Development of an analytical model for warm deep drawing of aluminum alloys

Development of an analytical model for warm deep drawing of aluminum alloys

j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 7 ( 2 0 0 8 ) 393–407 journal homepage: www.elsevier.com/locate/jmatp...

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j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 7 ( 2 0 0 8 ) 393–407

journal homepage: www.elsevier.com/locate/jmatprotec

Development of an analytical model for warm deep drawing of aluminum alloys Hong Seok Kim b,∗ , Muammer Koc¸ a , Jun Ni b a

NSF I/UCRC Center for Precision Forming (CPF), Department of Mechanical Engineering, Virginia Commonwealth University (VCU), Richmond, VA 23284, USA b Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109-2125, USA

a r t i c l e

i n f o

a b s t r a c t

Article history:

In this study, an analytical model was developed to investigate the effects of material, pro-

Received 31 March 2007

cess, and geometric parameters in the warm forming of aluminum alloys under simple

Received in revised form 9 June 2007

cylindrical deep drawing conditions. The model was validated with both existing experimen-

Accepted 15 June 2007

tal findings in the literature and FEA results. The effects of the main process parameters (i.e., temperature, forming rate, blank holder pressure (BHP), and friction between a blank and a tooling element) on formability were studied under a variety of warm forming conditions.

Keywords:

The developed model offers rapid, useful, and reasonably accurate results for the design of

Warm forming

warm forming process by predicting the deformation mechanism of the material and the

Lightweight material

relationships between limiting drawing ratio (LDR) and process parameters in isothermal

Analytical model

and non-isothermal heating conditions. It was demonstrated that significant formability

Deep drawing

improvement could be achieved when a large temperature gradient was realized between

Finite element analysis

die and punch, while a slight decrease of LDR was observed when tooling elements and a blank were heated up to same temperature levels. © 2007 Elsevier B.V. All rights reserved.

1.

Introduction

The realization of lightweight structures for transportation vehicles (aerospace and automotive) is a prominent way of improving fuel efficiency and reducing emissions. Because of their low density, comparable strength, and stiffness, lightweight materials such as aluminum and magnesium alloys offer great potential in replacing mild steel structures to reduce weight. Other important factors in selecting lightweight materials for engineering applications, compared to plastics and polymer matrix composites, include their ease of recycling, thermal properties, and dimensional stability and manufacturability (Avedesian and Baker, 1999). However, costeffective forming of lightweight sheet materials into desired functional complex shapes is extremely difficult with the con-



ventional forming technologies (i.e., stamping) because of the formability limitations of these materials at cold conditions. To improve the formability of lightweight sheet materials, warm forming processes have been widely investigated since the 1940s as an alternative manufacturing process. It was reported that with warm forming of lightweight materials (i.e., 200–350 ◦ C), up to 300% total elongation could be achieved (Shehata et al., 1978; Li and Ghosh, 2003; Ayres, 1979). Moreover, many preliminary studies proved a significant increase in formability with 5XXX and 6XXX series of aluminum and AZ31, AZ61 magnesium alloys by performing forming tests at elevated temperatures for simple deep drawing and rectangular cup forming models (Naka and Yoshida, 1999; Doege and Droder, 2001; Bolt et al., 2001; Moon et al., 2001). Bolt et al. (2001) conducted warm forming experiments on various aluminum

Corresponding author. Tel.: +1 734 763 7119. E-mail addresses: [email protected], [email protected] (H.S. Kim). 0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2007.06.046

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Nomenclature b Cb Fc Fp i j K n P re rp , rd r0 , r R0 , R R¯ te , tb tf , tub t0 , t u v ıW

time increment linear work hardening coefficient critical failure load drawing load nodal point on the blank time step strain hardening coefficient strain hardening exponent blank holder pressure radius of inner flange radii of punch and die cavity initial and current radii of a blank element initial and current radii of outer flange normal anisotropy parameter thickness of a blank element before and after bending thickness of a blank element before and after unbending initial and current thickness of a blank element punch speed radial velocity of a blank element strain work of an elementary volume

Greek letters εr , ε , εz plastic strain in radial, circumferential, and thickness directions ε0 material constant ε˙ r , ε˙  , ε˙ z plastic strain rate in radial, circumferential, and thickness directions ε¯˙ equivalent plastic strain rate b displacement of the neutral axis from central axis  coefficient of friction c , n radii of central and neutral axes p , d radii of punch and die profiles e, b drawing stress before and after bending  f ,  ub drawing stress before and after unbending  r ,   ,  z radial, circumferential, and thickness stress thickness stress at  = d  z−  z (+),  z (−) thickness stress in the tensile and compressive regions ¯ equivalent plastic stress ¯ b , ε¯ b equivalent stress and strain during bending ¯ e , ε¯ e equivalent stress and strain at the beginning of bending contact angle of the blank with the punch profile

alloys (1050, 5754-O and 6016-T4) between 100 and 250 ◦ C using box shaped and conical rectangular dies. They showed that increasing temperature increased formability (higher drawing ability and higher cup heights before fracture). It was also demonstrated that keeping the punch cool would help increasing the formability. Doege and Droder (2001) conducted very comprehensive experimental work including aluminum and magnesium alloys at different temperatures and forming

