A fast algorithm for strain prediction in tube hydroforming based on one-step inverse approach

Journal of Materials Processing Technology 211 (2011) 1898–1906

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Journal of Materials Processing Technology journal homepage: www.elsevier.com/locate/jmatprotec

A fast algorithm for strain prediction in tube hydroforming based on one-step inverse approach M.S. Chebbah a,∗ , H. Naceur b , A. Gakwaya c a b c

University of Biskra, P.O. Box 145, Biskra 07000, Algeria University of Valenciennes, Lab. LAMIH, 59313 Valenciennes, France Laval University, 1065 Avenue de la Médecine, G1V 0A6, Québec, Canada

a r t i c l e

i n f o

Article history: Received 5 October 2010 Received in revised form 21 April 2011 Accepted 14 June 2011 Available online 23 June 2011 Keywords: Tube hydroforming Shell element Inverse approach Large elastoplastic strains

a b s t r a c t This paper presents recent developments of a simplified finite element method called the inverse approach (IA) for the estimation of large elastoplastic strains and thickness distribution in tube hydroforming. The basic formulation of the IA, proposed by Guo et al. (1990), has been modified and adapted for the modeling of three-dimensional tube hydroforming problems in which the initial geometry is a circular tube expanded by internal pressure and submitted to axial feed at the tube ends. The application of the IA is illustrated through the analyses of numerical applications concerning the hydroforming of axisymmetric bulge, made from aluminum alloy 6061-T6 tubing, the hydroforming of square section hollow component and the hydroforming of a free Tee extrusion from welded low carbon steel LCS-1008 tubing. Verifications of the obtained results have been carried out using experimental results together with the classical explicit dynamic incremental approach using ABAQUS® commercial code to show the effectiveness of our approach. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Nowadays, tube hydroforming (THF) is still being one of the most important manufacturing techniques used in sheet metal forming. The increasing application of hydroforming techniques in the automotive and the aircraft industries as well as manufacturing components for sanitary use is due to its advantages compared to classical processes as stamping or welding. Tube hydroforming is a perfect technique for manufacturing tubes of complex shapes with high level of repeatability and offers an effective integration of structural components manufactured using a minimal space. Tube hydroforming has many other advantages compared to conventional stamping processes, according to Nikhare et al. (2010), these advantages include weight reduction, strength improvement and higher geometry accuracy of the final manufactured part. Historical overviews and discussion of future trends in tube hydroforming can be found in Ahmed and Hashmi (1997), Koc¸ and Altan (2001), Singh (2003) or Hartl (2005). In THF the main parameters that may affect the general feasibility of the final product are: tube material and dimensions, tools geometry, as well as the magnitude and the loading path of the hydraulic pressure and the axial feeding. If one of these parameters

∗ Corresponding author. Tel.: +213 7 74 02 81 42. E-mail addresses: [email protected], chebbah [email protected] (M.S. Chebbah). 0924-0136/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2011.06.012

is not carefully adjusted, defects such as bursting or wrinkling may appear according to Langa et al. (2009), Di Lorenzo et al. (2010). Since the recent past, the feasibility of a product was mainly carried out by engineers using trial-and-error iterative procedures that are expensive and error prone. Over the last decade, the numerical simulation of tube hydroforming using the finite element (FE) method has received significant attention as an alternative to the trial-and-error methods; however, there is still lack of a fast and robust control of hydroforming process parameters at the early design stage; where the classical based dynamic explicit method is not convenient to use due to large amount of data needed to carry out a simulation see Hosford and Caddell (2011). Ahmetoglu et al. (2000) provided fundamental issues related to material and lubrication requirements, material shaping capabilities, tool design and process control in tube hydroforming of low carbon steel and aluminum alloy 6061-T6 tubes. It has been achieved through the establishment of a consortium between part manufacturers, and material and equipment suppliers in the Ohio State in the US. Koc¸ et al. (2000) used a design of experiments technique in conjunction with FEM to facilitate the economical prediction and optimization of the height as a function of geometrical parameters subject to thinning of the wall thickness at the protrusion region. Results in their study suggest that comprehensive and detailed investigations of tribology in hydroforming should be conducted. Mac Donald and Hashmi (2001) used LS-DYNA 3D to compare bulge with a solid medium against a hydraulic medium. In their

M.S. Chebbah et al. / Journal of Materials Processing Technology 211 (2011) 1898–1906

