Towards modelling elastic-plastic deformation of a tube ... · Towards modelling elastic-plastic...

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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

Transcript of Towards modelling elastic-plastic deformation of a tube ... · Towards modelling elastic-plastic...

Page 1: Towards modelling elastic-plastic deformation of a tube ... · Towards modelling elastic-plastic deformation of a tube-shaped work-piece under axisymmetric load ... necessary changes

Towards modelling elastic-plastic deformation of a tube-shaped work-pieceunder axisymmetric loadKoos 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 GrotnesBV, 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 thetechnological 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 ableto describe the elastic and plastic behaviour during flare forming. For the analytical model classical theories are applied. Such theories areon the one hand those of Timoshenko, which are applied for the description of the elastic behaviour, and on the other hand a limit analysisfor the characterization of the plastic behaviour. For the numerical model a non-linear elastic-plastic finite-element simulation is carried outwith 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 modelledsufficiently by both approaches, but beyond that, tools for an optimised process design can be derived. These design tools can directly beintegrated 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, diemethodische 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 elastischenFormänderung angewandt wurden, zum anderen Grenzwertbetrachtungen für den Fall der plastischen Formänderung. Für die numerischeModellierung wird eine nichtlineare elasto-plastische Finite-Elemente-Simulation mit Hilfe der kommerziell verfügbaren Computer-SoftwareMARC-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 practicalrelevance

The mechanisms occurring during the expansion of atube-shaped work-piece are of particular interest for manyindustrial production processes. Among these processes,flare forming of metallic materials constitutes one the mostwidely applied variants. A well-known application can befound e.g. in the production process of wheel rims. In caseof the sheet-metal production route of wheel rims, a cylin-drical ring, made out of a bent and welded stroke of steel oraluminium plate, is flare formed by pressing two conicaltools into its open sides. By this type of flare forming acylinder with two upstanding sides is created. After flareforming 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 formingstep 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 offlare 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 ofcharacteristic process parameters. Additionally, these mod-els should still be valid, even when new materials (e.g. highstrength or dual phase steel) are used or when other shapesof the plate material (e.g. tailored or tailor-rolled blanks) areapplied, in order to provide sufficient information about thenecessary changes in the forming process.

Modelling the flare forming process

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

Metal forming

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The first one is a continuum-mechanics based model for ananalytical description of elastic and plastic deformation ofmaterials. 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-pieceduring forming is described by a set of equations for theelastic, elastic-plastic and plastic deformation under influ-ence of a force applied by the cone-shaped tools. As a resultof 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 tables1 and 2.

Elastic deformation. The first process phase during flareforming is characterised by an elastic deformation. Due tothe symmetry of the cylindrical ring and the symmetry ofthe load situation when pressing cones from both opensides, it is sufficient to consider only half of the cylinder forthe development of the model.

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

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

d2

dx2

(D

d2w

dx2

)+ E · h

r2w = Z (1)

Assume the thickness of the shell as constant and assumean equal radial distribution of the applied radial force andthe resulting bending momentum. According to [1], in thiscase 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 followingdefinitions:

β4 = 3(1 − v2)

r2 · h2(3)

D = E · h3

12(1 − v2)(4)

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

M0 = Q0 · tan(α) · 1

2· h (5)

The maximum elastic radial displacement is reachedwhen 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 thebending moment at the edge work as indicated in figure 1.This leads to the equation for the maximum radial forcewhich is applied by the cone:

Q0 =σv · R0

E· 2 · β3 · D(1 − β · tan(α) · 1

2 · 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 axial direction

r radial direction

T true

v yield

ϕ circumferential

Table 2. [Indices]

Table 1. [Symbols]

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The axial force can now be calculated by:

V = Q0 · tan(α) (8)

This finally is the force with which the hydraulic cylinderpresses the conical tools into the cylinder.

The friction between the tool and the cylinder is in a do-main of very high contact pressure because of a thin contactline. It is assumed that there is no deformation at the edgeof the ring and that a sharp angle is kept. Therefore, lubri-cation is difficult and dry friction is assumed. The frictionforce can be calculated by:

W ‘ = R · µdry (9)

The axial component of the friction force can be calculat-ed by:

W ‘x = Q0 · µdry (10)

The frictional conditions during flare forming can becharacterized by a coefficient of friction varying between0.3 < µ < 0.8 [12, 13].

