Sheet Metal Forming Analysis of Planar Anisotropic Materials with a Proper Numerical Scheme for the Blank Holding Force

Tae Hoon Choi, Choong Ho Lee and Hoon Huh*

Dept. of Mechanical Engineering Korea Advanced Institute of Science and Technology 373-1 Husondong, Yusonggu, Taejon 305-701, Korea

Abstract

A finite element formulation is derived for the incremental analysis the non-steady large deformation of sheet metal. A modified membrane element with a proper formulation is introduced to correctly enhance the flexural rigidity not only within an element but among elements. The strain energy term in the formulation is decomposed into the membrane energy term for mean stretching and the bending energy term for pure bending. Distribution of the blank holding force is calculated in each step according to the thickness in the flange region. The calculation employs a special relation between the thickness and the blank holding force. The formulation is then combined with an effective algorithm to deal with the contact between the material and the dies. The simulation examples demonstrate the validity and versatility of the computer code by showing the earing phenomenon in circular cup drawing and the variation of the blank holding force in rectangular cup drawing. In simulation, the thickness variation in the flange region redistributes the blank holding force during the deformation, which can predict a more precise contour line in the flange region and a more precise shape with the strain.

1. Introduction Sheet metal forming process has taken an important role in such industries as automobiles,

airplanes and electric appliances due to its various advantages. In conjunction with this situation, there has been a remarkable advances in the sheet forming analysis by the finite element method[1-5]. Since sheet metal undergoes large deformation and rotation from stretching and bending into a very complicated shape during the forming process, the finite element approach is inevitably necessary for the more precise sheet forming analysis. To obtain more useful results, the finite element method has to be supplemented by various numerical techniques, that is to say, the geometric description for tool and dies[6-8], the treatment of contact and friction between tool and dies and the solution procedure for the convergence. Nevertheless, when we consider the rapid development in industry, the computational efficiency is becoming very important issue with the above technologies in order to promote the interaction between the research and the reality.

Generally speaking, the finite element analysis for the sheet metal forming problems can be categorized by three groups: a membrane theory, a shell theory[9-12] and a continuum theory. The membrane theory has been widely used in finite element analysis because of its computational efficiency and better convergence than the shell theory. However, because the membrane element cannot include bending effect inherently, it is difficult to achieve objective precision. Therefore it is taken for granted that some modifications of the membrane element are required in the finite element formulation procedure to take the bending effect into account. As a suggestion for this need, Huh & Han[13] made a membrane element fortified with the advantage of a shell element by considering the bending effect. They decomposed the strain energy term(the plastic dissipation term) into the two terms: the term due to the mean stretching throughout the thickness and the term due to the bending deformation which was dependent upon the thickness. Employing this concept, they derived a variational formulation for the incremental analysis of the non-steady large deformation of sheet metal assumed as normal anisotropic. In their paper, the developed algorithm was proved to improve the precision of the numerical solutions for the bending-dominant sheet forming process without sacrificing the computational efficiency. However, sheet metal was assumed as normal anisotropic at that time, which resulted in inability to predict the earing phenomenon in the cylindrical cup deep drawing process. As the anisotropic characteristic has an *Corresponding authors

effect not only on the membrane energy, but also on the bending energy, it causes unignorable effects on the deformation of sheet metal consequently. Therefore, for the sake of the more realistic analysis, sheet metal should be assumed as planar anisotropic and the variational formulation should be modified appropriately.

Recently, there has been a few effort to search for the moderate scheme applying the blank holding force[14, 15]. The blank holding force is a very important process variable to affect the deformed shape of a product by controlling the inflow of material. It does not act on the whole part of the flange, but on the part in which the contact between material and die occurs. Therefore it must be influenced by the thickness variation of the sheet material in the flange. But, since the modified membrane element has the same external appearance as the ordinary membrane element, it is not able to apply the thickness variation of the sheet metal in the blank holder to the contact treatment and the equally distributed blank holding force should be imposed on sheet metal along the edge of the flange regardless of the contact status. But sheet metal does not contact with the blank holder at the boundary, nor the blank holding force is distributed uniformly along the boundary. Therefore a new scheme is requested in order to apply the blank holding force to the proper region.

In this study, the variational formulation using the modified membrane element is improved to analyze planar anisotopic sheet metal. A rigid plastic finite element formulation with planar anisotropy obeying Hill's quadratic yield criterion, derived by Kim & Yang[16] is adopted and the bending energy term with planar anisotropy is developed. And to impose the blank holding force properly, the scheme is improved so that the blank holding force imposed on the nodal point in the flange region is dependent on the calculated thickness and keeps in a state of equilibrium with the total blank holding force. The validity of the improved formulation is demonstrated with the simulation of the deep drawing processes of a cylindrical cup, a square cup and a rectangular cup. The computational results with a new scheme are compared with the experimental results and the results with the old scheme which apply the blank holding force to the nodal points along the edge. It can be noted that the computed results are in better agreement with the experimental results than the results from the old scheme in point of the boundary shape of sheet metal in the blank holder. Accordingly, this formulation is proved to apply the planar anisotropy to the modified membrane element adequately and impose the blank holding force with more reality preserving the computational efficiency.