speeds. It was observed that for AZ31B alloy maximum LDR was achievable between 175 and 210 ◦ C. However, due to the complex nature of the warm forming process including highly non-linear material behavior, continuously varying contact conditions, thermo-mechanically coupled characteristics, and multi-faceted interactions between material, process, and tooling factors, the design of warm forming process greatly relies on the experience of process engineers in addition to costly and lengthy experimentations. For cold forming conditions, comprehensive analytical models have been developed by researchers considering the relatively simple deep drawing cases. Chung and Swift (1952a,b) analytically investigated and experimentally validated the radial drawing process over a wide range of operating conditions taking into account thickness changes, bending and unbending effect, die profile friction, and tooling geometry. Yamada (1961) adopted the incremental strain theory based on a small strain formulation and proposed a finite difference method to calculate the stresses and strains in the flange region. Chang and Wang (1998) incorporated the effect of friction, BHP, and radial thickness variation into their radial drawing model, and developed separate radial drawing and plastic bending analysis modules to systematically analyze drawing and redrawing processes. For warm forming conditions, however, very little analytical work has been carried out, and no full account has been given due to the complex deformation conditions compared to cold forming cases. Naka et al. (2000) developed a simple analytical model for warm deep drawing using temperature and strain rate dependent material properties of an aluminum alloy without considering the thickness change of the sheet and plastic bending effect. They reported that the predicted drawing ability of cylindrical cups was in good agreement with the corresponding experimental results. In recent decades, as an accurate and efficient analysis and design tool, finite element analysis (FEA) techniques have been increasingly used for the simulation of material forming processes. The reliability of FEA for the analysis of warm forming has been verified through various case studies in recent investigations. Takuda et al. (2002) successfully predicted the improved drawing ability and the failure characteristics of aluminum alloy using a 2D rigid-plastic and heat conduction finite element method. Palaniswamy et al. (2004) compared the experimental results on warm cup drawing of AZ31 magnesium alloy with 2D and 3D FEA predictions. Similarly, the authors of this paper previously presented the thermomechanically coupled FEA results by comparing the predicted part depth values and strain distribution with experiments in various warm forming process conditions (Kim et al., 2006). In this study, in order to develop guidelines and extend the fundamental understanding of warm forming process, an attempt has been made to develop an analytical model considering a simple circular cup part based on the previous investigations reported in the literature (Chung and Swift, 1952a; Yamada, 1961; Chang and Wang, 1998; Naka et al., 2000). All effects of tooling geometry, anisotropy, friction, and blank

j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 7 ( 2 0 0 8 ) 393–407

holder pressure (BHP) were included in the model, and the material parameters for hardening constants dependent on temperature and strain rate were implemented to describe the warm forming conditions. The accuracy and effectiveness of the analytical model were verified through the comparisons with the experimental results (Naka and Yoshida, 1999). A finite element model was also developed for the same deep drawing process conditions to compare with the analytical model predictions. Thermo-mechanically coupled analyses were performed both in isothermal and non-isothermal heating conditions and a static implicit solution procedure was used to secure the reliability using a commercially available FEA software called ABAQUS/Standard. In isothermal condition, stress and strain distribution, minimum wall thickness, and LDR, under various warm forming processes, conditions were evaluated using both analytical and FEA models. In non-isothermal condition, the effect of temperature gradient between tooling elements on forming performance was analyzed. Detailed failure characteristic and favorable temperature condition for improved formability were explained and suggested. In addition, the effect of forming speed (v), friction (), and BHP on formability was investigated. In the next section, the derivation of the analytical model, assumptions, conditions, and solution procedure are explained in detail considering deformation characteristics in flange, bend (die radius), and cup wall regions in a typical axisymmetric deep drawing case. In the third section, isothermal and nonisothermal FEA models for the same conditions are described. Comparison with the experimental findings and discussion of results are presented in Section 4.

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2. Description of the analytical model for an axisymmetric warm deep drawing case Illustrated in Fig. 1 is the cylindrical deep drawing of a sheet blank. A round blank with an initial radius of R0 and thickness of t0 is clamped between a die and a blank holder. Then, a flat-faced punch moved down into the die cavity to form an axisymmetric cup part. As shown in Fig. 1b, the deformation process can be analyzed in five distinct regions. The regions from I to III successively undergo radial drawing (I) (i.e., flange), plastic bending and unbending under tension and frictional stresses (II), and side-wall stretching (III). The tangential sites of a cup denoted by U and N in Fig. 1b are the critical failure locations in isothermal (Tdie = Tpunch ) and non-isothermal (Tdie > Tpunch ) process conditions, respectively where the drawing stress reaches a maximum (Chung and Swift, 1952b; Takuda et al., 2002). In the remaining regions, the blank is subject to biaxial tension over the bottom of a cup (V), and stretched on the punch profile radius combined with a plastic bending (IV). Since the main tasks of this study are to predict the LDR (i.e., the maximum ratio of blank diameter to punch diameter which can be drawn into a cup without failure) under warm forming conditions, and to establish the relationship between the LDR and process parameters, the derivations are carried out from the edge of the flange to the beginning of the punch profile region (i.e., regions I–III) in isothermal conditions and to the end of the die profile region (i.e., regions I–II) in non-isothermal conditions. Then, the maximum punch loads obtained from the point U in isothermal cases and point N

Fig. 1 – Deep drawing for a circular cup (a) initial stage and (b) intermediate stage.