The total force to be applied in axial direction to the toolsis equal to the sum of the axial components of the deforma-tion force and the friction force multiplied by the circum-ference of the ring. This leads to:

Fel =2 · π · R0 · S · tan(α) · 2 · β3 · D(1 − β · tan(α) · 1

2 · h)

· (tan(α) + µdry)

(11)

In this equation the variable S is the stroke, the distancewhich the cone is pressed into the cylinder. The maximumstroke in the elastic domain is:

Sel max = R0 · σv

E· 1

tan(α)(12)

Plastic deformation. As the cones are pressed furtherinto the cylinder, the deformations become plastic. The de-scription of the deformation process during this plastic de-formation is based on further analysis of Onat [2]. This ar-ticle describes the plastic collapse of cylindrical shells un-der axially symmetrical loading which is similar to the flareforming process.

Because of the axial symmetry only the axial and radialcomponents of the displacements and velocities have to beconsidered. The equilibrium equations combined withTresca’s yield condition leads to:

Nϕ = Np (13)

−Mp < Mx < Mp (14)

These equilibrium equations match the conditions ofOnats first law which describes the flow rule that corre-sponds to the approximated yield curve for plastic collapse[2]. The plastic strain of the cylinder, caused by pressing inthe cones, is now defined by:

εx : εϕ : κ = −1 : 1 : 0 (15)

With this flow rule together with Tresca’s conditions,considering the material as perfect plastic and the theoriesfor stress and strain as described in [2, 3, 6] we find for thestresses in axial and radial direction:

σx = σϕ = σv (Nx �= 0) (16)

Using these equations for the stresses and the strains alimit analysis [3, 7] makes it possible to calculate the forcerequired for the plastic deformation. This type of analysisconstitutes a limit approach which determines the limit loadfor which the first plastic deformations occur. The limitanalysis basically consists of two theorems:1. Static theorem (lower limit), the existence of a so calledsafe static stress distribution (Kazincz and Kist [7]).2. Kinematical theorem (upper limit), the dual or comple-mentary law, which considers the surplus of the power ofthe external load compared to the plastic deformationpower.

This second theorem expresses that a minimum and satis-fying condition for a safety factor m < 1 is the existence ofa kinematical allowable deformation mechanism for whichthe power of the applied external load is bigger as the dissi-pated power by deformation. Therefore, m is nothing elsethan a safety factor according to the applied load. Now it ispossible to calculate the deformation force with equilibriumof power.

Metal forming

170 steel research 74 (2003) No. 3

Figure 1. Schematic illustration of the deformation during the flareforming process. The cone shaped tools are pressed from bothsides into the cylinder. The stroke of which the cone shaped toolsare pressed into the cylinder is equal to the variable S. The processhas two kinds of symmetry; in the middle of the axial length of thering a symmetry plane (B-B) and rotational symmetry (A-A).

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The cylinder needs to deform in a kinematical allowableform such that the second theorem of the limit analysis isfulfilled. The form of the tool forms a boundary conditionand the cylinder will deform over the cone. With

εx = ∂u

∂x, εϕ = w

rand κ = ∂2w

∂x2(17)

it is possible to describe the deformations in a kinemati-cal admissible form:

w = c · sin(α) (18)

u = −µ · σϕ

E· x2

2 · r(19)

εϕ = c · sin(α)

R0 + (x − x0) · tan(α)(20)

εx = ∂u

∂x= −c · cos(α) · µ · σϕ

E · r(21)

During the forming process there should exist balance ofpower. The power is transferred into the system by the hy-draulically driven tools. The absorbed power equals thesum of the power absorbed by deformation and friction. Thepower absorbing elements of deformation are:

1.) The plastic hinge (bending point).2.) The deformation of the cylinder in circumferential di-

rection.3.) The deformation of the cylinder in axial direction.

This leads to:

Pout = Pp.h. + Pϕ + Pf riction + Px (22)

The (absorbed) power is defined by:

P =∫

Vσ · ε · dV (23)

Power absorbed by the plastic hinge. During flareforming the area between the undeformed cylindrical work-piece zone and the deformed conical zone, forms a plastichinge, figure 1. The plastic hinge is a bending area in whichthe cylinder is bent open (flared).

Two kinds of plastic hinges can be found in literature.Firstly, a hinge that has no length finally forming a bendingpoint between the two lines (κ = ∞). Secondly, a hinge inwhich the bending radius is a boundary condition becauseof the shape of a bending tool. However, none of these twoplastic hinges is able to describe the plastic hinge of theflare forming process under investigation.