2. Planar anisotropic formulation with the modified membrane

Principle of virtual work

The principle of virtual work describes a form of the equilibrium equation and boundary conditions. At time t0, the shape of sheet material and the distribution of the effective strain are supposed to be given or determined already. The necessary and sufficient condition for the stress field in sheet metal to be given in equilibrium at time t0 + τ , is given from the principle of virtual work[17].

δ σ δεW ijijV= ∫ dV (1)

where is the second Piola-Kirchhoff's stress tensor and σ ij εij is the Lagrangian strain tensor. The second Piola-Kirchhoff's strain tensor is equivalent to Cauchy stress tensor for incompressible material in the convected coordinate system. Therefore, from now on it will denote Cauchy stress tensor for the convenience of using Hill's yield criterion. There has been a great number of finite element analyses using the rigid plastic formulation with the existing membrane element, which is sufficiently successful for the stretching dominant process. In the deep drawing process, however, since sheet metal undergoes bending and unbending around the die profile, the effect of bending cannot be neglected any more. Therefore, the effect of bending should be taken into account for the accurate analysis of the deep drawing process. When sheet metal is in the bending condition, the

strain tensor can be considered to consist of the two terms: δε δε δεij ij

mij

b= + (2) where is the membrane strain, which has the uniform distribution throughout the thickness due to the stretch deformation and is the bending strain, which has linear distribution throughout the thickness due to the bending deformation. By using Eqn(2), Eqn(1) can be expressed as:

ε ijm

εijb

δ σ δε σ δεW dVijijm

Vij

ijb

V= +∫ ∫0 0 dV

dA

(3)

which can be rewritten as[13]: δ σ δε δκW t dA Mij

ijm

Aij

ijA= +∫ ∫0

0 0 (4)

where is the bending moment obtained by integrating the stress multiplied by z throughout the thickness. And z is the distance from the neutral surface and the curvature can be expressed as a second derivative of vertical displacement at the neutral surface. At this time, there is a theoretical problem in making up

M ij

κ ij

κ ij numerically because it cannot be derived in the right way by bi-quadratic shape functions used in the membrane element. In order to obtain the curvature

for the finite element approximation, a higher order element must be used, however, which is less efficient computationally than a linear element. To maintain the computational efficiency with the bending effect considered, the curvature is approximated as the difference of the kink angle of the two neighboring membrane elements[13] as below.

κ ij

κ ii

i iLw w= −

1 1 2,( )

,( ) (5)

where Li is the distance between the centers of two neighboring elements and , are derivatives of normal displacements in the normal direction to the boundary. Although this method is dependent upon the shape and constitution of initial finite element mesh and includes the bending only in the normal direction across the neighboring elements, it has the physical reliability as an upper bound. Using Eqn(5), the bending energy term can be approximated as:

w ,( )11 w ,

( )12

M dA M dA M dijijA A A

δκ δκ δκ0 0 01

12

2∫ ∫ ∫≅ + A (6)

where M1 and M2 are the bending moment corresponding to κ1 and κ2 , defined in Eqn(5), respectively. The orthonormal coordinate in sheet plane, an axis of which is perpendicular to the boundary neighboring elements, will be called as bending associated coordinate.

Constitutive relation for the anisotropic materials There has been several yield functions for the last 3 decades such as Hill's quadratic[18], Gotoh's fourth order[19] and Balat's[20]. Although Hill's quadratic yield function has a few disadvantages[21], it is employed in this paper for its good performance with steel alloys which take a important part in the sheet metal forming processes. The effective stress and strain can be obtained from the work equivalence and incompressibility as:

[ ]σ σ σ σ σ=+ +

+ − + + + + +3

21 2 1 1 2

0 90 0 9090 0

20 90 0 90

245 0 90

2 1 2

( )( ) ( ) ( )( )

/

r r r rr r r r r r r r rx x y y xτ y (7a)

& ( )( )

( ) & & & ( ) & ( )( )( )

&/

ε ε ε ε ε=+ +

+ ++ + + + +

+ ++ +

⎡

⎣⎢

⎤

⎦⎥

23 1

1 2 1 11 2

0 90 0 90

0 90 0 900 90

20 90 90 0

2 0 90 0 90

45 0 90

21 2r r r r

r r r rr r r r r r r r r r

r r rx x y y xyγ (7b)

and by the help of the associated flow rule, the constitutive equation can be constructed from the given yield criterion as follows:

σσε

ε εxr r r rr r r r

r r=+ ++ +

+ +23

10 90 0 90

90 902

0 9090 90

( )( ) &

{( )& & }x y (8a)

σσε

ε εyr r r rr r r r

r r=+ ++ +

+ +23

10 90 0 90

0 02

0 900 0

( )( ) &

{( )& & }y x (8b)

τσ

εγxy xy

r r r rr r r

=+ ++ +

43 2 1

0 90 0 90

0 90 45

( )( )( ) &

& (8c)

where σ and &ε are defined in Eqn(7a) and Eqn(7b). It must be noted that the components of stress and strain are not expressed in the convected coordinate nor in the bending associated coordinate, but in the local Cartesian frame for anisotropic axes. The description for this constitutive equation in the convected coordinate was derived by Kim & Yang[16], which was used to formulate the membrane energy term. The constitutive equation in the bending associated coordinate is derived so forth, using Eqn(8) and tensor transformation rule.