Nomenclature initial 0 ı virtual h thickness P intensity of pressure point on the mid-surface, in distance z p, q s curvilinear coordinate z distance of point to the mid surface displacements in local coordinate u, v, w U, V, W displacements in global coordinate up displacements of p x, dx position vector, its variation n, t normal and tangent vectors in the mid surface membrane forces, bending moments N, M Fint , Fext internal and external forces in global coordinate [Bm ], [Bf ] membrane strain, bending strain tangent matrix [Kt ] [T] transformation local-global matrix Greek symbols r, ϕ, Z cylindrical coordinate 1 , 2 , 3 principal stretches , e curvature, membrane strains ε, ε¯ strain, equivalent strain , ¯ stress, equivalent stress

investigation, they examined the effect of varying friction between the bulging medium and the tube and the history of development of the bulge and stress conditions in the formed component. They concluded that the use of a solid bulging medium allows for greater branch height, less thinning of the branch top and less stress in the formed component when compared to the hydraulic bulging process. Kim et al. (2002) introduced a backward tracing technique to predict an appropriate pre-formed configuration and determine the initial tube dimensions from the desired final shape. The developed program was applied to a hydroforming process of a box expansion in order to get the uniform wall thickness after hydroforming, and the conceptual application has been proved to be successful on its effectiveness and feasibility. Kridli et al. (2003) investigated corner filling by 2D simulations, using ABAQUS/Standard, and experiments. They examined the effects of the strain-hardening exponent, initial tube wall thickness and die corner radii on corner filling and thickness distribution. They concluded that thickness distribution is a function of die corner and strain-hardening behavior and the initial tube wall thickness affects the pressure while maintaining the same thinning. Kwan and Lin (2003) used the FE program DEFORM-3D to investigate the cold hydroforming process of a T-shape tube. They examined the influences of the process parameters such as the internal pressure, the fillet radius, and counterforce on the minimum wall thickness of formed tube. They found a suitable range of the process parameters for producing an acceptable T-shape tube that fulfills the industrial demand. Jain et al. (2004) introduced “dual hydroforming” where the counter pressure as a new process parameter to achieve favorable tri-axial stress state during deformation process. They observed that the counter pressure provides back support to the tube material and excessive thinning and premature wrinkling could be prevented and thus, larger tube expansion could be achieved. Ray and Mac Donald (2004) used a fuzzy logic control algorithm in conjunction with LS-DYNA finite element code for simulation and optimization of the forming load path to avoid the failure of the tube. They sustain that by means of a minor modifications in the strain limit setting in the load con-

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trol algorithm; their method can be used to determine the optimal and feasible forming load paths for asymmetric or axisymmetric components with relatively complex geometries. Abrantes et al. (2005), used FEA program LS-DYNA to establish a basic understanding of both free bulge and calibration against closed dies in tube hydroforming process. They observed that formation of winkles, resulting in folding and unfolding of tube regions, causes a differenced springback effect along the longitudinal axle of the tube. From the equivalence curves it was possible for them, to program the parameters in order to obtain balanced axial displacements for the punch strokes. Kashani Zadeh and Mashhadi (2006) used ABAQUS FE code to quantify the effects of coefficient of friction, strain hardening exponent, and fillet radius on the parameters, protrusion height, thickness distribution, and clamping an axial forces. Yuan et al. (2006) investigated via FEM and experiments the hydroforming of automotive rectangular section structural components. They explored the effect of loading path on the failures and thickness distribution and the reasons were analyzed for the failures, such as bursting and folding. Mohammadi and Mosavi Mashadi (2009) determined the loading paths for copper joint hydroforming via FEM and a fuzzy controller. They determined rules to increase axial feeding by introducing the calibration indicator, in order to have a product with appropriate mechanical properties. The presented method can enable automatic determination of optimal loading paths for the hydroforming of complex structural parts in a short time. Alaswad et al. (2011) conducted a finite element study along with response surface methodology for design of experiment to construct models for three responses (bulge height, thickness, and wrinkle height) for X shape bi-layered tube hydroforming. They found that, tube geometry has an important influence on the shape of the hydroformed junction. They concluded that, the usage of a larger die corner radius leads to higher bulges and smaller wrinkles. However, critical thickness reduction can be avoided for large tube diameters if a big thickness is assigned for both layers. In all of the previous cited research works dedicated for the design and the analysis of tube hydroforming process parameters, generally the classical incremental approach based on dynamic explicit formulation is used by means of commercial codes such as LS-DYNA® or ABAQUS® . Although, the classical incremental methods can provide accurate solutions for complex forming problems, the simulations using these methods are very expensive in terms of solving CPU-time and also for engineer’s time to set up and run the problem: complex die CAD meshing, material data, initial tube mesh, etc. see Numisheet (2008). Therefore, since the last twenty years, significant research has been devoted to the development of alternate approaches allowing fast solutions of the forming problems. These methods have become valuable tools in the preliminary design stage of components or structures mainly used in the automotive industry. The one-step methods are based on the general assumption of knowledge of the final geometry of the 3D part and the total deformation theory of plasticity. The unknowns are material positions of points on the initial geometry as well as strains and thickness variation. The inverse approach developed by Guo, Batoz and their coauthors since 1990 (Guo et al., 1990), is very attractive since authors showed that the IA can estimate large strains in deep drawing with a very good accuracy compared to incremental analysis or to the experiments (Batoz et al., 1998; Guo et al., 2000). More recently Naceur et al. (2006, 2008) introduced new enhancements on the IA to take into account the loading path in deep drawing simulation in order to improve the stress state obtained at the end of forming. While the IA formulation has gained a great success and attracted many research groups for the fast simulation of sheet metal forming, unluckily its development has been only limited to the deep drawing simulation. Based on our knowledge only a very

