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Calculation of the forward tension in drawing processes
Journal of Materials Processing Technology 162–163 (2005) 551–557
Calculation of the forward tension in drawing processes E.M. Rubio a,∗ , A.M. Camacho a , L. Sevilla b , M.A. Sebasti´an a a
Department of Manufacturing Engineering, National Distance University of Spain (UNED), Juan del Rosal 12, Madrid, Spain b Department of Materials and Manufacturing Engineering, University of Malaga, Plza. EI Ejido, s/n M´ alaga, Spain
Abstract Drawing process is one of the most used metalforming process within the industrial field, particularly, in automotive and electric sectors. Then, different analytical, numerical, empirical and experimental methods have been developed in order to analyse it and to optimise it. However, exact solutions have not achieved yet due to the great number of factors involved in this type of processes and to the mathematical complexity that they present. In this work, the main variants of the drawing process have been studied by different methods. Concretely, wire drawing and plate drawing have been modelled and simulated by means of the slab method (SM) and finite element method (FEM). In addition, the results obtained in both cases have been compared with other solutions found in the literature about these themes, particularly, with Wistreich’s solutions in wire case and with Green and Hill and the upper bound technique ones in plate drawing case. © 2005 Elsevier B.V. All rights reserved. Keywords: Drawing process; Slab method; Finite element method; Coulomb friction
1. Introduction Drawing process is one of the most used metalforming process within the industrial field and, particularly, in automotive and electric sectors. The process consists of reducing or changing the crosssection of pieces such as wires, rods, bars or plates, making pass them through a die by means of a pulling force. Materials traditionally used in this kind of manufacturing processes are aluminium and copper alloys and steels. The main variables involved in this type of processes are: die semiangle, α, cross-sectional area reduction, r and friction coefficient, µ or m, for representing the friction along the die-workpiece interfaces, depending on if Coulomb friction or partial one is considered [1]. In general, the complexity of these processes and the great number of factors involved in them make very difficult to select the parameter values properly [2–4]. Then, different analytical, numerical, empirical and experimental methods have been developed in order to analyse the best combination of them [5–11].
∗
Corresponding author. E-mail address: [email protected] (E.M. Rubio).
0924-0136/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2005.02.122
Drawing process began to have theoretical models sufficiently developed thanks to the contributions of Davis and Dokos [12] and Hill and Tupper [13], as well as the empirical and experimental one carried out by several investigators such as Green and Hill [5] and Wistreich [11]. Nowadays, analytical methods still continue being studied and developed in spite of numerical methods allow obtaining solutions with high precision and detail levels in the analysis of this type of processes. Among the analytical methods more commonly used to analyse and simulate these processes are, for example, homogeneous deformation (HD), slip lines field (SLF), slab method (SM) and upper bound technique (UBT). And, as numerical method more widely spread in recent years, it is possible to mention the finite element method (FEM). In this work, slab method and finite element method have been applied to calculate the drawing force necessary to carry out a wire and a plate drawing process. Slab method has already been used in the analysis of other metalforming processes such as forging [14–17], rolling [10,18–20], extrusion [17,21] or deep drawing [22,23]. Slab method was initially developed by Sachs in 1928 [9]. It is a method very easy to apply and that does not require big calculation resources neither of time. It takes into account the homogeneous deformation of the material and the fric-
552
E.M. Rubio et al. / Journal of Materials Processing Technology 162–163 (2005) 551–557
tion between the workpiece material and the die-interfaces to estimate the necessary force (stress, energy or power) to carry out the process. Finite element method has been used as well in several studies about metalforming processes recently [24–29]. The main advantages of the FEM are: the capability of obtaining detailed solutions of the mechanics in a deforming body, namely, velocities, shapes, strains, stresses, temperatures, or contact pressure distributions; and the fact that a computer code, once written, can be used for a large variety of problems by simply changing the point data [30]. Its main disadvantages are the bigger calculation resources, the need of having a good knowledge of the used software programme and the investments in hardware and software equipment. In addition, the results obtained by the both mentioned methods have been compared with the obtained one by Rubio et al. [2–4] using the upper bound technique, by Green and Hill [5] using empirical methods, and with experimental works made by Wistreich [11].
2. Wire drawing 2.1. Modelling In order to make a stress local analysis of the problem by means of the slab method the plastically deformation zone has been modelled like it is shown in Fig. 1. There, it is possible to see a circular section wire of initial diameter Di which is drawn through a conical die of semiangle α, to reduce its diameter to Df at the exit of the die. The forward tension is σ z and the normal die pressure is p. The used die has a constant semiangle, α. The friction between the workpiece and the die interfaces is modelled by Coulomb’s coefficient, which is represented by µ and, in this case, is constant. The lubrication is considered so efficient
Fig. 1. Modelling of the plastically deforming zone for wire drawing.