Constitutive equation in the bending associated coordinate

Because , , and have to be calculated in the bending associated coordinate, Eqn(8) is transformed into this situation to derive the bending energy term. While the orthonormal basis vectors consist of p and q in the anisotropic Cartesian frame, , and are orthonormal basis vectors in the bending associated frame of the first element as shown in Fig.1. Fig.1 shows the configuration of two neighboring elements. As the components of and

κ1 κ 2 M1 M2

n11( ) n2

1( ) n31( )

λ μ in the convected coordinate frame are imported at each element, they are expressed as follows:

p b b= =p pii i

i (9a) q b b= =q qi

i ii (9b)

where bi are covariant basis vectors in the convected coordinate which have the form of

b xi i=

∂∂ξ

(10)

and and are given in the initial configuration since the rolling direction before the forming process and the basis vectors in the convected coordinate are known through the deformation.

p i q i

The initial values of and at every step are obtained incrementally from the values at the previous step by the assumption that the angle between anisotropic axis and the principal convected coordinate axis remains constant during the deformation[16]. Then the interior angle between and n is derived as:

p i q i

λ 11( )

θ λ= − •cos ( )( )111n = =− −

•cos ( ) cos ( )( ) ( )111 1

11λ λ •

∂∂ξ

ii

iib n r n (11)

The components of stress and strain in Eqn(8) are measured by λ and μ . By assuming that all the orthonormal basis vectors in the anisotropic coordinate frame and bending associated coordinate frame, so to speak , , and coexist in a plane, Eqn(8) can be expressed through the vector transformation rule as:

λ μ n11( ) n2

1( )

σσε

εθ θ

θ θ θ

θ θ

x xr r r r r rr r r r

r rr r r r

r r r

* *& ( ) & [ ( )cos cos sin

( )( )sin cos sin

( )cos sin

( )( )]

= + ++ +

+ +

++ +

+ +

++ +

23

1

1

2 1

0 90 0 9090

490

2 2

90 902

0 90

04

02 2

0 02

0 90

2 2

0 90 45

θ

(12)

where both of and are the stress and strain tensor components measured along nσx* & *εx 1 in the

bending associated coordinate frame.

3. Finite element approximation

With the use of the convected coordinate system, the derived virtual work energy formulation is more easily transformed into a form of finite element approximation. The orthogonal Cartesian coordinates of a material point at time t0 are described with Xi and the orthogonal Cartesian components of the displacement vector of a material point during the time interval are described with u

τi and. Then the natural Lagrangian strain in the natural convected coordinate

system has the form as follows:

E u x x u u uij i j i j i j= + +

12

[∂ ∂∂ξ ∂ξ

∂ ∂∂ξ ∂ξ

∂ ∂∂ξ ∂ξ

α α α α α α

]

X

(13)

The coordinates and displacements of a material point can be approximated in the following matrix form:

( )X H≡ =X X XT1 2 3 $ , ( )u≡ u u u

T1 2 3 HU= (14)

where H, a matrix having order 3 12 is a shape function and and U are the nodal point coordinate and displacement vectors. Then the natural Lagrangian strain is expressed as:

× $X

EX B U U B X U B UX B U U B X U B UX B U U B X U B U

=⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟=

+ ++ ++ +

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

EEE

T T T

T T T

T T T

11

22

12

1 1 1

2 2 2

3 3 3

12

$ $

$ $

$ $ (15)

where BB1, B2B and BB3 are all symmetric matrices given by

B H H B H H B H H H H1 1 1 2 2 2 3 1 2 2 1

12

= = = +∂ ∂∂ξ ∂ξ

∂ ∂∂ξ ∂ξ

∂ ∂∂ξ ∂ξ

∂ ∂∂ξ ∂ξ

T T T

, , (T

) (16)

After the algebraic manipulation, the membrane energy term in Eqn(4) is expressed as[16] δ σδ ε δW tdAm

VT m= =∫ ( ) ( )Δ0 U Q U (17)

In the natural convected coordinates, the metric tensor components of the deformed configuration and the basis vectors in the deformed body are denoted by gij and bi, respectively. In the Fig.1, considering an element P(1 or 2), the derivative of the displacement normal to the neighboring element boundary, can be expressed as w P

,( )1

w w w w w gp

P

PP P P

P

ii P

P

iij

j,( )

( )

( )( ) ( ) ( )

( )( )

( )

11

1 1 1= = ∇ = =• • •∂∂

∂∂ξ

∂∂ξn

n n b nr

b

u

(18)

