Analytical modeling of the extrusion process using the electrostatics concept

Accepted Manuscript Analytical modeling of the extrusion process using the electrostatics concept S.A. Tabatabaei, K. Abrinia, S.M. Tabatabaei, M. Shahabadi, M.K. Besharati PII: DOI: Reference:

S0167-6636(15)00076-9 http://dx.doi.org/10.1016/j.mechmat.2015.03.007 MECMAT 2395

To appear in:

Mechanics of Materials

Received Date: Revised Date:

7 August 2014 6 December 2014

Please cite this article as: Tabatabaei, S.A., Abrinia, K., Tabatabaei, S.M., Shahabadi, M., Besharati, M.K., Analytical modeling of the extrusion process using the electrostatics concept, Mechanics of Materials (2015), doi: http://dx.doi.org/10.1016/j.mechmat.2015.03.007

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Analytical modeling of the extrusion process using the electrostatics concept S. A. Tabatabaeia*, K. Abriniaa, S.M. Tabatabaeib, M. Shahabadic, M.K. Besharatia a

School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran b

c

MSc Graduated Student in Electrical Engineering, Iran

School of Electrical and Computer Engineering, College of Engineering, University of Tehran, Tehran, Iran

*Corresponding Author Address: Kargar Shomali, St., School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran, P.O. Box 14395-515; Email: [email protected], Tel:(+98)- 21-61114026, Fax: (+98) - 21-66461024

ABSTRACT Analytical and numerical modeling of the metal forming processes has been studied by many researchers during the past decades. So far, several methods have been introduced to model the material behavior during the forming processes. In this research, the analogy between the classical plasticity theorem and electrostatic field line formation has been investigated for the first time. It is shown that the velocity potential function of movement in the plasticity and the charge-free electrostatic field, both are in the form of Laplace’s relation. The aim of the current research is to model the metal forming processes using the equations governing the electrostatic fields. To demonstrate the effectiveness of the proposed method, the extrusion process is used as the benchmark. A three-dimensional CAD model of the deformation region is constructed and two different voltages are applied to the entry and exit sections based on the geometrical and extrusion parameters. Then, the Laplace’s relation is solved numerically and the relative extrusion pressure, particle flow path in the deforming region, and maximum plastic strain are calculated. To verify the proposed method, different sections are modeled and the results are compared with the literature. Finally, experimental tests are conducted for the extrusion process of the round billets to complex sections. Significant agreement has been observed between the experimental results and the analytical ones. It is shown that the extrusion process could be modeled easily and accurately using the mathematical formalism of the electrostatics. Key words: Electrostatics, EPLs, Extrusion, Analogy, Experiment, Upper bound theory.

1. INTRODUCTION The process of metal forming has been used in industry for a long time and several analytical and numerical solutions have been presented in the literature for the prediction of the process parameters. The primary purpose in analyzing the metal forming operations is to determine the required forming force/pressure, stress and strain distribution and predict the unknown process variables in order to achieve the optimum design of die/work piece. The modeling of the metal forming processes can be categorized as physical, numerical and analytical techniques. The numerical modeling of the metal forming processes is mostly based on the finite element (FE) methods (Surdon, and Chenot, 1986; Dyduch, 1992; Sosnowski, 1992; Dvorkin and Petocz, 1993; Hallquist, 1995). The finite element modeling methods suffer from convergence problem and mesh distortion (Braess and Wriggers, 2000) especially in complex geometries.

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Nomenclature α Velocity potential function Vx, Vy ,Vz

Velocity components

J

Total power consumption

m

Frictional factor

W
e

Power dissipations due to velocity

V0

Velocity of the material at

discontinuity at the entrance section

∇

Vector differential operator

Power dissipations due to velocity

W
x

the entry section

Velocity of the material at

Vf

discontinuity at the exit section

the exit section

E

Electric field in electrostatic field

W
f

Power dissipations due to friction

Se or A0

Area of the inlet section

ρ

Charge density in electrostatics

W
i

Power dissipations due to internal

Sx or Af

Area of the outlet section

Ex, Ey ,Ez

Electric field components

deformation

ε0

Vacuum permittivity coefficient

Yield strength/ mean effective stress

Y

of billet material

φ

εp

Electric potential (electric intensity)

d or L

Strain of particle at point P in the deforming

ε
eff

region

Length of the deforming zone Strain rate of the particle at point P in

P

T

∆T p

Time Traveling time of point P

the deforming region

The physical modeling is a simple technique for simulation of the actual metal forming processes using the model material and dies (Sofuoglu and Gedikli, 2004). The physical methods require special tools to model the metal forming processes as close as to the real conditions. However, any change in the geometry and boundary conditions needs extra simulations. Moreover, the main analytical solutions for modeling of the metal forming processes are the slip-line field method (Prager and Hodge, 1951), the upper (Hill, 1950) and lower (Hoffman and Sachs, 1953) bound theories. The slipline field theory is based on plane strain condition and cannot be applied in three-dimensional models (Hill, 1950). The upper and lower bound methods have been widely applied in different metal forming processes such as rolling, forging, ECAP/ ECAE, extrusion and many other metal forming processes (Kukureka et al., 1992; Kiuchi et al., 1981; Moon and Tyne, 2000; Altan et al., 2005; Pérez and Luri, 2008; Abrinia and Fazlirad, 2009; Khalili et al., 2012). The current analytical methods have been applied either in simple or less complex geometries and boundary conditions. The upper bound method is based on the definition of a kinematically admissible velocity field (KAVF) in the deforming region. Having defined the KAVF, the velocities, strain rates, strains, and consequently the forming force/pressure can be calculated. In the conventional upper bound method, there is no global technique for segmentation of the deforming region and proper definition of the KAVF. So far, analytical methods for analyzing the velocity fields in the solid mechanics have been investigated by Talbert and Avitzur (1996) using the streamline functions. Rejaeian and Aghaie-Khafri (2014) used the fluid mechanics and streamline function method to define the velocity fields in the ECAP process. They used the Eulerian and Lagrangian method to define the material flow in the deforming region and applied the upper bound theory to find the optimum velocity field. There is no straightforward and simple technique for proper definition of the KAVF, and particle flow path in the deforming zone of the metal forming processes.