Fig. 2. Stress local analysis for wire drawing.
that the friction coefficient will be low. The forces acting on an elemental frustum have been plotted in Fig. 2. The workpiece material can be considered as a rigidperfectly plastic material then, its yield stress, Y, will be a constant. In finite element method case, the drawing process modelling has been carried out by means of the finite element code called ABAQUS [31]. The workpiece has been meshed with the CGAX4R element. This type of elements belongs to the ABAQUS element library. It is a four-node bilinear, reduced integration and hourglass control element. An example of mesh with this type of element is shown in Fig. 3. This is exactly the mesh used in this work for wire drawing analysis. 2.2. Simulations algorithms The adimensional forward tension has been chosen as output variable. The main reason for making this election is this variable can be assimilated to the necessary energy to carry out the process (of course in no dimensional terms as well). This fact is very useful, especially, in the selection of machines and equipment to obtain a certain type of parts since simply multiplicand by the volume piece it is possible to
Fig. 3. FEM model by CGAX4R elements.
E.M. Rubio et al. / Journal of Materials Processing Technology 162–163 (2005) 551–557
553
Table 1 Parameters values range 2.5–16◦ 0–0.2 0.20–0.40
Die semiangle, α Coulomb friction coefficient, µ Cross-sectional area reduction, r
know the necessary power to carry out the process. In this work, it will be very useful, as well, in order to compare the results obtained using other different methods. In the slab method case, making the forces balance (Fig. 2) along z- and x-directions, neglecting differential quantities of second order and simplifying adequately, it is possible to write [9]: σzf 1+B = [1 − (1 − r)B ] Y B
(1)
where σ zf is the forward tension, Y, the yield stress, B = µ cot gα and r the cross-sectional area reduction given by: r=
π 2 2 4 (Di − Df ) π 2 4 Di
=1−
Df Di
2 (2)
In finite element method case, the value of the expression σ zf /Y, can be calculated by: Ff NFORC σzf = = (3) Y YAf YAf where Ff is the drawing force; Af the cross-sectional area at the die exit; Y the yield stress of the material and; NFORC a nodal variable obtained from the ABAQUS code by extrapolation of the stress values at the integration points. 2.3. Applications The adimensional drawing stress value has been calculated by the expressions (1) and (3) for the range of parameter values showed in Table 1. In the finite element method case, some properties of the material used in the process must be introduced in the preprocessing stage. Particularly, the used material is an aluminium which presents a rigid-perfectly plastic behaviour and the next characteristics collected in Table 2. In slab method case, this information is not required since all features of the material are represented in the yield stress Y, in this case constant.
2.4. Results and comparison with other previous solutions Fig. 4 shows the obtained results by both described methods for cross-sectional area reductions of 0.20; 0.30 and 0.40 and a Coulomb friction value equal to 0.10 and 0.20. In that figure, it is possible to see that FEM solutions are bigger than SM ones. This is due to slab method only takes into account the terms of homogenous deformation and the friction one while the FEM takes into account other terms. In Fig. 5, SM, FEM and Wistreich solutions have been presented for µ = 0.03 Coulomb friction coefficient and two cross-sectional area reduction. Concretely, r = 0.30 and 0.40. The main reason to select these values is that the experimental results found in the classical literature about these themes use this coefficient for them [11]. In the same way that in Fig. 4, FEM and Wistreich’s experimental solutions are bigger than SM ones, and, for this small µ value, lower and closed among them.
3. Plate drawing
Table 2 Properties of the aluminium used in the process Density, ρ Young’s modulus, E Poisson’s ratio, ν Yield stress in simple tension, Y Plane strain yield stress, S (=2k)
Fig. 4. Comparison between slab method and finite element method solutions for r = 0.20-0.30-0.40: (a) µ = 0.10 and (b) µ = 0.20.
2700 kg/m3 7 × 1010 Pa 0.33 2.8 × 107 Pa 1.155Y
3.1. Modelling In the plate drawing case, the plastically deformation zone for the analysis with the slab method has been modelled like it is shown in Fig. 6.
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Fig. 7. Stress local analysis for plate drawing.
Fig. 5. Comparison among slab method, finite element method and Wistreich solutions for µ = 0.03: (a) r = 0.30 and (b) r = 0.40.