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in non-isothermal cases (Fig. 1b) are compared with the critical value from the instability criterion (Swift, 1952). When the maximum load exceeds the critical value, the part is considered to fail. ¯ Assuming that the material exhibits normal anisotropy (R), the stress and strain rate relations can be written using Hill’s anisotropy yield criterion (Hill, 1950) as: ε¯˙ ¯ ¯ (1 + R)

=

ε˙ r ε˙  = ¯ r −  ) + (r − z ) ¯  − r ) + ( − z ) R( R(

=

ε˙ z (z −  ) + (z − r )

(1)

¯ and ε¯˙ are defined by the following forms: ¯ =



2

1 + R¯ 1 + 2R¯

2

¯ r −  ) (r − z ) + ( − z ) + R(

1 + R¯

 ε¯˙ =



1



2

2

2

¯ εz ) + (˙ε − R˙ ¯ εz ) + R(˙ ¯ εr − ε˙  ) (˙εr − R˙

(2)

2

(3) Fig. 2 – Stresses on an element in flange.

The sheet material is assumed to follow a generalized Swift’s power-hardening law: ¯ = K(ε0 + ε¯ )

n

(4)

In order to analyze forming performance at elevated temperatures and different forming rates, the material constants (i.e., K, n, and ε0 ) are made temperature and strain rate dependent by fitting the flow stress curves measured at a wide range of temperatures and strain rates, and the uniform strain rate of the material is assumed during forming. The equivalent strain rate (ε¯˙ ) of the material is estimated from the simple correlation with the punch speed (u). If the plain strain condition is assumed, the equivalent strain rate can be rewritten using equation (3) as:

 ε¯˙ =

¯  v  2(1 + R)   1 + 2R¯ r



(8)

where εr = ln(dr/dR) and ε = ln(r/R). Since the strain rates in each direction are expressed as: ε˙ r =

dv , dr

ε˙  =

v r

and

ε˙ z =

t˙ t

(9)

Combining it with Eq. (1) results in the following relations:

(6)

By taking the average of radial velocities of elements at several positions in the flange, the order of equivalent strain rate can be finally calculated with respect to the punch speed.

2.1.

1 dε = [1 − exp(ε − εr )] dr r

(5)

since ε˙  = (v/r). Here, v can be approximated as the following form: rp v≈− u r

Examining the geometry of deformation of an element at two different stages, the following strain compatibility equation can be derived:

¯ r − R ¯ v (1 + R) dv = ¯  − R ¯ r r dr (1 + R)

(10)

vt (r +  ) ¯  − R ¯ r r (1 + R)

(11)

t˙ = −

The detailed computational scheme accounting for the effect of BHP is developed using the finite difference method as a slight extension of the incremental strain theory proposed by Yamada (1961). Hence, the basic equations (7), (8), (10), and (11) are rewritten into finite difference forms as follows:

Radial drawing

The stresses on an element of the flange are shown in Fig. 2. If it is assumed that the blank holder force is distributed around the edge of the flange that thickens most (Chung and Swift, 1952a), the thickness stress ( z ) can be neglected. Then, the radial equilibrium equation reduces to: d(tr ) t = ( − r ) dr r

(7)

(tr )i,j = (tr )i−1,j +

(ε )i,j = (ε )i−1,j +

 vi,j = vi−1,j +

ti−1,j

( − r )i−1,j ri,j

(12)

1 [1 − exp (ε − εr )i−1,j ]ri,j ri−1,j

(13)

ri−1,j

¯ r − R ¯  (1 + R) ¯  − R ¯ r (1 + R)



vi−1,j

r i−1,j i−1,j

ri,j

(14)

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 ti,j = ti,j−1 −

(r +  ) ¯  − R ¯ r (1 + R)



vi,j−1 ti,j−1

i,j−1

ri,j−1

b

(15)

point is expressed as:

(r )0,j = where ri,j = ri,j − ri−1,j . When the solution is obtained up to the points (i, j − 1) and (i − 1, j), the radial stress ( r ), thickness (t), and circumferential strain (ε␪ ) can be calculated at the point (i, j) using the foregoing equations. Then, the strains εz , and εr , are determined based on the current thickness value and under the assumption of volume consistency of the material as follows:

εz = ln(t/t0 )

and

εr = −(ε + εz )

ε˙ i,j =

εi,j − εi,j−1 b

2 2 P (r0,j − re ) t0,j r0,j

 (ε )0,j = ln

(17)