The normal displacement of nodes in each element, , can be described as: w P( )

w P P( ) ( )= •n3 (19) where is a unit vector normal to the element surface. By the simple tensor manipulation,

can be expressed as follows: n 3

( )P

w P,( )1

w g g

g g

P P Pi

P ijj

P ijj

Pi

P

P ijj

PT

iP P ij

j

T

iP P

,( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ]

( ) ( ) ( ) [( )

1 1 3 1 3

1 3 1

= = ×

= × = ×

• • • • • •

• • • • • •

n n H U b n b n H U

n b U H n n b H n U

∂∂ξ

]3

∂∂ξ

∂∂ξ

∂∂ξ

(20)

Accordingly, the kink angle between two neighboring elements can be given as:

w w g gijj

T

iij

j

T

i,( )

,( ) ( ) ( ) ( ) ( )[( ) ( ( ) ( ) ( ( )]1

112

11

31

12

32− = × − ו • • •n b H n n b H n∂

∂ξ∂∂ξ

•U

]

(21)

where U U U U U U U=[ 1 2 3 4 5 6

T (22) whose subscribed index denotes the connectivity defined in the Fig.2.

By using Eqn(5) and (21) the bending energy term in Eqn(6) can be derived as: δ δ δ

δ δ δ

W M dA M dA

C dA C dA

bA A

A AT b

= +

= + =

∫ ∫∫ ∫

0 0

0 0

1 1 2 2

1 1 2 2

( ) ( )

( ) ( ) (

Δκ Δκ

Δκ Δκ Δκ Δκ U Q U) (23)

where

Q U B B Ub TA

e

NC

Ve( ) ( )= ∫∑ 4 40 1

1

dA (24)

In Eqn(24), N is the total number of boundary lines between elements in the whole domain, Av is the area inside the dashed line of Fig.2 and C is the stiffness modulus due to bending, which is introduced by Eqn(12). By the tensor manipulation, can be given as below. B4

B n b H n n b H n4 11

31

12

321

= × − ×⎡

⎣⎢

⎤

⎦⎥• • • •

Lg g

i

ijj

T

iij

j

T

i( ) ( ) ( ) (( ) ( ) ( ) ( )∂∂ξ

∂∂ξ

) (25)

In a sheet bending problem, it can be assumed that strain induced from the bending is linearly distributed along the thickness and a component of bending moment induces a corresponding components of strain and does not affect other components of strain. By using this assumption, C is obtained as:

C t r r r r r rr r r r

r rr r r r r r r

= + ++ +

+ +

++ +

+ ++

+ +

3

0 90 0 9090

490

2 2

90 902

0 90

04

02 2

0 02

0 90

2 2

0 90 45

181

12 1

σε

θ θ θ

θ θ θ θ θ

Δ( )[ ( )cos cos sin

( )( )sin cos sin

( )cos sin

( )(]

)

+

(26)

where t is the thickness of sheet metal. It can be noted that C is dependent on the cube of thickness and a function of which should be determined at every element by Eqn(11). By eliminating the virtual displacement , the variational expression of Eqn(4) is approximated to the following equation:

θδU

Q U Q U P U Fm b( ) ( ) ( )+ = (27) which is a nonlinear function of U. This nonlinear algebraic equation can be solved iteratively by Newton-Rapson method.

4. Implementation of blank holding force application

Since the membrane element looks to have neither thickness nor thickness variation to all

appearance, which is not able to realize the effective contact treatment. So far, in the membrane analysis all the part in the blank holder region has been regarded as keeping in contact with the blank holder in spite that is not true obviously. Thus, the contact condition cannot be the adequate criterion for applying the blank holding force. The most widely used method to apply the blank holding force with the membrane element is to impose the uniform traction boundary condition at the periphery of the flange as described in Fig.3 , however, which is not realistic and induces erroneous deformation of sheet metal during the deep drawing process. The blank holding force acting on the i-th node can be described with this scheme as below.

BHF D D

DBHFi

i i

ii

nb= +⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

−

=

−

∑

12

1

1

1 (27)

where Di is the distance from the i-th node to i+1-th node, nb is the number of nodes along the periphery in the blank holder region and BHF is the total blank holding force. Since the analysis for the cylindrical cup deep drawing process carried out with this scheme cannot predict concentration of blank holding force in the direction at an angle of 45 degrees with respect to the rolling direction due to the thickening of sheet metal in the prescribed direction, it produces the incorrect deformed shape and the exaggerated earing phenomenon. As a remedy, a new scheme for applying blank holding force in the membrane analysis is introduced. Because the modified membrane element has the same appearance and structure as the ordinary membrane element, the new scheme can be applied to the modified membrane element, too. The new scheme takes the equilibrium of reaction force of sheet metal with the blank holding force and the effect of thickness derivation during the

deformation into account. The abstract concept of this scheme is exhibited in Fig.4 , where t and tc represent virtual

material thickness calculated form the analysis and virtual compressed thickness, respectively. The equilibrium equation between the blank holder and sheet metal can be expressed as:

σ bhfc

e

nbc

t t d BHFe

( , ) ΩΩ∫∑

=

=1

(28)

where is the virtual compressive stress which is dependent upon t and tσ bhfc. And nbc is the

number of elements which are thicker than tc. As noted in Eqn(28), is dependent upon tσ bhfc and

consequently equilibrium between the reaction force in material and the blank holding force can be achieved with a proper distribution of . Employing the bi-section search method, tσ bhf

c is chosen at the equilibrium condition.

