2007 - Fourier Series Based Finite Element Analysis of Tube Hydro Forming - Generalized Plane Strain Model

8/9/2019 2007 - Fourier Series Based Finite Element Analysis of Tube Hydro Forming - Generalized Plane Strain Model

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Accepted Manuscript

Title: Fourier series based finite element analysis of tubehydroforming - generalized plane strain model

Authors: Yabo Guan, Farhang Pourboghrat

PII: S0924-0136(07)00634-6

DOI: doi:10.1016/j.jmatprotec.2007.06.044

Reference: PROTEC 11049

To appear in: Journal of Materials Processing Technology

Received date: 21-11-2006Revised date: 3-6-2007

Accepted date: 15-6-2007

Please cite this article as: Y. Guan, F. Pourboghrat, Fourier series based finite element

analysis of tube hydroforming - generalized plane strain model, Journal of Materials

Processing Technology (2007), doi:10.1016/j.jmatprotec.2007.06.044

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http://dx.doi.org/doi:10.1016/j.jmatprotec.2007.06.044http://dx.doi.org/10.1016/j.jmatprotec.2007.06.044http://dx.doi.org/10.1016/j.jmatprotec.2007.06.044http://dx.doi.org/doi:10.1016/j.jmatprotec.2007.06.044


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FOURIER SERIES BASED FINITE ELEMENT ANALYSIS OF TUBE

HYDROFORMING -

GENERALIZED PLANE STRAIN MODEL

Yabo Guan1, Farhang Pourboghrat

2

1Department of Neurosurgery, Medical College of Wisconsin,

Milwaukee, WI, 53226, USA

2Department of Mechanical Engineering, Michigan State University,

East Lansing, MI, 48824-1226, USA

Corresponding author:

Yabo GuanDepartment of Neurosurgery

Medical College of Wisconsin

9200 West Wisconsin AvenueMilwaukee, WI 53226, USA

Tel: 414-384-2000-ext 41387Fax: 414-483-4393E-mail: [email protected]

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Abstract In previous paper (Fourier series based finite element analysis of tube hydroforming --

an axisymmetric model. Engineering Computations. 23(7): 697-728, 2006), an axisymmetric

analysis of tube hydroforming process was discussed. In the present paper, a generalized plane

strain implicit formulation of the cross sectional expansion of an extruded aluminum tube with

pressurized fluid to fill a hydroforming die is presented. The cross-section of the tube is modeled

with thin straight and circular segments with constant thickness, and Fourier series are used to

approximate nodal displacements. The material of the tube is assumed to obey a rate-independent,

elastoplastic model that takes into account work hardening and normal anisotropy. At the tube-die

interface, frictional stress is assumed, based on Coulomb friction, to be proportional to the contact

pressure whenever relative sliding occurs. The kinematics relationships are derived based on thin

shell theory, and the equilibrium equation is derived based on virtual work principle. The axial feedis implemented by imposing either a compressive force or strain in the tube length direction.

Frictional boundary condition is introduced into the formulation in the form of a penalty function,

which imposes the constraints directly into the tangent stiffness matrix. The Newton-Raphson

iterative method is used to incrementally solve the resulting nonlinear equations. Two examples of

tube hydroforming problems are solved and numerical predictions of the deformed shape,

hydroforming pressure, and deformation strains are compared with experimental and ABAQUS

results.

Keywords: Aluminum; Tube Hydroforming; Finite Element; Contact Analysis; Fourier series;

Plane Strain

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1. Introduction

Tube hydroforming is receiving the greatest attention, especially in the auto industry, because existing multi-piece, stamped/welded assemblies in auto body and frame

structures could be potentially replaced with less expensive hydroformed parts that are

lighter, stronger and more precise. Well known hydroformed automotive applicationsinclude exhaust manifolds, radiator enclosure, dash assemblies, frame rail, and engine

cradles etc. [1, 2].

Prior to tube hydroforming, pre-bending and stretching operations take place to shape

the blank tube to fit into the hydroforming die. The analysis of this pre-forming isnecessary in order to accurately predict the formability of the tube during the

hydroforming process. Wu and Yu [3] simulated the multi-operation tube hydroforming

of an automotive structural part with explicit LS-Dyna3D commercial code. Using the

explicit finite element code LS-Dyna3D, Srinivasan et al. [4] provided additionalcorrelation of experimental and simulation results for tube hydroforming, and Liu et al.

[5] provided analytical and experimental examination of tube hydroforming limits. Kaya

et al. [6] performed plane strain analysis of crushing and expansions of tube cross-sections using the two-dimensional implicit finite element code DEFORM 2D. Kim et al.

[7] developed a rigid-plastic finite element method for the analysis of tube hydroforming

process. Hwang and Altan [8, 9] evaluated the quality of the tubes formed byhydroforming and crushing in a square die and rectangular die respectively. A two

dimensional model for the bend-stretch-pressure forming process was developed by

Corona [10]. Other numerical analyses of tube hydroforming performed recently can befound in Refs. [11-28].

Tube hydroforming is the process whereby a closed-section hollow part with varying

cross sections is formed by applying internal fluid pressure and axial compressive loads

to force a tubular metal blank to conform into the shape of a given die cavity. Althoughfinite element method has been used widely for simulating the process, unfortunately it

becomes very costly when three dimensional model of working piece is created and usedfor analysis directly. Therefore, cost-efficient two dimensional finite element method

capable of simulating various cross section shapes is desirable. In our previous study

[29], an axisymmetric tube hydroforming finite element analysis program was developed.