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This research is the supplementary material for the previous works of the authors to apply the equi-potential lines (EPLs) in conjunction with the upper bound theory. The aim of this research is to model the extrusion process in 3D and for any complex geometry using the electrostatics concept. The present authors used the idea of EPLs for accurate representation of the deforming zone with high order polynomial (Tabatabaei et al. 2013) and Bezier curves (Tabatabaei et al. 2014). It was postulated that the intersection of planes- passing through the corners/edges of the final section and gravity center point of the deforming zone- and the drawn EPLs can simulate the particle flow from the entry section to the final one. Then different kinds of curves such as 3rd order polynomial and Bezier curves were fitted from the intersection points. The obtained curves were used in the conventional upper bound theory to predict the extrusion pressure. This method is easily applicable for symmetric cross-sections in the extrusion process since the material flow can be followed by few planes passing through the EPLs and corners/edges of the final section. However, using the concept of intersection of drawn EPLs and planes for predicting the material flow in the deforming region neither is realistic nor applicable for the complex sections. In addition, the idea of EPLs was used to investigate the effect of internal profile of the deforming zone on the extrusion pressure (Tabatabaei et al. 2014). The internal profile in the extrusion of round billets to the square sections was modeled from EPLs method and then the experimental tests were performed using two different dies: a) conventional die; with linear section variation between the entry and exit sections , b) EPL die; obtained by connecting the drawn EPLs between the circular and square sections. The 3D cavity of the deforming region was modeled in Matlab, produced with CNC machining and finally the experimental tests were performed. It was shown that the EPL die had lower extrusion pressure compared to the conventional die. The concept behind such result was that the drawn EPLs between the entry and exit sections follow the least energy principle so the designed EPL die would lead to minimum extrusion pressure (Tabatabaei et al. 2014). The aim of this research was to propose a new method for analytical modeling of the metal forming processes without the above mentioned limitations of the physical, numerical and analytical methods. In the current study, the analogy between the metal forming process and the electrostatics was investigated for the first time. The constitutive relations in the classical plasticity theorem and the electrostatic equations were investigated to obtain the analogue relations and parameters between the two theories. To investigate the effectiveness of the electrostatic concept in the analytical modeling of the metal forming processes, the extrusion process was used as the benchmark. The intermediate sections in the deforming region of the extrusion process of complex sections; between the initial and final sections were predicted using the concept of equi-potential lines (EPLs). Moreover, a novel technique for particle flow path in the metal forming processes using the electrostatic concept was introduced. The particle flow path in the deforming region was estimated based on the streamlines orthogonal to the drawn EPLs. In despite of the conventional analytical methods a 3D-CAD model of the deformation region was constructed and used in the computational calculations of the extrusion power. In the following, a new formulation for strain calculation in the deformation zone was proposed. To validate the electrostatic method in the extrusion modeling, different sections varying from a simple circular and square to the complex I-, U-, and T- shaped sections were modeled using the proposed method and the results were compared with the literature. Finally, the experimental tests were performed

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for U- and non-symmetric I-shaped sections in the cold and hot extrusion conditions respectively and the results were compared with those of the electrostatic modeling.

2. THE UNDERLYING THEORY 2.1 ANALOGY BETWEEN THE METAL FORMING PROCESS AND ELECTROSTATIC In the classical plasticity theorem the incompressibility condition during the metal flow is assumed. This means that during the plastic deformation of the material the volume constancy in the deforming region should be preserved. In other words, if the material volume remains constant and the velocity field is non-spinning, the following equation holds true:

∇2α =

∂ 2α ∂x12

+

∂ 2α ∂x2 2

+

∂ 2α ∂x32

=0

(1)

Where, α is a velocity potential function of movement or strain and xi (i=1, 2, 3) are the coordinates (Xiaona and Fuguo, 2009). From the incompressibility criterion, the sum of the velocity variations in xyz directions should be zero:

ε
ii =

∂V x ∂V y ∂V z + + =0 ∂x ∂y ∂z

(2)

In the above relation, Vx, Vy and Vz are the components of the velocity in an arbitrary point in the deforming region. Relation (2) can be re-written as:  ∇. V = 0

(3)

Where, ∇ is the vector differential operator that means the gradient of velocity at any point of the material is zero and the particles in the deforming region move along the minimum work path (Yu, 2006). On the other hand, from the electrostatics, the governing equation for the electric field E; containing a charge density of ρ; can be expressed as follows (Lee et al., 2002; Hayt and Buck, 2005):

 ρ ∇.E =

(4)

ε0

Where, ε0 is the vacuum permittivity coefficient. Moreover, the potential value φ; is obtained by the following relation:

 E = −∇ϕ

(5)

Relation (5) means that the gradient of electric potential (electric intensity) is the exterior normal of equi-potential lines. The gradient of electric potential gives the minimum work path between the potential lines and is in accordance with the least-energy principle (Yu, 2006). Substituting (5) into (4) yields:

4

∇ 2ϕ = −

∇ρ

(6)

ε0

For the charge-free condition, the governing equation of electrostatics can be represented with Laplace’s relation:

∇ 2ϕ = 0

(7)

The relation (7) in an arbitrary point with xi (i=1, 2, 3) coordinates can be written as:

∇ 2ϕ =

∂ 2ϕ ∂ 2ϕ ∂ 2ϕ + + =0 ∂x12 ∂x2 2 ∂x32

(8)

As a result, the velocity potential function of movement or strain in the plasticity and the charge-free electrostatic field, both are in the form of Laplace’s relation (relations (1) and (8)). So, in the plasticity theorem the particle flow path in the deforming region can be modeled with the electrostatics. Moreover, for a charge-free electric field using the relation (4), gives:

 ρ  ∇.E = , ρ = 0 → ∇. E = 0

(9)

ε0

By comparing the relations (3) and (9), it could be stated that the velocity of the particles in the plasticity is equivalent to the electric potential in the electrostatics. So, the constitutive relations for the plasticity and electrostatics are in the form of Laplace’s relation. In addition, the velocity parameter in the plasticity is analogous to the electric potential variable in the electrostatics. As a result, to model the metal forming processes using the electrostatics, the constitutive relation for the material behavior can be modeled with Laplace’s relation. To solve the Laplace’s relation, proper boundary conditions should be defined based on the geometrical information and forming parameters. Moreover, the obtained parameters can be applied directly in any analytical methods that are based on the velocity field concept. In the other words, any boundary conditions, strain and forming energy calculations in the conventional plasticity theorem that are based on the velocity field definition, should be adapted with the electric potential in the electrostatics. As it mentioned, one of the main analytical solutions for modeling of the metal forming processes using the KAVF is the upper bound theory (UB) (Hill, 1950). In the next section, it will be shown that the abovementioned analogy can be applied in the upper bound theory.