Generally speaking, the whole part of an element is not thicker than tc. Therefore, to carry out the integration in Eqn(28), the region should be predicted precisely in which the thickness is larger than tc, and in other words, the contact between die and sheet occurs. To realize this, the thickness dimensions at every nodal points are obtained by least square method from the thickness distribution at the integration points which is calculated in the analysis. It was assumed in this paper that the thickness in an element has a linear distribution. By this assumption, the four node bilinear quadrilateral element can by categorized into 6 types according to the contact condition of sheet metal in the blank holder as shown in Fig.5. The region thought to be in contact with blank holder was hatched. The integration is carried out by dividing this predicted region into triangles as shown in Fig.5 and was obtained at every integration points in the triangular elements as below:

σ bhf

σ ε εbhf bhf nk= +( 0 ) (29) where k, n are material hardening constant and exponent respectively, and ε 0 is pre-strain which is decided by the initial yield stress. In other words, it was assumed that the compressive deformation of sheet metal due to the blank holder behaves identically with the tensile deformation. In Eqn(29), is the compressive strain, which is defined as: ε bhf

ε bhf

c

tt

= ⎛⎝⎜

⎞⎠⎟

ln (30)

In accordance with the previous scheme, the reaction force in sheet material due to the blank holding force can be calculated with the assumed tc and becomes equivalent to the blank holding force at the specific tc after the several iterations by the bi-section search algorithm.

The reaction force calculated in the triangular element was distributed to the nodal points of four node bilinear quadrilateral element by assuming that the reaction force was imposed onto the centroid of triangular element and using the shape function of the four node element.

BRF BRF N s ti iα α

α α= × ( , ) (31) where is the reaction force calculated in the BHFα α − th triangular element contained in a four node element, is the reaction force distributed to the i-th nodal point of the four node

element and N

BHFiα

i is the shape function of four node element. Since with any , the

total sum of the reaction force at the every nodal points of four node element is equivalent to the reaction force in the triangular element. Moreover, and which were the local coordinates in the four node element corresponding to the centroid of the triangular element was obtained from the general coordinates, and by the least square method as shown in Fig.6.

Ni s ti

( , )α α ==∑ 1

1

4α

sα tα

xα yα

5. Numerical results

To validate the modified scheme applying the blank holding force, the deep drawing processes

of a cylindrical cup, a square cup and a rectangular cup have been simulated. Deep drawing of a cylindrical cup from a circular blank

The material used in the computation is aluminum-killed steel whose stress-strain characteristic is expressed as σ ε= 5192 0 247. . ( /kgf mm2) and r0, r45 and r90 are 2.01, 1.443 and 2.571. The process variables used in the simulations are as follows:

Sheet thickness: t = 0.8mm Punch radius and die radius: 5mm Diameter of punch: 60mm Diameter of sheet blank: 120mm Blank holding force: 1000kgf Coulomb coefficient of friction: 0.12

The finite element mesh system is constructed with 347 nodal points and 318 elements. By the

geometric symmetry, only a sector of a sheet blank, 1/4 of the domain, is considered. Fig.7 shows the deformed mesh configurations at the stroke of 40mm. The deformed meshes demonstrate their stability against the distortion or numerical buckling and show the earing phenomenon due to the anisotropy.

The calculation completes the simulation at the stroke of about 50mm and the CPU time takes 9120sec with HP C100 workstation. The variation of the drawing load with respect to the punch displacement is calculated and shown in Fig.8. Both results of the modified scheme with the distributed blank holding force in the blank holder region and the pre-existent scheme are in good agreement with the experimental load[12], while that is more precise than this near the largest value of the load. The difference in the two analyses is due to the small change of the deformed shape according to the scheme imposing the blank holding force. In order to show the change of the deformed shape, the corresponding boundary shapes of deformed meshes with respect to both schemes are shown in Fig.9. The result form the old scheme applying the blank holding force along the edge of the flange, exhibits exaggerated earing phenomenon, since it cannot predict the thickening of sheet metal along the direction at an angle of 45 degree with respect to the rolling direction and the concentration of blank holding force. The results from the modified scheme is in better agreement with the experimental results. During the analysis with the modified scheme, the region in which the blank holding force is applied, changes as shown in Fig.10. It can be known the blank holding force is concentrated due to the thickness variation and prevents the inflow of sheet metal in the part of thicker region. In order to confirm the reliability of the predicted contacting region obtained from the numerical analysis, it is compared with the results from the author’s own experiment with the cylindrical punch whose radius is 25mm in Fig.11. It can be seen that the contacting region is underestimated in the numerical analysis, which is due to the ignorance of the compression of the thickness by the blank holding force. Deep drawing of a square cup from a square blank