For flare forming the bending radius is assumed to beconstant and independent of the penetration depth of thetool. Besides that the stress over the bending radius is as-sumed to be constant and absolutely plastic. The energyneeded for the deformation of the plastic hinge is now ex-pressed as:

E =∫

Vσ · ε · dV (24)

Based on symmetry there is no bending in circumferentialdirection, neither there occurs a variation in the circumfer-ential radius:

Mϕ = 0 → κϕ = 0 (25)

With the substitution of the general equation for the strainin a plate, ε = z · κ [1, 4, 5] and with the infinitesimal vol-ume, dV = dx · R0 · dϕ · dz , equation (24) can be trans-ferred to:

Ev =∫ 2π

0

∫x

R0 · x · dϕ ·∫

hσ · z · κ · dz (26)

The aforementioned integral can be calculated by substi-tuting the domains (0, h

2 ) and (h2 , 0) and the associated

stresses σ and –σ respectively. The power that is needed forthe deformation of the plastic hinge can now be calculatedby differentiation of the energy to the time:

Pp.h. = d E

dt= 2π · R0 · dx

dt· σ · κ · 1

4· h2 (27)

in which:

dx

dt= c (28)

κ = d2w

dx2= 1

R(29)

The bending radius R during plastic deformation can becalculated by application of the bending formulas and theplastic moment according to [18] and [19]. This results in arelatively large bending radius, and a minimum power con-sumption for the deformation of a plastic hinge. The rela-tively large bending radius is confirmed by experiments[22] and by the finite element calculations to be presentedlater in this article. The influence of the deformation of theplastic hinge is very small, less then 1 % of the total of pow-er needed, and therefore will be neglected in further calcu-lations.

Power absorbed by radial deformation. The power re-quired for radial deformation is calculated from (23). Thestrain is known from equation (20) and the infinitesimalvolume dV = 2π · r(x) · h · dx is a ring with an infinitesi-mal height. This results in:

Pϕ = 2π · h · σϕ · c · sin(α) · S (30)

Power absorbed to overcome friction. The power re-quired to overcome the friction asks for another kind ofequation. Friction can be modelled as a shear stress andtherefore equation (23) is written as:

Pf riction =∫

Aτ · x · d A (31)

Metal forming

steel research 74 (2003) No. 3 171

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The infinitesimal surface is defined by d A = 2π · r(x) · dxwith r(x) as radius of the ring and dx as width. The shearstress is a function of the pressure by τ = µ · p . The hoopstress in the (deformed) cylinder has a direct relation withthe pressure exerted by the tool:

p = σϕ · h

r(32)

By substituting the above mentioned equations into (31)the power required to overcome the friction can be ex-pressed as:

Pf riction = 2π · h · σϕ · µ · c · cos(α) · S (33)

Power absorbed by axial deformation. The power re-quired for the axial deformation is calculated in a similarway as the power needed for radial deformation was calcu-lated. Substituting the axial strain (21) and the same infini-tesimal volume as with friction, the power needed for axialdeformation is expressed by:

Px = 2π · h · σx · µ · σϕ

E· c · cos(α) · S (34)

Power input by external force. To complete the balanceof power, the power exerted by the hydraulic cylinder needsto be calculated. The power of the hydraulic cylinder is:

Pin = F · c (35)

Solving the system. All contributions of the power bal-ance contain the deformation speed c , hence dividing by cresults in the equilibrium equation:

F = 2π · h · σν · S ·[sin(α) + µ · cos(α)

{1 + σv

E

}](36)

Strain hardening can be included in the model by takingfor σv the Hollomon equation [8, 9, 10]:

σT = K · [ϕ0 + ln(εϕ + 1)]n (37)

Elasto plastic deformation. To complete the model ofthe flare forming process, the elastic-plastic deformationstage of the ring needs to be considered as well. In this partof the deformation cycle the stress distribution changesfrom pure elastic to the fully plastic stress distribution. Weshall assume that during this stage the force will be almostconstant, because of the change of elastic- into plastic de-formation at constant yield stress and the change of contactsurface (line- to surface contact).

Combining the three parts of the model gives the com-plete model as depicted in figure 2.

During the elastic-plastic stage of the forming process thelarge radius of the smooth deformation over half of thewidth of the cylinder evolves into a small local bendingzone, with a small bending radius, a so-called plastic hinge.

Dimensionless form and nomograms. A dimensionlessgraph of the analytical model is derived constructed ac-cording to the Buckingham π-theorem [23]. From this graphthe deformation force can be found at any penetration dis-tance of the cone and therefore directly integrated in theprocess design. Although the resulting nomograms can offcourse be defined for the elastic domain, the plastic domainis of special interest because of the occurring maximumforces.

