A Hybrid Numerical Method for Modeling the Extrusion Process

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ScienceDirect Materials Today: Proceedings 2 (2015) 4748 – 4757

Aluminium Two Thousand World Congress and International Conference on Extrusion and Benchmark ICEB 2015

A hybrid numerical method for modeling the extrusion process S. A. Tabatabaeia,*, K. Abriniab a b

School of Mechanical Engineering, University of Tehran, Kargar Shomali, St., P.O. Box 14395-515, Iran School of Mechanical Engineering, University of Tehran, Kargar Shomali, St., P.O. Box 14395-515, Iran

Abstract Analytical and numerical modelling 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 behaviour during the forming processes. In this research, the analogy between the metal forming process and the electrostatics is investigated. It is shown that the governing relations in the plasticity theory of upper bound are analogous to the electrostatic equations. The forward extrusions of round billets to complex sections from the published papers were analyzed using the electrostatic method and the calculated results were compared to the simulated and experimental data in the literature. Finally, some experiments were performed for the extrusion of a round billet to a complex section. It was shown that the extrusion process could be modelled easily and accurately using the electrostatics mathematical formulation. © 2014 Elsevier ElsevierLtd. Ltd.All All rights reserved. © 2015 rights reserved. Selection andPeer-review Peer-review under responsibility of Conference Committee of Aluminium Two Thousand World Congress and Selection and under responsibility of Conference Committee of Aluminium Two Thousand World Congress and International Conference onConference Extrusion and 2015 onBenchmark Extrusion ICEB and Benchmark ICEB 2015. International Keywords: Numerical; Extrusion; Electrostatics; Upper bound.

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 modelling of the metal forming processes can be categorized as physical, numerical and analytical. The numerical modelling of the metal forming processes has been mostly based on the finite element (FE) methods [1-5]. The finite element * Corresponding author.Tel:+98-21-61114026; fax: +98-21-66461024. E-mail address: [email protected]

2214-7853 © 2015 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility of Conference Committee of Aluminium Two Thousand World Congress and International Conference on Extrusion and Benchmark ICEB 2015 doi:10.1016/j.matpr.2015.10.008

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modelling methods suffer from convergence problem and mesh distortion [6] and are time consuming, especially for the complex geometries. The physical modelling is a simple technique for observing the actual metal forming flow patterns using the model material and dies [7]. The physical methods require special tools to model the metal forming. However, any change in the geometry and boundary conditions needs extra tests or simulations. Moreover, the main analytical solutions for modelling the metal forming processes are the slip-line field method [8], and the upper [9] and lower [10] bound theories. The slip-line field theory is based on plane strain condition and can not be applied in three-dimensional models [9]. 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 [11-17]. The upper bound method is based on the definition of a kinematically admissible velocity field (KAVF) in the deforming region. Having defined the KAVF, it is possible to calculate the forming force/pressure using velocities, strain rates, and strains. In the conventional upper bound method, there is no global or generalized technique for segmentation of the deforming region and proper definition of the KAVF. 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 [18-21]. The aim of this research is to model the extrusion process in 3D and for any complex geometry using the electrostatics concept without the above mentioned limitations of the physical, numerical and analytical methods. In the current study, the analogy between the conventional plasticity theorem and electrostatics was investigated. The constitutive relations in the classical plasticity theorem and the electrostatic equations were investigated and analogue relations and parameters between the two theories were obtained. To investigate the effectiveness of the electrostatic concept in the analytical modelling of the metal forming processes, the extrusion process was used as the benchmark. A 3D-CAD model of the deformation region is constructed and used in the computational calculations of the extrusion power. To validate the electrostatic modelling of the extrusion process, different sections varying from a simple hexagon to the complex I-, U-, and T- shaped sections were modelled using the proposed method and the results were compared with the data given in the literature. Finally, the experimental tests were performed for a non-symmetric I-shaped section in the warm extrusion condition and the results were compared with the electrostatic modelling. 2. Theory 2.1 Analogy between the plasticity theorem and electrostatics 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:

’2D

w 2D wx12



w 2D wx22



w 2D wx32

0

(1)

where, D is a velocity potential function of movement or strain and xi (i=1, 2, 3) are the coordinates [22]. From the incompressibility criterion, the sum of the velocity variations in xyz directions should be zero:

H ii

wVx wVy wVz   wx wy wz

0

(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

where, ‫ ׏‬is the vector differential operator that means the gradient of velocity at any point of the material is zero.