There, it is possible to see a rectangular cross-sectional area plate with an initial height of hi which is drawn through an edged die of semiangle α, to reduce its height to hf at the exit of the die. The drawing process is carried out under plane strain conditions then, the workpiece width, w (the plate dimension along the y-axis), can be considered constant during the whole process. The forward tension is given by σ z and the normal die pressure is p, that is constant on its surfaces. Equal to the wire case, the used die has a constant semiangle, α. The friction between the workpiece and the die
interfaces is of Coulomb kind, µ, and constant. The lubrication is considered so efficient that the friction coefficient will be low. The forces acting on a differential elemental of the workpiece have been plotted in Fig. 7. The workpiece material can be considered as a rigidperfectly plastic material then, its yield stress, Y, will be a constant. In finite element method case, the drawing process simulation has been carried out by ABAQUS [31] as in the wire case. The workpiece has been meshed with the CPE4R element. This is a four-node bilinear, plane strain, quadrilateral, reduced integration and hourglass control element. An example of mesh with this type of element is shown in Fig. 8. This is exactly the mesh used in this work. 3.2. Simulations algorithms In plate case, the adimensional forward tension has been chosen as output variable by the same reasons explained in the wire one. Using Fig. 7 for making the forces balance it is possible to write the equations of the slab method for the drawing process as [9]: B σzf 1+B hf = 1− (4) S B hi Taking into account that, under plane strain, cross-sectional area reduction can be given by r = 1−hf /hi , then the expression (4) will be: 1+B σzf = (1 − (1 − r)B ) S B
Fig. 6. Modelling of the plastically deforming zone for plate drawing.
Fig. 8. FEM model by CPE4R elements.
(5)
E.M. Rubio et al. / Journal of Materials Processing Technology 162–163 (2005) 551–557
Applying FEM in the plate drawing case, the expression, σ zf /2k, can be calculated by: σzf Ff NFORC = = (6) 2k 2kAf 2kAf where Ff is the drawing force; Af the cross-sectional area at the die exit; k, the shear yield stress and; NFORC a nodal variable obtained from the ABAQUS code by extrapolation of the drawing stress values at the integration points. 3.3. Applications The adimensional forward tension value has been calculated by the expressions (5) and (6) for the range of parameter values showed in Table 1. In the finite element method case, the properties of the material collected in Table 2 have been introduced in the pre-processing stage like in wire case. With the slab method, this information is not required since all features of the material are represented in the shear yield stress, k. 3.4. Results and comparison with other previous solutions Fig. 9 shows the obtained results by both described methods for cross-sectional area reductions of 0.20; 0.30 and 0.40 and a Coulomb friction value equal to 0.10 and 0.20. In that figure, it is possible to see that FEM solutions are bigger than SM ones by the same reasons that in the wire case. The results for plate drawing under plane strain and Coulomb friction conditions obtained using the slab method
Fig. 9. Comparison between slab method and finite element method solutions for r = 0.20-0.30-0.40: (a) µ = 0.10 and (b) µ = 0.20.
555
and the finite element method have been compared with the obtained by Green and Hill (G&H) using empirical methods [5]; since most of the experimental values available at the moment are for axisymmetric case. In addition, all results were compared with the obtained ones in previous works applying the upper bound technique (UBT) [2–4]. The G&H results were obtained empirically by Green and Hill for different cross-sectional area reductions, die semiangles from 5◦ to 15◦ and Coulomb friction coefficients smaller than 0.10 [5]. They gave the next expression in order to fit the results to a theoretical model: σzf p r = [(1 + µ ctg α) − µ(0.2 + 0.08r ctg2 α)] (7) 2k 2k 1 − r Outside of the intervals described above, the expression (7) is not correct and has not to be used. Then, the application in this case, has been only made for the values mentioned. To obtain the results by means of the upper bound technique, first of all, the plastically deformation zone was modelled by an only triangular rigid zone as it is shown in Fig. 10. This model fits better than others [3] to the results found on the classical references about these themes [1,5,11,13]. Based on this model the simulation algorithm can be written as: σzf ABν12 + BCν23 = µ 2k hf ν3 − sin α+µ cos α ν2
(8)
where: AB and BC are the discontinuity lines, vij , the relative speed between the i and j zones, and vi , the speed of the zone
Fig. 10. Triangular rigid model in the UBT case.
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i. The rest of the variables within the expression (8) have been previously described.