The dimensionless form of the force needed for plasticdeformation as a function of the penetration depth of thetool is:

F

2π · h · S · σT= sin(α) + µ ·

(1 + σT

E

)· cos(α) (38)

This dimensionless form results in the nomogram offigure 3.

Metal forming

172 steel research 74 (2003) No. 3

Figure 2. Force-path-diagram and stress patterns of the flare form-ing process.

Figure 3. Nomogram for the force required for plastic deformationduring the flare forming process.

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Numerical model

The second model of the flare forming process is a finiteelement model. A numerical model makes it possible tovary parameters which cannot or can hardly be varied withthe analytical model, for example the usage of tailoredblanks with various sheet thickness over the width of thecylinder.

For modelling the flare forming process with a finite ele-ment program a non linear FEM-code is necessary. This isbecause the flare forming process is a plastic formingprocess and large deformations and rotations occur. Thecommercially available software Marc-Autoforge wasfound to be most suited for this purpose.

Building the model. In the FEM-program the conicalflaring tool is modelled as a rigid body. The formingprocess is assumed to be isothermal due to large amounts ofcooling liquid used. The cylinder material is modelled aselastic-plastic material. In the model the tool moves into thecylinder with a defined velocity.

Symmetry. The flare forming process is symmetrical intwo different ways. First the tools are pressed into thecylinder from both sides, this results in a symmetry plane atthe middle of the axial length of the cylinder. This symme-try plane can be applied to the model by a cinematic bound-ary condition in the middle of the cylinder. The second formof symmetry is rotational symmetry, both, the tools and thecylinder are cylindrical and therefore rotational symmetri-cal.

By applying both forms of symmetry to the model it ispossible to simulate the flare forming process in a two di-mensional analysis, which saves elements and calculationtime.

Elements. The cylinder, modelled in two dimensions asrectangular box, is divided into Quad-4 elements [4]. Forsome specific calculations, for example with a ring out of atailored blank, Trim-3 elements are used.

Material properties. The flow curves of the material areadded to the material database of the program. Other mate-rial properties, such as Youngs modules, Poissons ratio, areadded for each flow curve as appropriate. As the flare form-ing process is considered to be isothermal and the deforma-tion speeds vary only slightly we didn’t use the possibilityto define material properties for various speeds and temper-atures.

Friction. The friction between the tool and the cylindercan be described with various friction cases and various co-efficients of friction. Since lubrication during flare formingis poor, slip-stick friction with a coefficient of friction of0.15 was chosen. This choice appeared to show a good cor-relation with experimental results.

Results of FEM-simulation. Force-path diagram. A ma-jor result of the FEM calculation is the force-path diagramfor the tool, figure 4. This diagram is constructed from the

path increments of the tool and the calculated force on thetool. It was observed that the force- path diagram was ratherfluctuative possibly due to the numerical calculationprocess. To create a clear graph the original data aresmoothed over a few time steps.

From model to the real process

Validation and verification of models

During an earlier investigation of the flare formingprocess [22] accurate measurements were done at the flarepress. The results of these experiments are used for the val-idation of the two models.

The analytical and the numerical results are comparedwith the aforementioned experimental data. The dry frictioncoefficient, during the elastic deformation, is set to 0.55.For the friction coefficient during plastic deformation thefriction coefficient is set to 0.15. These values give a goodmatch with the measurements and they are equal to the val-ues mentioned in various literature [12, 13].

In figure 5 a force-path diagram is drawn for both of themodels and for the experiment. In the figure the three re-gions of elastic, elasto-plastic, and plastic deformation canbe recognised. The first part of the graph where elastic de-formation occurs shows a steep increase of the press force.Then the deformations become elastic-plastic, the plastichinge is formed. In this stage the force becomes nearly con-stant. Finally when the plastic hinge starts to move in axialdirection, the deformations become plastic and the force ris-es again almost linear with the path. The most remarkabledifference between the models and the experiment is thatthe force rises more quickly in the beginning of the plasticdeformation. The latter phenomenon is included only in thenumerical model. After a while it disappears and the forcesof the models and the experiment are all equal.

The forces calculated by the numerical model and theforces measured by the experiment during elastic deforma-tion are exactly equal, but the start of the elastic-plasticzone is more abrupt in the models as it is in the experiment.