2.2 ANALOGY BETWEEN THE UPPER BOUND THEORY AND ELECTROSTTIC In this section, the analogues parameter between the electrostatic field and classical plasticity theorem is applied in the upper bound calculations of the forming energy. The upper bound value for the forming energy (Abrinia and Makaremi, 2007) can be calculated as follows:

J = W
e + W
x + W
f + W
i

(10)

Where W
e is the power due to the velocity discontinuity at the entrance section:

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Y W
e = 3

Y ∆Ve dS e = 3

∫∫ Se

1 1

∫∫ 0

0

Vx2 

+ V y2

1 2

+ (Vz − V0 )  dS e t =0 2

(11)

In the relation (11) Y is the yield strength/mean effective stress of the extruded material and Se is the area of the entrance section. The limits of the integrals vary from 0 to 1 due to normalization procedure performed for the geometrical parameters in the cylindrical coordinate system (Abrinia and Fazlirad, 2009).

W
x

is the power due to the velocity discontinuity at the exit section:

Y W
x = 3

∫∫

Y 3

∆V x dS x =

Sx

1

∫∫

1

1

 (V x2 + V y2 + (V z − V0 ( S e / S x )) 2  2 dS x  t =1 0

0

where S x is the area of outlet section. The power due to friction between working material and die surface

(12)

W
f

can be calculated as:

Y W
f = m 3

∫∫

(Vx2

+ V y2

1 2 2 + Vz )u =1 dS f

(13)

In relation (13), m is the friction factor and S f is the contact area. Finally,

W
i

the power due to the internal

deformation is calculated as follows: 1

 1 ∂Vx 2 2 2Y ∂V 1 ∂V 1 ∂V ∂V 1 ∂V ∂V W
i = (( ) +( )2 + ( z )2 ) + ( ( x + ))2 + ( ( y + z ))2 + ( ( z + x ))2  dV  ∫ ∫ ∫ ∂y ∂z 2 ∂y ∂x 2 ∂z ∂y 2 ∂x ∂z 3 0 0 0  2 ∂x  1

1

∂V y

1

∂Vy

(14) It was shown that the velocity parameter in the plasticity theorem is equivalent to the electric potential in the electrostatics. So, the relations (11)-(14) in the electrostatics can be re-written as follows:

Y W
e = 3

∫∫

Y W
x = 3

Se

Y ∆E edS e = 3

∫∫

∆ E x dS x =

Sx

Y W
f = m 3

∫∫

( E x2

1 1

∫∫

0 0

Y 3

+ E 2y

1

 E x2 

∫∫

+

0

+

E 2y

1 2

+ ( E z − E0 )  dS e t =0 2

(15) 1

1

 ( E x2 + E y2 + ( E z − E 0 (Se / S x )) 2  2 dS x  t =1 0

1 2 2 Ez )u =1 dS f

(16)

(17) 1

2  E
xx2 + E
2yy + E
zz2 2Y

2 + E
2 + E
2  dV Wi = ( ) + E  xy yz zx  2 3 ∫0 ∫0 ∫0   1 1 1

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(18)

As a result, to estimate the forming power using the electrostatics it is only needed to solve the Laplace’s relation in the deforming region and obtain the electric potential and its rates along the xyz directions and then calculate the four integrals at the inlet, outlet, inner, and contact regions using the relations (15) to (18).

3. APPLICATION OF THE ANALOGY BETWEEN THE METAL FORMING

AND

ELECTROSTATICS TO THE EXTRUSION PROCESS To demonstrate the effectiveness of the modeling procedure using the concept of electrostatic field, the extrusion process was selected as the case study. In the extrusion process, the material is formed in the die cavity that is between the inlet and outlet sections. In this research, a three-dimensional CAD model of the deformation region is constructed using the commercial CAD tools. Then, two different voltages are applied to the entry and exit sections based on the geometrical and extrusion parameters such as area reduction in the extrusion (Ra), and relative length of the deforming zone (L/R; L is the die length and R stands for the billet radius). Then, the Laplace’s relation (section 2.1; relation (7)) is solved numerically in 3D. The electric potential values as well as their rates are extracted from the numerical solution of the Laplace’s relation. Finally, the extrusion power is calculated using the relations (15) to (18). It should be noted that in the streamlined dies where the drawn streamlines in the deforming zone based on the electrostatic field are normal to the entry and exit surfaces, there is no velocity discontinuity. Therefore, the components of the extrusion power due to velocity discontinuities at the entry and exit sections of streamlined dies (relations (15) and (16)) will be zero (Ponalagusamy et al., 2005). In other cases, all of the components of the extrusion power including those due to velocity discontinuity should be accounted for (relations (15) to (18)). In the following section, the boundary conditions, intermediate sections between the initial and final cross sections, particle flow path in the deforming region, and strain calculation using the concept of electrostatics will be discussed.

3.1 BOUNDARY CONDITIONS From the volume constancy in the extrusion process, the boundary conditions could be extracted. It is supposed that the areas of the inlet and outlet sections are A0 and Af, respectively. Moreover, the material enters the deforming region with the velocity of V0 and exists with Vf in z direction that is the extrusion direction. So, the following relation could be concluded:

Vf =

A0 V0 Af

(19)

Normally, in the numerical modeling of the extrusion process the initial velocity is regarded as unit (Abrinia and Makaremi, 2007). On the other hand, it was shown that the velocity in the plasticity is equivalent to the electric potential in the electrostatics (section 2). Therefore, the following boundary conditions could be applied to the entry and exit sections in the electrostatic modeling:

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V0 = 1 , Ez (z 0 ) = V0 → Ez (z 0 ) = 1 and E z ( ∆z + z0 ) = V f → Ez ( ∆z + z0 ) =

A0 Af

(20)

In the extrusion process the inlet and outlet surfaces are parallel with a distance of d (∆z=d). To implement the length of the deforming region in the extrusion process it can be supposed that the entry and exit surfaces form a parallel plate capacitor. The electric potential of the parallel capacitor can be replaced with a voltage; φ by accounting for the length of the deforming region as follows (Hayt and Buck, 2005):

Ez ( z ) =

ϕ

∆z + z0

−ϕ

∆z

z0

, ∆z = d

(21)

So, in the electrostatic modeling of the extrusion process two different voltages should be applied to the entry and exit surfaces using the relations (20) and (21). 3.2 INTERMEDIATE SECTIONS It was shown that velocity potential function of movement in the plasticity formulation and the governing equation of charge-free electrostatic field are in the form of Laplace’s relation (section 2). To implement the analogy in the extrusion process, the initial billet and final section were considered. Two different voltages were assigned to the surfaces of the initial and final sections, respectively. Then the Laplace’s relation was solved between the two conductors in MATLAB and points of equal voltages were connected to create same-voltage contours. Here, the term “equi-potential lines (EPLs)” is used for the same-voltage contours. Figure 1 shows the EPLs as the intermediate shapes between the initial and final sections. Totally, twenty sections were considered between the initial and final shapes. The drawn EPLs can be used in accurate representation of the deforming zone using the Bezier and polynomial curves (Tabatabaei et al. 2013; 2014). In addition, the EPLs show the minimum work path between the entry and exit cross-sections so they can be used in 3D design of the deforming region in the extrusion process as well (Tabatabaei, et.al 2014).