4. Discussion The adimensional forward tension has been calculated by means of the slab method and the finite element method. in general, it is possible to affirm that using both SM and FEM the necessary forces to carry out the process will be bigger if the cross-sectional area reductions increases. FEM solutions are always bigger than SM ones. This is due to SM only takes into account the terms of homogenous deformation and the friction one while FEM takes into account other terms. Then, SM always underestimates the necessary drawing stress to carry out the process. SM/FEM adimensional forward tension curves are decreasing and quite similar for die semiangles smaller than 10◦ . Over this value, the slope of the FEM curves are also constant and the SM ones continue decreasing. In wire drawing case, a more specific analysis can be made seeing Fig. 4. It shows that the solutions obtained by both methods are quite close especially for high cross-sectional area reductions, low Coulomb coefficient and low die semiangles. Particularly, solutions are comparable for die semiangles between 6◦ and 10◦ and they diverge for bigger ones. Even, for die semiangles near 14◦ , FEM solutions obtained for r = 0.20 and 0.30 also coincide with SM ones obtained for bigger reductions, concretely, for r = 0.30 and 0.40 respectively. In addition, both solutions have been compared with the experimental ones obtained by Wistreich. Fig. 5 confirms that the three solutions are very near especially for small semiangles and low friction coefficients. The differences between solutions decrease as the reduction increases. Besides, FEM and Wistreich solutions coincide for r = 0.40, µ = 0.03 in the interval of the die semiangles from 4◦ to 10◦ . In plate drawing case, most of the made comments above can be repeated now for the comparison between FEM and
Fig. 11. Comparison among slab method, finite element method, upper bound technique and Green and Hill solutions for r = 0.30: (a) µ = 0.03 and (b) µ = 0.10.
SM solutions. Only, it could be added that solutions are lightly more next now. The results have been compared with the empirical ones carried out by Green and Hill [5], and the analytical ones calculated using the UBT [2–4]. In this case, experimental values were not available. This is because the investigation works are generally centered in the wire drawing; mainly due to the importance that they acquire in industrial sectors such as the electric one. Seeing Fig. 11 it can be said that for die semiangles from 10◦ to 14◦ the four solutions are very next and this is especially true for values from 10◦ to 12◦ .
5. Conclusions Drawing process has been studied by means of different methods in recent years because of its great importance in the industrial sector. In this work, the main types of the drawing process has been analysed by means of the slab method and the finite element method. The obtained solutions have been tested with other ones found in the literature about this theme. Concretely, in the wire drawing case, with the experimental results given by Wistreich [11] and, in the plate drawing case, with the empirical ones proposed by Green and Hill [5] and with the obtained applying by the UBT. The solutions comparison confirm that FEM is a method more accurate than SM because the obtained results with it are nearer to the real results. Besides, as it is shown in Fig. 12, FEM provides very intuitive simulations in which can be seen the forward tension not only at the die exit but in the deformation zone as well. However, FEM needs more calculation resources and a good code knowledge by the users. SM is easy to apply and does not require big calculation resources neither the time but solutions are usually not
Fig. 12. Drawing process simulations by finite element method.
E.M. Rubio et al. / Journal of Materials Processing Technology 162–163 (2005) 551–557
acceptable. SM underestimates the drawing stress value to carry out a certain process, since it only takes into account the homogeneous deformation of the material and the friction between the workpiece material and the die-interfaces. SM forward tension curves only are similar to experimental and FEM ones for low reductions carried out in dies with low semiangles and under low friction conditions.
References [1] G.W. Rowe, Principles of Industrial Metalworking Processes, Edward Arnold, London, 1986. [2] E.M. Rubio, M.A. Sebasti´an, A. Sanz, J. Mater. Process. Technol. 143/144 (2003) 539–545. [3] E.M. Rubio, R. Domingo, J.M. Arenas, C. Gonz´alez, J. Mater. Process. Technol. 155-156 (2004) 1220–1226. [4] E.M. Rubio, A.M. Camacho, R. Domingo, C. Gonz´alez, in: R. Teti (Ed.), CIRP Intelligent Computation in Manufacturing Engineering, vol. 4, Salerno, Italy, June 2004. [5] A.P. Green, R. Hill, J. Mech. Phys. Solids 1 (1953) 31–36. [6] B. Avitzur, Metal Forming: Processes and Analysis, McGraw-Hill, 1968. [7] B. Avitzur, Handbook of Metal Forming Processes, Wiley, New York, 1983. [8] B. Avitzur, Metal Forming: The Application of Limit Analysis, Marcel Dekker, New York, 1980. [9] G. Sachs, Trans. Faraday Soc. 24 (1928) 84–92. [10] G.Y. Tzou, J. Mater. Process. Technol. 114 (1) (2001) 41–50. [11] J.G. Wistreich, Process. Inst. Mech. Eng. 169 (1955) 654–670. [12] E.A. Davis, S.J. Dokos, J. Appl. Mech. 66 (1944) 193. [13] R. Hill, S.J. Tupper, J. Iron Steel Inst. 159 (1948) 353–359. [14] A.G. Mamalis, G.L. Petrossian, D.E. Manolakos, J. Mater. Process. Technol. 96 (1–3) (1999) 112–116.