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The deformation gradient tensors at points q0 and q referenced in C can be defined by Eqs. (2) and (3). dx0q = [F 0 ]−1 dx = [xp,x − up,x ;

xp,y − up,y ;

n0 /3 ]dx

(2)

dxq = [F z ]−1 dx = [xp,x + zn,x ;

xp,y + zn,y ;

n]dx

(3)

We calculate the deformation gradient inverse tensor describing the movement between positions q0 and q referenced in C as follows: dxq0 = [F]−1 dxq

(4)

The inverse Cauchy-Green left tensor can be obtained by: [B]−1 = [F]−T [F]−1

(5)

Eigenvalues of the left Cauchy-Green tensor [B] and the assumption of incompressibility give the three principal stretches 1 , 2 , 3 and their direction transformation matrix [M]. Finally, the logarithmic strains are obtained: Fig. 1. 3D shell kinematics in tube hydroforming.



few works have been done to the extension of the IA to the modeling and simulation of tube hydroforming. In a simple application limited to the round tubes, Nguyen et al. (2003a,b) used the IA to analyze cylindrical tube hydroforming, where the final configuration is axisymmetric. They used a constant strain triangular (CST) membrane element together with a criterion based on a forming limit diagram to predict the critical sections where local necking can occur. More recently, Fu et al. (2009) proposed a study on onestep simulation for the bending process of extruded profiles. They showed that, the stretch bending process of rectangular aluminum extrusions and three-point bending of stainless steel extrusions can be simulated successfully. Chebbah et al. (2010) proposed a specific methodology based on the coupling between the IA and a Response Surface Method based on diffuse approximation using an axisymmetric membrane/bending shell element. The originality of the present work consists of the introduction of a new modified version of the inverse approach proposed initially by Guo et al., 1990, for the deep-drawing. New enhancements have been introduced to the algorithm in order to deal with hydroforming of tubes of general shapes (not only those of axisymmetric shape). Starting from a 3D final tube mesh, the algorithm is able to determine material positions of points in the initial tube. In the 3D case, the final shape of the desired tube is discretized by triangular shell elements DKT12 (Guo et al., 1990). The application of the inverse approach is illustrated through the analyses of some numerical applications. Verifications of the obtained results have been carried out using experimental results together with the classical incremental dynamic explicit approach by means of ABAQUS® commercial code, to show the effectiveness of our approach.

[ε] =

εx εxy 0

εxy εy 0

0 0 εz



 = [M]

ln 1 0 0

0 ln 2 0

0 0 ln 3

 [M]T

(6)

2.2. Constitutive equations In the present Inverse Approach the elasto-plastic deformation is assumed to be independent of the loading path. The constitutive relations remain in the framework of the deformation theory of plasticity in which Hill’s anisotropic criterion (Hill, 1984) is employed to describe the plastic flow. A planar anisotropic sheet is considered and the small elastic strains are assumed to have the same anisotropic directions as the plastic strains. The total constitutive relations can be written by:  = Es [P]−1 ε,

Es =

¯ ε¯

(7)

where , ¯ ε¯ are the equivalent yield stress and equivalent plastic strain, respectively, Es denotes the secant modulus of the uniaxial stress–strain curve. The matrix [P] is expressed in terms of the mean planar anisotropic coefficient r¯ defined by the Lankford coefficients as following:

⎡ 1

⎢ ⎢ r¯ ⎣ 1 + r¯

[P] = ⎢ −

0

r¯ =

−

⎤

r¯ 1 + r¯

0

1

0

0

2(1 + 2¯r ) 1 + r¯

1 (r0 + 2r45 + r90 ) with 4

r˛ =

⎥ ⎥ ⎥, ⎦

εP

(˛+900 ) εPz

(8)

2. Formulation of the inverse approach The 00 direction is defined as the rolling direction. 2.1. Kinematics of a large transformation In the inverse approach only the configurations of the initial tube of cylindrical form C0 and the final 3D workpiece C are considered. Using the Kirchhoff assumption, the initial and final position vectors of a material point can be expressed on C (Fig. 1): x0q = x0p + z 0 n0 = xp − up + z 0 n0 xq = xp + zn where x = (x, y, z) is the position vector at the mid-surface.