The material used in the computation is cold rolled steel whose stress-strain characteristic is expressed as σ ε= +584 8 0 00803 0 2608. ( . ) . ( /kgf mm2) and r0, r45 and r90 are 1.465, 1.319 and 1.884 which was measured by the author’s own experiment. The process variables used in the simulations are as follows:

Sheet thickness: t = 0.774mm Blank size: 160mm 160mm ×Blank holding force: 3000kgf Coulomb coefficient of friction: 0.15

The schematic view of the punch and the die used in both of the simulations and the experiment

is shown in Fig.12. The finite element mesh system consists of 587 nodal points and 546 elements. By the geometric symmetry, 1/4 of the whole blank, is considered. Fig.13 shows the deformed mesh configurations computed by the modified scheme and the old scheme at the stroke of 40mm. The deep drawing process of a square cup has so severe geometric constraint that sheet metal in the wall of a cup cannot move across the side, which prevents from the directional variation due to the anisotropy. That is the reason why the deformed shapes from the modified scheme and the old scheme exhibits little difference in the analyses of a square cup.

The calculation completes the simulation at the stroke of about 50mm and the CPU time takes 26380sec with HP C100 workstation. The variation of the drawing load with respect to the punch displacement is calculated and shown in Fig.14. The drawing load from the modified scheme with the distributed blank holding force in the blank holder region is in good agreement with the experimental load. The results from the old scheme shows a little difference due to the change of the method applying blank holding force. The corresponding boundary shapes of deformed meshes computed from both schemes are shown in Fig.15. Both results are in good agreement with the experimental results and alike with each other. Therefore, it can be remarked that the method applying blank holding force in the analysis of a square cup deep drawing process does not have an effect on the deformation. But, to verify this, more analysis must be carried out with the various dimensions of punch and die and diverse sizes of blank. During the analysis with the modified scheme, the region in which the blank holding force is applied, changes as shown in Fig.16. This result is compared with the experimental results as shown in Fig.17. Deep drawing of a rectangular cup from a rectangular blank

The material used in the computation is cold rolled steel whose stress-strain characteristic is expressed as σ ε= +58 78 0 0009 0 274. ( . ) . ( /kgf mm2) and r0, r45 and r90 are 1.833, 1.434 and 2.016. The process variables used in the simulations are as follows:

Sheet thickness: t = 0.69mm Blank size: 130mm 170mm ×Blank holding force: 1600kgf Coulomb coefficient of friction: 0.11

The schematic view of the process is shown in Fig.18. The finite element mesh system consists

of 529 nodal points and 484 elements. By the geometric symmetry, 1/4 of the whole blank, is considered. Fig.19 shows the deformed mesh configurations computed by the modified scheme and the old scheme at the stroke of 40mm. It can be seen that the meshes flow into the cavity along the longer side in the results from the old scheme. But, at that time the results from the modified scheme shows that the blank in the flange remains outside the cavity. This difference is due to only the method variation applying the blank holding force.

The calculation completes the simulation at the stroke of about 55mm and the CPU time takes 23160sec with HP C100 workstation. The variation of the drawing load with respect to the punch displacement is calculated and shown in Fig.20. The drawing load from the modified scheme with the distributed blank holding force in the blank holder region is in good agreement with the experimental load[23], while the results from the old scheme shows considerable difference from the experimental results. This remarkable difference represents the effect of blank holding force.

In order to show the change of the deformed shape, the corresponding boundary shapes of deformed meshes with respect to both schemes are shown in Fig.21. The old scheme cannot predict the thickening of sheet metal along the rolling direction and the concentration of blank holding force, which causes the overestimated inflow of sheet metal. The results from the modified scheme

is in better agreement with the experimental results. During the analysis with the modified scheme, the region in which the blank holding force is applied, changes as shown in Fig.22. It can be known the blank holding force is localized along the rolling direction due to the thickness variation and prevents the inflow of sheet metal in the part of thicker region. The thickness strain distribution along the rolling direction(the longer side) at the stroke of 30mm is shown in Fig.23. The thickness strain from the modified scheme is smaller than the one from the old scheme near the flange region, which is in better agreement with the experiment. And the thickness strain distribution along the transverse direction(the shorter side) at the stroke of 30mm is shown in Fig.24. It can be noted that the thickness strain is not influenced by the change of the method applying the blank holding force.