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Figure 1 Equi-Potential lines between the round billets and different final sections in 3-D and 2-D views, (a) Square section, (b) T- shaped and (c) Non-symmetric U- shaped section.

3.3 PARTICLE FLOW PATH IN THE DEFORMING ZONE In this part, the electrostatics concept was used for prediction of the particle flow path in the deforming zone of the extrusion process. To investigate the mechanics of plastic deformation in the metal forming processes, the field variables such as displacement, velocity, and strain should be analyzed (Liu et al., 2006). It was shown that in the solid mechanics the velocity potential function and velocity stream function both are in the form of Laplace’s relation (Liu et al., 2006). Moreover, the equi-potential lines of velocity potential function are orthogonal to the streamlines of velocity stream function. This condition is analogous to the slip-line theory in the plasticity (Hill, 1950), (Liu et al., 2006). It can be stated that in the electrostatic modeling of the metal forming processes, a similar pattern can be drawn between the electric potential lines and equi-potential lines. Since, the Laplace’s relation is the governing equation  in the electrostatics and the relation (5) (section 2.1; E = −∇ϕ ) shows that the gradient of electric potential is the exterior normal of equi-potential lines but in the opposite direction. In the extrusion process the material flow is from the inlet to the exit section, so the calculated boundary conditions in the electrostatics should be applied reversely. As it was shown in relation (20), the higher electric potential should be applied to the final section. For this condition, the direction of the electric potential vectors will be from the exit to the inlet section. Therefore, by applying the higher voltage/electric potential to the entry section the effect of opposite direction in relation (5) will be compensated. To extract the particle flow path in the electrostatic modeling of the extrusion process, a 3D shape of the deforming region was modeled. Two different voltages were applied to the upper and lower surfaces (entry and exit sections) of

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the deforming region using the relations (20) and (21). Then the Laplace’s relation (relation (7)) was solved in 3D and EPL’s were drawn. The streamlines orthogonal to the equi-potential lines were obtained. Each streamline showed the minimum work path between the inlet and outlet sections (Xiaona and Fuguo, 2009). Figure 2 shows the 3D streamlines between the initial and final sections in the extrusion process of a round billet to a square section.

Figure 2 3D streamlines between the circle and square sections.

3.4 STRAIN CALCULATION The electrostatics is a stationary or time independent phenomenon. So, it is assumed that the strain from the electrostatic method can be calculated from the effective strain rate multiplied by the time (Rejaeian and AghaieKhafri, (2014)). To calculate the strains, based on the strain rates, the traveling time of the particles in the deforming region should be known. However, using the geometrical information and boundary conditions, it is possible to estimate the forming time in the extrusion process. It is supposed that the minimum distance for particle flow in the deformation region is L, and the velocities of V0 and Vf are the known boundary conditions at the entry and exit sections. Moreover, the particles flow in z direction that is the extrusion direction. So, the minimum time for particle flow in the deforming region can be obtained from: T ≥

L V f − V0

(22)

From the analogy between the classical plasticity theorem and electrostatics, the velocity can be replaced with electric potential as following: T ≥

L E z ( z 0 + ∆ z ) − E z ( z0 )

(23)

Moreover, the traveling time of a particle between the two consecutive points in the deformation region is proportional to the traveling distance in z direction. The above assumptions yield:

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ε p  ε
eff

ε
eff

P

P

× ∆Tp ,

∂V y 2 ∂V x 2 ∂V 1 ∂V x 2 1 ∂V x ∂V y 2 1 ∂V y ∂V z 2 1 ∂V (( ) +( ) + ( z )2 ) + ( ( )) + ( ( )) + ( ( z + )) at( x p , y p , z p ) + + 2 ∂x ∂y ∂z 2 ∂y ∂x 2 ∂z ∂y 2 ∂x ∂z

=

(24) From the abovementioned analogy and replacing the velocity with the electric potential the relation (24) can be written as:

ε
eff

P

 1
2
2
2
2
2
2   ( E xx + E yy + E zz ) + E xy + E yz + E zx  at( x p , y p , z p ) 2  



(25)

and ∆Tp = T ×

zp

L

,

T ≥

L → E z ( z 0 + ∆ z ) − E z ( z0 )

∆Tp 

zp

E z ( z 0 + ∆ z ) − E z ( z0 )

z 0 = 0 , ε
0 = 0

where, zp is the coordinate of the particle in the deforming zone along the z direction, z0 = 0 and ε
0 = 0 are the known boundary conditions at the entry section. As a result, from the electrostatic modeling of the extrusion process the strain values at any point of the deforming region can be easily calculated using the relation (25). 4.

ADVANTAGES OF THE MODELING THE EXTRUSION PROCESS USING ELECTROSTATICS CONCEPT

In this part the advantages of the electrostatic modeling of the extrusion process regarding to the conventional upper bound theory and FE method was discussed. The main advantages of the extrusion modeling using the concept of electrostatics are as follows: •

No need for discretizing the deforming zone.

•

Accurate definition for particle flow path in the deforming region.

•

Reasonable results for the extrusion pressure and strain values (section.6).

It was shown that by proper discretization of the deforming zone and consequently better definition of the kinematically admissible velocity field (KAVF); accurate results from the upper bound solution could be obtained (Tabatabaei and Abrinia, et.al 2014). The base of the segmentation process is the constancy of the material flow. It means that the area of the exit segment multiplied by the extrusion ratio should be equal to the area of the entry segment. In addition, the discretization method has to be performed in a way that the development of blind section between the inlet and outlet sections be avoided. In other words, the material entering the entry cross section should come out from the exit section without intersecting its circumferences. Different methods have been proposed for

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section division in the extrusion process that are tedious and time consuming processes and also not correct for some cases. These methods could not be used for complicated sections (figure 3).

Figure 3 Discretizing methods for different sections: A) I-Shaped section (Han et al.,1986); B) Non-symmetric U-shaped section(Celik and Chitkara, 2002): (a) Entry section, (b) Exit section, and (c) Off-centric positioning; (C) Complex geometry (Abrinia and Davarzani, 2012): (a) Non-symmetric section with first quarter, (b) Segments in the exit section , (c) Segments in the entry section; (D) U-shaped section with broken lines for segmentation (Abrinia and Ghorbani, 2012): (a) Exit section, (b) Entry section.