Fourier series interpolation functions, which reduce the size of the global stiffness matrixand the number of variables considerably, were employed for approximating the

displacements. Some simplifications were assumed. The principal geometrical

assumption is that the representative meridian of the tube is initially straight. Thisassumption however could be relaxed by using a curved, instead of a straight segment to

represent the initial geometry of the tube. All segments making up the meridian areassumed to be relatively thin and of constant thickness. The deformation of the tube isassumed not to vary along its cross-section, hence, the analysis could be considered to be

axisymmetric. The axisymmetric hydroforming program (AXHD) written based on this

formulation is very efficient in predicting the deformations for the free-forming stage oftube hydroforming under simultaneous action of internal pressure and displacement

stroke. Failure model (FLC) based on shear instability was also incorporated into the

code to predict the onset of fracture for the steel tube. The hoop and axial strains

anuscript

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predicted with AXHD code compared excellently with those from ABAQUS using plane

stress axisymmetric (SAX1) and four-node shell (S4R) elements.

In the present study, the principal geometrical assumptions of the model are that thecross-section of the round extruded tube can be modeled with circular segments, and all

segments are relatively thin and of constant thickness. The axial feed is implemented by

imposing either a compressive force or strain in the tube length direction. The material ofthe tube is assumed to be elastoplastic and to obey a plasticity model that takes into

account rate-independent, work hardening and normal anisotropy. The boundary friction

condition is introduced into the formulation in the form of a penalty function, which

imposes the constraint directly into the tangent stiffness matrix. The Newton-Raphsonalgorithm is used to solve the nonlinear equations. The PSHD (Plane Strain

Hydroforming) program has been written based on the above formulations.

The layout of present paper is as following: In Section 2 the thin shell model is

described. The kinematics assumptions and principal strain formulations are discussed inSection 3. The constitutive model and contact algorithm are described in Sections 4 and 5

respectively. Section 6 describes the equilibrium equation formulations based on the

virtual work principle (VWP) and the application of Newton-Raphson iterative method tosolve the resulting non-linear equations. Finally, in Section 7 two examples are provided

in support of the section-analysis finite element model, where numerical predictions of

the deformed shape, hydroforming pressure, and deformation strains are compared withexperimental measurements and the nonlinear finite element code ABAQUS.

2. Thin Shell Theory

The deformation of the mid-surface of an element will be considered based on thin

shell theory. Figure 1 shows the shell mid-surface at the reference time ot and currenttime tas it bends and stretches.

Pourboghrat et al. [30] derived the principal curvatures and stretches of a shell ofrevolution undergoing axisymmetric deformation using both total and updated

Lagrangian formulations. Below, the principal curvature and stretch of a thin shell

undergoing generalized plane-strain deformation will be derived. The generalized planestrain assumption implies that strain in the third principal direction (in this case the tube

length direction) could be specified as a constant value. When this constant value is

chosen to be zero, conventional plane strain assumption will result.

2.1 Principal Curvature and Stretch (updated Lagrangian)

After bending and stretching, the principal mid-surface curvature, k1, of a shell

element at the current configuration, t (=ot+t), could be calculated from the known

information about the element at the reference configuration (timeot, see Figure 1), as

follows:

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~~~~

NwAuRr ++= (1)

where~

R is the reference configuration at timeot and u and w are incremental

displacements defined in Figure 1. In Eq. (1), the unit tangent vector ~A and the unit

principal normal vector~

N to the mid-surface of the reference configuration are defined

as

SR

S

RA

,~

~

~

=

= (2)

1

,~

~

K

AN

S= or,1

,~

~

K

NA S= (3)

whereK1 is the centerline curvature at the reference configuration. To calculate centerline

curvature at the current configuration, k1,the unit tangent vector

=

~~~ aaa and the unit

principal normal vector of the mid-surface of the shell

=

~~~ nnn must be known. Using

Eq. (1), the tangent vector~a could be calculated as

SS

SS

sSNwNwAuAuRr

S

ra

,~~,

,~~,

,~,~

~

~ ++++==

= (4)

By substituting from Eqs. (2) and (3) into (4), and after re-arranging, the following

expression results:

~~~1,

~1,

,~

~

~

)()1( NdAcNuKwAwKurS

ra SS

S+=+++==

= (5)

The principal incremental stretch of the mid-surface in the radial direction calculated

from the magnitude of the base vector~a in Eq. (5) is

( ) ( )[ ]2

1

21

21

221 1 uKwwKudcaaa S,S,

~~~+++=+=== (6)

The current length of the mid-surface of the shell in the radial direction, ds, is calculated

from the reference length, dS, and 1 as follows:

ds = 1dS (7)

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The unit principal normal vector of the surface of the current shell,~n , is

~ ~

~1

d A c N

n

+= (8)

which, from Eqs. (5), (6) and (8), shows that 0~~

= an . The current principal curvature of

the shell, k1, could now be found as

S,~~s,~s,~s,~~nanrnak

===

2

1

1

1(9)

where~a is given by Eq. (4) and

Sn

,~ can be derived from Eq. (8):

1 , , 1 1 1,~ ~ ~~ ~ ~ ~

2~ ,1

( ) ( )

S S S

S

d n d A c N K c A K d N d A c N n

dS

+ += = (10)

In Eq. (10), S,1 is assumed to vanish within an element and the above expressionsimplifies as

( ) ( )

1

11

+++= ~

S,~

S,

S,~

NdKcAcKdn (11)

By substituting from Eqs. (5), (6) and (11) into Eq. (9), the current centerline curvature ofthe shell, k1, can be found:

3

1

2

11

1

+=

Kdccdk

S,S,(12)

3. Kinematics of the Circular Segment

Using the updated Lagrangian formulation, exact expressions for membrane strains,

normal vector rotation, and principal curvatures of plane strain shell element werederived in Section 2 (Shell Element Model). By using in these expressions the values of

displacements and curvatures of the shell at previous time increment (i.e., ot t= ), onewould recover the incrementalvalues of strains. However, by using in these expressionsthe values of displacements and curvatures of the shell at the initial time (i.e., 0t= ) one

would recover the totalvalues of strains. In this paper, the difference between the totalstrains at time t (current) and o t (previous) is used to calculate the incremental strains,

i.e., ( ) ( )oY ,t Y , t = , where Ycorresponds to the location of a material point onthe cross section of the tube.