(3)

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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 [23]:

’.E

U H0

(4)

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

E ’M

(5)

Substituting (5) into (4) yields:

’2M



’U

(6)

H0

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

’2M

0

(7)

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

’2M

w 2M wx12



w 2M wx2 2



w 2M wx32

0

(8)

As a result, the velocity potential function of movement (the strain) in the plasticity and the charge-free electrostatic field, both are in the form of Laplace’s relation (relations (1) and (8)). Moreover, in a charge-free electric field, relation (4) yields:

’.E

U ,U H0

0 o ’.E

0

(9)

By comparing 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. As a result, to model the metal forming processes using the electrostatics, the constitutive relation for the material behaviour can be modelled with Laplace’s relation. To solve the Laplace’s relation, proper boundary conditions should be defined based on the geometrical information and forming parameters. In the next section, it will be shown that the above-mentioned analogy can be applied in conjunction with the upper bound theory. 2.2 Application of the analogy between the plasticity theorem and electrostatics using the upper bound formulation In this section, the analogue parameters between the electrostatic field and classical plasticity theorem are applied in the upper bound calculations of the forming energy. The upper bound value for the forming energy [24] can be calculated as follows:

J We  Wx  W f  Wi

(10)

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where J is the total power consumption in the forming process and We is the power due to the velocity discontinuity at the entrance section:

We

Y 3

³³ 'V dS e

Y 3

e

Se

1 1

³³

ªVx2 ¬

0 0

 V y2

1 2

 (Vz  V0 ) º¼ dSe t 0 2

(11)

In elation (11), Y is the mean effective stress of the extruded material and Se is the area of the entrance section. Wx is the power due to the velocity discontinuity at the exit section:

Y 3

Wx

³³ 'V dS x

Y 3

x

Sx

1

1 1

³³

ª(Vx2  V y2  (Vz  V0 ( Se / S x ))2 º 2 dS x ¼t 1 0¬

0

(12)

where S x is the area of the outlet section. The power due to friction between working material and die surface W f can be calculated as:

Wf

m

Y 3

³³

ሺVx2

 Vy2

1 2 2  Vz ሻu 1 dS f

(13)

In relation (13), m is the friction factor. Finally, Wi the power due to the internal deformation is calculated as follows: 1

Wi

·2 2Y 1 1 1 § 1 wVx 2 wVy 2 wVz 2 1 wV wV 1 wV wV 1 wV wV (( ) ( ) ( ) )  ( ( x  y ))2  ( ( y  z ))2  ( ( z  x ))2 ¸ dV ³ 0 ³0 ³0 ¨ 2 wx wy wz wx wy 2 wy 2 wz 2 wx wz ¹ 3 ©

(14)

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

We

Wx

Y 3

³³ 'E dS e

Y 3

e

Se

Y 3

³³

'E x dS x

Sx

Y

³³

ሺ E x2

1 1

³³

0 0

Y 3

 E y2

1

ª E x2 ¬

³³ 0



E y2

1 2

 ( E z  E0 ) º¼ dSe t 0 2

1

1

ª( E x2  E y2  ( Ez  E0 (Se / S x ))2 º 2 dS x ¼t 1 0¬

1 2 2  Ez ሻu 1 dS f

Wf

m

Wi

· 2Y 1 1 1 § E  E  E ( )  E xy2  E yz2  Ezx2 ¸ dV ¨ ³ ³ ³ ¸ 2 3 0 0 0 ¨© ¹

3

2 xx

2 yy

(15)

2 zz

(16)

(17) 1 2

(18)

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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 relations (15) to (18). 3. Electrostatic modelling of the extrusion process 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)-(18). In the following section, the boundary conditions in the electrostatic modelling of the extrusion process will be discussed. 3.1 Boundary conditions in the electrostatic modelling of the extrusion process 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 modelling of the extrusion process, the initial velocity is regarded as unit [24]. On the other hand, it was shown that the velocity in the plasticity is equivalent to the electric potential in the electrostatics. Therefore, the following boundary conditions could be applied to the entry and exit sections in the electrostatic modelling:

V0 1 , E z (z 0 ) V0 o E z (z 0 ) 1 and E z ( 'z  z0 ) V f o E z ( 'z  z0 )

A0 Af

(20)

In the extrusion process, the inlet and outlet surfaces are assumed to be parallel with a distance of d (Δz=d) in between. 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 [25]:

Ez ( z )

M

'z  z0

M

'z

z0

, 'z

d

(21)

Hence, in the electrostatic modelling of the extrusion process, two different voltages should be applied to the entry and exit surfaces using the relations (20) and (21). In the electrostatic modelling of the extrusion process, the deforming region is modelled 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

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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. Moreover, unlike the finite element modelling 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. 4. Experiment The experimental tests were performed in the extrusion process of non-symmetric I-shaped section (Fig.1). The round billets of Al-6063 with the diameter of 25.2 mm and length of 30.15 mm were used in the extrusion through the non-symmetric I-shaped section at the elevated temperature. The I-shaped section had corner radii of 0.7 mm and the reduction of area in the extrusion process was 60%. The true stress-strain curves of Al 6063 obtained from the standard compression test (ASTM E9-09) is shown in Figure 2 [26]. The flat faced dies were made from hot worked 2343 steel material and had a bearing of 4 mm and a conical part with a semi-cone angle equal to 5° at the back to ease the martial exit. In the warm 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.