By substituting the strain rate in each direction into equation (3), and combining it with the strain-hardening characteristic in equation (4), the equivalent stress () ¯ and strain (¯ε) values are obtained. Then, using equation (2), the circumferential stress (  ) can be finally determined. The calculation is performed from the edge of the flange to the beginning of the die profile region with appropriate boundary conditions. Since the friction force on the flange is assumed to apply only around the rim, the radial stress at this

(18)

where r0,j and t0,j denote the current radius and thickness of the edge of the flange respectively. In addition, since the radial drawing stress ( r ) is usually small when compared to the circumferential stress ( ␪ ) at the edge of the flange, the initial strains at the rim can be determined by assigning appropriate initial radial velocity to the rim (v0,j ) and using stress and strain rate relationship in equation (1) as:

(16)

The strain rate at the current position (i, j) is approximated from the strain difference between two time steps as:

397

r0,j−1 + v0,j b R0

¯ z )0,j = − (εr )0,j = R(ε

2.2.

R¯ 1 + R¯

 ,

(ε )0,j

(19)

Plastic bending

As shown in Fig. 3, when the element in the flange reaches the die profile region, it starts to deform by bending along the curved line of the die radius. Due to the tensile stress at both ends before bending, the radius of the neutral axis (n ) becomes smaller than that of the central axis (c ), and the thickness decreases depending on the ratio of the original thickness to the die profile radius. For simplicity, the deformation in the elastic regime is neglected in this study. Based on the stress distribution shown in Fig. 3b, the equilibrium equation for forces in the thickness direction can be

Fig. 3 – Schematic of plastic bending (a) deformation in bending and tension and (b) stresses on an element in bending.

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derived as dz r − z =0 −  d

(20)

Substituting for  r from Eqs. (27) and (28) into Eq. (21), and rearranging and integrating it yields the drawing stress after bending ( b )

In the die corner region, since we assumed that the circumferential strain is negligible (Chang and Wang, 1998), Eqs. (2) and (3) can be rewritten as

1 b = tb

¯ =

d¯ε =

1 + R¯



1 + 2R¯

+ Cb

      ln  

(22)

n

¯ b = ¯ e + Cb (¯εb − ε¯ e )



    ln(/n )  ¯ e + Cb   1 + 2R¯ 1 + R¯

n−1

1 + R¯

+

1 + 2R¯ ¯ (1 + R)

z (−) = z− −



2

Cb



 ln

1 + R¯



1 + 2R¯

¯ (1 + R)

2

¯ 2(1 + 2R)

Cb

 d + tb

d + tb n

¯ e ln





2



− ln

 n

(27)



 ln d n

d

2



 − ln n

2  (28)

By equating Eqs. (27) and (28) at  = n , the following relation is obtained:

2 ln



n

d (d +tb )

1+R¯



1 + 2R¯

¯ e +

¯ (1 + R)



2

¯ 2(1 + 2R)

Cb ln

d + tb d

= z−

(29)



d2

 ( +t )2 d d b +

n

2

 + t

d b n



1 + 2R¯



2

1 + 2R¯

¯ b (1 + R)C 8

¯ (1 + R)



¯ e +

2



1 + 2R¯

¯ (1 + R)

¯ e −



2

1 + 2R¯

Cb

Cb





 1 − ln d n



 + tb 1 − ln d n





2

+

(2n2 − d2 − (d + tb ) )e



4

tb (tb + 2d )

(33)

The stress and thickness values after bending are obtained by equating Eqs. (30) and (33), and using geometric constraints in Fig. 3 given by tb =

2 

c tb

× ln

(26)

¯ e ln

¯ 2(1 + 2R)

(32)

1 + R¯

× ln

(25)

Integrating Eq. (25) in the tensile and compressive regions respectively yields the thickness stresses as follows:



¯ d¯ε

b = e +

+ Cb = Kn(ε0 + ε¯ e )

(31)

where ıV =  d d per unit width. By substituting Eqs. (21), (22), and (32) into Eq. (31), the drawing stress after bending ( b ) can be obtained as

Here, the constant Cb can be determined by taking the instantaneous strain-hardening rate at the current strain level as:

z (+) =

ıW

(24)

Re-writing Eq. (20) using Eqs. (21), (22), and (24) yields

1 + 2R¯

(30)

c tb ı

␦W = ␦V

Strain hardening is assumed to be linear (Chung and Swift, 1952a), hence the relation between equivalent stress (¯ b ) and strain (¯εb ) during bending is defined as







(23)

1 + R¯

1 + 2R¯

On the other hand, the increase in drawing stress during bending can be also calculated using the strain energy as follows: b = e +

n

 

¯ ¯ e (1 + R)



n2

 + t

¯ 2 (1 + R) d b ln + z− (n − d ) ¯ d 2(1 + 2R)





dz 1 = d 



(21)

since dεr = ln

n ln

d (d + tb )





1 + 2R¯ |r − z | 1 + R¯



n te n + b

2.3.

and

n − d = −b +

tb 2

(34)

