistant, Metal Forming Institute, University of Technology, Aachen/ Germany; **Full Professor and Associate Professor, Pro- duction Technology and Industrial Organisation, Delft University of Technology/ The Netherlands; ***Manager of technology, Fontijne Grotnes BV, Vlaardingen/ The Netherlands.

    The complex mechanisms occurring during the expansion of a tube-shaped work-piece are of particular interest for many industrial produc- tion processes. In case of the expansion of the outer end of a tube, so-called flare forming, most of the process-design parameters are main- ly based on experience and empirical knowledge. To improve the methodological basis of process design and consequently to increase the technological efficiency of the process itself, two different types of models are developed and compared in this paper. The first one is a con- tinuum-mechanics based analytical model, the second one is a numerical model based on finite-element simulation. Both models are able to describe the elastic and plastic behaviour during flare forming. For the analytical model classical theories are applied. Such theories are on the one hand those of Timoshenko, which are applied for the description of the elastic behaviour, and on the other hand a limit analysis for the characterization of the plastic behaviour. For the numerical model a non-linear elastic-plastic finite-element simulation is carried out with the commercially available FEM-software MARC-Autoforge. Both models are validated and verified with experimental data and evalu- ated regarding their applicability under real industrial process conditions. Finally, it is not only concluded that flare forming can be modelled sufficiently by both approaches, but beyond that, tools for an optimised process design can be derived. These design tools can directly be integrated in a CAD-system.

    Ein Beitrag zur Modellierung der elasto-plastischen Umformung eines rohrförmigen Werkstückes unter axialsymmetrischer Bean- spruchung. Die komplexen Mechanismen, die beim Aufweiten rohrförmiger Bauteile auftreten, sind für eine Vielzahl industrieller Ferti- gungsprozesse von besonderer Bedeutung. Betrachtet man zum Beispiel die Prozessvariante des Aufweitens von Rohrenden, das soge- nannten flare forming, so basiert die Gestaltung dieses Prozesses im Wesentlichen auf empirisch begründeten Kenntnissen. Im vorliegen- den Bericht werden zwei Modellierungsansätze zur Beschreibung des Aufweitens von rohrförmigen Bauteilen mit dem Ziel vorgestellt, die methodische Basis der Prozessgestaltung und damit konsequenterweise auch die technologische Effizienz des daraus resultierenden Pro- zesses zu verbessern. Dabei beruht das erste Modell auf einem kontinuumsmechanischen Beschreibungsansatz, das zweite auf einem nu- merischen Ansatz auf der Grundlage einer Finite-Elemente-Berechnung. Beim analytischen Modell kommen klassische Theorien der Me- chanik zur Anwendung; dies sind im vorliegenden Fall zum einen die Theorien von Timoshenko, die für die Beschreibung der elastischen Formänderung angewandt wurden, zum anderen Grenzwertbetrachtungen für den Fall der plastischen Formänderung. Für die numerische Modellierung wird eine nichtlineare elasto-plastische Finite-Elemente-Simulation mit Hilfe der kommerziell verfügbaren Computer-Software MARC-Autoforge durchgeführt. Beide Modelle werden anhand von experimentell ermittelten Daten verfiziert und hinsichtlich ihrer Anwend- barkeit unter industriellen Bedingungen bewertet. Aus den vorliegenden Ergebnissen kann nicht nur eine hohe Übereinstimmung von Mo- dell und Experiment abgeleitet werden, sondern darüber hinaus können hieraus auch neuartige Hilfsmittel für die Prozessgestaltung und -optimierung entwickelt werden. Diese Gestaltungshilfsmittel lassen sich dabei unmittelbar in ein CAD-System integrieren.

    168 steel research 74 (2003) No. 3

    Flare forming – a process with high practical relevance

    The mechanisms occurring during the expansion of a tube-shaped work-piece are of particular interest for many industrial production processes. Among these processes, flare forming of metallic materials constitutes one the most widely applied variants. A well-known application can be found e.g. in the production process of wheel rims. In case of the sheet-metal production route of wheel rims, a cylin- drical ring, made out of a bent and welded stroke of steel or aluminium plate, is flare formed by pressing two conical tools into its open sides. By this type of flare forming a cylinder with two upstanding sides is created. After flare forming the wheel rim is roll formed into its final shape.