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3.1 Kinematics Assumptions

The hydroforming of a tube, shown in Figure 2a, proceeds by calculating the

deformation of its cross section (YZ-plane), while applying a compressive force or strainin the length direction (

X-axis) to simulate the axial feed. The cross section of the tube,

Figure 2b, could be a closed or open section comprised of straight and circular segments.

Only circular segment kinematics will be discussed in the present paper for the roundextrusion tube hydroforming application. Detail of kinematics formulation for straight

segment can found in a previous paper [29].

All segments making up the cross-section of the tube are assumed to be thin and of

constant thickness. To meet the thin shell assumption, the radius of the circular segmentshould be greater than eight times the thickness. Based on thin shell theory described in

Section 2, deformation (i.e., strain, rotation) and curvature expressions will be derived

only for circular segments as functions of displacements ( wu, ), and their derivatives

( ssssss wwuu ,,,, ,,, ). Details of these derivations are given in appendix A.

3.2 Principal Strains

For the circular segment shown in Figure 3, the local coordinates of the segment are

defined by the angle and the through-thickness axis z. The initial geometry of thesegment is specified by: 1) the coordinates of the center of the arc, cY and cZ , 2) the

angle 0 , which locates the line where 0= , 3) the centerline radius R, 4) the anglespanned by the arc , and 5) the thickness of the segment t. As in the straight segment,

the mid-surface is atz=0 and is indicated by the dashed line.

The rotation of a through-thickness line, , is derived in appendix A (Eq. A7) to be:

R

wu

,= (13)

The true axial strain at any point along the tube cross section was given by

+= Yzxx0 (14)

where and are defined as

)cos(sinzcosusin)wR(Zc +++++= (15a)

)sin(cossincos)( +++= zuwRYc (15b)

where += 0 . The engineering tangential strain is given by

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zee +=0 (16)

where membrane portion of the strain, 0e , and the local curvature are derived in

appendix A (Eqs. A3 and A12) to be:

2

,

2

,,0

21

21

++

+

+=

Ruw

Rwu

Rwue (17)

and

2,

,

)(1R

wuR

R

wu

= (18)

The true strain is then calculated from )1( en += l .

3.2 Constraints

Since the cross-section is made up of several independent segments, with their ownlocal coordinate systems and variables, it is necessary to enforce compatibility of

deformations at junctions of two or more segments. Two constraint equations are used to

ensure compatibility of displacements and one equation to ensure compatibility ofrotation between two segments. Therefore, at a junction where M segments come

together, 3(M-1) constraint equations need to be enforced. At a junction between two

straight segments having orientation angles 1 and 2 and components 11, wu and 22 , wu ,the two displacement compatibility conditions can be written in terms of the displacementcomponents in the Y and Z directions as follows:

011112222 =+ sinwcosusinwcosu (19)

011112222 =+ coswsinucoswsinu (20)

The rotation constraint requires that the angle between segments at a junction remainunchanged. For the current example, if the rotations of the two members at the junction

are 1 and 2 , compatibility condition is

012 = (21)

4. Constitutive Equation

The elastic-plastic, rate-independent constitutive model implemented in the

generalized plane strain tube hydroforming analysis code assumes isotropic hardeningand is based on Pourboghrat et al. [30]. The uniaxial stress-plastic strain curve of the

material is assumed to have the following power-law form:

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( )noK + (22)where is the effective stress and is the effective plastic strain. Parameters K, n and

0 are material constants that are calculated by curve fitting Eq. (22) to stress-strain datafrom a uniaxial tensile test. The elastic strain increment is related to the stress increment

through the equations of linear, isotropic elasticity with Youngs modulus E andPoissons ratio . The yield function given below allows for anisotropic yielding of thematerial:

( )22 2

20

1

x s x sR

R

+ + = =

+(23)

whereR is the normal anisotropy parameter andx ands indicate axial and hoop direction,respectively.

.

During the loading, Hooke's law is used to calculate stress below the elastic limit; i.e.,

y , where y is the initial yield stress of the tube material obtained from a uniaxialtensile test. Beyond the elastic limit; i.e., y> , the co-rotational time derivative of

stress (Jaumann stress rate) is calculated, for a given strain rate, from an elastic-plastic

constitutive equation:

( ) ( )~ ~ ~ ~

~~ ~~ ~ ~

: ::

: :

L P P LL D

h P L P

= +

(24)

Here

~

and )DD(Dp

~

e

~~+= are the Jaumann rate of stress and strain rate tensors,

respectively,~ is the stress tensor, )(h = is the plastic hardening parameter, ~L is

the fourth order elastic tensor and

=

~~~P , where

~~~: = , is the second order tensor representing the unit normal to

the flow potential surface. The effective plastic strain rate, associated with Eq. (24), is

calculated from the following expression:

~ ~ ~

~ ~ ~

: :

: :

P L D

h P L P =

+

.(25)

The fourth order elastic tensor )L(L ijkl~

= used in this work is the standard tensor for the

isotropic elasticity, which has only two independent components.