(1)

2.3. DKT12 shell formulation The known three-dimensional workpiece is discretized by flat triangular shell elements called DKT12 presented by Batoz and Dhatt (1992) of constant thickness having three corner nodes and three mid-side nodes (Fig. 2). This element is obtained by assembly of the element of membrane CST (u, v at corner nodes) with the discrete Kirchhoff triangular plate element DKT6 (w at corner nodes and s at mid-side nodes).

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Fig. 2. 3D mesh mapping onto the initial cylindrical tube surface. Fig. 3. Positions of a point p in the initial configuration during iterations.

The virtual strains of membrane are expressed in terms of virtual displacement components ıu and ıv (linear approximations) along the local coordinates x, y as follows: ıeT = ıu,x ıv,y ıu,y + ıv,x ; with uTnm = . . . ui vi . . .;



1 [Bm ] = 2A

y23 0 x32

0 x32 y23

ıe = [Bm ]ıunm

(9)

2A = (y31 x21 − x31 y21 ),

0 x31 y31

y12 0 x21

0 x21 y12



e Wint = ıU n F eint ,

F eint

ı = [Bf ]ıunf

(10)

i = 1, 2, 3

∗ = ı ∗ = −ıw ∗ The virtual rotations ıˇnx are zero since the sk ,nk rotations at mid-nodes are known. We have only two dof per node. We finally obtain the matrix [Bf ] as (see Batoz and Dhatt, 1992, Section 6.4.3 for details):

⎡ 1⎢ [Bf ] = ⎢ A⎣

S4 C4 − S6 C6 −S4 C4 + S6 C6 −C42 + S42 + C62 − S62

. . . . . . . . .

S5 C5 − S4 C4 −S5 C5 + S4 C4 −C52 + S52 + C42 − S42

. . S6 C6 − S5 C5 . . . . −S6 C6 + S5 C5 . . . −C62 + S62 + C52 − S52

⎤ ⎥ ⎥ ⎦

where A is the area of triangular element, Ck = xji /Lk ; Sk = Yji /Lk (k = 4, 5, 6 for ji = 21, 32, 13) and Lk2 = xji2 + yji2 . The principal of virtual work express on the known final 3D configuration leads to: W=

elt

e (Wint

e − Wext )

=0

(12)

(13)

The global components of the internal force vector are defined by:

,

For the virtual curvatures of the DKT6 element, the rotation components ıˇx and ıˇy of the normal (rotations from z to x and y) are linearly expressed with semi-C0 approximations in terms of the rotations at the mid-nodes:

with uTnf = . . . wi . . . i . . .;

ıεT  dv ve

ıε = ([Bm ] + z[Bf ])ıun

xij = xi − xj

ıT = ıˇx,x ıˇy,y ıˇx,y + ıˇy,x ;



e Wint =

with

i = 1, 2, 3

y31 0 x13

e is the internal virtual work and W e is the external where Wint ext virtual work related to the tool actions.

T

T

i = 1, 2, 3

(14)

T

with = [T ] ([Bm ] N + [Bf ] M)A The element external forces are due to the hydraulic pressure and can be computed using the equilibrium conditions. The tool actions are replaced by a normal pressure force because in tube free hydroforming, the deformed configuration is coaxial with the initial round tube. The nodal external force is specified through a normal pressure of intensity P. At each node there are three equilibrium equations and four unknowns: three displacements U, V, W and the force intensity P (Fig. 3). kT k k k Fint = Fx(ext) Fy(ext) Fz(ext)  = Pk nkx nky nkz 

(15)

where nkx nky nkz  is the average normal in the wall of final tube to the node i. Then we can give the pressure nodal Pk normal with the wall of final tube for each node: k k k Pk = nkx nky nkz  · Fx(int) Fy(int) Fz(int) 

T

(16)

But in our case of tube, the radial displacement U, in the cylindrical coordinates (r, ϕ, Z) for each node is known. As a consequence, we can use this assumption to reduce the number of displacements unknowns to two displacements for each node (ϕ and W). And the displacements in the Cartesian coordinate are related with the displacements in the cylindrical coordinate at any iteration i as follows: Upi = Ur cos ϕ − Upi Vpi = Ur sin ϕ + Vpi

(11)

ıU n = ıUi ıVi ıWi ,

with

Upi = R0 (cos(ϕ − ϕi ) − cos ϕ) Vpi = R0 (− sin(ϕ − ϕi ) + sin ϕ)

(17)

The nonlinear system of equations with only two degree of freedom (ϕ and W) per node is solved using a Newton–Raphson

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Fig. 5. Part geometries (axisymmetric bulge).