    During an analysis of the existing process-design meth- ods it became obvious that especially for the flare forming step most of the design parameters are mainly based on ex- perience and empirical knowledge. However, to improve

    the methodological basis of process design and consequent- ly to increase the technological efficiency of the process it- self, a comprehensive phenomenological understanding of flare forming is necessary.

    For this purpose, different types of models will be devel- oped in this paper. These models will enhance the calcula- tion of the required forces on the conical tools, the occur- ring strain in the ring, and the study of the influences of characteristic process parameters. Additionally, these mod- els should still be valid, even when new materials (e.g. high strength or dual phase steel) are used or when other shapes of the plate material (e.g. tailored or tailor-rolled blanks) are applied, in order to provide sufficient information about the necessary changes in the forming process.

    Modelling the flare forming process

    To predict the forming behaviour during the flare forming process two different models are developed and compared.

    Metal forming

  • The first one is a continuum-mechanics based model for an analytical description of elastic and plastic deformation of materials. The second one is a non-linear elastic-plastic fi- nite-element model. This paragraph describes the two mod- els and the underlying theoretical considerations.

    Analytical model

    The geometrical change of the cylindrical work-piece during forming is described by a set of equations for the elastic, elastic-plastic and plastic deformation under influ- ence of a force applied by the cone-shaped tools. As a result of this approach, the typical deformation during flare form- ing can be analytically described in the context of charac- teristic process parameters. Symbols are explained in tables 1 and 2.

    Elastic deformation. The first process phase during flare forming is characterised by an elastic deformation. Due to the symmetry of the cylindrical ring and the symmetry of the load situation when pressing cones from both open sides, it is sufficient to consider only half of the cylinder for the development of the model.

    During the elastic deformation there are two major forces: a radial force which bends the cylinder open and a friction force which works between the cone and the cylin- der.

    Flare forming is similar to the deformation of a circular cylindrical shell loaded symmetrically with respect to its axis as described by Timoshenko [1]. The differential equa- tion for the deflection w is:

    d2

    dx2

    ( D

    d2w

    dx2

    ) + E · h

    r2 w = Z (1)

    Assume the thickness of the shell as constant and assume an equal radial distribution of the applied radial force and the resulting bending momentum. According to [1], in this case the general solution for the radial displacement is giv- en by:

    w = e −βx

    2 · β3 · D [β · M0{sin(β · x) − cos(β · x)} − Q0 · cos(β · x)]

    (2)

    In this solution D and β are constants with the following definitions:

    β4 = 3(1 − v 2)

    r2 · h2 (3)

    D = E · h 3

    12(1 − v2) (4)

    The radial force is applied eccentrically on the very inner edge of the cylinder and therefore induces a bending mo- ment depending of the cone angle:

    M0 = Q0 · tan(α) · 12 · h (5)

    The maximum elastic radial displacement is reached when the yield stress is reached. This happens when:

    wmax = σv · R0 E

    (6)

    When the cones are pressed into the cylinder a radial out- ward displacement is taken positive, the radial force and the bending moment at the edge work as indicated in figure 1. This leads to the equation for the maximum radial force which is applied by the cone:

    Q0 = σv · R0

    E · 2 · β

    3 · D ( 1 − β · tan(α) · 12 · h

    ) (7)

    Metal forming

    steel research 74 (2003) No. 3 169

    [ y ]

    Symbol Meaning Unit

    c Given displacement mm

    D membrane stiffness Nmm

    E Youngs modules N/mm2

    E Energy Nmm

    F Force N

    S Stroke mm

    h thickness mm

    M Normalized plastic moment Nmm/mm

    N Normalized membrane force N/mm

    P Power W

    Q Normalized radial force N/mm

    R Normalized resultant force N/mm

    R0 Radius (original) mm

    r radius mm

    u axial displacement mm

    V Volume mm3

    W Normalized friction force N/mm

    w radial displacement mm

    Z external axial force N

    α flare angle °

    ε Strain -

    κ Radius of curvature mm

    σ Stress N/mm2

    τ Shear stress N/mm2

    µ Friction coefficient -

    ν Poisson’s ratio -

    Indices Meaning

    0 plastic

    dry dry

    o origin

    p plastic

    x a