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5. Contact Algorithm

The tube hydroforming simulation requires modeling the frictional contact betweenthe tube and the die. The contact analysis is complex because it requires accurate

tracking of the motion of multiple bodies, and the motion due to the interaction of these

bodies after the contact. The numerical objectives are to detect the motion of the bodies,apply a constraint to avoid penetration, and finally apply appropriate boundary

conditions to simulate the frictional contact behavior. Each of these objectives will be

separately described next.

5.1 Contact Detection

Many contact search algorithms including methods for global search and local search

using sheet (mesh) normal or tool normal were proposed in the sheet metal forming

simulation literature [31-39]. In the present paper, a global search method using tubesurface normal was employed.

To detect contact between the tube and the die, evenly spaced contact nodes are

initially defined along the tube cross section (e.g., at os , 1 os s s+ , etc.). During thecontact analysis, the displacement of each contact node is checked for surface

penetration, by determining whether it has crossed into the die. For this purpose, the

calculation of the tube surface normal is required, since it is used to determine which

segment on the die is closest to a potential contact node on the tube cross section. For

example, as shown in Figure 4, the closest segment on the die (i.e., ii BB 1 or 1+iiBB ) to

the contact node ( kA ) on the tube can be determined using the following cross-product

algorithm:

If: k i k i 1 (A B ) (A B ) 0n nuuuuur uuuuuuur

+ < (26)then, 1+iiBB will be the die segment associated with the contact node kA .

5.2 Projection Algorithm

A nodal position produced by the trial solution may penetrate the die. By using thecross-product algorithm, the closest segment on the die corresponding to the contact node

can be found. The nodal coordinates are then modified by a projection scheme such that

the contact node just touches the die surface. There are two ways to bring the penetratedcontact node back to the die surface. As shown in Figure 5, PQ is assumed to be the die

segment associated with the penetrated contact node A, point B is the intersection point

between the normal vector and PQ, and O is the original location of the contact node.Based on the following vector equation, the coordinate of point B could be calculated:

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ABOAPBOP +=+ (27a)

PQ

PQPB 1= , 2 AB nuuur (27b)

where 21, are scalar parameters. Once 21, are solved for from Eqs. (27a) and (27b),

the coordinate of point B could be determined.

5.3 Implementation of Contact Constraints

A contact node projected on the die surface at time tt + , is constrained to move inthe tangent direction defined by the trial solution, *u

%. The constraint on the

displacement vector ( , )u u w%

, for contacting nodes is then:

0u n =%

(28)

5.4 Separation of a Node in Contact

After a node on the tube comes into contact with the die surface, it is possible for it toseparate from the die surface in a subsequent iteration or deformation increment.

Mathematically, a node should be separated from the die when the calculated reaction

force at the node becomes tensile, as it would imply that the node on the tube is beingpulled by a tensile force to keep it in contact with the die. In contrast, when the reaction

force on the node is compressive, the node continues to stay in contact with the die.

When contact occurs, a reaction force associated with the contact node balances the

internal stress of the element sharing this node. When separation occurs, this reactionforce behaves as a residual force (as the force on a free node should be zero). This

requires that the internal stresses in the deformable body be redistributed.

6. Equilibrium Equation

The equilibrium equation is satisfied using the virtual work principle (VWP). Incontrast to traditional finite element method, nodal displacements in this formulation are

approximated with Fourier series, which makes the implementation of contact constraints

and boundary conditions more challenging. The boundary friction condition is introduced

into the formulation in the form of penalty functions, which imposes those constraintsdirectly into the tangent stiffness matrix. The Newton-Raphson algorithm is then used to

solve the nonlinear equilibrium equations. In the following sections the VWP will be

discussed for bending, pressure loading and frictional contact modeling.

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6.1 Tube Bending Problem

The principle of virtual work for tube bending could be expressed as following:

( ) 011

xj

J

j

j

I

iA

ii

s

i

s

i

x

i

x TCdAi ii =++ ==

(29)

where the integral on the left hand side represents the virtual internal work ( IntW ), the

term on the right hand side represents the virtual external work ( 0Ext xW T ) due to

axial feed, indexIcorresponds to the total number of segments defining the cross-section,

index J corresponds to the total number of constraint equations, j are Lagrange

multipliers, jC are the constraint equations (from Eqs. (19-21)), Tis the applied tension

(compression) due to axial feed, iii

dzdsdA = is the area of a straight segment, andii

ii dzdRdA = is the area of a circular segment,x ands indicate axial and hoop direction,respectively.

To solve Eq. (29), the displacement components ,i iw u for each circular segment are

approximated using the following Fourier series expressions:

0

1 1

cos sin

i iN Ni i i ii i

n ni in n

n nw

= == + +

(30a)

0

1 1

cos sini iN N

i i i ii in ni i

n n

n nu

= == + +

(30b)

After substituting from Eqs. (30a-b) into the principle of virtual work, Eq. (29), anonlinear expression of the following form will result:

0,,,c~

=dTdd ZY f (31a)

where

{ }jxininiinini ,,,,,,,c 000~

= (31b)

Equation (31a) should be incrementally solved for the unknown vector, c%, given input

values for incremental bending curvatures, Y Zd ,d , and incremental axial force, dT,

applied to the ends of the tube (along X-direction) for the purpose of axial feeding. Since

Eq. (31a) is highly nonlinear, it is numerically solved using the Newton-Raphson method.