As it shown in figure (3), there is no global technique for segmentation process in the methods used in previous works. Any change in the geometry requires additional computations. Moreover, the mapping of segments from the initial section to the final one is arbitrary. On the other hand, even for the mapped segments the x and y coordinates of the corresponding point on the exit section should be computed based on the geometrical interpretations (Abrinia and Makaremi, 2007). There is no need to emphasize how difficult the process is for more complex sections. However, in the electrostatic modeling of the extrusion process, the deforming region is modeled in 3D and by applying voltages to the inlet and outlet surfaces, and solving the Laplace’s relation, it is possible to draw the streamlines. The drawn streamlines give an accurate representation for the particle flow path in the deforming region. So, there is no need for the definition of the position/velocity vector similar to the conventional upper bound theory. In addition, the discretizing process to map the segments from the entry section to the exit section will be eliminated.

12

As a result, by electrostatic modeling of the extrusion process and elimination of the segmentation process the computational time/cost will be decreased. It is only needed to solve the Laplace’s relation and calculate the extrusion power and strains using the relations (17) and (18), (21), and (25). Moreover, unlike the finite element modeling methods, there is no mesh deformation or mesh distortion (especially in complex features) using the concept of electrostatics. It is only needed to solve the Laplace’s relation numerically. It will be shown in section.6 that from the electrostatic modeling of the extrusion process reasonable results for the forming pressure and strain values can be obtained.

5. EXPERIMENTAL PROCEDURE The experimental tests were performed in the extrusion process of U-shaped and non-symmetric I-shaped sections (figure 4) that were different from the reported sections in the literature. The examined sections in the experiments have no axis of symmetry and are considered as complex sections. For the case of non-symmetric I-shaped section, even the thicknesses in the legs and flanges are different which makes it very difficult for modeling using the conventional analytical methods. Commercial lead (Table 1) was used in the extrusion of the round billets to the Ushaped section at room temperature and Al-6063 was used in the extrusion through the non-symmetric I-shaped section at the elevated temperature. The lead billets had an initial diameter of 24.8 mm and length of 30.3 mm. The diameter and length of the Al 6063 billets were 25.2 mm and 30.15 mm, respectively. Both of the dies had a bearing and conical part at the back to ease the martial exit. The bearing lengths and conical angles were 2 mm and 4° for the U-shaped section and 4 mm and 5° in the non-symmetric I-shaped section. The I-shaped section had a corner radius of 0.7 mm. The reduction of area in the extrusion process of the U- and non-symmetric I-shaped sections were 70% and 60%, respectively.

Table 1 Mechanical properties of pure commercial lead from compression test.

Strength

Strain hardening

Effective yield

Modulus of

Poisson’s

Density

coefficient

exponent

stress

elasticity (GPa)

ratio

(Kg/m3)

K (MPa)

n

(MPa)

15

0.35

11340

37.24

0.299

26

The true stress-strain curves of the lead and Al 6063 obtained from the standard compression tests are shown in figure 5. The flat faced dies were made from hot worked 2343 steel material. An oiled based lubricant was used in the experiments for the extrusion of the lead material in the U-shaped section. In the hot extrusion of the Al 6063, graphite powder was used between the billet and die to decrease the frictional forces. A ring shaped ceramic heater heated up the I-shaped die during the experiment. The tests were performed at 264 °C.

13

Figure 4 The configuration of the dies in the experiments: (a) U-shaped section (Ra=70%), and (b) Non-symmetric I-shaped section (Ra=60%); (The dimensions are in millimeter).

Figure 5 The true stress-strain curve from the standard compression test (a) Pure commercial Lead material (b) Al 6063

14

The ram speed during the experiments was 1 mm/min and 5mm/min for U- and non-symmetric I-shaped sections. The experimental setups are shown in figure 6.

Figure 6 Experimental setups: (a) Cold extrusion; U-shaped section (b) Hot extrusion; Non-symmetric I-shaped section.

6.

RESULTS AND DISSCUSION

To show the effectiveness of the proposed method in this research, different models were investigated. Firstly, the extrusion of round billets to the square and circular sections were investigated. Then, three complex features from the literature with the known experimental values were modeled. Finally, the results of the two different complex sections were compared analytically and experimentally.

6.1 EXTRUSION OF A ROUND BILLET TO A SQUARE SECTION In this part, the results for the extrusion of a round billet to a square section using the electrostatic and finite element modeling were investigated. The mechanical properties of “lead” material are shown in Table 1. The area reduction (Ra) and relative length of the deformation region (L/R; R=12.5 mm; R stands for the billet radius) were 60% and 0.90, respectively. The particle flow path in the deforming zone, strains, and relative extrusion pressure were compared.

6.1.1

FEM

A commercial FE code (Abaqus/Explicit) was used to perform the simulations. The simulations were performed using 3D models in which eight-node linear brick elements (C3D8R) were used for the billet material and four-node 3D bilinear rigid quadrilateral elements (R3D4) for the dies. The material of the billet is “lead” with the mechanical properties given in Table 1. The coulomb coefficient of friction was taken as 0.20 and the penalty method considered for contact modeling between the billet and the die. Figure 7 shows the FEM results for the extrusion of a square section from a round billet. The maximum extrusion pressure was obtained as 66.50 MPa.

15

Figure 7 FEM results for the extrusion of a square section from a round billet: (a) Extruded part, (b) Extrusion pressure (MPa) vs. ram displacement (mm).

6.1.2

ELECTROSTATIC MODELING

The 3D model of the deformation region was modeled in a commercial CAD tool and then meshed (Figure 8). Two different voltages according to the relations (20), (21) were applied to the circular and square sections and Laplace’s relation (relation (7)) was solved. The electric potential and its rates along the xyz directions were extracted at the centroid of each element and multiplied by the element area/volume to calculate the extrusion power due to the friction and internal deformation using relations (15) to (18). The equi-potential lines at the middle surface of the billet that is parallel to the XZ plane (figure 7) is shown in figure 9. The arrows of the electric potential that are  orthogonal vectors to the equi-potential lines ( E = −∇ϕ ) are shown as well (section 3.3). The arrows point to the particle flow path direction.

16

Figure 8 Mesh configuration for the deformation region in the extrusion of a round billet to the square section; Ra=0.60%, L/R=0.90 : (a) Extremely fine (b) Extra fine (c) Finer (d) Fine (e) Normal

Figure 9 (a) Equi-potential lines between the entry and exit sections at the middle plane of the billet; extrusion from the round billet to the square section (b) Electric potential arrows orthogonal to the equi-potential lines; as the particles flow directions.

6.1.3

COMPARISON OF THE FEM AND ELECTROSTATIC MODELING

Figure 10 shows the particle flow path at the middle surface of the billet (parallel to the XZ plane; figure 7) from FE and electrostatic modeling.