Fig. 4. The computational procedure using the one-step inverse approach.

procedure. An approximation of the solution at iteration i, leads to: [K iT ]U = F ext (U i ) − F int (U i ) = −R(U i ) [K iT ]

(18)

i

where = [∂R (U)/∂U] In the system (18) the force corresponding to the rotation ϕ is assumed to be the moment: M = Ft · r and r is the first coordinate of node in cylindrical system in the final configuration. Uki = . . . ϕki Wki . . . where i is the iteration and k is the number of node. Two criteria can be used in the IA. The first one is the relative displacement norm criterion given by: err =

U{U} U{U}

(19)

where {U} is the displacement increment vector during one iteration, {U} is the total displacement vector. A second convergence criterion can also be used in the IA, it is based on relative residual norm, given by err =

Ri {Ri } R0 {R0 }

(20)

where {Ri } is the residual vector at the ith iteration, {R0 } is the initial residual vector i.e. at iteration 0. During all our application the first convergence criterion has been used with a tolerance of 10−6 . 2.4. Computational procedure The computational procedure is as follows (Fig. 4). First, the final configuration depicting the geometry of the workpiece is discretized in DKT12 membrane-bending elements. In the analyses presented in this work, the incremental approach by means of the ABAQUS code is first used, and the deformed configuration obtained at a given instant is considered as the final configuration for the inverse analysis (only for validation and comparison between the two methods). The initial guess representing the guessed nodal

positions is first obtained by a radial projection of the nodes in the final configuration onto the initial round tube. The initial guess allows starting the inverse process using Eq. (6) to estimate the strains. Using the uniaxial stress–strain curve, we determine the equivalent stress and the secant modulus. After that, the stresses in each element are computed using the constitutive relation (7). After computing the internal and external forces and assembling the element tangent stiffness matrix, a first estimate of displacement increment is obtained by the resolution of the equilibrium equations (system (18)). After transformation of displacements to the Cartesian coordinate, the initial positions can be updated to start a new iteration. The iterative process stops when the displacement norm is below or equal to a desired precision. 3. Numerical results In this section we will present the simulation results of final tube wall thickness distribution for calibration forming of axisymmetric bulge, of square section hollow component and free T-branch forming for different branch height. These results are compared with the results of incremental explicit method of ABAQUS commercial code and discussed. The loading path used in incremental analyses for all applications is the optimal in terms of thickness distribution. For all following applications, we have considered the residual forces norm of 10−6 for the convergence criteria. To reduce the simulation time in incremental method case, the forming time used in the simulation was sped up by a factor of 1000 with apparently no ill effect for all our applications cases (see also Koc¸ et al., 2000). 3.1. Hydroforming of axisymmetric bulge The analyses of aluminum alloy tubes under calibration hydroforming conditions are carried out, and we consider the same tube as used in the work of Ahmetoglu et al. (2000). This tube is of 177.8 mm length and has a 63.6 mm outside diameter and 1.65 mm initial wall thickness; the geometric parameters of the profile of the Die are given in Fig. 5. The axisymmetric initial tube is made of an Aluminum alloy AA6061-T6 with a Young’s modulus E = 70665 MPa, a Poisson’s ratio = 0.3 and density = 2700 kg/m3 . The equivalent uniaxial stress–strain curve follows the Holloman model: ¯ = 431.01ε−0.1126 . Experiments with the aluminum alloy 6061-T6 were run with an axial feed of 12.7 mm and a variable pressure given in Ahmetoglu et al. (2000). The axial feeding force is applied to the upper side only, while the other side is completely clamped. During the experimental process, pressure reaches its maximum value of 57 MPa.

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Table 1 Comparison of thickness variations.

ABAQUS-Explicit Inverse Approach Experiment

hmin [%]

hmax (%)

CPU times

−13.7 −13.76 −14.56

1.15 0.18 1.21

1 min 9 s 2.5 s

Fig. 6. Comparison of thickness variations along the tube profile.

In this application because the tube is axisymmetric and the axial loading is asymmetric, only a quarter of the tube is modeled by 2000 S3R shell elements (50 elements along the meridian direction) in ABAQUS code. The deformed configuration obtained at the end of the incremental analysis is considered as the final configuration for the inverse analysis and it is modeled using DKT12 shell element. Fig. 6 shows the thickness variation along the meridian profile. We can observe that our results are in good agreement with the results obtained using ABAQUS Dynamic Explicit code. One can observe also, that the thickness curve obtained by the incremental explicit-method is closer to the experimental result especially at the die inner radius at 80 mm, than the one obtained using the IA. This can be explained by the fact, that in the IA the deformation is independent of the loading path. Thickness distribution onto the final hydroformed tube is also given in Fig. 7. We can observe a good agreement between the two solutions. More quantitatively summary of thinning results for both two approaches are given in Table 1. As we can see the maximum thinning is estimated with very good accuracy compared to the result we have obtained using ABAQUS Explicit.