The final form of the Newton-Raphson iterative method used for solving c%

looks as

following:

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~~~~IntExt FFRcdK ==

(32)

where

K ( c cIntW2=

% %) is the second variation of the virtual internal work with

respect to c%

,~cd is the incremental c

%, and

~R is the residual.

~ExtF ( cExtW=

%) is the

variation of the virtual external work with respect to c%

, and~IntF ( cIntW=

%) is the

variation of the virtual internal work with respect to c%

. The nodal force~

F (

~

~

~

c

uFInt

= )

can be calculated from~IntF and

~

~

c

u.

6.2 Pressure Loading Model

Pressure loading is modeled as an external force to expand the tube. The virtual

external work done by a pressurep applied to the inside of the tube is equal to

~ ~1 1

~

( ) ( )

i ip p

iI IP i i i i i

Ext p p

i iA A

uW p n u dA p n c dA

c

%= =

= = (33)where )(

~

iu is the incremental virtual displacement vector having two components

( )i

w and ( )iu , and in is the unit outward normal to the tube surface ipA .

The principle of virtual work for bending and pressure loading of a tube then becomes

( )I1 1 1

( )i i

i ip

J Ii i i i i o i i

x x s s j j x i p

i j iA A

dA C T p n u dA = = =

+ + = + %

(34)

The variation of the virtual external work due to internal pressure loading is:

( )1

~

i

iI

P i i

Ext p

i A

uF p n dA

c

= %

%(35)

Due to the follower forces effect [40], the load stiffness matrix is

2

21

~ ~ ~

( ) ( ) ( )

ip

i i iIP i i

pExt

i A

n u uK p n dA

c c c

= + % %

1~ ~

( ) ( )

ip

i iIi

p

i A

n up dA

c c

= % (36)

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Equations (35) and (36) will appear on the right- and left-hand sides of the Newton-

Raphson expression, Eq. (32), as follows

~~ ~ ~ ~

p p

Ext Ext Ext Int K K d c R F F F = = + (37)

Once the above formulation was implemented into the numerical analysis code, it wasfound that the load stiffness matrix

p

ExtK has no or little effect on the solution, which

drove us to further simplify the formulation. To that end, by using the following

approximation,~

( ) ( )i i in u w , the whole formulation was greatly simplified. The

details of the proof could be found in appendix B. Since 2 2~

( ) 0iw c = , there is alsono contribution to the stiffness matrix as a result of

p

ExtK , appearing on the left hand side

of the Newton-Raphson expression, Eq. (37). When both methods were implemented into

the numerical code, the results turned out to be almost identical.

6.3 Frictional Contact Modeling

The most challenging task when developing a numerical code for metal forming

processes is to model frictional contact. To model the tooling-workpiece frictional

contact correctly, the following two conditions were continuously monitored during theequilibrium iteration:

(1)Penetration of the contact nodes into the die, and(2)

Nodal contact forces becoming tensile at the contact boundary (separation).

Once the penetration of the contact nodes into the die has been detected, the

penetrated nodes were returned to the die surface and constrained to stay on the diesurface for the remainder of the equilibrium iterations. The nodes, which were returned to

the die surface, were constrained to move only tangent to the die surface and only

condition 2 stated above could cause the contacting node to be separated from the die

surface. Figure 6 shows the schematic of a typical contact check during the Newton-Raphson equilibrium iteration. The external work done by the frictional contact is added

to the virtual work principle Eq. (34) as following:

( )i

I0

i

1 1 1 i 1A

( ) ( )i i i

I J I i i i i i i i i i

x x s s j j x i pA

i j i

dA C T Pn u dA u dA

= = = =+ + = + +

% %

(38)

where i is the traction on the surface of the tube and ( )i

u%

is the virtual incremental

displacement of the contacting nodes.

In order to improve convergence, a special algorithm was introduced. For each trial

set of contacting and non-contacting nodes, equilibrium iteration was performed. Afterequilibrium was satisfied, the nodes were reexamined for non-penetration condition.

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Releasing or projecting certain nodes then updated the contact set and another

equilibrium iteration was initiated.

During contact iteration, the trial displacements were first updated according to theNewton-Raphson procedure and the non-penetration contact condition was then applied

to these trial values by projecting the contact nodes to the die surface along the normal

vector. The modified trial solutions were then used for Newton-Raphson iteration. Withinthis force equilibrium iteration, the internal force was calculated. The signs of the sheet

normal force at contact nodes were checked so that the nodes having non-compressive

(tensile) force were released.

7. Numerical Results

The PSHD (Plane Strain Hydroforming) program has been written based on the above

formulations and examples of generalized plane strain tube hydroforming with analuminum tube were solved and verified with ABAQUS finite element code, and when

available, compared with experimental results. The results of these simulations will be

presented next.

7.1 Hydroforming of a Round Aluminum Tube into a Square Die

The following example was chosen in order to illustrate the capability of the new

formulation to model the hydroforming of a round aluminum tube into a square die, and

to also study:

(a)Effect of die-tube friction coefficient on predicted strains,(b)Sensitivity of the contact solution to the number of segments used, and(c)Effect of pre-bending on tube hydroforming.

For case (a), the predicted shape of the hydroformed tube for an applied internal pressure,

and deformation strains were verified by direct comparison with experimental data. Forcase (b), the shape of the hydroformed tube was verified against ABAQUS simulation

results. Verification of case (c) was not possible since no experimental data was available

and we were unable to get converged solution with ABAQUS.