6. Conclusions

A modified membrane finite element formulation with planar anisotropy is derived for the

incremental analysis of sheet metal forming processes and the new method applying the blank holding force is suggested in order to achieve the more realistic deformed shapes. In this scheme, the nodal force due to the blank holding force is decided by the thickness distribution. The present analysis has been carried out for deep drawing processes of a cylindrical cup, a square cup and a rectangular cup and compared with the results from the pre-existent scheme applying blank holding force to demonstrate the validity of the new scheme. It can be known from the comparison that the new scheme controls the exaggerated earing phenomenon in the analysis of a cylindrical cup drawing process and imposes the blank holding force on the proper region in the blank holder. And using this method, the prediction of the more precise deformed shape in the forming processes, is expected. References 1. J. H. KIM, S. I. OH and S. KOBAYASHI, Analysis of stretching of sheet metal with

hemispherical punch. Int. J. Mach. Tool Des. Res. 18, pp. 209 (1978) 2. H. ISEKI and T. MUROTA, Analysis of deep drawing of non-axisymmetric cup by the finite

element method. Advanced Technology of Plasticity (Proc. of 1st ICTP) 1, pp. 678 (1984) 3. W. J. CHUNG, Y. J. KIM and D. Y. YANG, Rigid-plastic finite element analysis of hydrostatic

bulging of elliptic diaphragms using Hill’s new yield criterion. Int. J. Mech. Sci. 31, pp. 193 (1989)

4. P. KECK, M. WILHELM and K. LANGE, Application of the finite element method to the simulation of sheet forming processes: comparison of calculations and experiments. Int. J. Numer. Engng. 30, pp. 193 (1989)

5. H. Y. JIANG and D. LEE, Numerical simulation of sheet metal forming process based on large deformation shell elements. in Proc. NUMIFORM 92, pp. 485 (1992)

6. J. H. CHENG and N. KIKUCHI, An analysis of metal forming processes using large deformation elastic-plastic formulations. Comput. Meth. Appl. Mech. Engng. 49, pp.71 (1985)

7. D. Y. YANG, W. J. CHUNG and H. B. SHIM, Rigid-plastic finite element analysis of sheet metal forming processes with initial guess generation. Int. J. Mech. Sci. 32,pp. 687 (1990)

8. Y. T. KEUM, E. NAKAMACHI, R. H. WAGONER and J. K. LEE, Compatible description of tool surface and FEM meshes for analyzing sheet forming operations. Int. J. Numer. Meth. Engng. 30, pp. 1471 (1990)

9. E. MASSONI, M. BELLET, J. L. CHENOT, J. M. DETRAUX and C. DE BAYBEST, A finite element modeling for deep drawing of thin sheet in automotive industry. Advanced Technology of Plasticity (Proc. of 2nd ICTP), pp. 719 (1987)

10. J. L. BATOZ, P. DUROUX, Y. Q. GUO and J. M. DETRAUX, An efficient algorithm to estimate the large strains in deep drawing. in Proc. NUMIFORM 89, pp. 383 (1989)

11. S. C. TANG, Application of membrane theory to automotive sheet metal forming analysis. Proc.

NUMIFORM 92, pp. 549 (1992) 12. D. Y. YANG, H. B. SHIM and W. J. CHUNG, Comparative investigation of sheet metal forming

processes by the elastic-plastic finite element method with emphasis on the effect of bending. Eng. Comput. 7, pp. 274 (1990)

13. H. HUH, S. S. HAN and D. Y. YANG, Modified membrane finite element formulation considering bending effects in sheet metal forming analysis. Int. J. Mech. Sci. 36, pp. 659 (1994)

14. D. W. JUNG, I. S. SONG and D. Y. YANG, An improved method for application of blank holding force considering sheet thickness in deep-drawing simulation of planar anisotropic sheet. J. of Materials Processing Technology 52, pp. 472 (1995)

15. D. Y. YANG, D. W. LEE, S. W. LEE and J. W. YOON, Effective blankholder gap control algorithm for implicit and explicit codes in sheet metal forming processe. NUMISHEET 96, pp. 32 (1996)

16. D. Y. YANG and Y. J. KIM, A rigid-plastic finite element formulation for the analysis of general deformation of planar anisotropic sheet metals and its application. Int. J. Mech. Sci. 28, pp. 825 (1986)

17. L. MALVERN, Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, Englewood Cliffs, NJ (1969)

18. R. HILL, The mathematical theory of plasticity. Clarendon Press, Oxford (1950) 19. M. GOTOH, A finite element analysis of the rigid-plastic deformation of the flange in a deep-

drawing process based on a fourth-degree yield function-II. Int. J. Mech. Sci. 22, pp. 367 (1980)

20. F. BARLAT, Plastic behavior and stretchability of sheet metals. part I : A yield function for orthotropic sheets under plane stresss conditions. Int. J. Plasticity. 5, pp. 51 (1989)

21. W. F. HOSFORD, Comments in anisotropic yield criteria. Int. J. Mech. Sci. 27, pp. 423 (1985) 22. H. HUH and S. S. HAN, Numerical simulation of rectangular cup drawing processes with

drawbeads, NUMIFORM 95, pp. 723 (1995)

pq

n11( )

n21( )

n31( )

n12( )

n32( )

n22( )

θ

θ

Fig. 1. Configuration of two neighboring elements in the bending associated coordinate.