Fig. 7. Comparison of thickness distribution.

Fig. 8. Principal strain in mid of each element along the curvilinear length.

In Fig. 8, we show the logarithmic strains along the curvilinear length of the tube. These strains are given for the case of DKT12 shell element by the relation 6. We can remark that our results are in agreement with those obtained with ABAQUS code, except soft perturbation between the results of ε22 principal strain. 3.2. Hydroforming of square section hollow component Fig. 9 shows the shape and dimensions of a square section hollow component to be formed. The side length of the square section is 45 mm and the transition radius is 6 mm. A tube with 40 mm in diameter and 2 mm in thickness was used in this simulation. The material of the tube is a stainless steel, its mechanical properties are as following: The material model of the tube blank is ¯ = 628 ε−0.32 , yielding stress y = 368 MPa, Young’s modulus of 208 GPa (Yuan et al., 2006), Poisson’s ratio = 0.3 and density = 7800 kg/m3 . The total axial displacement is 8 mm in each side of the tube and the maximum calibration pressure is 120 MPa, the relation between stroke and pressure is considered as linear. For symmetry reasons of geometry and loading, only 1/8 of the tube is modeled by 750 S3R shell elements (25 elements along the meridian direction) in ABAQUS code. The same mesh is used also in IA using the DKT12 shell elements for comparison purpose. The results we obtained with IA in terms of thickness distribution are −16.08% of thinning and +4.62% of thickness increase (Fig. 10). These results are in good agreement with the results we

Fig. 9. Shape and dimensions of part.

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Fig. 12. Principal strain in mid of each element along the Arc ADB.

3.3. Hydroforming of T-shaped part

Fig. 10. Comparison of thickness distribution.

Table 2 Comparison of thickness variations.

ABAQUS-Explicit Inverse Approach

hmin [%]

hmax (%)

CPU times

−19.1 −16.08

8.35 4.62

45 s 1s

have obtained using ABAQUS Explicit. Our model well captured the variation in thickness along a section of the tube that caused the problem of calibration in a Die of square section. This variation can be explained by a rotation of positions of points in the initial configuration around the tube axis. Instead, in axisymmetric Die we do not see this variation (Fig. 7). The principal results we have obtained are summarized in Table 2. Refer to Fig. 11, from the wall thickness plots it was observed that the simulation using the IA and the incremental method wall thickness distributions and variation trend were in good agreement for both curvilinear length (ADB and CD), the maximum error between the two methods is 6.35%. In Fig. 12, we show the logarithmic strains along the Arc ADB of the tube. We can remark that our results are in accord with those obtained with ABAQUS code for principal strain ε22 , except soft perturbation between the results of principal strain ε11 .

T-shaped part is one of the cases where more of material feeding, in the circumferential direction of the tube in expansion zone, is necessary to carry out the part. The essential objective of this application is to demonstrate the importance of verifying the equilibrium en the circumferential direction of the final configuration in the IA. In this test a cylindrical low carbon steel (LCS 1008) tube of 45 mm (outer diameter), 169.12 mm length and 2 mm thickness was used as the blank. The diameter of the T-branches was equal to that of the main tube as presented in Fig. 13. A power law plasticity model ¯ = 484 ε−0.19 was used for the simulation and the other material properties as: Young’s modulus of 200 GPa, Poisson’s ratio of 0.3 and density of 7800 kg/m3 .

Fig. 13. Geometrical parameters for a Tee-shaped part.

Fig. 11. Comparison of thickness variation (Arcs: ADB and CD).

Fig. 14. Comparison of thickness distribution (stroke of 27.65 mm).

M.S. Chebbah et al. / Journal of Materials Processing Technology 211 (2011) 1898–1906

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Table 3 Comparison of thickness variations. hmin (%) ABAQUS-Explicit Inverse Approach

−0.9 −1.39

hmax (%)