7.1.1 Tube Hydroforming Experiment

An 8.0 (203 mm) long aluminum 6061-T4 tube with an outer radius of 1.0 (25.4mm) and a thickness of 0.049 (1.24 mm) was hydroformed into a square die with a side

length of 2.0 (50.8 mm) using a maximum internal pressure of 3040 psi (21 Mpa). Themechanical properties of the aluminum 6061-T4 tube obtained from a uniaxial tensile test

are shown in Table 1.

Figure 7a shows the hydroformed tube and its cross section. The hoop strain

distribution of the hydroformed tube at the maximum pressure of 3040 psi was

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experimentally measured using circular grids etched on the straight tube. Table 2 shows

measured hoop strains as a function of the angular position along the cross section of thehydroformed tube, as shown in Figure 7b. It can be seen, from Table 2, that the maximum

hoop strain is measured to be between 7-11%, which occurs in the 0-10 degree zone.

Near the boundary between the curved and flat portion of the tube (i.e., 10-20 degree

zone), hoop strains are about 7%. The tube was axially compressed a total of 0.1181 (3mm) from each end during the hydroforming process, which resulted in a measured axial

strain of -3%.

7.1.2 Tube Hydroforming Simulation with PSHD Code

The tube hydroforming process was simulated using the new formulation to illustrate

its capability to model frictional contact. Due to symmetry, only one quarter of the tube

was modeled, as shown in Figure 7b, to reduce computational time. The tube was

modeled using 8 circular segments, and for each segment 6 Fourier series terms wereused (i.e., N=6). The best number of terms to use in the simulation was determined based

on whether or not a converged solution was obtained. It was found that N=4 to 6 (total of

9 to 13 coefficients) would result in numerically stable solutions. Divergence of solutionmore commonly occurred when larger values of N were used. Sometimes using larger

values of N caused numerical instability during the contact search algorithm. To obtain

accurate contact solution, 20 evenly distributed contact nodes were used for each segment.The number of Gauss integration points used along the length of the segment was 12,

while that used through the thickness was 3.

The tube hydroforming simulation was carried out incrementally. That is, the total

pressure was divided into several hundred steps, and at each step 10 psi of pressure was

applied to the tube until the total pressure of 3040 psi was reached. The size of theincremental pressure loading was decided based on whether or not a converged solutionwas obtained. It was found that at early stages of the deformation it is better to take very

small pressure loading increments, but larger increments was taken once the tube became

fully plastic. Figure 7b shows the predicted deformed shape of the tube at 2000 psi and3040 psi. The friction coefficient used in this simulation was 0.1. It can be seen that at

3040 psi, the predicted cross section of the tube slightly underestimates the actual shape

of the tube.

The convergence difficulties occurring in this example using the implicit codeindicated the sensitivity of the mesh normal direction method. It is difficult to define the

tube normal accurately at each node during each increment while the accurate die normal

can be determined from the die geometry. Although it was shown that mesh normalcontact search algorithm was cost efficient and robust in the dynamic explicit FE code for

sheet forming simulation [39], it was recommended to use tool normal instead in order to

obtain better simulation accuracy [41].

7.1.3 Effect of Die Friction Coefficient

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The predicted hoop strain variation as a function of three different friction

coefficients are shown in Table 2, and compared with experimental data. In this work theactual friction coefficient was not measured. Instead, the sensitivity of predicted hoop

strain as a function of friction coefficient, as it increased from 0.0 to 0.3, was studied with

the PSHD program. It was found that by increasing the friction coefficient the strain

magnitude decreases only where the tube contacts the die, and there is little or no effecton the hoop strain where the tube is freely expanding. At 0= the maximum hoopstrain occurs in the 0-10 degree zone and the magnitude of the strain matches the

experimental one.

7.1.4 Sensitivity of the Contact Solution to the Number of Segments Used

A fictitious tube hydroforming problem was devised to numerically study thesensitivity of the contact solution to the number of segments used to model the tube. In

this problem, a larger square die with a side length of 3.0 (76.2 mm) was used so that the

tube with a radius of 1.375 (35 mm) and a thickness of 0.1378 (3.5 mm) had to firstexpand before making contact with the die. Since no experimental data was available, for

verification purposes this tube hydroforming example was also simulated with the

commercial code ABAQUS. In this numerical example, the aluminum tube (Table 1) washydroformed with a maximum pressure of 7000 psi. Beyond this pressure, the code could

not converge to a solution due to tensile instability. The goal of this exercise was to

determine the minimum number of circular elements required in order to predict a

comparable deformed tube shape as ABAQUS.

Due to the symmetry of the die, only one half of the tube was modeled for the

hydroforming simulation, as shown in Figure 8a. The ABAQUS model used 26 8-node

shell elements with reduced integration (S8R5) for a more efficient simulation. In all

ABAQUS simulations the number of elements to be used was decided by trial and error.That is, tube hydroforming simulation was performed with a few elements and the

number of elements was then increased until no additional improvement (changes) in the

simulation results occurred. Figure 8a shows the predicted shape of the hydroformedtube with 1 and 4 circular segments, using the current formulation. In the simulation, 4

Fourier series terms were used to approximate displacements (i.e., N=4), 8 Gaussintegration points were used along the length of the element, and 3 integration points

were used through the thickness of the tube. A friction coefficient of 0.1 was also used

for the simulation. Figure 8a also shows the predicted shape of the hydroformed tube byABAQUS. When using only 1 segment, although very CPU-efficient, the PSHD model

was stiff and had difficulty capturing the expansion and the true size of the contact area

between the tube and the die. However, by using 4 segments the model, similar toABAQUS, was able to capture the deformation of the tube. Using more than 4 segments

only slightly changed the result, however, the CPU time increased dramatically.