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[15] F. Fereshteh-Saniee, M. Jaafari, J. Mater. Process. Technol. 125/126 (2002) 334–340. [16] H.H. Hsu, G.Y. Tzou, J. Mater. Process. Technol. 140 (1–3) (2003) 604–609. [17] L. Capan, O. Baran, J. Mater. Process. Technol. 102 (1–3) (2000) 230–233. [18] H. Gao, S.C. Ramalingam, G.C. Barber, G. Chen, J. Mater. Process. Technol. 124 (1/2) (2002) 178–182. [19] M.N. Huang, G.Y. Tzou, S.W. Syu, J. Mater. Process. Technol. 140 (1–3) (2003) 598–603. [20] X. Duan, T. Sheppard, J. Mater. Process. Technol. 130/131 (2002) 245–249. [21] A.S. Wifi, M.N. Shatla, A. Abdel-Hamid, J. Mater. Process. Technol. 73 (1–3) (1998) 97–107. [22] D. Delarbre, P. Montmitonnet, J. Mater. Process. Technol. 91 (1–3) (1999) 95–104. [23] M.A. Hassan, N. Takakura, K. Yamaguchi, J. Mater. Process. Technol. 139 (1–3) (2003) 408–413. [24] D. Petrescu, S.C. Savage, P.D. Hodgson, J. Mater. Process. Technol. 125/126 (2002) 361–368. [25] H. Cho, H.H. Jo, S.G. Lee, B.M. Kim, Y.J. Kim, J. Mater. Process. Technol. 130/131 (2002) 416–420. [26] L. Li, J. Zhou, P. Duszczyk, Modelling, Simulat. Mater. Sci. Eng. 11 (2003) 401–416. [27] S.W. Lo, Y.H. Lu, Wire drawing dies with prescribed variations of strain rate, J. Mater. Process. Technol. 123 (2) (2002) 212– 218. [28] T. Pepelnjak, K. Kuzman, J. Mater. Process. Technol. 80/81 (1998) 413–420. [29] T. Sheppard, X. Duan, Modell. Simulat. Mater. Sci. Eng. 10 (2002) 597–610. [30] S. Kobayashi, S.I. Oh, T. Altan, Metal Forming and the Finite Element Method, Oxford University Press, Inc., 1989. [31] Hibbitt, Karlsson and Sorensen, ABAQUS v6.4, User’s Manuals, ABAQUS, Inc., USA, 2003.
Calculation of the forward tension in drawing processes E.M. Rubio a,∗ , A.M. Camacho a , L. Sevilla b , M.A. Sebasti´an a a
Department of Manufacturing Engineering, National Distance University of Spain (UNED), Juan del Rosal 12, Madrid, Spain b Department of Materials and Manufacturing Engineering, University of Malaga, Plza. EI Ejido, s/n M´ alaga, Spain
Abstract Drawing process is one of the most used metalforming process within the industrial field, particularly, in automotive and electric sectors. Then, different analytical, numerical, empirical and experimental methods have been developed in order to analyse it and to optimise it. However, exact solutions have not achieved yet due to the great number of factors involved in this type of processes and to the mathematical complexity that they present. In this work, the main variants of the drawing process have been studied by different methods. Concretely, wire drawing and plate drawing have been modelled and simulated by means of the slab method (SM) and finite element method (FEM). In addition, the results obtained in both cases have been compared with other solutions found in the literature about these themes, particularly, with Wistreich’s solutions in wire case and with Green and Hill and the upper bound technique ones in plate drawing case. © 2005 Elsevier B.V. All rights reserved. Keywords: Drawing process; Slab method; Finite element method; Coulomb friction
1. Introduction Drawing process is one of the most used metalforming process within the industrial field and, particularly, in automotive and electric sectors. The process consists of reducing or changing the crosssection of pieces such as wires, rods, bars or plates, making pass them through a die by means of a pulling force. Materials traditionally used in this kind of manufacturing processes are aluminium and copper alloys and steels. The main variables involved in this type of processes are: die semiangle, α, cross-sectional area reduction, r and friction coefficient, µ or m, for representing the friction along the die-workpiece interfaces, depending on if Coulomb friction or partial one is considered [1]. In general, the complexity of these processes and the great number of factors involved in them make very difficult to select the parameter values properly [2–4]. Then, different analytical, numerical, empirical and experimental methods have been developed in order to analyse the best combination of them [5–11].
∗
Corresponding author. E-mail address: [email protected] (E.M. Rubio).