CPU times

45.95 55.89

13 min 5 s 15 s

The forming load was internal hydraulic pressure along with incremental axial compressive load (tube end axial feed); the leading path is as presented in the work of Koc¸ et al. (2000). Only the so-called free forming is considered, where the internal pressure reaches a maximum of 44 MPa. The opposite axial punches move toward each other with the same velocity fixed to 3 mm/s and the maximum stroke of 34.56 mm. The interface between the tube and the die was modeled with an automatic surface-to-surface contact algorithm for the case of incremental method. By taking advantage of the symmetry, it was possible to model 1/4th of the T-branch. In the analyses presented in this work, the incremental approach by means of the ABAQUS code is first used (using 4000 S3R triangular shell element, 50 elements along the meridian direction), and the deformed configuration obtained at a given instant is considered as the final configuration for the inverse analysis (using 4000 DKT12 shell element). Fig. 14 shows the comparison of thickness distribution for stroke of 27.65 mm, we can observe a good results correlation between the two methods especially in the top side of the branch. In Table 3 we summarized the thinning results for both two approaches. We can see that the maximum thinning is estimated with very good accuracy compared to the result we have obtained using ABAQUS Explicit. These results are obtained with very small CPU time using the inverse approach. For more details we present in Fig. 15 the distribution of thickness on the arc AB for different case of stroke (6.91, 13.82, 20.74 and 27.65 mm). For each stroke the protrusion height (Hp ) obtained by the incremental approach is 28.69, 35.37, 42.82 and 49.81 mm, respectively. Although the error of the maximum thickening between the two methods is increase with the augmentation of stroke, in the case of 27.65 mm, this error rich only 6.8%. This maximum error obtained by the IA where the elements are under a big geometric distortion (blue areas, Fig. 14) (for interpretation of the references to color in the text, the reader is referred to the web version of the article.). This is certainly related to local effects that are due to the history of bending/unbending moment through the radius are not taken into account in our model, let us remember that is based on a one step.We have not present the cases obtained with more than the stroke of 27.65 mm because after this later the

Fig. 15. Comparison of thickness distribution (Arc AB).

Fig. 16. Principal strain in mid of each element along the Arc AB (stroke of 27.65 mm).

convergence criteria (the residual forces norm less to 10−6 ) is note verified in our model. In Fig. 16, we show the logarithmic strains along the Arc AB of the tube for the stroke of 27.65 mm. The deformed configuration obtained for the stroke of 27.65 mm considered as the final configuration for the inverse analysis. The mesh obtained by the inverse approach is similar (structured mesh) to the initial mesh used for the incremental method, as presented in Fig. 17.

Fig. 17. Deformed configuration used and initial configuration obtained by IA (stroke of 27.65 mm).

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4. Conclusions This work focuses on the simulation of tube hydroforming process in which the initial geometry is a cylindrical tube submitted to an internal pressure and axial feeding at the tube ends. A 3D membrane/bending shell element formulation based on the inverse approach for tube hydroforming have been developed in this paper. The present model can be used to estimate the strain and thickness distributions in the workpieces hydroformed from an initial round tube. The formulation considers the initial and final configurations that are coaxial. For validation purposes, experimental results based on literature review are used while the incremental explicit dynamic approach was conducted for the simulation of all applications using ABAQUS software. Application of tube hydroforming process simulation has been performed to validate the proposed 3D model, and the obtained results are in a good agreement with both experimental results and results obtained using classical incremental dynamic explicit approach by ABAQUS. The general CPU computing time using the proposed FE inverse model is very small (even negligible) in comparison with the computation time of incremental explicit dynamic approach using ABAQUS. Globally, the results of numerical applications obtained confirm that the model based on IA for the hydroforming simulation is efficient and particularly suited as tool to verifying the feasibility of product.

References Abrantes, J.P., Szabo-Ponce, A., Batalha, G.F., 2005. Experimental and numerical simulation of tube hydroforming (THF). Journal of Materials Processing Technology 164–165, 1140–1147. Ahmed, M., Hashmi, M.S.J., 1997. Estimation of machine parameters for hydraulic bulge forming of tubular components. Journal of Materials Processing Technology 64 (1–3), 9–23. Ahmetoglu, M., Sutter, K., Li, X.J., Altan, T., 2000. Tube hydroforming: current research, applications and need for training. Journal of Materials Processing Technology 98 (2), 224–231. Alaswad, A., Olabi, A.G., Benyounis, K.Y., 2011. Integration of finite element analysis and design of experiments to analyse the geometrical factors in bi-layered tube hydroforming. Materials and Design 32, 838–850. Batoz, J.L., Dhatt, G., 1992. Modélisation des structures par éléments finis, vol. 3. Editions Hermès, Paris, pp. 387–393 (in French). Batoz, J.L., Guo, Y.Q., Mercier, F., 1998. The inverse approach with simple triangular shell elements for large strain predictions of sheet metal forming parts. Engineering Computations 15 (7), 864–892. Chebbah, M.S., Naceur, H., Hecini, M., 2010. Rapid coupling optimization method for tube hydroforming process. Journal of Engineering Manufacture 224 (2), 245–256. Di Lorenzo, R., Ingarao, G., Chinesta, F., 2010. Integration of gradient based and response surface methods to develop a cascade optimisation strategy for Yshaped tube hydroforming process design. Advances in Engineering Software 41, 336–348. Fu, L., Dong, X., Wang, P., 2009. Study on one-step simulation for the bending process of extruded profiles. International Journal of Advanced Manufacturing Technology 43 (11–12), 1069–1080. Guo, Y.Q., Batoz, J.L., Detraux, J.M., Duroux, P., 1990. Finite element procedures for strain estimations of sheet metal forming parts. International Journal for Numerical Methods in Engineering 30, 1385–1401.