Figure 8b shows predicted hoop strain distribution as a function of the internal

pressure by the new formulation. It could be seen that predicted strains are constantthroughout the tube as long as no contact occurs between the tube and the die. However,

as soon as a finite size contact region develops, e.g., at 7000 psi, strain distribution

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changes from being constant. In fact, strains are largest around the contact region, e.g.,

1.5Z= .7.1.5 Effect of Pre-Bending on Tube Hydroforming

As mentioned in the Introduction section, it is common in tube hydroforming industry

to pre-bend the tube before hydroforming it. To study the effect of pre-bending, the

previous tube hydroforming example (7.1.4) was used with 4 segments, but this time the

tube was first bent to a maximum curvature of 0.04/y in = before hydroforming it

with a maximum pressure of 7000 psi. Similar to previous examples, this combined

bending/hydroforming problem was solved incrementally. That is, by applying a smallcurvature at each step, the incremental tube-bending problem was solved until the total

curvature of 0.04 /y in = was reached. Then, by applying a small pressure the

incremental tube hydroforming process was solved, until the maximum pressure of 7000

psi was reached. As in the previous example (7.1.4), once pressure exceeded 7000 psi,the code could not converge to a solution due to tensile instability.

Figure 9a shows the deformed shape of the tube after bending and hydroforming.

Compared to Figure 8a, the tube-die contact is asymmetric with respect to 0Z= . Figure9b shows the hoop strain distribution in the tube. Initially, strain distribution corresponds

to that of a bent tube, i.e., positive strains where 0 1.5Z , and negative strainswhere 1.5 0Z < . As the tube is further pressurized, the strain distribution continuesto grow, due to the superposition of a positive hoop strain. A comparison between

Figures 8b and 9b at the maximum pressure of 7000 psi shows that the maximum strain is

larger when the tube is bent first and then hydroformed, i.e., 26% vs. 22%. However, the

minimum strain is lower in the bent/hydroformed tube, i.e., 12% vs. 16%.

8. Conclusions

A generalized plane strain assumption allows the user to specify a compressive load

or axial strain to each end of the tube to simulate axial feeding. Based on this formulation,a tube hydroforming code (PSHD) was written and several examples of this process were

investigated. Only a few circular segments and 4-6 Fourier series terms to approximate

displacement were required to model the cross section of the tube and accurately predictthe final deformed shape and strain distribution of the hydroformed tube. Numerical

codes such as the one described in this paper are useful engineering tools for quick and

efficient simulation of tube stretching, tube bending and tube hydroforming at earlystages of the process design.

Acknowledgements

The authors wish to thank the National Science Foundation for the partial support of

this project through the grant DMI 0084992 in conjunction with Alcoa through the

GOALI program.

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Appendix A

According to the total Lagrangian formulation discussed in Section 2 (Shell Element

Model), we derived the membrane strain, rotation of normal vector and the current

principal centerline curvature used in Section 3 (Kinematics Assumptions).

1. Membrane strain

Forcircular arc segment, the undeformed shell is non-flat and has an initial curvature K1,

then rearranging Eq.(6) and approximating the square root , we obtain

]2222[2

11 ,1

22

1

2

,,11

22

1,

2

,1 SSSSS uwKuKwwuKwKwKuu ++++++ ,

then membrane strain is

)()()( ,,,,, wwuuwKuwuKwue SSSSS +++++ 1222

1

22

2

1

2

1(A1)

Consider the circular arc segment shown in Figure 3. Since dSRd = , then

=

=

)()(1)(

1KRS

, Eq. (A1) becomes:

)()(2

1)(

2

1,1,11,1

222

1

2

,

2

,

2

1 wwuKuwKKuKwuKwuKe +++++

)()](2)()[(2

1 ,1,,

222

,

2

,

2

1 wuKwuuwwuwuK +++++=

22

2

1

2

1

+

++

=

R

uw

R

wu

R

wu ,,,(A2)

As shown in Figure 3, if the positive w direction is in the opposite direction of the normal

to the mid-surface vector,~n , then Eq. (A2) becomes:

2

,

2

,,

2

1

2

1

++

+

+=

R

wu

R

wu

R

wue

(A3)

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18

which is used in Section 2 as Eq.(17) and Eq.(4.3) in Brush and Almroth [42].

Based on the following assumptions, we could derive the normal vector rotation and

current principal centerline curvature for both straight segment and circular arc segment.

Assumption 1: The rotation angle is small,

Assumption 2: Bending deformation is dominant and stretching is negligible, namely,

0,0 ,, SSs uu or 11 =dS

ds .

2. Rotation of normal vector

We assumed the angle between the current normal vector~n and the S(arc length) is

, the angle between the current normal vector~n and normal vector

~

N at the reference

is .

Then, we have 2

=+

, and -sincos ==~~

An (A4)

Substituting Eq. (8) in Section 2 into Eq. (A4), we obtain:

, 1 , 1

2 21 1 , 1 , 1

sin(1 ) ( )

S S

S S

w K u w K ud

u K w w K u

+ += = =

+ + +(A5)

Forcircular arc segment( )01 K , according to assumptions 1 and 2, Eq. (A5) becomes:

SwuK ,1 += Since RddS= andR

K1

1 = , we obtain

R

wu ,

+= (A6)

For the coordinates shown in Figure 3, we have

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R

wu ,

= (A7)

which is used in Section 2 as Eq. (13) and Eq. (4.4) in Brush and Almroth [42].