17

Figure 10 Particle flow path at the middle plane of billet (figure 7); extrusion process of a round billet to a square section: (a) FEM, (b) Electrostatic modeling (c) Superimposed FEM and electrostatic modeling (The length of deforming zone is shown).

As it is shown in figure 10, there is a reasonable similarity between the FE and electrostatic modeling for the particle flow path in the deformation region (L/R=0.90; R=12.5 mm). In the FEM, the material continuously is formed in the deforming zone so several nodes outside of the deforming band are visible in figure 10. Moreover, the number of the nodes in the deforming region from the FEM is less that the number of particles from the electrostatic modeling that can be attributed to the mesh quality in the FEM. It means that to find the flow path of each node in the FEM, very fine meshes should be defined in the billet material. Table 2 compares the relative extrusion pressure between the electrostatic modeling, FEM and experimental data.

18

Table 2 Comparison of the relative extrusion pressure (P/Y; Y: yield stress of the material) between the electrostatic modeling, FEM and experimental data; extrusion of a round billet to a square section (Ra=60%, L/R=0.90, m=1).

Parameter

Method

P/Y

Electrostatic modeling

Extremely fine mesh

2.882

Extra fine mesh

2.876

Finer mesh

2.868

Fine mesh

2.855

Normal mesh

2.851

Maximum plastic strain

FEM

2.55

Experiment

Electrostatic

(Abrinia et al., 2013)

modeling

2.60

0.986

FEM

0.959

As could be seen, there is an acceptable agreement between the results from electrostatics, FEM and experiment. To investigate the mesh size effect, the relative extrusion pressure is reported for different mesh configurations in the electrostatic modeling. The effect of mesh refinement shows that the finest and coarsest meshes have only 1.35% deviation in predicting the relative extrusion pressure. In other words, the results from the electrostatics theory have the least sensitivity to the mesh size. From the sensitivity analysis, the trivial differences between the fine and rough mesh configurations can be attributed to the accurate representation of the area/volume of the deforming region using the fine meshes. That condition yields a bit higher estimation for internal and frictional powers (relations (17) and (18)). The maximum plastic strain from FEM and electrostatic modeling has 2.9% difference which is reasonable.

6.2 EXTRUSION PROCESS OF DIFFERENT SECTIONS FROM LITERATURE In this part the extrusion process of round billets to circular, U-, I- and T- shaped sections (Chitkara and Celik, 2000; 2001; 2002) were modeled using the proposed concept in this research. The results were compared with the analytical and experimental results of (Chitkara and Celik, 2000; 2001; 2002). Figure 11, 12 shows the meshed configuration of the deformation region for the circular, U-, I- and T- shaped sections. The streamlines between the initial and final sections are shown as well. The ruled surface dies from an initially round billet were used in U-, I- and T- shaped sections (Chitkara and Celik, 2000; 2001; 2002).

19

Figure 11 Meshed configurations of the deformation zone and streamlines from the electrostatic method for different extrusion ratios (L/R); extrusion from a circular billet to the circular section (Celik and Chitkara, 2000); (Ra=60%, m=0.2, 0.40): (a) L/R=1, (b) L/R=1.5, (c) L/R=2.

Figure 12 Meshed configurations of the deformation zone and streamlines from the author’s method for different complex sections (Celik and Chitkara, 2001; 2002): (a) U-shaped (Ra=69%) (b) I-shaped (Ra=68.6%) and (c) T-shaped section (Ra=61.7%).

In Table 3 the relative extrusion pressures in the extrusion process of round billets to the circular sections are compared with the upper bound method of (Celik and Chitkara, 2000) in wet and dry frictional conditions. In the upper bound calculation of (Celik and Chitkara, 2000) the off-centric value between the initial and final sections were assumed as zero and the area reduction in the extrusion processes was 60%.

20

Table 3 Comparison of the relative extrusion pressure (P/Y; Y: yield stress of the material) between the conventional upper bound method (Celik and Chitkara, 2000) and electrostatic modeling; extrusion of round billets to circular sections (Ra=60%, m=0.20, 0.40, L/R=1, 1.5, 2).

Frictional condition

Method

P/Y @ wet condition; m=0.20

P/Y @ dry condition; m=0.40

Current

Upper bound method

Difference

Current

Upper bound method

Difference

study

(Celik and Chitkara, 2000)

(%)

study

(Celik and Chitkara, 2000)

(%)

1.29

1.39

-7.62

1.55

1.71

-9.31

1.39

1.46

-4.73

1.80

1.82

-1.21

1.51

1.55

-2.74

2.05

2.09

-1.88

Relative die length (L/R): 1

Relative die length (L/R): 1.5

Relative die length (L/R): 2

As could be seen there is an acceptable agreement between the upper bound method (Celik and Chitkara, 2000) and electrostatic modeling of the extrusion process from circular billet to the circular shape. The maximum difference between the two methods and wet contact condition is about 7%. Such difference in the dry contact condition becomes around 9% that is reasonable. It should be noted that the electrostatics method predicts lower total power/pressure consumption, for all the values of relative die length compared to the conventional upper bound method of (Celik and Chitkara, 2000). In Table 4 the relative extrusion pressures in U-, I- and T- shaped sections are compared with the experimental results for two different frictional conditions. The values of the friction factor for the wet and dry conditions (with/out lubricant) were considered as m=0.2 and 0.4, respectively (Celik and Chitkara, 2001; 2002). Table 4 Comparison of the relative extrusion pressure (P/Y; Y: yield stress of the material) for the results of the electrostatic modeling and experimental data; extrusion of the U-, I- and T-shaped sections (Celik and Chitkara, 2001; 2002).

Frictional condition

Method

P/Y @ wet condition; m=0.20

P/Y @ dry condition; m=0.40

Current

Experiment

Difference

Current

Experiment

Difference

study

(Celik and Chitkara, 2001;

(%)

study

(Celik and Chitkara, 2001;

(%)

2002)

2002)

U-shaped (Ra=69%, L/R=1.73)

2.87

2.81

2.24

3.69

3.37

9.67

I-shaped (Ra=68.6% L/R=1.73)

2.86

2.79

2.55

3.67

3.35

9.56

T-shaped (Ra=61.7% L/R=1.73)

1.81

1.89

-4.07

2.36

2.33

1.28

21

From Table 4 it could be seen that there is a good agreement between the results of the authors’ method and the experimental data of (Celik and Chitkara, 2001; 2002). For U- and I-shaped sections, the present method has less than 3% deviation from the experiment in the wet condition. For the dry condition, such deviation becomes less than 10%. For the T-shaped section, the difference between the analytical and experimental results for both contact conditions is less than 5%. Almost for all contact conditions the proposed technique predicts an upper bound value for the extrusion pressure. However, for T-shaped section and wet contact condition the calculated extrusion pressure is less than the experiment that may be attributed to the unknown conditions during the experiment. The relative extrusion pressure for U- and I- and T-shaped sections at dry contact condition is compared between the author’s method and conventional upper bound method in Table 5. The difference between the upper bound methods in Table 5 is related to the segmentation of the deforming region.