Fig. 1. The configuration of the non-symmetric I-shaped section in the experiments (Ra=60%); (The dimensions are in millimetre).

Fig. 2. The true stress-strain curves for Al 6063 from the standard compression test [26].

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An Instron 4028 hydraulic press and experimental setup used to perform the tests are shown in Figure 3. The ram speed during the experiments was 5mm/min.

Fig. 3. Experimental setup in the warm extrusion of non-symmetric I-shaped section.

5. Results and discussion To show the effectiveness of the proposed method in this research, different shaped profiles were investigated. Firstly, the extrusion of round billets to the hexagonal section [27] was investigated. Then, three complex features from the literature with the known experimental values were modelled [28-30]. Finally, the results of the nonsymmetric I-shaped section were compared analytically and experimentally. 5.1 Validation of the electrostatic modelling of the extrusion process with the data from the literature Figure 4, 5 shows the meshed configuration of the deformation region for the hexagonal, U-, I- and T- shaped sections. The streamlines between the initial and final sections are shown as well.

Fig. 4. Meshed configurations of the deformation zone and streamlines from the author’s method for different relative die lengths; extrusion process from a round billet to a hexagonal section [27] (Ra=60%): a) L/R=1 (b) L/R=1.5 (c) L/R=2

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Fig. 5. Meshed configurations of the deformation zone and streamlines from the author’s method for different complex sections [28-30]: a) U-shaped (Ra=69%) (b) I-shaped (Ra=68.6%) and (c) T-shaped section (Ra=61.7%).

In Tables 1 and 2, the relative extrusion pressures for the extrusion of hexagonal, U-, I- and T- shaped sections are compared with the theoretical and experimental results of [27-30] for different frictional conditions. Table 1. Comparison of the relative extrusion pressure (P/Y; Y: yield stress of the material) between the conventional upper bound method (UB) [27] and electrostatic modelling for different frictional conditions; extrusion of round billets to hexagonal sections (Ra=60%, m=0.10, 0.20, 0.40, L/R=1, 1.5, 2).

Frictional condition Method Relative die length (L/R): 1 Relative die length (L/R): 1.5 Relative die length (L/R): 2

P/Y @ m=0.10

P/Y @ m=0.20

P/Y @ m=0.40

Current study

UB [27]

Current study

UB [28]

Current study

UB [28]

1.19

1.36

1.29

1.39

1.55

1.71

1.22

1.34

1.39

1.46

1.80

1.82

1.27

1.37

1.51

1.55

2.05

2.09

Table 2. 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 [28-30].

Frictional condition Method

P/Y @ wet condition; m=0.20 Current study

Experiment [28-30]

P/Y @ dry condition; m=0.40

Difference (%)

Current study

Experiment [28-30]

Difference (%)

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

2.37

2.33

1.64

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As could be seen from Table 1, there is an acceptable agreement between the upper bound method [27-28] and electrostatic modelling of the extrusion process from circular billet to the hexagonal shape. Based on the results, the electrostatic method predicts lower pressures for all contact conditions and relative die lengths compared to the conventional upper bound methods [27-28]. As seen in Table 2, there is a good agreement between the results of the authors’ method and the experimental data of [28-30]. 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 the T-shaped section and the wet contact condition, the calculated extrusion pressure is less than the experiment which may be attributed to the unknown conditions during the experiment. 5.2 Experimental and electrostatic results for the extrusion of non-symmetric I-shaped section In this part, the results between the authors’ method and experiment are compared. Different contact conditions were assumed between the billets and tools at different zones: a) deforming region b) container, and c) bearing. The friction factor in the extrusion of Al 6063 billets were considered as m=1 (pure shear) in the deforming region because of the dead metal zone. The friction factors between the container and bearing was considered as m=0.15 and m=0.35, respectively [31]. The extrusion pressure vs. ram displacement and the extruded part from the experiment are shown in Figure 6.

Fig. 6. Experimental results in the extrusion of Al 6063 billet through the non-symmetric I-shaped section at 264 °C: (a) Extrusion pressure vs. ram displacement (b) Extruded part.