Die friction and unbending

When a sheet metal is sliding over the die profile radius, there will be an increase of drawing stress due to the frictional sheer stress depending on the coefficient of friction () and contact pressure. If the changes in thickness are neglected and the contact angle of the blank over the die profile radius is approximated as 90◦ , the increased drawing stress at the vertical wall ( f ) can be calculated based on the force equilibrium as follows: f = b

re ( /2) e rd

(35)

Although the unbending process is more complicated than the bending process due to the complex strain history through the thickness, it can be treated in the same way with

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the bending since the work done by unbending is not very substantial. Finally, the drawing force (Fp ) in isothermal conditions (Tdie = Tpunch ) can be found at the punch corner region taking into account the clearance between the die cavity and the punch as follows:

 Fp = 2 rp tub

r ub d rp

 (36)

The above equation is also used for the calculation of drawing force at the die corner region in non-isothermal cases (Tdie > Tpunch ) since the thickness change of the sheet in the cup wall region (region III in Fig. 1b) is not included in this study.

2.4.

rf = rp − p (1 − sin )

and

rf = rf +

t sin 2

(39)

Under the assumption of plane strain condition (ε = 0), the equivalent stress and strain are rewritten from Eqs. (2) and (3) as

¯ =



 ¯ r −  ) = 1 + 2R(

1 + 2R¯

1 + R¯

(r − z )

(40)

1 + R¯ d¯ε = −  dεz 1 + 2R¯

(41)

Instability criterion

In order to determine the LDR, as a measure of formability in deep drawing process, the maximum drawing force obtained in the previous section needs to be evaluated whether it exceeds the critical load or not at the critical failure site. To describe the stress states more accurately at the punch corner region where failure occurs in isothermal conditions, the equilibrium equations are derived based on the simplified analysis of an axisymmetric shell (Wan et al., 2001) as shown in Fig. 4. In non-isothermal cases, the same derivations can be used neglecting the thickness change of the cup wall region. The normal equilibrium equation is derived as: p=

where rf and rf are defined as:

t ( p sin + r r ) rp

t ( p sin + r rf ) rf p

r =

1 + R¯ + (tp sin /rf p )

1



  1 + 2R¯ 1 + (tp sin /rf p ) + (trf /rf p )

K(ε0 + ε¯ )

n

(42)

According to the instability criterion by Swift (1952), the critical failure condition can be determined by the following relation: ∂¯ = ∂¯ε



1 + 2R¯

1 + R¯



(43)

(37)

where p = p + (t/2), r = r + (t/2) sin . Re-writing Eq. (37) at the critical failure location (rf ) denoted by U in Fig. 1, the thickness stress ( z = −p) is determined as z = −

Substituting Eq. (38) into Eq. (40) and using Eq. (4) gives the drawing stress

Then, by inserting Eq. (4) into (43), the equivalent plastic strain at failure (¯εc ) is obtained: 1 + R¯ ε¯ c = n  − ε0 1 + 2R¯

(44)

(38) The drawing force in the cup wall can be defined as F = 2 rp tr

(45)

By substituting Eqs. (42) and (44) into the above equation and approximating = 90◦ , the critical punch force can be determined as 2 rp t Fc = 1 + R¯



1 + R¯



n+1

1 + (tp /rf p ) + (trf /rf p )

1 + 2R¯



Fig. 4 – Stresses acting on a shell element.

1 + R¯ + (tp /rf p )

Knn

(46)

¯ ¯ 0 ) based on Eqs. (16) where t = t0 exp(−n + ( 1 + 2R/(1 + R))ε and (41). Finally, by comparing the maximum force in Eq. (36) with the critical loading condition in Eq. (46), the LDR can be determined as a measure of deep drawing ability. The effect of friction, BHP, temperature, and strain rate on LDR can be investigated by using different coefficients of friction, BHP values, and material models obtained at a wide range of temperatures and strain rates. The detailed calculation scheme is illustrated in Fig. 5.

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Fig. 5 – Calculation scheme of the analytical model.

Fig. 6 – Axisymmetric FE model for deep drawing of a circular cup.

j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 7 ( 2 0 0 8 ) 393–407

3.

Finite element model

FEA was carried out for the deep drawing process using an implicit FEA package, ABAQUS/Standard, to investigate the effects of process parameters, such as temperature, forming speed, friction, and BHP, on forming performance and to compare the predicted results with analytical and experimental investigations. As illustrated in Fig. 6, the FE model was developed with a 2D continuum element (CAX4RT) based on the axisymmetric geometry and loading condition assuming isotropic material behavior. To further simplify the analysis, heating and cooling devices for the tooling elements (i.e., die, blank holder, and punch) were not designed, and tooling distortion due to temperature changes was ignored by using rigid body constraints. Uniform temperature distribution was directly assigned on the tooling surfaces to describe warm forming condition. The sheet was modeled with five elements in the thickness direction to include non-linear bending and frictional shear effects, and the element size was initially 0.3 mm in the radial direction. For the process parameters, the Coulomb friction coefficients in a range of 0.1–0.5 were prescribed between a blank and tools, and uniform BHP of 1–10 MPa was applied on the top surface of a blank holder plate. The contact heat transfer coefficient for non-isothermal simulations was assumed to be uniform (1400 W/m2 K) regardless of the temperature and pressure at the interface based on the study of Takuda et al. (2002).