Table 5 Comparison of the relative extrusion pressure (P/Y; Y: yield stress of the material) for the results of the electrostatic modeling and conventional upper bound method; extrusion of the U-, I- and T-shaped sections at dry contact condition (m=0.40) (Celik and Chitkara, 2001; 2002; Abrinia and Davarzani, 2012; Abrinia and Ghorbani, 2012).

Method

Current study

Upper bound method (Celik and

Upper bound method (Abrinia

Upper bound method (Abrinia

Chitkara, 2002 , (Chitkara and

and Davarzani, 2012)

and Ghorbani, 2012)

Celik, 2001)

U-shaped (Ra=69%, L/R=1.73)

3.69

4.05

3.93

3.71

I-shaped (Ra=68.6% L/R=1.73)

3.67

3.95

3.80

3.68

T-shaped (Ra=61.7% L/R=1.73)

2.36

2.54

2.52

2.49

As could be seen from Table 5, the author’s method has better results than the conventional upper bound formulation with close results to the experiment (Table 4). Moreover, there is no need for discretizing the deforming zone using the electrostatic modeling. Finally, using the electrostatic modeling the maximum plastic strain for U- , I and T-shaped sections were obtained as 1.0845, 0.8297, and 0.8190, respectively.

6.3 EXPERIMENATL AND ANALYTICAL MODELING OF COMPLEX SECTIONS In this part the results between the authors’ method and experiment are compared. Different contact conditions were assumed between the billets and dies at different zones: a) deforming region b) container, and c) bearing. The friction factor in the extrusion of the lead and Al 6063 billets were considered as m=1 in the deforming region because of the dead metal zone. The friction factor between the lead material, container and bearing was considered as m=0.1 and m=0.25 due to the liquid lubricant. For Al 6063 billets and powder lubricant the friction factors between the container and bearing was considered as m=0.15 and m=0.35, respectively. The extrusion pressures vs. ram displacement from the experiments are shown in figure 13.

22

Figure 13 Extrusion pressure vs. ram displacement from the experiments: (a) Extrusion of the lead billet to the U-shaped section (b) Extrusion of the Al 6063 billet through the non-symmetric I-shaped section at 264 °C.

The extrusion pressure for both sections rises to the peak point where the maximum forming pressure of the material occurs. Then, the extrusion pressure decreases and the extruded parts exit almost at a constant pressure. The maximum extrusion pressures were 115.43 MPa and 324.06 MPa for U-shaped and non-symmetric I-shaped sections. The extruded parts from the experiments are shown in figure 14.

(a)

(b)

Figure 14 Extruded parts from the experiments: (a); Extruded lead billet to the U-shaped section in the cold extrusion (b) Extruded Al 6063 billet to the non-symmetric I-shaped section in the hot extrusion.

23

To calculate the extrusion pressure from the electrostatic modeling, the relations (15) to (18) were used. In these relations, the effect of temperature was applied to the compressive yield stress of the Al 6063 that was regarded here as 75 MPa at 264 °C (Astrop, 2002). From Table 1, we recall that the effective compressive yield strength of lead was 26 MPa at room temperature (25°C). The results for the authors’ and experimental methods for the U- and non-symmetric I-shaped sections are compared in Table 6.

Table 6 Comparison of the relative extrusion pressure (P/Y) from the electrostatic modeling and experiments; extrusion process of round billets to the U-, and non-symmetric I- shaped sections.

Profile

Method

Current study

Experiment

Difference (%)

U-shaped section

4.752

4.440

7.02

Non-symmetric I-shaped section

4.692

4.321

8.60

From Table 6, it could be seen that there is an acceptable agreement between the author’s method and experimental results. The difference between the theory and experiment for U- and non-symmetric I-shaped sections are 7.02% and 8.60%, respectively that shows the effectiveness of the proposed technique in the modeling of the complex sections. This validation opens ways to investigation of the applicability of the electrostatics in analytical modeling of the other metal forming processes. It is the belief of the authors that the analogue parameters for friction and strain hardening effects in the classical plasticity theorem can be derived in the electrostatics method which needs further investigations.

CONCLUSION

In this research, the analogy between the classical plasticity theorem and electrostatic concept was investigated. It was shown that the governing equation in both theories was in the form of Laplace’s relation. In addition, the velocity parameter in the classical plasticity theorem was equivalent to the electric potential in the electrostatics. The extrusion process was used as the case study and the effectiveness of the electrostatic modeling was investigated for the different sections. The relative extrusion pressure, particle flow path in the deformation zone, and strains were calculated from the electrostatics and compared with the results of published papers and also with the performed experimental tests. There was a reasonable agreement between the results. Based upon the theoretical and experimental works carried out in this paper the following conclusions were reached: •

It was shown that the idea of electrostatics opened a new avenue in the analysis of the metal forming processes.

24

•

Using the proposed method, the computations for the analysis of the metal forming/extrusion process was considerably decreased.

•

Application of the electrostatic method in the extrusion modeling, gives an accurate prediction for the material flow in the deformation region.

•

Unlike the conventional upper bound method, there is no need for segmentation process when analyzing complex shapes with blind sections using the electrostatic equations.

•

The mapping of the points from the entry section to the final section is accurate in the electrostatic modeling.

•

The constitutive equations and relations for modeling of the metal forming processes using the concept of electrostatics are much simpler than the other methods.

•

Unlike the FE based methods, there is no mesh deformation during the solution of the Laplace’s relation in the electrostatics theory.

•

There is no limitation in modeling of the complex features in the extrusion process using the concept of electrostatics.

REFERENCES Abrinia, K., Davarzani, H., 2012. A universal formulation for the extrusion of sections with no axis of symmetry. J. Mater. Process. Technol. 212, 1355-1366.

Abrinia, K., Farahmand, P., Parchami-Sarghin, M., 2013. Formulation of a new generalized kinematically admissible velocity field with a variable axial component for the forward extrusion of shaped sections. Int. J. Adv. Manuf. Technol. DOI 10.1007/s00170-013-5365-3.

Abrinia, K., Fazlirad, A., 2009. Three-dimensional analysis of shape rolling using a generalized upper bound approach. J. Mater. Process. Technol. 209 (7), 3264-3277.