1

12

34

5

6

A V 2

Fig. 2. Virtual element in the neighboring element and its connectivity.

i-th nodeBHFi

Di

Di−1

Fig. 3. Scheme applying the blank holding force along the edge boundary.

tc

blank.holder

blank.holder

sheet metal

Fig. 4. Virtual compressed thickness in the modified scheme applying the blank holding force.

type 1 type 2 type 3

type 4 type 5 type 6

Fig. 5. Types according to the contact condition of a four node bilinear quadrilateral element.

( , )x yα α

BRFα

BRFiα

Fig. 6. Distribution of Nodal force from a triangular element to a four node element.

(a) (b)

Fig.7. Deformed mesh configurations computed by the modified scheme and the old scheme at the stroke of 40mm: (a) modified scheme; (b) old scheme.

0

1000

2000

3000

4000

5000

6000

7000

0 10 20 30 40 50 6

Punch Stroke (mm)

Punc

h Lo

ad (k

gf)

0

Modified b.h.f.Edge b.h.f.Experiment[12]

Fig. 8. Comparison of the drawing load between computed results and the experimental results in the deep drawing of a circular cup.

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70

Rolling direction coordinate(mm)

Tran

sver

se d

irect

ion

coor

dina

te(m

m) Modified b.h.f.

Edge b.h.f.Experiment[12]

H=15.8mm

H=30.4mm

Fig. 9. Comparison of the boundary shape contour between the computed results and experimental results in the deep drawing of a circular cup.

H=10mm H=20mm H=30mm H=40mm

Fig. 10. Change of the region with respect to the stroke in which the blank holding force is applied in the analysis of a deep drawing process of a circular cup.

H=20mm H=30mm

(a)

(b)

Fig. 11. Comparison of the contact region in the blank holder with respect to the stroke between the experiment and the numerical results: (a) experimental results; (b) numerical results by the modified scheme.

Fig. 12. Schematic view of the process of the deep drawing process of a square cup.

(a) (b)

Fig. 13. Deformed mesh configurations computed by the modified scheme and the old scheme at the stroke of 40mm: (a) modified scheme; (b) old scheme.

0

2000

4000

6000

8000

10000

0 10 20 30 40 50 6

Punch Stroke(mm)

Punc

h Fo

rce(

kgf)

0

Modified b.h.f.Edge b.h.f.Experiment

Fig. 14. Comparison of the drawing load between computed results and the experimental results in the deep drawing of a square cup.

0

20

40

60

80

100

0 20 40 60 80 100

Rolling Direction(mm)

Tran

sver

se D

irect

ion(

mm

)

Modified b.h.f. Edge b.h.f. Experiment

Fig. 15. Comparison of the boundary shape contour between the computed results and experimental results in the deep drawing of a square cup.

H=10mm H=20mm H=30mm H=40mm Fig. 16. Change of the region with respect to the stroke in which the blank holding force is applied in the analysis of a deep drawing process of a square cup.

H=20mm H=30mm H=40mm Fig. 17. Change of the contact region in the blank holder with respect to the stroke in the experiment.

Fig. 18. Schematic view of the process of the deep drawing process of a rectangular cup.

(a) (b)

Fig. 19. Deformed mesh configurations computed by the modified scheme and the old scheme at the stroke of 40mm: (a) modified scheme; (b) old scheme.

0

1000

2000

3000

4000

5000

6000

0 10 20 30 4

Punch Stroke (mm)

Punc

h Lo

ad (k

gf)

0

Modified b.h.f.Edge b.h.f.Experiment

Fig. 20. Comparison of the drawing load between computed results and the experimental results in the deep drawing of a rectangular cup.

0

10

20

30

40

50

60

70

80

90

0 10 20 30 40 50 60 70 80 90

Rolling Direction(mm)

Tran

sver

se D

irect

ion

(mm

)

Modified b.h.f.Edge b.h.f.Experiment

H=10mm

H=20mm

H=30mm

Punch

Fig. 21. Comparison of the boundary shape contour between the computed results and experimental results in the deep drawing of a rectangular cup.

H=10mm H=20mm H=30mm Fig. 22. Change of the region with respect to the stroke in which the blank holding force is applied in the analysis of a deep drawing process of a rectangular cup.

- 0.2

- 0.1

0.0

0.1

0.2

0.3

0.4

0 10 20 30 40 50 60 70 80 90

Initial Distance from the Center (mm)

Thic

kness S

train

Modified b.h.f.Edge b.h.f.Experiment

Fig. 23. Comparison of the thickness strain distribution along the longer side between the computed results and experimental results in the deep drawing of a rectangular cup. (punch stroke: 30mm)

-0.050

0.000

0.050

0.100

0 10 20 30 40 50 60 70

Punch Stroke (mm)

Punc

h Lo

ad (k

gf)

Modified b.h.f.Edge b.h.f.Experiment

Fig. 24. Comparison of the thickness strain distribution along the longer side between the computed results and experimental results in the deep drawing of a rectangular cup. (punch stroke: 30mm)