0924-0136/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2005.02.122
Drawing process began to have theoretical models sufficiently developed thanks to the contributions of Davis and Dokos [12] and Hill and Tupper [13], as well as the empirical and experimental one carried out by several investigators such as Green and Hill [5] and Wistreich [11]. Nowadays, analytical methods still continue being studied and developed in spite of numerical methods allow obtaining solutions with high precision and detail levels in the analysis of this type of processes. Among the analytical methods more commonly used to analyse and simulate these processes are, for example, homogeneous deformation (HD), slip lines field (SLF), slab method (SM) and upper bound technique (UBT). And, as numerical method more widely spread in recent years, it is possible to mention the finite element method (FEM). In this work, slab method and finite element method have been applied to calculate the drawing force necessary to carry out a wire and a plate drawing process. Slab method has already been used in the analysis of other metalforming processes such as forging [14–17], rolling [10,18–20], extrusion [17,21] or deep drawing [22,23]. Slab method was initially developed by Sachs in 1928 [9]. It is a method very easy to apply and that does not require big calculation resources neither of time. It takes into account the homogeneous deformation of the material and the fric-
552
E.M. Rubio et al. / Journal of Materials Processing Technology 162–163 (2005) 551–557
tion between the workpiece material and the die-interfaces to estimate the necessary force (stress, energy or power) to carry out the process. Finite element method has been used as well in several studies about metalforming processes recently [24–29]. The main advantages of the FEM are: the capability of obtaining detailed solutions of the mechanics in a deforming body, namely, velocities, shapes, strains, stresses, temperatures, or contact pressure distributions; and the fact that a computer code, once written, can be used for a large variety of problems by simply changing the point data [30]. Its main disadvantages are the bigger calculation resources, the need of having a good knowledge of the used software programme and the investments in hardware and software equipment. In addition, the results obtained by the both mentioned methods have been compared with the obtained one by Rubio et al. [2–4] using the upper bound technique, by Green and Hill [5] using empirical methods, and with experimental works made by Wistreich [11].
2. Wire drawing 2.1. Modelling In order to make a stress local analysis of the problem by means of the slab method the plastically deformation zone has been modelled like it is shown in Fig. 1. There, it is possible to see a circular section wire of initial diameter Di which is drawn through a conical die of semiangle α, to reduce its diameter to Df at the exit of the die. The forward tension is σ z and the normal die pressure is p. The used die has a constant semiangle, α. The friction between the workpiece and the die interfaces is modelled by Coulomb’s coefficient, which is represented by µ and, in this case, is constant. The lubrication is considered so efficient
Fig. 1. Modelling of the plastically deforming zone for wire drawing.
Fig. 2. Stress local analysis for wire drawing.
that the friction coefficient will be low. The forces acting on an elemental frustum have been plotted in Fig. 2. The workpiece material can be considered as a rigidperfectly plastic material then, its yield stress, Y, will be a constant. In finite element method case, the drawing process modelling has been carried out by means of the finite element code called ABAQUS [31]. The workpiece has been meshed with the CGAX4R element. This type of elements belongs to the ABAQUS element library. It is a four-node bilinear, reduced integration and hourglass control element. An example of mesh with this type of element is shown in Fig. 3. This is exactly the mesh used in this work for wire drawing analysis. 2.2. Simulations algorithms The adimensional forward tension has been chosen as output variable. The main reason for making this election is this variable can be assimilated to the necessary energy to carry out the process (of course in no dimensional terms as well). This fact is very useful, especially, in the selection of machines and equipment to obtain a certain type of parts since simply multiplicand by the volume piece it is possible to
Fig. 3. FEM model by CGAX4R elements.
E.M. Rubio et al. / Journal of Materials Processing Technology 162–163 (2005) 551–557
553
Table 1 Parameters values range 2.5–16◦ 0–0.2 0.20–0.40
Die semiangle, α Coulomb friction coefficient, µ Cross-sectional area reduction, r
know the necessary power to carry out the process. In this work, it will be very useful, as well, in order to compare the results obtained using other different methods. In the slab method case, making the forces balance (Fig. 2) along z- and x-directions, neglecting differential quantities of second order and simplifying adequately, it is possible to write [9]: σzf 1+B = [1 − (1 − r)B ] Y B
(1)
where σ zf is the forward tension, Y, the yield stress, B = µ cot gα and r the cross-sectional area reduction given by: r=
π 2 2 4 (Di − Df ) π 2 4 Di
=1−
Df Di
2 (2)
In finite element method case, the value of the expression σ zf /Y, can be calculated by: Ff NFORC σzf = = (3) Y YAf YAf where Ff is the drawing force; Af the cross-sectional area at the die exit; Y the yield stress of the material and; NFORC a nodal variable obtained from the ABAQUS code by extrapolation of the stress values at the integration points. 2.3. Applications The adimensional drawing stress value has been calculated by the expressions (1) and (3) for the range of parameter values showed in Table 1. In the finite element method case, some properties of the material used in the process must be introduced in the preprocessing stage. Particularly, the used material is an aluminium which presents a rigid-perfectly plastic behaviour and the next characteristics collected in Table 2. In slab method case, this information is not required since all features of the material are represented in the yield stress Y, in this case constant.