Abrinia, K., Ghorbani, M., 2012. Theoretical and Experimental Analyses for the Forward Extrusion of Nonsymmetric Sections. Mater. Manuf. Process. 27, 420-429. Abrinia, K., Makaremi, M., 2007. A new three dimensional solution for the extrusions of sections with larger dimensions than the initial billet. J. Mater. Process. Technol. 205, 259–271. Altan, B.S., Purcek, G., Miskioglu, I., 2005. An upper bound analysis for equal-channel angular extrusion. Mater. Process. Technol. 168(1), 137-146.

25

Astrop, K. I., 2002. Plastic deformation at moderated temperatures of 6XXX-series Aluminum alloys. In: PhD thesis. The Norwegian University of Science and Technology.

Braess, H., Wriggers, P., 2000. Arbitrary Lagrangian Eulerian finite element analysis of free surface flow. Comput. Methods in Appl. Mech. Eng. 190, 95-109. Celik, K.F., Chitkara, N.R. , 2000. Off-centric extrusion of circular rods through streamlined dies, CAD/CAM applications, analysis and some experiments. Int. J. Mech. Sci. 42, 295–320.

Celik, K.F., Chitkara, N.R. , 2002. Extrusion of non-symmetric U- and I-shaped sections, through ruled-surface dies: numerical simulations and some experiments. Int. J. Mech. Sci. 44, 217-246.

Chitkara, N.R., Celik, K.F., 2001. Extrusion of non-symmetric T-shaped sections, an analysis and some experiments. Int. J. Mech. Sci. 43, 2961–2987.

Dvorkin, N. E., Petocz, E. G., 1993. An effective technique for modelling 2D metal forming processes using an Eulerian formulation. Engineering computations. 10, 323-336.

Dyduch, M., Habraken, A.M. , Cescotto, S., 1992. Automatic adaptive remeshing for numerical simulations of metalforming, Comput.Methods in Appl. Mech. Eng. 101, 283-298.

Hallquist, J.O., Wainscott, B., Schweizerhof, K., 1995. Improved simulation of thin-sheet metal forming using LSDYNA3D on parallel computers, Comput. Methods in Appl. Mech. Eng .50, 144-157. Han, C.H., Yang, D.Y., Kiuchi, M., 1986. New formulation for three-dimensional extrusion and its application to extrusion of clover sections. Int. J. Mech. Sci. 28(4), 201-218. Hayt, W., Buck, J., 2005. Engineering Electromagnetics. McGraw-Hill Science/Engineering/Math; 7 edition.

Hill, R., 1950. The Mathematical Theory of Plasticity. Oxford at the Clarendon Press. Hoffman, O., Sachs, 1953. Introduction to the theory of plasticity for engineers, Mc Graw-Hill, New York.

Khalili Meybodi, A., Assempour, A., Farahani, S., 2012. A general methodology for bearing design in non-symmetric T-shaped sections in extrusion process, J. Mater. Process. Technol. 212, 249–261

Kiuchi, M., Kishi, H., Ishikawa, M., 1981. Study on non-symmetric extrusion and drawing. In: Proceedings of the 22nd International Machine Tool Design Research Conference, vol. 523.

26

Kukureka, S.N., Craggs, G., Ward, I.M., 1992. Analysis and modelling of the die drawing of polymers. J. Mater. Sci. 27, 3379-3388.

Lee, S.R., Lee, Y.K., Park, C.H., Yang, D.Y., 2002. A new method of preform design in hot forging by using electric field theory. Int. J. Mech. Sci. 44, 773-792.

Liu, Y., Li, F., Wang, S., Jack Hu, S., 2006. Preform Design of Powder Metallurgy Turbine Disks Using EquiPotential Line Method. J. Manuf. Sci. E-T ASME. 128, 677-682. Moon, Y.H., Van Tyne, C.J., 2000. Validation via FEM and plasticine modeling of upper bound criteria of a processinduced side-surface defect in forgings. Mater. Process. Technol. 99, 185-196.

Pérez, C.J. L., Luri, R., 2008. Study of the ECAE process by the upper bound method considering the correct die design, Mech. Mater. 40, 617–628. Ponalagusamy, R., Narayanasamy, R., Srinivasan, P., 2005. Design and development of streamlined extrusion dies a Bezier curve approach. J. Mater. Process. Technol. 161, 375–380.

Prager, W., Hodge, P.G., 1951. Theory of perfectly plastic solids, Wiley, New York. Rejaeian, M., Aghaie-Khafri, M., 2014. Study of ECAP based on stream function, Mech. Mater. 76, 27–34.

Sofuoglu, H., Hasan Gedikli, H., 2004. Physical and numerical analysis of three dimensional extrusion process, Comp. Mater. Sci. 31, 113–124.

Sosnowski , W., 1992. Comparative study on sheet metal forming processes by numerical modelling and experiment. J. Mater. Process. Technol. 34 (1-4), 109-116.

Surdon, G., Chenot, J. L., 1986. Finite element calculations of three-dimensional hot forging, International Conference on Numerical Methods in Industrial Forming Processes Numiform'86, ed. by K. Mattiasson et al. A. A. Balkema, Rotterman. 287-292.

Tabatabaei S.A., Abrinia K., Besharati Givi M.K., et al., 2013. Application of the equi-potential lines method in upper bound estimation of the extrusion pressure, Mater. Manuf. Process. 28(3): 271 – 275.

Tabatabaei, S.A., Abrinia, K., Besharati, M.K., 2014. Application of equi-potential lines method for accurate definition of the deforming zone in the upper-bound analysis of forward extrusion problems, Int. J. Adv. Manuf. Technol. 72, 1039–1050.

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Tabatabaei, S.A., Abrinia, K., Besharati, M.K., et al. 2014. A novel method for 3D-die design in the extrusion process using equi-potential lines, Key Engineering Materials. 585, 67-75.

Talbert, S.H., Avitzur, B., 1996. Elementary mechanics of plastic flow in metal forming. J. Wiley. Xiaona, W., Fuguo, L., 2009. A quasi-equi potential field simulation for preform designs of P/M superalloy disk. Chin. J. Aeronaut. 22(1), 81–86. Yu, H.S., 2006. General Elastic-Plastic Theorems. In: Plasticity and Geotechnics. Springer, US, pp. 40-68.

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Analogy between the metal forming process and electrostatics is investigated.
Extrusion process is modeled based on the electrostatic field.
The intermediate sections in the deforming region are predicted using the electrostatic concept.
Electrostatic method accurately predicts the material flow in the deforming region.
There is no mesh deformation in the electrostatic modelling of extrusion process.
Electrostatic modelling of the extrusion process is fast and accurate.

48