It should be noted that the rotation of a through-thickness line, , was considered andcontributed to the hoop strain, however, there is no transverse strain considered in the

formulation.

3. The current principal centerline curvature

Forcircular arc segment, using Eq. (A5) and finite rotation assumption, we have

R

wu ,sin = (A8)

According to Flugge [43], we have the change in curvature:

R

=

&(A9)

By differentiating Eq. (A8), we obtain:

cos

,

,R

wu

d

d ===& (A10)

where

2,2 )(1sin1cosR

wu

== (A11)

Substituting from Eqs. (A10), (A11) into Eq. (A9), we have

2,

,

)(1R

wuR

R

wu

= (A12)

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Which is used in Section 3 as Eq. 18.

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21

Appendix B

Here we will prove the equivalence of the two formulations in the pressure modeling, i.e.:

( ) ( )1 1

~ ~

i i

i iI I

P i i i

Ext p pi iA A

u w

F p n dA p dAc c

== %% (B1)In Eq. (B1), the displacement vector iu

%has two components, iu and iw . For a point

at the mid-surface, i.e., z=0, of a circular segment, Eqs. (15a-b) will be:

( )sin cosi i icZ R w u = + + + (B2.a)

( )cos sini i i

cY R w u = + + (B2.b)

where += 0 .

The unit outward normal to the surface ipA ,in , is then defined as

),(

),(

=

ii

ii

in (B3)

where

sin ( )cos cos sini i i

i iw uR w u

= + + (B4.a)

cos ( )sin sin cosi i i

i iw uR w u

+ (B4.b)1/ 2

2 2

( , ) ( ) [( ) ]

i i i ii i iw u

u R w R w

= + + + + (B4.c)

We also have,

~ ~ ~

( ) ( ) ( ),

i i iu

c c c

% (B5)

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where

sin cosi i iw u + (B6.a)

cos sini i iw u (B6.b)Combining (B3), (B4.a-c), (B5) and (B6.a-b), we have

~ ~ ~ ~ ~

( ) 1 ( ) ( ) ( ) ( ) ( )

i i i i i i ii i i

i

u w u u w w un R w u

c R w c c c c

+ + + %

~

( )

iw

c

(B7)

Thus, we have proved that (B1) is correct. Since 0)(

~

2

2

=

c

wi, there will be no change in

the stiffness matrix and pressure will only show up on the right hand side of the Newton-

Raphson expression, Eq. (37).

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Figure 1. Shell mid-surface at reference timeot and current time t.

u

w

uw

e1

e2

Z

X

z

z

time t

R~ r

~

AN

a

n ds

dS time to

ure

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script

(a)

(b)

Figure 2. (a) The assumed global coordinate system for the tube, (b) The rectangular

cross section of the tube is defined with 4 straight and 4 circular arc segments.

Figure 3. Geometry of a circular segment.

Arc

Segment

Straight

SegmentYBending

Surface

Y

Tube

Zc

Yc

Z

Y

X

z

R

t

u

w

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Figure 4. The associated die segment with contact node.

Figure 5. The projection Algorithm.

Tube Segment

Die Segment

Ak

n

Bi-1Bi

Bi+1

O

P

B

Q

A

n

1

2

Die Segment

Tube Segment

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(a) (b) (c) (d)

Figure 6. A schematic of tool-workpiece contact check; (a) shows the tube and the

tooling, (b) shows initial penetration of some of the contact nodes as internal pressure

increases from P0 to P1, (c) shows how those nodes are returned to the tooling surface,and finally (d) shows how the equilibrium shape is reached after several iterations.

P0 P1 P1 P1

Tooling SurfaceTube

Node on

The Tube

Symmetry Line

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(a)

(b)

Figure 7. A 6111-T4 aluminum alloy tube was hydroformed with a maximum pressure of3040 psi into a square die, (a) deformed tube and its cross section, and (b) predicted

intermediate tube shapes compared with the actual tube cross section.

Actual

cross section

Actual

cross section

Actual

cross section

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(a)

(b)Figure 8. A tube expanded into a symmetric square die,

(a) deformed tube shape, (b) strain distributions.

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(a)

(b)Figure 9. A tube bent and then expanded into a symmetric square die,

(a) deformed tube shape, (b) strain distributions.

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Table 1. Material properties of aluminum 6061-T4 tube

Material

Type

YoungsModulus

(psi)

Poisson

Ratio

YieldStress

(psi)

R-valueK-value

(psi)N-value 0

Aluminum6061-T4

1.03E+7 0.33 18,730 0.82 69,183 0.2646 0.0

Table 2. Predicted and measured strains at the maximum pressure of 3040 psi

Hoop StrainsAxial

Strains

FrictionCoefficient

0~10degree

10~20degree

20~30degree

30~45degree

=0.0 6.5%~12.1% 6.7%~6.8% 6.9%~7.2% 6.7%~7.0% -3.25%

=0.1 6.4%~6.8% 6.8%~7.1% 6.8%~7.1% 6.8%~7.3% -3.34%=0.3 6.3%~6.7% 6.7%~7.0% 6.7%~7.0% 6.7%~7.3% -3.25%

Experiment 7-11% 6-7% 6-7.5% 6-8% -3%

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Documents

2007 - Fourier Series Based Finite Element Analysis of Tube Hydro Forming - Generalized Plane Strain Model