2.4. Results and comparison with other previous solutions Fig. 4 shows the obtained results by both described methods for cross-sectional area reductions of 0.20; 0.30 and 0.40 and a Coulomb friction value equal to 0.10 and 0.20. In that figure, it is possible to see that FEM solutions are bigger than SM ones. This is due to slab method only takes into account the terms of homogenous deformation and the friction one while the FEM takes into account other terms. In Fig. 5, SM, FEM and Wistreich solutions have been presented for µ = 0.03 Coulomb friction coefficient and two cross-sectional area reduction. Concretely, r = 0.30 and 0.40. The main reason to select these values is that the experimental results found in the classical literature about these themes use this coefficient for them [11]. In the same way that in Fig. 4, FEM and Wistreich’s experimental solutions are bigger than SM ones, and, for this small µ value, lower and closed among them.
3. Plate drawing
Table 2 Properties of the aluminium used in the process Density, ρ Young’s modulus, E Poisson’s ratio, ν Yield stress in simple tension, Y Plane strain yield stress, S (=2k)
Fig. 4. Comparison between slab method and finite element method solutions for r = 0.20-0.30-0.40: (a) µ = 0.10 and (b) µ = 0.20.
2700 kg/m3 7 × 1010 Pa 0.33 2.8 × 107 Pa 1.155Y
3.1. Modelling In the plate drawing case, the plastically deformation zone for the analysis with the slab method has been modelled like it is shown in Fig. 6.
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Fig. 7. Stress local analysis for plate drawing.
Fig. 5. Comparison among slab method, finite element method and Wistreich solutions for µ = 0.03: (a) r = 0.30 and (b) r = 0.40.
There, it is possible to see a rectangular cross-sectional area plate with an initial height of hi which is drawn through an edged die of semiangle α, to reduce its height to hf at the exit of the die. The drawing process is carried out under plane strain conditions then, the workpiece width, w (the plate dimension along the y-axis), can be considered constant during the whole process. The forward tension is given by σ z and the normal die pressure is p, that is constant on its surfaces. Equal to the wire case, the used die has a constant semiangle, α. The friction between the workpiece and the die
interfaces is of Coulomb kind, µ, and constant. The lubrication is considered so efficient that the friction coefficient will be low. The forces acting on a differential elemental of the workpiece have been plotted in Fig. 7. The workpiece material can be considered as a rigidperfectly plastic material then, its yield stress, Y, will be a constant. In finite element method case, the drawing process simulation has been carried out by ABAQUS [31] as in the wire case. The workpiece has been meshed with the CPE4R element. This is a four-node bilinear, plane strain, quadrilateral, reduced integration and hourglass control element. An example of mesh with this type of element is shown in Fig. 8. This is exactly the mesh used in this work. 3.2. Simulations algorithms In plate case, the adimensional forward tension has been chosen as output variable by the same reasons explained in the wire one. Using Fig. 7 for making the forces balance it is possible to write the equations of the slab method for the drawing process as [9]: B σzf 1+B hf = 1− (4) S B hi Taking into account that, under plane strain, cross-sectional area reduction can be given by r = 1−hf /hi , then the expression (4) will be: 1+B σzf = (1 − (1 − r)B ) S B
Fig. 6. Modelling of the plastically deforming zone for plate drawing.
Fig. 8. FEM model by CPE4R elements.
(5)
E.M. Rubio et al. / Journal of Materials Processing Technology 162–163 (2005) 551–557
Applying FEM in the plate drawing case, the expression, σ zf /2k, can be calculated by: σzf Ff NFORC = = (6) 2k 2kAf 2kAf where Ff is the drawing force; Af the cross-sectional area at the die exit; k, the shear yield stress and; NFORC a nodal variable obtained from the ABAQUS code by extrapolation of the drawing stress values at the integration points. 3.3. Applications The adimensional forward tension value has been calculated by the expressions (5) and (6) for the range of parameter values showed in Table 1. In the finite element method case, the properties of the material collected in Table 2 have been introduced in the pre-processing stage like in wire case. With the slab method, this information is not required since all features of the material are represented in the shear yield stress, k. 3.4. Results and comparison with other previous solutions Fig. 9 shows the obtained results by both described methods for cross-sectional area reductions of 0.20; 0.30 and 0.40 and a Coulomb friction value equal to 0.10 and 0.20. In that figure, it is possible to see that FEM solutions are bigger than SM ones by the same reasons that in the wire case. The results for plate drawing under plane strain and Coulomb friction conditions obtained using the slab method
Fig. 9. Comparison between slab method and finite element method solutions for r = 0.20-0.30-0.40: (a) µ = 0.10 and (b) µ = 0.20.