4.

Results and discussion

The reliability of the developed analytical model was validated through the comparison with the experimental findings available in the literature and FEA results. Using the analytical and FEA models, the warm forming behavior of an aluminum alloy, Al5083, was investigated over a wide range of warm forming process conditions. First, in isothermal conditions, where blank and tooling elements are heated up to the same temperature levels, stress and strain distributions, minimum thickness values at failure, and interactions between LDR and various process parameters (i.e., temperature, BHP, punch speed, and friction) were examined both analytically and numerically. Then, in non-isothermal condition, where blank temperature changes temporally and spatially depend on tooling temperatures, the influence of a temperature gradient between tooling elements on warm forming behavior was evaluated. The comparison of deformation mechanisms is also made between isothermal and non-isothermal conditions to explain the differences in forming performance. The tooling dimension and material properties were determined based on the experimental set up by Naka and Yoshida (1999). The flow stress of Al 5083 was measured in the tensile tests at a wide range of temperatures (20, 80, 100,150, 200, and 250 ◦ C) and strain rates (5.56 × 10−5 , 5.56 × 10−4 , 5.56 × 10−3 , 5.56 × 10−2 , and 5.28 × 10−1 ) as illustrated in Fig. 7, and used in the analytical and FEA models by fitting the curves up to the maximum tensile strength. The stress–strain

401

relationships at other temperatures and strain rates were interpolated based on the given experimental measurements in Fig. 7. A blank thickness was initially 1 mm, and the radius of flat-bottomed punch and die cavity was 18 and 20 mm, respectively. Both die and punch had a profile radius of 4 mm.

4.1.

Isothermal conditions

Fig. 8 shows the stress and strain distributions of the flange region obtained from the analytical and FEA models at a part depth of 10 mm. In most of the flange region, the predicted and calculated values of stress and strain are very close to each other (i.e., <5%). However, some deviations in stress values can be found in the inner flange region of Fig. 8a. The reason seems that the plastic bending over the die profile region affects part of the inner flange region in the FEA model, while it is not reflected in the analytical model. As shown in Fig. 8a, the circumferential compressive stress ( ␪ ) is greatest at the edge of the flange while the radial stress ( r ) is a minimum. As the blank material approaches to the inner flange region, the radial stress increases and the circumferential stress becomes less compressive. Since the existence of a high circumferential compression in the outer flange region causes wrinkling, the proper level of BHP needs to be applied to prevent wrinkling during forming. However, in warm forming, this compressive stress tends to decrease due to the decreased flow stress level of the material at elevated temperatures. Hence, it can be implied that relatively smaller BHP values can be used in warm forming processes to eliminate wrinkling when compared to room temperature forming cases. In terms of the strain distribution (Fig. 8b), the overall ranges of the predicted values are not very severe since the failure by localized thinning occurred around the punch corner. The blank becomes thicker as the blank radius increases due to the increasing circumferential compression. The minimum thickness values around the punch corner are analytically calculated as one of the formability measurements, and compared with the simulation results at various warm forming process conditions. As shown in Fig. 9, in general, the analytical results overestimates the minimum thickness values since thickness decreases over the punch profile radius and in the clearance between the die and the punch are ignored to simplify the analysis. However, the magnitude of errors is within 10% in all process conditions. In addition, it should be noted that the minimum thickness values are not very sensitive to the temperature and forming speed levels in both analytical and FEA models. Maximum 8% of thickness variation is observed as temperature changes from 20 to 250 ◦ C. Hence, it can be concluded that the formability improvement cannot be expected by increasing temperature of all tooling elements to the same level. Fig. 10 shows the LDR values obtained from the analytical and FEA models at two different punch speed conditions. To determine the LDR in simulations, blank radii were progressively increased from 32 to 45 mm by 0.5 mm until the part failed. Then, the largest blank radius which can be drawn into a cup without failure was used for the calculation of LDR (i.e., the maximum ratio of blank diameter to punch diam-

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Fig. 7 – Stress–strain relationship of Al5083 at six different temperatures and strain rates (Naka and Yoshida, 1999).

eter which can be drawn into a cup without failure). In this study, the failure is considered only in terms of necking. However, it should be noted that wrinkling is also one critical failure in an actual forming process. The compressive stress in the flange increases with increasing blank diameter and

its effect becomes prominent at room temperature due to the higher material strength level compared to warm forming cases. When wrinkling occurs, material flow into the die cavity is restrained and the increased stretching effect in the punch corner region leads to localized necking. At elevated

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Fig. 8 – Stress and strain distributions in flange at T = 250 ◦ C. (a) Stress distribution and (b) strain distribution (BHP = 1 MPa;  = 0.1; v = 10 mm/min; DR = 2.11).