The maximum extrusion pressure was obtained as 324.06 MPa. To calculate the extrusion pressure from the electrostatic modelling, relations (15)-(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 [26]. The authors’ findings and the experimental results for the non-symmetric I-shaped section are compared in Table 3. Table 3. Comparison of the relative extrusion pressure (P/Y) from the electrostatic modelling and experiments; extrusion process of round billets to non-symmetric I- shaped section. Profile Non-symmetric I-shaped section

Current study

Method Experiment

Difference (%)

4.69

4.32

8.60

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From Table 3, 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 non-symmetric I-shaped section is 8.60% that shows the effectiveness of the proposed technique in the modelling of the complex sections. This validation opens ways to the investigation of the applicability of the electrostatics in analytical modelling of the other metal forming processes. Conclusion Based upon the theoretical and experimental works carried out in this paper, the following conclusions were reached: x It was shown that the idea of electrostatics opened a new avenue in the analysis of the metal forming processes. x Unlike the conventional upper bound method, there is no need for segmentation process when analyzing complex shapes with blind sections. x The constitutive equations and relations for modelling the metal forming processes using the concept of electrostatics are much simpler than the other methods. x There is no limitation in modelling the complex features in the extrusion process using the concept of electrostatics. References [1] G. Surdon, J. L. Chenot, International conference on numerical methods in industrial forming processes numiform'86, ed. (1986) 287-292. [2] M. Dyduch, A. M. Habraken, S. Cescotto, Comput.Methods in Appl. Mech. Eng. 101 (1992) 283-298. [3] W. Sosnowski , J. Mater. Process. Technol. 34 (1-4) (1992)109-116. [4] N. E. Dvorkin, E. G. Petocz, Engineering computations. 10 (1993) 323-336. [5] J. O. Hallquist, B. Wainscott, K. Schweizerhof, Comput. Methods in Appl. Mech. Eng .50 (1995) 144-157. [6] H. Braess, P. Wriggers, Comput. Methods in Appl. Mech. Eng. 190 (2000) 95-109. [7] H. Sofuoglu, H. Hasan Gedikli, Comp. Mater. Sci. 31 (2004) 113–124. [8] W. Prager, P.G. Hodge, Theory of perfectly plastic solids, Wiley, New York, 1951. [9] R. Hill, The Mathematical Theory of Plasticity. Oxford at the Clarendon Press, 1950. [10] O. Hoffman, Introduction to the theory of plasticity for engineers, Mc Graw-Hill, New York, 1953. [11] S.N. Kukureka, G. Craggs, I.M. Ward, J. Mater. Sci. 27 (1992) 3379-3388. [12] M. Kiuchi, H. Kishi, M. Ishikawa, In: Proceedings of the 22 nd International Machine Tool Design Research Conference, vol. 523, 1981. [13] Y. H. Moon, C. J. Van Tyne, Mater. Process. Technol. 99 (2000) 185-196. [14] B. S. Altan, G. Purcek, I. Miskioglu, Mater. Process. Technol. 168(1) (2005) 137-146. [15] C. J. L. Pérez, R. Luri, Mech. Mater. 40 (2008) 617–628. [16] K. Abrinia, A. Fazlirad, J. Mater. Process. Technol. 209 (7) (2009) 3264-3277. [17] A. Khalili Meybodi, A. Assempour, S. Farahani, J. Mater. Process. Technol. 212, (2012) 249–261 [18] S.A. Tabatabaei, K. Abrinia, M.K. Besharati Givi, et al., Mater. Manuf. Process. 28(3), (2013) 271 – 275. [19] S.A. Tabatabaei, K. Abrinia, M.K. Besharati Givi, Int. J. Adv. Manuf. Technol. 72, (2014)1039–1050. [20] S.A. Tabatabaei, K. Abrinia, M.K. Besharati Givi, et al., Key Engineering Materials. 585, (2014) 67-75. [21] S.A. Tabatabaei, M.K. Besharati Givi, K. Abrinia, M. H. Rostamlou, Int. J. Adv. Manuf. Technol.80 (1), (2015) 209-219. [22] W. Xiaona, L. Fuguo, Chin. J. Aeronaut. 22(1) (2009) 81–86. [23] S. R. Lee, Y. K. Lee, C. H. Park, et al., Int. J. Mech. Sci. 44 (2002) 773-792. [24] K. Abrinia, M. Makaremi, J. Mater. Process. Technol. 205 (2007) 259–271. [25] W. Hayt, J. Buck, Engineering electromagnetics. McGraw-Hill Science/Engineering/Math; 7th edition, 2005. [26] K I. Astrop, In: PhD thesis. The Norwegian University of Science and Technology, 2002. [27] J. S. Gunasekera, S. Hoshino, J. Manuf. Sci. Eng. 104(1) (1982) 38-45. [28] K. F. Celik, N. R. Chitkara, Int. J. Mech. Sci. 42 (2000) 295–320. [29] N. R. Chitkara, K. F. Celik, Int. J. Mech. Sci. 43 (2001) 2961–2987. [30] K. F. Celik, N. R. Chitkara, Int. J. Mech. Sci. 44 (2002) 217-246. [31] J. A. Schey, Tribology in metalworking, American Society for Metals, 1983.