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and the finite element method have been compared with the obtained by Green and Hill (G&H) using empirical methods [5]; since most of the experimental values available at the moment are for axisymmetric case. In addition, all results were compared with the obtained ones in previous works applying the upper bound technique (UBT) [2–4]. The G&H results were obtained empirically by Green and Hill for different cross-sectional area reductions, die semiangles from 5◦ to 15◦ and Coulomb friction coefficients smaller than 0.10 [5]. They gave the next expression in order to fit the results to a theoretical model: σzf p r = [(1 + µ ctg α) − µ(0.2 + 0.08r ctg2 α)] (7) 2k 2k 1 − r Outside of the intervals described above, the expression (7) is not correct and has not to be used. Then, the application in this case, has been only made for the values mentioned. To obtain the results by means of the upper bound technique, first of all, the plastically deformation zone was modelled by an only triangular rigid zone as it is shown in Fig. 10. This model fits better than others [3] to the results found on the classical references about these themes [1,5,11,13]. Based on this model the simulation algorithm can be written as: σzf ABν12 + BCν23 = µ 2k hf ν3 − sin α+µ cos α ν2
(8)
where: AB and BC are the discontinuity lines, vij , the relative speed between the i and j zones, and vi , the speed of the zone
Fig. 10. Triangular rigid model in the UBT case.
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i. The rest of the variables within the expression (8) have been previously described.
4. Discussion The adimensional forward tension has been calculated by means of the slab method and the finite element method. in general, it is possible to affirm that using both SM and FEM the necessary forces to carry out the process will be bigger if the cross-sectional area reductions increases. FEM solutions are always bigger than SM ones. This is due to SM only takes into account the terms of homogenous deformation and the friction one while FEM takes into account other terms. Then, SM always underestimates the necessary drawing stress to carry out the process. SM/FEM adimensional forward tension curves are decreasing and quite similar for die semiangles smaller than 10◦ . Over this value, the slope of the FEM curves are also constant and the SM ones continue decreasing. In wire drawing case, a more specific analysis can be made seeing Fig. 4. It shows that the solutions obtained by both methods are quite close especially for high cross-sectional area reductions, low Coulomb coefficient and low die semiangles. Particularly, solutions are comparable for die semiangles between 6◦ and 10◦ and they diverge for bigger ones. Even, for die semiangles near 14◦ , FEM solutions obtained for r = 0.20 and 0.30 also coincide with SM ones obtained for bigger reductions, concretely, for r = 0.30 and 0.40 respectively. In addition, both solutions have been compared with the experimental ones obtained by Wistreich. Fig. 5 confirms that the three solutions are very near especially for small semiangles and low friction coefficients. The differences between solutions decrease as the reduction increases. Besides, FEM and Wistreich solutions coincide for r = 0.40, µ = 0.03 in the interval of the die semiangles from 4◦ to 10◦ . In plate drawing case, most of the made comments above can be repeated now for the comparison between FEM and
Fig. 11. Comparison among slab method, finite element method, upper bound technique and Green and Hill solutions for r = 0.30: (a) µ = 0.03 and (b) µ = 0.10.
SM solutions. Only, it could be added that solutions are lightly more next now. The results have been compared with the empirical ones carried out by Green and Hill [5], and the analytical ones calculated using the UBT [2–4]. In this case, experimental values were not available. This is because the investigation works are generally centered in the wire drawing; mainly due to the importance that they acquire in industrial sectors such as the electric one. Seeing Fig. 11 it can be said that for die semiangles from 10◦ to 14◦ the four solutions are very next and this is especially true for values from 10◦ to 12◦ .
5. Conclusions Drawing process has been studied by means of different methods in recent years because of its great importance in the industrial sector. In this work, the main types of the drawing process has been analysed by means of the slab method and the finite element method. The obtained solutions have been tested with other ones found in the literature about this theme. Concretely, in the wire drawing case, with the experimental results given by Wistreich [11] and, in the plate drawing case, with the empirical ones proposed by Green and Hill [5] and with the obtained applying by the UBT. The solutions comparison confirm that FEM is a method more accurate than SM because the obtained results with it are nearer to the real results. Besides, as it is shown in Fig. 12, FEM provides very intuitive simulations in which can be seen the forward tension not only at the die exit but in the deformation zone as well. However, FEM needs more calculation resources and a good code knowledge by the users. SM is easy to apply and does not require big calculation resources neither the time but solutions are usually not
Fig. 12. Drawing process simulations by finite element method.
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acceptable. SM underestimates the drawing stress value to carry out a certain process, since it only takes into account the homogeneous deformation of the material and the friction between the workpiece material and the die-interfaces. SM forward tension curves only are similar to experimental and FEM ones for low reductions carried out in dies with low semiangles and under low friction conditions.
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