temperatures, on the other hand, friction effect becomes dominant in restraining the blank movement of the flange because wrinkling tendency is reduced due to the decreased material strength. As expected from the minimum thickness values in Fig. 9, a slight decrease of LDR (∼10%) is observed in the temperature range of 20–250 ◦ C under isothermal condition (Fig. 10). Since a similar trend has been obtained in the experimental study by Sugamata et al. (1987), it can be confirmed that the isothermal heating conditions are not favorable to increase the formabil-

ity in circular cup part forming cases. The prediction errors between two models are 5–10%. Figs. 11 and 12 illustrate the effects of BHP and friction. A monotonic decrease of the LDR with increasing BHPs and friction coefficients is observed in both room and warm temperature conditions. As BHP increases from 1 to 10 MPa, about a 7% and a 12% decrease of the LDR values are observed in room and warm temperature conditions, respectively. In the case of the friction variation, the absolute decrease of the LDR are 26% at T = 20 ◦ C and 32% at T = 250 ◦ C with the varying friction

Fig. 9 – Effect of temperature on cup wall thickness at different temperatures (BHP = 1 MPa;  = 0.1; v = 10 mm/min; DR = 2.11).

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Fig. 10 – Effect of temperature on LDR under different punch speeds (BHP = 1 MPa;  = 0.1).

Fig. 12 – Effect of friction on LDR at different temperatures (v = 10 mm/min; BHP = 1 MPa).

coefficient of 0.1–0.5. Therefore, it is recommended that the friction and BHP should be kept as low as possible to achieve increased formability. However, as mentioned earlier, a compromise is required when determining these factors to prevent other types of failure, such as wrinkling and surface defects. Even though the analytical and FEA models developed in this section for the analysis of warm forming in isothermal conditions have not been directly compared with the experimental results, their reliability can be reasonably deduced from the previous investigation by the authors (Kim et al., 2006, 2004), and the accurate validation results of the nonisothermal FEA in the next section performed with the same tooling geometry and material model with the isothermal FEA. In addition, it is proved that the analytical model can provide an efficient analysis tool for warm forming compared to costly and lengthy experimental trials and FEA. Only less than 30 s is required to finish one calculation in the analytical model, while it takes about 30 min for one simulation in the case of the FE model.

4.2.

Fig. 11 – Effect of BHP on LDR at different temperatures (v = 10 mm/min;  = 0.1).

Non-isothermal conditions

In order to apply the analytical model to the non-isothermal forming cases, the blank temperature contacting with the flange and punch profile radius was assumed to be the same with the die and punch temperature. For the temperature of the sheet between these two regions, the value in the middle range of die and punch temperature is assigned to realize the gradual temperature change of the blank. In the case of FEA, the blank was initially set 25 ◦ C and allowed to heat up by heat transfer from the tooling elements. The BHP was 1 MPa, and a temperature dependent friction coefficient between 0.05 and 0.25 for a temperature of 20 and 300 ◦ C was used based on the experimental results by Naka et al. (2000). In Fig. 13, the LDR values predicted from the nonisothermal FEA and analytical models in this study are compared with the experimental and analytical results by Naka and Yoshida (1999) and Naka et al. (2000) at two different temperatures and various forming rates. The LDR values predicted from the non-isothermal FEA model matches best with the experimental measurements. The maximum prediction error is less than 3% when the punch speed is 500 mm/min. The analytical model developed in this study also shows accurate prediction results. The error is within 6%. In both room and warm temperature condition, the LDR decreases with increasing punch speed. However, the variation is more sensitive in warm forming condition due to the increased strain rate sensitivity. The remarkable formability improvement can be achieved especially when larger temperature gradient between die and punch (Tdie > Tpunch ) is realized together with the slow punch speed. In the rectangular cup part forming analyses (Kim et al., 2006), similar results were obtained. Hence, it can be concluded that the favorable heating mode for warm forming is to introduce a large temperature gradient between the die and punch (i.e., cooled punch, heater die, and blank holder). This result can be also explained based on the analytical model developed in this paper. The critical failure load (Fc ) and the maximum drawing force (Fp ) in Eqs. (36) and (46),

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Fig. 15 – Effect of temperature on maximum drawing load (BHP = 1 MPa;  = 0.1).

Fig. 13 – Comparison of LDR at various temperatures and punch speeds (a) Tdie = 25 ◦ C, Tpunch = 25 ◦ C and (b) Tdie = 180 ◦ C, Tpunch = 25 ◦ C.

Ghosh, 2003). The Al5083 alloy used in the study also shows the similar trend (Fig. 14). To see the combined effect of these two factors, the maximum load at the failure site is calculated with temperature in Fig. 15 using the K and n values in Fig. 14. It is found that the maximum load is significantly reduced at elevated temperature. On the other hand, when the punch is kept cold at the room temperature level, the critical failure loads are the same in isothermal and non-isothermal forming conditions since there is no change in the material properties of the sheet contacting with the punch corner region. Hence, the formability can be improved in the heated die, blank holder, and the cooled punch conditions by reducing the maximum load at the failure site and keeping the critical failure load to the higher level. The deformation characteristic in non-isothermal condition is quite different from that of the isothermal forming case. For an in-depth evaluation, the comparison of thickness distributions is made between these two cases in Fig. 16.

which determine the forming performance, can be regarded as functions of material parameters and tooling geometry. When the forming temperature increases under the given tooling geometry, the values of strength coefficient (K) and hardening exponent (n) of Aluminum alloys generally decrease (Li and

Fig. 14 – Material characteristic of Al5083 alloy at various temperatures (strain rate = 5.56 × 10−3 s−1 ).