Guo, Y.Q., Batoz, J.L., Naceur, H., Bouabdallah, S., Mercier, F., Barlet, O., 2000. Recent developments on the analysis and optimum design of sheet metal forming parts using a simplified inverse approach. Computers & Structures 78 (1–3), 133–148. Hartl, Ch., 2005. Research and advances in fundamentals and industrial applications of hydroforming. Journal of Materials Processing Technology 167, 383–392. Hill, R., 1984. A theory of the yielding and plastic flow of anisotropic metals. Proceedings of the Royal Society of London, Series A 193 (1033), p.281–p.297. Hosford, W.F., Caddell, R., 2011. Metal Forming: Mechanics and Metallurgy. Edition Cambridge University Press, 331 pages, ISBN: 978-1-107-00452-8. Jain, N., Wang, J., Alexander, R., 2004. Finite element analysis of dual hydroforming processes. Journal of Materials Processing Technology 145 (1), 59–65. Kashani Zadeh, H., Mashhadi, M.M., 2006. Finite element simulation and experiment in tube hydroforming of unequal T shapes. Journal of Materials Processing Technology 177 (1–3), 684–687. Kim, J., Kang, B.-S., Hwang, S.-M., 2002. Preform design in hydroforming by the three-dimensional backward tracing scheme of the FEM. Journal of Materials Processing Technology 130-131, 100–106. Koc¸, M., Allen, T., Jiratheranat, S., Altan, T., 2000. The use of FEA and design of experiments to establish design guidelines for simple hydroformed parts. International Journal of Machine Tools & Manufacture 40 (15), 2249–2266. Koc¸, M., Altan, T., 2001. An overall review of the tube hydroforming (THF) technology. Journal of Materials Processing Technology 108 (3), 384–393. Kridli, G.T., Bao, L., Mallick, P.K., Tian, Y., 2003. Investigation of thickness variation and corner filling in tube hydroforming. Journal of Materials Processing Technology 133 (3), 287–296. Kwan, C.T., Lin, F.C., 2003. Investigation of T-shape tube hydroforming with finite element method. International Journal of Advanced Manufacturing Technology 21 (6), 420–425. Langa, L., Li, H., Yuanb, S., Danckert, J., Nielsenc, K.B., 2009. Investigation into the preforming’s effect during multi-stages of tube hydroforming of aluminum alloy tube by using useful wrinkles. Journal of Materials Processing Technology 209, 2553–2563. Mac Donald, B.J., Hashmi, M.S.J., 2001. Three-dimensional finite element simulation of bulge forming using a solid bulging medium. Finite Element in Analysis and Design 37 (2), 107–116. Mohammadi, F., Mosavi Mashadi, M., 2009. Determination of the loading path for tube hydroforming process of a copper joint using a fuzzy controller. International Journal of Advanced Manufacturing Technology 43, 1–10. Naceur, H., Guo, Y.Q., Ben-Elechi, S., 2006. Response surface methodology for design of sheet forming parameters to control springback effects. Computers & Structures 84 (26–27), 1651–1663. Naceur, H., Ben-Elechi, S., Batoz, J.L., Knopf-Lenoir, C., 2008. Response surface methodology for the rapid design of aluminum sheet metal forming parameters. Materials and Design 29, 781–790. Nikhare, C., Weiss, M., Hodgson, P.D., 2010. Die closing force in low pressure tube hydroforming. Journal of Materials Processing Technology 210 (15), 2238–2244. Nguyen, B.N., Johnson, K.I., Khaleel, M.A., 2003a. Analysis of tube free hydroforming using an inverse approach with FLD-based adjustment of process parameters. Journal of Manufacturing Science and Engineering 125, 135–140. Nguyen, B.N., Johnson, K.I., Khaleel, M.A., 2003b. Analysis of tube hydroforming by means of an inverse approach. Journal of Manufacturing Science and Engineering 125, 369–377. Numisheet, 2008. In: Hora, P. (Ed.) Part A: Proceedings of the 7th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes, ETH Zurich, September 1–5, 2008. Interlaken, Switzerland. Ray, P., Mac Donald, B.J., 2004. Determination of the optimal load path for tube hydroforming processes using a fuzzy load control algorithm and finite element analysis. Finite Elements in Analysis and Design 41, 173–192. Singh, H., 2003. Fundamentals of Hydroforming. Society of Manufacturing Engineers, Dearborn, MI, USA, 219 pages, ISBN-13: 9780872636620. Yuan, S.J., Han, C., Wang, X.S., 2006. Hydroforming of automotive structural components with rectangular sections. International Journal of Machine Tools & Manufacture 46 (11), 1201–1206.