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steel research 74 (2003) No. 3 173

Figure 4. An example of a force-path diagram calculated with thenumerical model.

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During the elastic deformation the forces calculated by theanalytical model are a somewhat higher as the forces meas-ured in the experiment.

In the elastic-plastic zone the forces of both models areequal, but in the experimental results it takes more time forthe force to become constant, it has a kind of overshoot, andin the experiment the force doesn’t become totally constantat all.

The starting of the plastic deformation is for both theo-retical models almost the same, but happens a little earlierthan in the experimental result. In the fully plastic zone theforce in the experiment rises quickly. In both of the theoret-ical models the forces rise as well, but not as quickly as inthe experiment, although the force of the numerical modelrises more quickly as that of the analytical model.

This quick rising of the force in the experiment can be ex-plained by the forming of a relatively small radius of cur-vature at the end of elastic-plastic zone which stands for awhile in the plastic zone. This radius of curvature exists fora while and makes the cylinder come free of the tool at the

front of the ring [21]. This is the result of more deformationof the material at the inside of the cylinder than on the out-side of the wall of the ring during the elastic-plastic hingeforming of the ring. The inner material is more deformedand has a higher flow stress due to strain hardening. Afterthe deformation in the plastic hinge the material has to bendback in a straight line, over the contact surface of the tool,for which a higher force is needed.

The partial loss of contact of cylinder and cone, onlymodelled by the numerical model, figure 6, might explainthe difference between the force calculated by the numeri-cal model and the experiment.

When the deformed part of the cylinder comes in full con-tact with the tool again, the forces calculated by the variousmodels are in full agreement.

Influence of process parameters

With the models process parameters can be varied andtheir influence on the forming process, the product design,and the production process can be studied. Numerical sim-ulations give fast and reliable results and cost much lessthan experiments. Doing so it becomes possible to optimisethe flare forming process. By varying the process parame-ters and by plotting the resulting force-path diagrams in asingle diagram the influence of the parameters can be stud-ied, figure 7.

Process design tools

The main application area for the discussed models of theflare forming process is the optimisation and development

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Figure 5. Force-path diagram of the experiment, the analytical mod-el and the numerical model. The three zones of deformation can beclearly recognised.

Figure 6. Finite element simulation of the flare forming process; dis-tribution of Von Mises stress. Partial loss of contact of cylinder andcone(numerical model). The frames A-D are marked in the force-path diagram.

Figure 7. Force-path of a base model with one parameter varied.Varied parameters: sheet thickness, cone angle, cylinder diameter,tool shape (convex/concave), tailored blank, material and friction.The drawn lines are derived from the analytical model, the dottedlines come from the numerical model.

Page 8: Towards modelling elastic-plastic deformation of a tube ... · Towards modelling elastic-plastic deformation of a tube-shaped work-piece under axisymmetric load ... necessary changes

of the wheel rim production process. Flare forming is how-ever also used in many other products. For the flare form-ing process in wheel rim production the analytical modelmeets the demands in most cases. The numerical model canbe more accurate for some specific problems, but it needsfar more calculation time. To make the analytic model morecomfortable and fast to use it is implemented in TK Solver.By this way the product and the process design engineersare able to use the analytical model.

Conclusions

In this article the physical backgrounds of the flare form-ing process are studied to make a further optimisation of aflare forming process as used in the rim wheel productionpossible. This study leads to following conclusions:

1) Flare forming can be modelled with an analytical mod-el as well as with a numerical model, both models are pre-sented in this publication. The models describe the elastic,elastic-plastic and plastic deformation during the flareforming process and give a good match with experimentaldata.

2) With the models force-path diagrams can be calculatedto be used as a optimisation tool in engineering the rim andthe flare forming process in rim production.

3) The analytical model calculates the force-path diagrammuch faster as the numerical model. The numerical modelis more accurate in the case where the cylinder looses con-tact with the cone. The numerical model also enhances vari-ations of the geometry like tailored blanks.

4) With the models of the flare forming process the influ-ence of various process parameters to the forming behav-iour and to the force-path diagram can be calculated and op-timised.

5) Based on the models, different engineering tools suchas a computer program in TK Solver and dimensionlessgraphs, were developed.

Acknowledgements

This research project has been carried out in coorporationwith Fontijne Grotnes BV Vlaardingen, The Netherlands.The authors also would like to express their thanks toRUAG Components Research and Development Altdorf,Switzerland, for providing the finite element simulationsoftware and computers and for their participation.

(A2003027; received on December 20,2002)

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