Fig. 16 – Comparison of thickness distribution between isothermal and non-isothermal forming cases (v = 10 mm/min; BHP = MPa;  = 0.1).

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comparison with the experimental results reported in the literature (Naka and Yoshida, 1999). The detailed deformation characteristic of the material and favorable heating condition in warm forming can be successfully analyzed and suggested with reasonably small prediction error with the experiments. However, since the current analytical model does not fully account for the complex strain history of the material and heat transfer from the tooling elements to the blank, further studies on these issues are required to determine the optimal temperature condition in each tooling region, hence, to maximize the formability. Some discrepancy observed between experiments, and FEA and analytical models can be also attributed to the inaccurate material property definition and the incomplete assumption of process parameters. The stress and stain curves obtained from the uniaxial tension tests have a limitation in predicting the biaxial deformation characteristics of the material. In addition, the friction coefficient and contact heat transfer coefficient have not been fully evaluated as a function of various process parameters (i.e., temperature and contact pressure). Tooling distortion by temperature change is also ignored. Therefore, for more reliable results, studies on the accurate material modeling and many unknown contact properties are necessary.

5.

Conclusions and future work

For the purpose of providing guidelines and further extending basic understanding of warm forming process, analytical and numerical models were developed in this study as rapid and cost effective prediction tools. The dependence of warm forming performance on temperature, punch speed, BHP, and friction, identified as main factors influencing the formability significantly, was investigated under various warm forming process conditions. In summary, the followings can be concluded:

Fig. 17 – Deformed shaped and failure at isothermal and non-isothermal conditions (a) Tdie = 180 ◦ C, Tpunch = 180 ◦ C and (b) Tdie = 180 ◦ C, Tpunch = 25 ◦ C.

In isothermal condition, the thickness strains are mainly concentrated around the critical punch corner region as mentioned earlier. However, those from the non-isothermal simulation are not very significant at the same part depth (10 mm). In addition, it is noted that the edge of the flange moved more into the die cavity in non-isothermal condition. Hence, it can be seen that relatively higher ductility of the flange at elevated temperatures and increased material strength around punch corner at a lower temperature helps to increase formability by delaying the localized thinning. As shown in Fig. 17, in non-isothermal condition, the failure eventually occurred around the die corner region as different with the isothermal forming case, and the deeper part depth value of 15 mm can be achieved. In summary, the prediction capability of the FEA and analytical models can be validated in this section through the

(1) An analytical model was developed to evaluate deep drawing process at elevated temperatures and under different BHP and friction conditions using a temperature and strain rate dependent material model. The results of calculations were shown to be in good agreement with the corresponding FEA predictions and experimental results. The required calculation time to finish one calculation was less than 30 s; hence, its cost effectiveness could be verified. However, since the constant temperature and uniform strain rate conditions were assumed for the analysis of each distinct region (i.e., flange, die profile, and punch profile), further developments integrating non-isothermal effects in the same deformation regions are required for a wide range of application to industrial cases. (2) A thermo-mechanically coupled FEA model was developed using an implicit software package called ABAQUS/Standard. The prediction error of the model was found to be less than 3% based on the comparison of LDR with experimental measurements. The slight deviations of predictions were mostly due to incomplete material modeling and inaccurate assumption of contact conditions between tooling and blank. For more reliable results, accurate stress-strain relationship under various

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loading conditions and complete experimental data on the anisotropic behavior and yield locus of the material are required. In addition, contact factors such as friction and heat transfer coefficients need to be evaluated as a function of temperature and contact pressure. (3) LDR values were not very sensitive to forming temperatures in isothermal condition, while a remarkable increase of formability was observed when the punch was kept at the room temperature level. Hence, it is concluded that the formability of aluminum alloys can be enhanced by introducing appropriate a temperature gradient on the work piece. Additional studies to find out the optimal heating condition of tooling elements are necessary to maximize formability. (4) Detailed deformation characteristics were compared between isothermal and non-isothermal conditions. In the former case, the critical failure location, where limit strain developed, was the punch corner region. However, in the latter case, relatively uniform straining and thinning was observed at the same part depth. It seems that the increased temperature of flange region delays the onset of localized thinning and shifts the failure site to the die corner region due to the improved ductility of flange material and the increased flow stress of punch corner region. (5) BHP and friction showed significant effects on formability. Lower BHP and friction coefficient were preferred to achieve increased formability due to the decreased restrain force of the material. However, these factors need to be carefully determined in practical application to prevent other types of failure such as wrinkling and surface defects.

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